This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
P > m1 ðbÞ > > þ i¼1 exp½½kni ðuÞðbÞ t½CRm > ni1 ðt; uÞ > > : Qn ðbÞ j¼niþ1 ½1 exp½½kj ðuÞ t
for n\m; for n ¼ m;
for n [ m;
for t 0; u ¼ 1; 2; . . .; z; ð3:26Þ (v)
for a series–parallel system Rk;l1 ;l2 ;...;lk ðt; Þ ¼ ½1; Rk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; Rk;l1 ;l2 ;...;lk ðt; zÞ;
ð3:27Þ
where Rk;l1 ;l2 ;...;lk ðt; uÞ ¼ 1
for t\0;
Rk;l1 ;l2 ;...;lk ðt; uÞ ffi 1
v X
pb
k Y
"
" 1 exp
kij ðuÞ
ðbÞ
## t
ð3:28Þ
j¼1
i¼1
b¼1
li X
for t 0; u ¼ 1; 2; . . .; z; (vi) for a parallel–series system Rk;l1 ;l2 ;...;lk ðt; Þ ¼ ½1; Rk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; Rk;l1 ;l2 ;...;lk ðt; zÞ
ð3:29Þ
where Rk;l1 ;l2 ;...;lk ðt; uÞ ¼ 1 Rk;l1 ;l2 ;...;lk ðt; uÞ ffi
v X b¼1
for t\0; " # li k Y Y ðbÞ pb 1 ½1 exp½½kij ðuÞ t i¼1
ð3:30Þ
j¼1
for t 0; u ¼ 1; 2; . . .; z; (vii) for a series-‘‘m out of k’’ system m m Rm k;l1 ;l2 ;...;lk ðt; Þ ¼ ½1; Rk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; Rk;l1 ;l2 ;...;lk ðt; zÞ;
where
ð3:31Þ
86
3 Complex Technical Systems, Reliability
Rm k;l1 ;l2 ;...;lk ðt; uÞ ¼ 1 Rm k;l1 ;l2 ;...;lk ðt; uÞ
for t\0;
ffi1
v X
pb
r1 ;r2 ;...;rk ¼0 i¼1 r1 þr2 þþrk m1
b¼1
"
li Y
1
" li k Y Y
1 X
# ri ðbÞ
exp½½kij ðuÞ t
j¼1
#1ri
ðbÞ
exp½½kij ðuÞ t
for t 0; u ¼ 1; 2; . . .; z;
j¼1
ð3:32Þ or m
m
m
Rk;l1 ;l2 ;...;lk ðt; Þ ¼ ½1; Rk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; Rk;l1 ;l2 ;...;lk ðt; zÞ;
ð3:33Þ
where m
Rk;l1 ;l2 ;...;lk ðt; uÞ ¼ 1 m
Rk;l1 ;l2 ;...;lk ðt; uÞ ffi
v X
for t\0;
"
" k Y
" 1
r1 ;r2 ;...;rk ¼0 i¼1 r1 ;r2 þþrk m
b¼1
1 X
pb
li Y
li Y
##ri exp½½kij ðuÞðbÞ t
i¼1
#1ri
ðbÞ
exp½½kij ðuÞ t
for t 0; m ¼ k m; u ¼ 1; 2; . . .; z;
j¼1
ð3:34Þ (viii) for an ‘‘mi out of li ’’-series system h i m1 ;m2 ;...;mk m1 ;m2 ;...;mk 1 ;m2 ;...;mk Rm k;l1 ;l2 ;...;lk ðt; Þ ¼ 1; Rk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; Rk;l1 ;l2 ;...;lk ðt; zÞ ;
ð3:35Þ
where 1 ;m2 ;...;mk Rm k;l1 ;l2 ;...;lk ðt; uÞ ¼ 1
1 ;m2 ;...;mk Rm k;l1 ;l2 ;...;lk ðt; uÞ ffi
v X b¼1
pb
for t 2 k 6 Y 6 61 4 i¼1
3 1 X r1 ;r2 ;...;rli ¼0 r1 þr2 þ...þrli mi 1
7 7 exp½rj ½kij ðuÞðbÞ t7 5 j¼1
li Y
½1 exp½½kij ðuÞðbÞ t1rj for t 0; u ¼ 1; 2; . . .; z; ð3:36Þ or m ;m ;...;m m ;m ;...;m m ;m ;...;m Rk;l11 ;l22;...;lk k ðt; Þ ¼ 1; Rk;l11 ;l22;...;lk k ðt; 1Þ; . . .; Rk;l11 ;l22;...;lk k ðt; zÞ ;
ð3:37Þ
3.2 Reliability of Multistate Systems at Variable Operation Conditions
87
where m ;m ;...;m
Rk;l11 ;l22;...;lk k ðt; uÞ ¼1
for t\0;
m ;m ;...;m
Rk;l11 ;l22;...;lk k ðt; uÞ ffi
v X b¼1
pb
2
k 6 Y 6 6 4 i¼1
1 X
li Y ½1 exp½½kij ðuÞðbÞ trj
r1 ;r2 ;...;rli ¼0 j¼1 r1 þr2 þþrli mi
3
7 7 exp½ð1 rj Þ½kij ðuÞðbÞ t7 5
for t 0; mi ¼ li mi ;
i ¼ 1; 2; . . .; k; u ¼ 1; 2; . . .; z; ð3:38Þ (ix) for a series-consecutive ‘‘m out of k: F’’ system m m CRm k;l1 ;l2 ;...;lk ðt; Þ ¼ ½1; CRk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; CRk;l1 ;l2 ;...;lk ðt; zÞ;
ð3:39Þ
where CRm k;l1 ;l2 ;...;lk ðt; uÞ ¼ 1 for t\0; m X ðbÞ CRm pb ½CRm k;l1 ;l2 ;...;lk ðt; uÞ ffi k;l1 ;l2 ;...;lk ðt; uÞ
for t 0; u ¼ 1; 2; . . .; z;
b¼1
ð3:40Þ ðbÞ and ½CRm k;l1 ;l2 ;...;lk ðt; uÞ ; b ¼ 1; 2; . . .; m; are given by
h iðbÞ CRm ðt; uÞ k;l1 ;l2 ;...;lk 8 1 > > h P ii > Q h > i > > kij ðuÞt 1 ki¼1 1 exp lj¼1 > > > > h P iðbÞ > ðbÞ ih m < lk ¼ exp j¼1 kkj ðuÞ t CRk1;l1 ;l2 ;...;lk ðt; uÞ > iðbÞ > ðbÞ iih m > Pm1 h h Plkj > > þ exp k ðuÞ t CR ðt; uÞ > kjv j¼1 kj1;l1 ;l2 ;...;lk v¼1 > > > h h P ii > > li ðbÞ : Qk 1 exp ½k ðuÞ t i¼kjþ1
v¼1
iv
for k\; m; for k ¼ m;
for k [ m;
ð3:41Þ for t 0; u ¼ 1; 2; . . .; z;
88
3 Complex Technical Systems, Reliability
(x) for a consecutive ‘‘mi out of li : F’’-series system h i m1 ;m2 ;...;mk m1 ;m2 ;...;mk m1 ;m2 ;...;mk CRk;l1 ;l2 ;...;lk ðt; Þ ¼ 1; CRk;l1 ;l2 ;...;lk ðt; 1Þ; . . .; CRk;l1 ;l2 ;...;lk ðt; zÞ ;
ð3:42Þ
where m1 ;m2 ;...;mk
CRk;l1 ;l2 ;...;lk ðt; uÞ ¼ 1 m1 ;m2 ;...;mk
CRk;l1 ;l2 ;...;lk ðt; uÞ ffi
m X b¼1
for t\0; pb
k Y ðbÞ i ½CRm i;li ðt; uÞ
for t 0; u ¼ 1; 2; . . .; z;
ð3:43Þ
i¼1
ðbÞ i and ½CRm i;li ðt; uÞ ; i ¼ 1; 2; . . .; k; b ¼ 1; 2; . . .; m; are given by
h iðbÞ i CRm i;li ðt; uÞ 8 1 > > h > ðbÞ ii Qli h > > > 1 1 exp k ðuÞ t ij > j¼1 > > > h ih iðbÞ > < ðbÞ mi ¼ exp ½kili ðuÞ t CRi;li 1 ðt; uÞ > h iðbÞ > P ðbÞ ih > > mi 1 i > þ j¼1 exp kili j ðuÞ t CRm > i;li j1 ðt; uÞ > > > h h ii > > : Ql i 1 exp ½kiv ðuÞðbÞ t v¼li jþ1
for li \mi ; for li ¼ mi ; ð3:44Þ
for li [ mi ;
for t 0; u ¼ 1; 2; . . .; z: Remark 3.1 The formulae for the reliability functions stated in Proposition 3.1 are valid for the considered systems under the assumption that they do not change their structures at different operation states zb ; b ¼ 1; 2; . . .; m: This limitation can be simply omitted by the replacement in these formulae the system’s structure shape constant parameters n; m; k; mi ; li ; respectively by their changing at different operation states zb ; b ¼ 1; 2; . . .; m; equivalent structure shape parameters ðbÞ ðbÞ nðbÞ ; mðbÞ ; kðbÞ ; mi ; li ; b ¼ 1; 2; . . .; m: For the exponential complex technical systems, considered in Proposition 3.1, we determine the mean values lðuÞ and the standard deviations rðuÞ of the unconditional lifetimes of the system in the reliability state subsets fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; the mean values lðuÞ of the unconditional lifetimes of the system in the particular reliability states u; u ¼ 1; 2; . . .; z; the system risk function rðtÞ and the moment s when the system risk function exceeds a permitted level d respectively defined by (3.7)–(3.12), after substituting for Rðt; uÞ; u ¼ 1; 2; . . .; z; the coordinates of the unconditional reliability functions given respectively by (3.17)–(3.44).
3.3 Asymptotic Approach to Reliability of Large Multistate Systems
89
3.3 Asymptotic Approach to Reliability of Large Multistate Systems at Variable Operation Conditions In the case of large systems, the determination of the exact reliability functions of the complex systems and the system risk functions, sometimes, leads us to very complicated formulae that are often useless for reliability practitioners. One of the important techniques that can be useful in this situation is the asymptotic approach [19, 24, 46–48, 52] to system reliability evaluation. In this approach, instead of the preliminary complex formula for the system reliability function, after assuming that the number of system components tends to infinity and finding the limit reliability of the system, we can obtain its simplified form. Moreover, in the case of large systems, the possibility of combining the results of the reliability joint models of complex technical systems and the results concerning the limit reliability functions of the considered systems is possible. In this way, the results concerned with the asymptotic approach to estimation of non-repairable multistate systems at variable operation conditions may be obtained. The main results concerning the asymptotic approach to multistate large systems reliability with ageing components in the constant operation conditions are comprehensively presented in the work [19] and some of these results’ extentions to the systems operating at the variable conditions can be found in [46–48, 51–52]. In order to combine the results on the reliability of multistate systems related to their operation processes with the results concerning the limit reliability functions of the multistate systems, and to obtain the results on the asymptotic approach to the evaluation of the large multistate systems reliability at the variable operation conditions, we assume the following definition [51]. Definition 3.1 A reliability function <ðt; Þ ¼ ½1; <ðt; 1Þ; . . .; <ðt; zÞ;
t 2 ð1; 1Þ;
ð3:45Þ
where <ðt; uÞ ¼
v X
pb ½<ðt; uÞðbÞ ;
u ¼ 1; 2; . . .; z;
ð3:46Þ
b¼1
is called a limit reliability function of a complex multistate system with the reliability function sequence Rn ðt; Þ ¼ ½1; Rn ðt; 1Þ; . . .; Rn ðt; zÞ;
t 2 ð1; 1Þ; n 2 N;
ð3:47Þ
where Rn ðt; uÞ ffi
v X b¼1
pb ½Rn ðt; uÞðbÞ ;
u ¼ 1; 2; . . .; z;
ð3:48Þ
90
3 Complex Technical Systems, Reliability
if there exist normalizing constants aðbÞ n ðuÞ [ 0;
bðbÞ n ðuÞ 2 ð1; 1Þ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v;
such that ðbÞ ðbÞ lim ½Rn ðaðbÞ ¼ ½<ðt; uÞðbÞ n ðuÞt þ bn ðuÞ; uÞ
n!1
for all t from the sets of continuity points C½<ðuÞðbÞ of the functions ½<ðt; uÞðbÞ ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v: Hence, for sufficiently large n, the following approximate formulae are valid Rn ðt; Þ ¼ ½1; Rn ðt; 1Þ; . . .; Rn ðt; zÞ;
t 2 ð1; 1Þ;
ð3:49Þ
where Rn ðt; uÞ ffi
v X
" pb <
b¼1
!#ðbÞ
ðbÞ
t bn ðuÞ ðbÞ
an ðuÞ
;u
;
t 2 ð1; 1Þ; u ¼ 1; 2; . . .; z: ð3:50Þ
The following propositions concerned with the large series–parallel and parallel–series exponential systems operating at the variable operation states are exemplary results that can be worked out on the basis of the results included in [19, 24, 48, 52] for the large systems considered in the book. Proposition 3.2 If components of the multistate series–parallel regular system at the operation states zb ; b ¼ 1; 2; . . .; v; i.e., the system with the structure shape parameters such that k ¼ knðbÞ ;
l1 ¼ l2 ¼ ¼ lk ¼ lðbÞ n ;
b ¼ 1; 2; . . .; v; n 2 N;
have the exponential reliability functions given by (3.15)–(3.16) are homogeneous, i.e.,
ðbÞ kij ðuÞ ¼ ½kðuÞðbÞ ;
i ¼ 1; 2; . . .; knðbÞ ; j ¼ 1; 2; . . .; lðbÞ n ; b ¼ 1; 2; . . .; v;
then the system unconditional multistate reliability function is given by the approximate formulae, respectively, in the following cases of the system structure shape at the particular operation states: (i)
ðbÞ
ðbÞ
kn ¼ n; ln [ 0; Rðt; Þ ¼ ½1; Rðt; 1Þ; . . .; Rðt; zÞ
3.3 Asymptotic Approach to Reliability of Large Multistate Systems
91
where Rðt; uÞ ffi 1
v X
pb exp½n exp½½kðuÞðbÞ lðbÞ n t
for t 2 ð1; 1Þ;
b¼1
ð3:51Þ
u ¼ 1; 2; . . .; z; (ii) knðbÞ ! kðbÞ ; lðbÞ n ! 1; Rðt; Þ ¼ ½1; Rðt; 1Þ; . . .; Rðt; zÞ where ( Rðt; uÞ ffi
1 1
for t\0;
Pv
b¼1
pb ½1
ðbÞ ðbÞ exp½½kðuÞðbÞ ln tk
for t 0;
ð3:52Þ
u ¼ 1; 2; . . .; z:
Proposition 3.3 If components of the multistate parallel–series regular system at the operation states zb ; b ¼ 1; 2; . . .; v; i.e., the system with the structure shape parameters such that k ¼ knðbÞ ;
l1 ¼ l2 ¼ ¼ lk ¼ lðbÞ n ;
b ¼ 1; 2; . . .; v; n 2 N;
have the exponential reliability functions given by (3.15)–(3.16) are homogeneous, i.e.,
ðbÞ kij ðuÞ ¼ ½kðuÞðbÞ ;
i ¼ 1; 2; . . .; knðbÞ ; j ¼ 1; 2; . . .; lðbÞ n ; b ¼ 1; 2; . . .; v;
then the system unconditional multistate reliability function is given by the approximate formulae, respectively, in the following cases of the system structure shapes at the particular operation states: (i)
ðbÞ
ðbÞ
kn ¼ n; ln ! lðbÞ ; lðbÞ [ 0; Rðt; Þ ¼ ½1; Rðt; 1Þ; . . .; Rðt; zÞ where ( Rðt; uÞ ffi
1 Pv
b¼1 pb
for t\0; ðbÞ
lðbÞ
exp½nð½kðuÞ tÞ for t 0;
u ¼ 1; 2; . . .; z:
ð3:53Þ
92
3 Complex Technical Systems, Reliability ðbÞ
ðbÞ
(ii) kn ! kðbÞ ; ln ! 1; Rðt; Þ ¼ ½1; Rðt; 1Þ; . . .; Rðt; zÞ where Rðt; uÞ ffi
v X
ðbÞ k pb ½1 exp½lðbÞ n exp½½kðuÞ t
ðbÞ
for t 2 ð1; 1Þ;
b¼1
u ¼ 1; 2; . . .; z:
ð3:54Þ
It is possible to obtain similar and more general results for other considered in the book multistate systems after some modification of the results included in [19, 24].
3.4 Renewal and Availability of Multistate Systems at Variable Operation Conditions The models and methods presented in the previous sections of this chapter can be applied for determining the reliability functions of the considered complex multistate systems at variable operation conditions, the mean values and variances of their lifetimes in the reliability state subsets and in the particular reliability states in the case when they are non-repairable. Combining these results with the results of the classical renewal theory it is possible to obtain the renewal and availability characteristics for repairable complex technical systems with ignored and non-ignored time of renovation. In this section are determined the distributions, the expected values and the variances of the times until the successive exceedings of the reliability critical state and the distributions, the expected values and the variances of the number of exceedings of the reliability critical state at a fixed moment of time for the considered complex technical systems in the case when they are repairable and the time of their renovation is ignored. The distribution functions, the expected values and the variances of the times are also determined until the successive renovations and the distribution functions, the expected values and the variances of the times until the successive exceeding of the reliability critical state and the distributions, the expected values and variances of the number of renovations up to a fixed moment of time and the distributions, the expected values and variances of the number of exceedings of the reliability critical state up to a fixed moment of time, the steady availability coefficients and the availability coefficients in a fixed time interval for the considered systems in the case when they are repairable and the time of their renovation is non-ignored.
3.4 Renewal and Availability of Multistate Systems
93
3.4.1 Multistate Systems with Ignored Time of Renovation We assume here that the considered systems after exceeding the critical reliability state are repaired and that the time of their renovation is very small in comparison to their lifetimes in the reliability state subsets, which is not worse than the critical reliability state, and hence we may omit it. Under this assumption, it is possible to obtain the results for the repairable systems with ignored time of renovation formulated in the following proposition [25, 31]. Proposition 3.4 If components of the multistate repairable system with ignored time of renovation have the exponential reliability functions at the operation states zb ; b ¼ 1; 2; . . .; v; given by (3.13)–(3.14) or respectively by (3.15)–(3.16) and the system reliability critical state is r; r 2 f1; 2; . . .; zg; then: (a) the time SN ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, has approximately normal distribution pffiffiffiffi NðNlðrÞ; N rðrÞÞ; i.e., t NlðrÞ F ðNÞ ðt; rÞ ¼ PðSN \tÞ ffi FNð0;1Þ pffiffiffiffi ; t 2 ð1; 1Þ; r 2 f1; 2; . . .; zg; N rðrÞ ð3:55Þ (b) the expected value and the variance of the time SN ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, are respectively given by E½SN ðrÞ ffi NlðrÞ;
D½SN ðrÞ ffi Nr2 ðrÞ;
r 2 f1; 2; . . .; zg;
ð3:56Þ
(c) the number Nðt; rÞ of exceeding by the system the reliability critical state r up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 1 1 0 0 BðN þ 1ÞlðrÞ tC BNlðrÞ tC qffiffiffiffiffiffi A FNð0;1Þ @ qffiffiffiffiffiffiA; PðNðt; rÞ ¼ NÞ ffi FNð0;1Þ @ t t rðrÞ lðrÞ rðrÞ lðrÞ N ¼ 0; 1; . . .; r 2 f1; 2; . . .; zg;
ð3:57Þ
(d) the expected value and the variance of the number Nðt; rÞ of exceeding by the system the reliability critical state r up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; rÞ ffi
t ; lðrÞ
Dðt; rÞ ffi
t l3 ðrÞ
where lðrÞ and rðrÞ are given by:
r2 ðrÞ;
r 2 f1; 2; . . .; zg;
ð3:58Þ
94
3 Complex Technical Systems, Reliability
(i) for a series system: lðrÞ ¼
v X b¼1
2
r ðrÞ ¼
Zþ1
pb P n
i¼1
1 ðbÞ
ki ðrÞ
;
2tRn ðt; rÞ dt ½lðrÞ2 ;
ð3:59Þ
ð3:60Þ
0
where Rn ðt; rÞ is given by (3.18) for u ¼ r; r 2 f1; 2; . . .; zg; (ii) for a parallel system: lðrÞ ¼
Zþ1
Rn ðt; rÞ dt;
ð3:61Þ
0
½rðrÞ2 ¼
Zþ1
2tRn ðt; rÞ dt ½lðrÞ2 ;
ð3:62Þ
0
where Rn ðt; rÞ is given by (3.20) for u ¼ r; r 2 f1; 2; . . .; zg; (iii) for an ‘‘m out of n’’ system: lðrÞ ¼
Zþ1
Rm n ðt; rÞ dt;
ð3:63Þ
0
2
½rðrÞ ¼
Zþ1
2 2tRm n ðt; rÞ dt ½lðrÞ ;
ð3:64Þ
0
where
Rm n ðt; rÞ
is given by (3.22) for u ¼ r; r 2 f1; 2; . . .; zg; or lðrÞ ¼
Zþ1
m
Rn ðt; rÞ dt;
ð3:65Þ
0
ð2Þ
½rðrÞ ¼
Zþ1
m
2tRn ðt; rÞ dt ½lðrÞ2 ;
0 m
where Rn ðt; rÞ is given by (3.24) for u ¼ r; r 2 f1; 2; . . .; zg;
ð3:66Þ
3.4 Renewal and Availability of Multistate Systems
95
(iv) for a consecutive ‘‘m out of n: F’’ system:
lðrÞ ¼
Zþ1
CRm n ðt; rÞ dt;
ð3:67Þ
0
2
½rðrÞ ¼
Zþ1
2 2tCRm n ðt; rÞ dt ½lðrÞ ;
ð3:68Þ
0
where CRm n ðt; rÞ is given by (3.26) for u ¼ r; r 2 f1; 2; . . .; zg; (v) for a series–parallel system: lðrÞ ¼
Zþ1
Rk;l1 l2 ;...;lk ðt; rÞ dt;
ð3:69Þ
0
2
½rðrÞ ¼
Zþ1
2tRk;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ2 ;
ð3:70Þ
0
where Rk;l1 ;l2 ;...;lk ðt; rÞ is given by (3.28) for u ¼ r; r 2 f1; 2; . . .; zg; (vi) for a parallel–series system:
lðrÞ ¼
Zþ1
Rk;l1 l2;...;lk ðt; rÞ dt;
ð3:71Þ
0
2
½rðrÞ ¼
Zþ1
2tRk;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ2 ;
ð3:72Þ
0
where Rk;l1 ;l2 ;...;lk ðt; rÞ is given by (3.30) for u ¼ r; r 2 f1; 2; . . .; zg; (vii) for a series-‘‘m out of k’’ system:
lðrÞ ¼
Zþ1
Rm k;l1 ;l2 ;...;lk ðt; rÞ dt;
ð3:73Þ
0
2
½rðrÞ ¼
Zþ1
2 2tRm k;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ ;
ð3:74Þ
0
where
Rm k;l1 ;l2 ;...;lk ðt; rÞ
is given by (3.32) for u ¼ r; r 2 f1; 2; . . .; zg; or
96
3 Complex Technical Systems, Reliability
lðrÞ ¼
Zþ1
m
Rk;l1 ;l2 ;...;lk ðt; rÞ dt;
ð3:75Þ
0
2
½rðrÞ ¼
Zþ1
m
2tRk;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ2 ;
ð3:76Þ
0 m
where Rk;l1 ;l2 ;...;lk ðt; rÞ is given by (3.34) for u ¼ r; r 2 f1; 2; . . .; zg; (viii) for an ‘‘mi out of li ’’-series system: lðrÞ ¼
Zþ1
1 ;m2 ;...;mk Rm k;l1 ;l2 ;...;lk ðt; rÞ dt;
ð3:77Þ
0
2
½rðrÞ ¼
Zþ1
2 1 ;m2 ;...;mk 2tRm k;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ ;
ð3:78Þ
0 1 ;m2 ;...;mk where Rm k;l1 ;l2 ;...;lk ðt; rÞ is given by (3.36) for u ¼ r; r 2 f1; 2; . . .; zg; or
lðrÞ ¼
Zþ1
m ;m ;...;m
Rk;l11 ;l22;...;lk k ðt; rÞ dt;
ð3:79Þ
0
½rðrÞ2 ¼
Zþ1
m ;m ;...;m
2tRk;l11 ;l22;...;lk k ðt; rÞ dt ½lðrÞ2 ;
ð3:80Þ
0 m ;m ;...;m
where Rk;l11 ;l22;...;lk k ðt; rÞ is given by (3.38) for u ¼ r; r 2 f1; 2; . . .; zg; (ix) for a series-consecutive ‘‘m out of k: F’’ system:
lðrÞ ¼
Zþ1
CRm k;l1 ;l2 ;...;lk ðt; rÞ dt;
ð3:81Þ
0
2
½rðrÞ ¼
Zþ1
2 2tCRm k;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ ;
ð3:82Þ
0
where CRm k;l1 ;l2 ;...;lk ðt; rÞ is given by (3.40)–(3.41) for u ¼ r; r 2 f1; 2; . . .; zg; (x) for a consecutive ‘‘m out of l: F’’-series system:
3.4 Renewal and Availability of Multistate Systems
lðrÞ ¼
Zþ1
m ;m ;...;m
97
CRk;l11 ;l22;...;lk k ðt; rÞ dt;
ð3:83Þ
m1 ;m2 ;...;mk
ð3:84Þ
0
½rðrÞ2 ¼
Zþ1
2tCRk;l1 ;l2 ;...;lk ðt; rÞ dt ½lðrÞ2 ;
0 m ;m ;...;m
where CRk;l11 ;l22;...;lk k ðt; rÞ is given by (3.43)–(3.44) for u ¼ r; r 2 f1; 2; . . .; zg:
3.4.2 Multistate Systems with Non-Ignored Time of Renovation We assume here that the considered systems after exceeding the critical reliability state are repaired and that the time of their renovation is not very small in comparison to their lifetimes in the reliability state subsets, which is not worse than the critical reliability state, and hence we may not omit it. Under this assumption, it is possible to obtain the results formulated in the following proposition [25, 31]. Proposition 3.5 If components of the multistate repairable system with nonignored time of renovation have the exponential reliability functions at the operation states zb ; b ¼ 1; 2; . . .; v; given by (3.13)–(3.14) or respectively by (3.15)–(3.16), the system reliability critical state is r; r 2 f1; 2; . . .; zg; and the successive times of the system’s renovations are independent and have an identical distribution function with the expected value l0 ðrÞ and the variance r20 ðrÞ; then: (a) the time SN ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðNlðrÞ þ ðN 1Þl0 ðrÞÞ; Nr2 ðrÞ þ ðN 1Þr20 ðrÞÞ; i.e.,
F
ðNÞ
ðt; rÞ ¼ PðSN ðrÞ\tÞ ffi FNð0;1Þ
! t NðlðrÞ þ l0 ðrÞÞ þ l0 ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Nðr2 ðrÞ þ r20 ðrÞÞ r20 ðrÞ
t 2 ð1; 1Þ; r 2 f1; 2; . . .; zg;
ð3:85Þ
98
3 Complex Technical Systems, Reliability
(b) the expected value and the variance of the time SN ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, are respectively given by E½SN ðrÞ ffi NlðrÞ þ ðN 1Þl0 ðrÞ; r 2 f1; 2; . . .; zg;
D½SN ðrÞ ffi Nr2 ðrÞ þ ðN 1Þr20 ðrÞ; ð3:86Þ
(c) the number Nðt; rÞ of exceeding by the system the reliability critical state r up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 1 0 BðN þ 1ÞðlðrÞ þ l0 ðrÞÞ t l0 ðrÞC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; rÞ ¼ NÞ ffi FNð0;1Þ @ A tþl0 ðrÞ 2 ðrÞ þ r2 ðrÞÞ ðr 0 lðrÞþl0 ðrÞ 1 0 BNðlðrÞ þ l ðrÞÞ t l0 ðrÞC FNð0;1Þ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 A; tþl0 ðrÞ 2 ðrÞ þ r2 ðrÞÞ ðr 0 lðrÞþl ðrÞ
ð3:87Þ
0
N ¼ 0; 1; . . .; r 2 f1; 2; . . .; zg; (d) the expected value and the variance of the number Nðt; rÞ of exceeding by the system the reliability critical state r up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; rÞ ffi
t þ l0 ðrÞ ; lðrÞ þ l0 ðrÞ
Dðt; rÞ ffi
t þ l0 ðrÞ ðlðrÞ þ l0 ðrÞÞ3
ðr2 ðrÞ þ r20 ðrÞÞ;
ð3:88Þ
r 2 f1; 2; . . .; zg; (e) the time SN ðrÞ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi approximately normal distribution NðNðlðrÞ þ l0 ðrÞÞ; Nðr2 ðrÞ þ r20 ðrÞÞÞ; i.e.,
F
ðNÞ
ðt; rÞ ¼ PðSN ðrÞ\tÞ ffi FNð0;1Þ
r 2 f1; 2; . . .; zg;
! t NðlðrÞ þ l0 ðrÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Nðr2 ðrÞ þ r20 ðrÞÞ
t 2 ð1; 1Þ; ð3:89Þ
(f) the expected value and the variance of the time SN ðrÞ until the Nth system’s renovation, for sufficiently large N, are respectively given by
3.4 Renewal and Availability of Multistate Systems
E½SN ðrÞ ffi NðlðrÞ þ l0 ðrÞÞ;
99
D½SN ðrÞ ffi Nðr2 ðrÞ þ r20 ðrÞÞ;
r 2 f1; 2; . . .; zg; ð3:90Þ
(g) the number Nðt; rÞ of the system’s renovations up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 0
1
BðN þ 1ÞðlðrÞ þ l0 ðrÞÞ tC PðNðt; rÞ ¼ NÞ ffi FNð0;1Þ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A t 2 2 lðrÞþl0 ðrÞðr ðrÞ þ r0 ðrÞÞ 0 1 B NðlðrÞ þ l0 ðrÞÞ t C FNð0;1Þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A; t 2 2 lðrÞþl ðrÞðr ðrÞ þ r0 ðrÞÞ
N ¼ 0; 1; . . .; r 2 f1; 2; . . .; zg;
0
ð3:91Þ (h) the expected value and the variance of the number Nðt; rÞ of system’s renova-tions up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; rÞ ffi
t ; lðrÞ þ l0 ðrÞ
Dðt; rÞ ffi
t ðlðrÞ þ l0 ðrÞÞ3
ðr2 ðrÞ þ r20 ðrÞÞ;
ð3:92Þ
r 2 f1; 2; . . .; zg; (i)
the steady availability coefficient of the system at the moment t; t 0; for suffi-ciently large t, is given by
Aðt; rÞ ffi
(j)
lðrÞ ; lðrÞ þ l0 ðrÞ
t 0; r 2 f1; 2; . . .; zg;
ð3:93Þ
the steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; for sufficiently large t, is given by 1 Aðt; s; rÞ ffi lðrÞ þ l0 ðrÞ
Z1 Rðt; rÞ dt;
t 0; s [ 0; r 2 f1; 2; . . .; zg;
ð3:94Þ
s
where lðrÞ and rðrÞ are given respectively for the considered system by corresponding formulae selected from (3.59)–(3.84) and Rðt; rÞ should be replaced by that corresponding to this system formula selected from (3.17)– (3.44) for u ¼ r:
100
3 Complex Technical Systems, Reliability
3.5 Safety of Multistate Systems at Variable Operation Conditions We assume that the changes of the system operation process ZðtÞ states have an influence on the system multistate components Ei ; i ¼ 1; 2; . . .; n; safety and the ðbÞ ðbÞ ðbÞ system safety structure as well. We mark by T1 ðuÞ; T2 ðuÞ; . . .; Tn ðuÞ the system components E1 ; E2 ; . . .; En conditional lifetimes in the safety states subset fu; u þ 1; zg; u ¼ 1; 2; . . .; z; and by T ðbÞ ðuÞ the system conditional lifetimes in the safety states subset fu; u þ 1; zg; u ¼ 1; 2; . . .; z; while the system is at the operation state zb ; b ¼ 1; 2; . . .; v: Further, we define the conditional safety function of the system multistate component Ei ; i ¼ 1; 2; . . .; n; while the system is at the operation state zb ; b ¼ 1; 2; . . .; v; by the vector [33, 50] h i ½si ðt; ÞðbÞ ¼ 1; ½si ðt; 1ÞðbÞ ; . . .; ½si ðt; zÞðbÞ ; ð3:95Þ where ðbÞ
½si ðt; uÞðbÞ ¼ PðTi ðuÞ [ tjZðtÞ ¼ zb Þ
ð3:96Þ
for t 2 h0; 1Þ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v; and the conditional safety function of the multistate system while the system is at the operation state zb ; b ¼ 1; 2; . . .; v; by the vector [33, 50] ½sðt; ÞðbÞ ¼ ½1; ½sðt; 1ÞðbÞ ; . . .; ½sðt; zÞðbÞ ;
ð3:97Þ
½sðt; uÞðbÞ ¼ PðT ðbÞ ðuÞ [ tjZðtÞ ¼ zb Þ
ð3:98Þ
where
for t 2 h0; 1Þ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m: The system conditional lifetimes ðbÞ
ðbÞ
T ðbÞ ðuÞ ¼ TðT1 ðuÞ; T2 ðuÞ; . . .; TnðbÞ ðuÞÞ at the operation states zb ; defined for u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m; n 2 N; are ðbÞ ðbÞ dependent on the system components conditional lifetimes T1 ðuÞ; T2 ðuÞ; . . .; ðbÞ
Tn ðuÞ; at the operation state zb and the coordinates of the system conditional multistate safety functions h iðbÞ ½sðt; uÞðbÞ ¼ sð½s1 ðt; uÞðbÞ ; ½s2 ðt; uÞðbÞ ; . . .; ½sn ðt; uÞðbÞ Þ at the operational state zb ; defined for t 2 h0; 1Þ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m; n 2 N; are dependent on the components, conditional safety function ½s1 ðt; uÞðbÞ ; ½s2 ðt; uÞðbÞ ; . . .; ½sn ðt; uÞðbÞ at the operation state zb :
3.5 Safety of Multistate Systems at Variable Operation Conditions
101
The safety function ½si ðt; uÞðbÞ is the conditional probability that the component ðbÞ Ei lifetime Ti ðuÞ in the safety state subset fu; u þ 1; . . .; zg is greater than t, while the process ZðtÞ is at the operation state zb : Similarly, the safety function ½sðt; uÞðbÞ is the conditional probability that the system lifetime T ðbÞ ðuÞ in the safety state subset fu; u þ 1; . . .; zg is greater than t, while the process ZðtÞ is at the operation state zb : Consequently, we mark by TðuÞ the system unconditional lifetime in the safety states subset fu; u þ 1; zg; u ¼ 1; 2; . . .; z; and we define the system unconditional safety function by the vector sðt; Þ ¼ ½1; sðt; 1Þ; . . .; sðt; zÞ;
ð3:99Þ
where sðt; uÞ ¼ PðTðuÞ [ tÞ
for t 2 h0; 1Þ; u ¼ 1; 2; . . .; z;
ð3:100Þ
In the case when the system operation time h is large enough, the system unconditional safety function is given by [33, 50] sðt; uÞ ffi
v X
pb ½sðt; uÞðbÞ
for t 0; u ¼ 1; 2; . . .; z;
ð3:101Þ
b¼1
where ½sðt; uÞðbÞ ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m; are the coordinates of the system conditional safety functions defined by (3.97)–(3.98) and pb ; b ¼ 1; 2; . . .; m; are the system operation process limit transient probabilities given by (2.22). In safety analysis of large multistate systems at the variable operation conditions, we assume the following definitions. Definition 3.2 A safety function Sðt; Þ ¼ ½1; Sðt; 1Þ; . . .; Sðt; zÞ;
t 2 ð1; 1Þ;
ð3:102Þ
where Sðt; uÞ ¼
v X
pb ½Sðt; uÞðbÞ ;
u ¼ 1; 2; . . .; z;
ð3:103Þ
b¼1
is called a limit safety function of a complex multistate system with the safety function sequence sn ðt; Þ ¼ ½1; sn ðt; 1Þ; . . .; sn ðt; zÞ;
t 2 ð1; 1Þ; n 2 N;
where sn ðt; uÞ ffi
v X b¼1
pb ½sn ðt; uÞðbÞ ;
u ¼ 1; 2; . . .; z;
102
3 Complex Technical Systems, Reliability
if there exist normalizing constants aðbÞ n ðuÞ [ 0;
bðbÞ n ðuÞ 2 ð1; 1Þ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v;
such that ðbÞ ðbÞ lim ½sn ðaðbÞ ¼ ½Sðt; uÞðbÞ n ðuÞt þ bn ðuÞ; uÞ
n!1
for all continuity points t of the safety functions ½Sðt; uÞðbÞ ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v: Hence, for sufficiently large n, the following approximate formulae are valid: sn ðt; Þ ¼ ½sn ðt; 0Þ; sn ðt; 1Þ; . . .; sn ðt; zÞ;
t 2 ð1; 1Þ;
ð3:104Þ
where sn ðt; uÞ ffi
v X
" pb S
b¼1
!#ðbÞ
ðbÞ
t bn ðuÞ ðbÞ
an ðuÞ
;u
;
t 2 ð1; 1Þ; u ¼ 1; 2; . . .; z: ð3:105Þ
From the definitions and agreements introduced in this chapter, it follows that all results concerning reliability analysis of multistate systems at the variable operation conditions determined in Sects. 3.2–3.4 can be directly used for safety analysis of the multistate systems at the variable operation conditions after replacing the appropriate notions of the multistate reliability analysis by the corresponding notions of the multistate safety analysis.
3.6 Applications 3.6.1 Reliability, Renewal and Availability of Exemplary System at Variable Operation Conditions The exemplary technical system reliability analysis considered here at the constant operation conditions is performed in Sect. 1.4.1, whereas its varying in time operation process is analyzed in Sect. 2.3.1. In Sect. 2.3.1, it is fixed that the exemplary system reliability structure and its subsystems and components reliability depend on its changing in time operation states. Considering the assumptions and agreements of these sections, we assume that its subsystems St ; t ¼ 1; 2; ðtÞ are composed of four-state, i.e. z = 3, components Eij ; t ¼ 1; 2; having the conditional reliability functions given by the vector
3.6 Applications
103
h iðbÞ h iðbÞ h iðbÞ h iðbÞ ðtÞ ðtÞ ðtÞ ðtÞ ; Rij ðt; Þ ¼ 1; Rij ðt; 1Þ ; Rij ðt; 2Þ ; Rij ðt; 3Þ
b ¼ 1; 2; 3; 4;
with the exponential co-ordinates ðtÞ
ðtÞ
ðtÞ
½Rij ðt; 1ÞðbÞ ¼ exp½½kij ð1ÞðbÞ ; ðtÞ
ðtÞ
½Rij ðt; 2ÞðbÞ ¼ exp½½kij ð2ÞðbÞ ;
ðtÞ
½Rij ðt; 3ÞðbÞ ¼ exp½½kij ð3ÞðbÞ ; different at various operation states zb ; b ¼ 1; 2; 3; 4; and with the intensities of departure from the reliability state subsets f1; 2; 3g; f2; 3g; f3g; respectively h
ðtÞ
kij ð1Þ
iðbÞ
;
h iðbÞ ðtÞ kij ð2Þ ;
h iðbÞ ðtÞ kij ð3Þ ;
b ¼ 1; 2; 3; 4:
The influence of the system operation states changing on the changes in the system reliability structure and its components reliability functions is as follows. At the system operation state z1 ; the system is composed of the series–parallel subsystem S1 with the structure shown in Fig. 2.8, containing two identical series subsystems (k ¼ 2), each composed of three components ðl1 ¼ 3; l2 ¼ 3Þ with the exponential reliability functions partly identified in Chap. 5 and given below. In both series subsystems of subsystem S1 there are respectively: ð1Þ
• the components Ei1 ; i ¼ 1; 2; with the conditional reliability function coordinates ð1Þ
ð1Þ
½Ri1 ðt; 1Þð1Þ ¼ exp½0:0008t; ½Ri1 ðt; 2Þð1Þ ¼ exp½0:0009t; ð1Þ ½Ri1 ðt; 3Þð1Þ ¼ exp½0:0009t; i ¼ 1; 2; ð1Þ
• the components Ei2 ; i ¼ 1; 2; with the conditional reliability function co-ordinates ð1Þ
ð1Þ
½Ri2 ðt; 1Þð1Þ ¼ exp½0:0011t; ½Ri2 ðt; 2Þð1Þ ¼ exp½0:0011t; ð1Þ ½Ri2 ðt; 3Þð1Þ ¼ exp½0:0012t; i ¼ 1; 2; ð1Þ
• the components Ei3 ; i ¼ 1; 2; with the conditional reliability function co-ordinates ð1Þ
ð1Þ
½Ri3 ðt; 1Þð1Þ ¼ exp½0:0011t; ½Ri3 ðt; 2Þð1Þ ¼ exp½0:0011t; ð1Þ ½Ri3 ðt; 3Þð1Þ ¼ exp½0:0011t i ¼ 1; 2: Thus, at the operational state z1 ; the system is identical with subsystem S1 that is a four-state series–parallel system with its structure shape parameters k ¼ 2; l1 ¼ 3; l2 ¼ 3; and according to (1.36)–(1.37) and (1.68)–(1.69), its conditional reliability function is given by
104
3 Complex Technical Systems, Reliability
h i ½Rðt; Þð1Þ ¼ 1; ½Rðt; 1Þð1Þ ; ½Rðt; 2Þð1Þ ; ½Rðt; 3Þð1Þ ;
t 0;
ð3:106Þ
where ½Rðt; 1Þ
ð1Þ
¼ R2;3;3 ðt; 1Þ ¼ 1 ¼1
2 Y
"
" 2 Y
1
3 Y
i¼1
"
j¼1 3 X
1 exp
i¼1
# ð1Þ ½Rij ðt; 1Þð1Þ
##
ð1Þ ½kij ð1Þð1Þ t
j¼1
¼ 1 ½1 exp½½0:0008 þ 0:0011 þ 0:0011t2
½Rðt; 2Þð1Þ
¼ 1 ½1 exp½0:003t2 ¼ 2 exp½0:003t exp½0:006t; " # 2 3 Y Y ð1Þ ¼ R2;3;3 ðt; 2Þ ¼ 1 1 ½Rij ðt; 2Þð1Þ "
¼1
2 Y
"
i¼1
1 exp
j¼1
3 X
i¼1
ð3:107Þ
##
ð1Þ ½kij ð2Þð1Þ t
j¼1
¼ 1 ½1 exp½½0:0009 þ 0:0011 þ 0:0011t2 ¼ 1 ½1 exp½0:0031t2
½Rðt; 3Þð1Þ
¼ 2 exp½0:0031t exp½0:0062t; " # 2 3 Y Y ð1Þ ð1Þ ¼ R2;3;3 ðt; 3Þ ¼ 1 1 ½Rij ðt; 3Þ ¼1
" 2 Y i¼1
"
i¼1
1 exp
j¼1
3 X
ð3:108Þ
##
ð1Þ ½kij ð3Þð1Þ t
j¼1
¼ 1 ½1 exp½½0:0009 þ 0:0012 þ 0:0011t2 ¼ 1 ½1 exp½0:0032t2 ¼ 2 exp½0:0032t exp½0:0064t:
ð3:109Þ
The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets f1; 2; 3g; f2; 3g; f3g at the operation state z1 ; calculated from the results given by (3.107)–(3.109), according to (3.8)–(3.9), respectively are: l1 ð1Þ ffi 505; r1 ð1Þ ffi 365:87;
l1 ð2Þ ffi 483:87; r1 ð2Þ ffi 360:66;
l1 ð3Þ ffi 468:73; r1 ð3Þ ffi 349:41;
ð3:110Þ ð3:111Þ
3.6 Applications
105
and further, using (3.10) and (3.110), the mean values of the conditional lifetimes in the particular reliability states 1, 2, 3 at the operation state z1 ; respectively are: l1 ð1Þ21:13;
l1 ð2Þ ffi 15:14;
l1 ð3Þ ffi 468:7:
ð3:112Þ
At the system operation state z2 ; the system is composed of the series–parallel subsystem S2 with the structure shown in Fig. 2.9, containing four identical series subsystems (k ¼ 4), each composed of two components ðl1 ¼ 2; l2 ¼ 2; l3 ¼ 2; l4 ¼ 2Þ with the exponential reliability functions given below. In all series subsystems of subsystem S2 there are respectively: ð2Þ
• the components Ei1 ; i ¼ 1; 2; 3; 4; with the conditional reliability function co-ordinates ð2Þ
ð2Þ
½Ri1 ðt; 1Þð2Þ ¼ exp½0:0013t; ½Ri1 ðt; 2Þð2Þ ¼ exp½0:0014t; ð2Þ ½Ri1 ðt; 3Þð2Þ ¼ exp½0:0015t; i ¼ 1; 2; 3; 4; ð2Þ
• the components Ei2 ; i ¼ 1; 2; 3; 4; with the conditional reliability function coordinates ð2Þ
ð2Þ
½Ri2 ðt; 1Þð2Þ ¼ exp½0:0015t; ½Ri2 ðt; 2Þð2Þ ¼ exp½0:0016t; ð2Þ ½Ri2 ðt; 3Þð2Þ ¼ exp½0:0017t; i ¼ 1; 2; 3; 4:
Thus, at the operation state z2 ; the system is identical with subsystem S2 that is a four-state series–parallel system with its structure shape parameters k ¼ 4; l1 ¼ 2; l2 ¼ 2; l3 ¼ 2; l4 ¼ 2 and according to (1.36)–(1.37) and (1.68)–(1.69), its conditional reliability function is given by h i ½Rðt; Þð2Þ ¼ 1; ½Rðt; 1Þð2Þ ; ½Rðt; 2Þð2Þ ; ½Rðt; 3Þð2Þ ; t 0; ð3:113Þ where ½Rðt; 1Þ
ð2Þ
¼ R4;2;2;2;2 ðt; 1Þ ¼ 1 ¼1
4 Y i¼1
"
"
" 4 Y
2 Y ð2Þ 1 ½Rij ðt; 1Þð2Þ
i¼1
j¼1
2 X ð2Þ 1 exp ½kij ð1Þð2Þ t
#
##
j¼1
¼ 1 ½1 exp½½0:0013 þ 0:0015t4 ¼ 1 ½1 exp½0:0028t4 ¼ 4 exp½0:0028t 6 exp½0:0056t þ 4 exp½0:0084t exp½0:0112t;
ð3:114Þ
106
½Rðt; 2Þ
3 Complex Technical Systems, Reliability
ð2Þ
¼ R4;2;2;2;2 ðt; 2Þ ¼ 1 ¼1
" 4 Y
"
" 4 Y
1
i¼1
2 Y
# ð2Þ ½Rij ðt; 2Þð2Þ
j¼1
2 X ð2Þ 1 exp ½kij ð2Þð2Þ t
i¼1
##
j¼1
¼ 1 ½1 exp½½0:0014 þ 0:0016t4 ¼ 1 ½1 exp½0:003t4 ¼ 4 exp½0:003t 6 exp½0:006t þ 4 exp½0:009t exp½0:012t; ð3:115Þ ½Rðt; 3Þð2Þ ¼ R4;2;2;2;2 ðt; 3Þ ¼ 1 ¼1
4 Y
"
" 1 exp
i¼1
4 Y
" 1
i¼1 2 X
2 Y
# ð2Þ
½Rij ðt; 3Þð2Þ
j¼1
##
ð2Þ ½kij ð3Þð2Þ t
j¼1
¼ 1 ½1 exp½½0:0015 þ 0:0017t4 ¼ 1 ½1 exp½0:0032t4 ¼ 4 exp½0:0032t 6 exp½0:0064t þ 4 exp½0:0096t exp½0:0128t:
ð3:116Þ The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets f1; 2; 3g; f2; 3g; f3g at the operation state z2 ; calculated from the results given by (3.114)–(3.116), according to (3.8)–(3.9), respectively are: l2 ð1Þ ffi 744:05;
l2 ð2Þ ffi 694:44;
l2 ð3Þ ffi 651:04;
ð3:117Þ
r2 ð1Þ ffi 426:12;
r2 ð2Þ ffi 397:76;
r2 ð3Þ ffi 372:86;
ð3:118Þ
and further, using (3.10) and (3.117), the mean values of the conditional lifetimes in the particular reliability states 1, 2, 3at the operation state z2 ; respectively are: l2 ð1Þ ffi 49:61;
l2 ð2Þ ffi 43:4;
l2 ð3Þ ffi 651:04:
ð3:119Þ
At the system operational state z3 ; the system is a series system with the structure shown in Fig. 1.12, composed of two series–parallel subsystems S1 and S2 illustrated respectively in Figs. 2.8 and 2.9. Subsystem S1 with the structure shown in Fig. 2.8, consists of two identical series subsystems (k ¼ 2) each composed of three components (l1 ¼ 3; l2 ¼ 3) with the exponential reliability functions given below. In both series subsystems of subsystem S1 there are respectively: ð1Þ
• the components Ei1 ; i ¼ 1; 2; with the conditional reliability function co-ordinates
3.6 Applications
107
ð1Þ
ð1Þ
½Ri1 ðt; 1Þð3Þ ¼ exp½0:0009t; ½Ri1 ðt; 2Þð3Þ ¼ exp½0:001t; ð1Þ ½Ri1 ðt; 3Þð3Þ ¼ exp½0:001t; i ¼ 1; 2; ð1Þ
• the components Ei2 ; i ¼ 1; 2; with the conditional reliability function co-ordinates ð1Þ
ð1Þ
½Ri2 ðt; 1Þð3Þ ¼ exp½0:0012t; ½Ri2 ðt; 2Þð3Þ ¼ exp½0:0012t; ð1Þ ½Ri2 ðt; 3Þð3Þ ¼ exp½0:0013t; i ¼ 1; 2; ð1Þ
• the components Ei3 ; i ¼ 1; 2; with the conditional reliability function co-ordinates ð1Þ
ð1Þ
½Ri3 ðt; 1Þð3Þ ¼ exp½0:0011t; ½Ri3 ðt; 2Þð3Þ ¼ exp½0:0012t; ð1Þ ½Ri3 ðt; 3Þð3Þ ¼ exp½0:0012t; i ¼ 1; 2: Thus, at the operation state z3 ; subsystem S1 is a four-state series–parallel system with its structure shape parameters k ¼ 2; l1 ¼ 3; l2 ¼ 3; and according to (1.36)–(1.37) and (1.68)–(1.69), its conditional reliability function is given by h ið3Þ h ið3Þ h ið3Þ h ið3Þ ð1Þ ð1Þ ð1Þ ð1Þ ; t 0; R ðt; Þ ¼ 1; R ðt; 1Þ ; R ðt; 2Þ ; R ðt; 3Þ ð3:120Þ where ð1Þ
ð3Þ
½R ðt; 1Þ
¼ R2;3;3 ðt; 1Þ ¼ 1 ¼1
" 2 Y
" 2 Y
"
1
i¼1
1 exp
# ð1Þ ½Rij ðt; 1Þð3Þ
j¼1
3 X
i¼1
3 Y
##
ð1Þ ½kij ð1Þð3Þ t
j¼1
¼ 1 ½1 exp½½0:0009 þ 0:0012 þ 0:0011t2 ¼ 1 ½1 exp½0:0032t2 ¼ 2 exp½0:0032t exp½0:0064t; " # 2 3 Y Y ð1Þ ð3Þ ð3Þ ð1Þ 1 ½Rij ðt; 2Þ ½R ðt; 2Þ ¼ R2;3;3 ðt; 2Þ ¼ 1 "
¼1
2 Y i¼1
"
i¼1
1 exp
j¼1
3 X
ð3:121Þ
##
ð1Þ ½kij ð2Þð3Þ t
j¼1
¼ 1 ½1 exp½½0:001 þ 0:0012 þ 0:0012t2 ¼ 1 ½1 exp½0:0034t2 ¼ 2 exp½0:0034t exp½0:0068t;
ð3:122Þ
108
3 Complex Technical Systems, Reliability
½Rðt; 3Þ
ð1Þ
¼ R2;3;3 ðt; 3Þ ¼ 1 ¼1
" 2 Y
" 2 Y
"
1
i¼1
1 exp
i¼1
3 Y
# ð1Þ ½Rij ðt; 3Þð1Þ
j¼1
3 X
##
ð1Þ ½kij ð3Þð1Þ t
j¼1
¼ 1 ½1 exp½½0:001 þ 0:0013 þ 0:0012t2 ¼ 1 ½1 exp½0:0035t2 ¼ 2 exp½0:0035t exp½0:007t:
ð3:123Þ
Subsystem S2 with the structure shown in Fig. 2.9, consists of four identical series subsystems (k ¼ 4), each composed of two components ðl1 ¼ 2; l2 ¼ 2; l3 ¼ 2; l4 ¼ 2Þ with the exponential reliability functions given below. In all series subsystems of subsystem S2 there are respectively: ð2Þ
• the components Ei1 ; i ¼ 1; 2; 3; 4; with the conditional reliability function co-ordinates ð2Þ
ð2Þ
½Ri1 ðt; 1Þð3Þ ¼ exp½0:0009t; ½Ri1 ðt; 2Þð3Þ ¼ exp½0:001t; ¼ exp½0:001t; i ¼ 1; 2; 3; 4;
ð2Þ
½Ri1 ðt; 3Þð3Þ
ð2Þ
• the components Ei2 ; i ¼ 1; 2; 3; 4; with the conditional reliability function co-ordinates ð2Þ
½Ri2 ðt; 1Þð3Þ ¼ exp½0:0012t; ¼ exp½0:0013t;
ð2Þ
½Ri2 ðt; 2Þð3Þ ¼ exp½0:0012t;
ð2Þ
½Ri2 ðt; 3Þð3Þ
i ¼ 1; 2; 3; 4:
Thus, at the operation state z3 ; subsystem S2 is a four-state series–parallel system with its structure shape parameters k ¼ 4; l1 ¼ 2; l2 ¼ 2; l3 ¼ 2; l4 ¼ 2; and according to (1.36)–(1.37) and (1.68)–(1.69), its conditional reliability function is given by ½Rð2Þ ðt; Þð3Þ ¼ ½1; ½Rð2Þ ðt; 1Þð3Þ ; ½Rð2Þ ðt; 2Þð3Þ ; ½Rð2Þ ðt; 3Þð3Þ ;
t 0;
ð3:124Þ
where ð2Þ
ð3Þ
½R ðt; 1Þ
¼ R4;2;2;2;2 ðt; 1Þ ¼ 1 ¼1
" 4 Y i¼1
" 1 exp
" 4 Y
1
i¼1 2 X
2 Y
# ð2Þ ½Rij ðt; 1Þð3Þ
j¼1
##
ð2Þ ½kij ð1Þð3Þ t
j¼1
¼ 1 ½1 exp½½0:0009 þ 0:0012t4 ¼ 1 ½1 exp½0:0021t4 ¼ 4 exp½0:0021t 6 exp½0:0042t þ 4 exp½0:0063t exp½0:0084t;
ð3:125Þ
3.6 Applications
ð2Þ
ð3Þ
½R ðt; 2Þ
109
¼ R4;2;2;2;2 ðt; 2Þ ¼ 1 ¼1
" 4 Y
" 1 exp
i¼1
" 4 Y
1
i¼1 2 X
2 Y
# ð2Þ ½Rij ðt; 2Þð3Þ
j¼1
##
ð2Þ ½kij ð2Þð3Þ t
j¼1
¼ 1 ½1 exp½½0:001 þ 0:0012t4 ¼ 1 ½1 exp½0:0022t4 ¼ 4 exp½0:0022t 6 exp½0:0044t þ 4 exp½0:0066t exp½0:0088t;
ð3:126Þ ð2Þ
ð3Þ
½R ðt; 3Þ
¼ R4;2;2;2;2 ðt; 3Þ ¼ 1
" 4 Y
1
i¼1
¼1
" 4 Y i¼1
" 1 exp
2 X
2 Y
# ð2Þ ½Rij ðt; 3Þð3Þ
j¼1
## ð2Þ ½kij ð3Þð3Þ t
j¼1
¼ 1 ½1 exp½½0:001 þ 0:0013t4 ¼ 1 ½1 exp½0:0023t4 ¼ 4 exp½0:0023t 6 exp½0:0046t þ 4 exp½0:0069t exp½0:0092t:
ð3:127Þ Considering that the system at the operation state z3 is a four-state series system composed of subsystems S1 and S2 ; after applying (1.22)–(1.23) and (3.120)– (3.123) and (3.124)–(3.127), its conditional reliability function is given by ½Rðt; Þð3Þ ¼ ½1; ½Rðt; 1Þð3Þ ; ½Rðt; 2Þð3Þ ; ½Rðt; 3Þð3Þ ;
t 0;
ð3:128Þ
where ½Rðt; 1Þð3Þ ¼ R2 ðt; 1Þ ¼ ½Rð1Þ ðt; 1Þð3Þ ½Rð2Þ ðt; 1Þð3Þ ¼ 8 exp½0:0053t 12 exp½0:0074t þ 8 exp½0:0095t 2 exp½0:0116t 4 exp½0:0085t þ 6 exp½0:0106t 4 exp½0:0127t þ exp½0:0148t;
ð3:129Þ ½Rðt; 2Þð3Þ ¼ R2 ðt; 2Þ ¼ ½Rð1Þ ðt; 2Þð3Þ ½Rð2Þ ðt; 2Þð3Þ ¼ 8 exp½0:0056t 12 exp½0:0078t þ 8 exp½0:01t 2 exp½0:0122t; 4 exp½0:009t þ 6 exp½0:0112t 4 exp½0:0134t þ exp½0:0156t;
ð3:130Þ
110
3 Complex Technical Systems, Reliability
½Rðt; 3Þð3Þ ¼ R2 ðt; 3Þ ¼ ½Rð1Þ ðt; 3Þð3Þ ½Rð2Þ ðt; 3Þð3Þ ¼ 8 exp½0:0058t 12 exp½0:0081t þ 8 exp½0:0104t 2 exp½0:0127t; 4 exp½0:0093t þ 6 exp½0:00116t 4 exp½0:0139t þ exp½0:0162t:
ð3:131Þ The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets f1; 2; 3g; f2; 3g; f3g at the operation state z3 ; calculated from the results given by (3.128)–(3.131), according to (3.8)–(3.9), respectively are: l3 ð1Þ ffi 405:56;
l3 ð2Þ ffi 383:04;
l3 ð3Þ ffi 370:67;
ð3:132Þ
r3 ð1Þ ffi 264:58;
r3 ð2Þ ffi 250:39;
r3 ð3Þ ffi 241:78;
ð3:133Þ
and further, using (3.10) and (3.132), the mean values of the conditional lifetimes in the particular reliability states 1, 2, 3at the operation state z3 ; respectively are: l3 ð1Þ22:52;
l3 ð2Þ ffi 12:37;
l3 ð3Þ ffi 370:67:
ð3:134Þ
At the system operation state z4 ; the system is a series system with the scheme shown in Fig. 1.12, composed of subsystems S1 and S2 illustrated respectively in Figs. 2.8 and 2.9, whereas subsystem S1 is a series–parallel system and subsystem S2 is a series-‘‘2 out of 4’’ system. Subsystem S1 consists of two identical series subsystems (k ¼ 2), each composed of three components (l1 ¼ 3; l2 ¼ 3Þ with the exponential reliability functions the same as at the operation state z3 : Thus, according to (3.120)– (3.123), subsystem S1 unconditional reliability function at the operation state z4 ; is given by ½Rð1Þ ðt; Þð4Þ ¼ ½1; ½Rð1Þ ðt; 1Þð4Þ ; ½Rð1Þ ðt; 2Þð4Þ ; ½Rð1Þ ðt; 3Þð4Þ ;
t 0;
ð3:135Þ
where ½Rð1Þ ðt; 1Þð4Þ ¼ 2 exp½0:0032t exp½0:0064t;
ð3:136Þ
½Rð1Þ ðt; 2Þð4Þ ¼ 2 exp½0:0034t exp½0:0068t;
ð3:137Þ
½Rðt; 3Þð4Þ ¼ 2 exp½0:0035t exp½0:007t:
ð3:138Þ
Subsystem S2 consists of four identical series subsystems (k ¼ 4), each composed of two components ðl1 ¼ 2; l2 ¼ 2; l3 ¼ 2; l4 ¼ 2Þ with the exponential reliability functions given below and is a series-‘‘2 out of 4’’ system (m ¼ 2). In all series subsystems of subsystem S2 there are respectively:
3.6 Applications
111 ð2Þ
• the components Ei1 ; i ¼ 1; 2; 3; 4; with the conditional reliability function co-ordinates ð2Þ
ð2Þ
½Ri1 ðt; 1Þð4Þ ¼ exp½0:0013t; ½Ri1 ðt; 2Þð4Þ ¼ exp½0:0014t; ð2Þ ½Ri1 ðt; 3Þð4Þ ¼ exp½0:0015t; i ¼ 1; 2; 3; 4; ð2Þ
• the components Ei2 ; i ¼ 1; 2; 3; 4; with the conditional reliability function co-ordinates ð2Þ
ð2Þ
½Ri2 ðt; 1Þð4Þ ¼ exp½0:0015t; ½Ri2 ðt; 2Þð4Þ ¼ exp½0:0016t; ð2Þ ½Ri2 ðt; 3Þð4Þ ¼ exp½0:0018t; i ¼ 1; 2; 3; 4: Thus, at the operation state z4 ; subsystem S2 is a four-state series-‘‘2 out of 4’’ system, with its structure shape parameters k ¼ 4; m ¼ 2; l1 ¼ 2; l2 ¼ 2; l3 ¼ 2; l4 ¼ 2; and according to (1.40)–(1.41) and (1.72)–(1.73), its conditional reliability function is given by ½Rð2Þ ðt; Þð4Þ ¼ ½1; ½Rð2Þ ðt; 1Þð4Þ ; ½Rð2Þ ðt; 2Þð4Þ ; ½Rð2Þ ðt; 3Þð4Þ ;
t 0;
ð3:139Þ
where ð2Þ
½R ðt; 1Þ
ð4Þ
¼
R24;2;2;2;2 ðt; 1Þ
" #ri 4 Y 2 Y ð2Þ ð4Þ ½Rij ðt; 1Þ
1 X
¼1
r1 ;r2 ;r3 ;r4 ¼0 i¼1 r1 þr2 þr3 þr4 1
"
2 Y ð2Þ 1 ½Rij ðt; 1Þð4Þ
j¼1
#1ri
j¼1
¼1
1 X
4 Y
"
2 X ð2Þ exp ri ½kij ð1Þð4Þ t
r1 ;r2 ;r3 ;r4 ¼0 i¼1 r1 þr2 þr3 þr4 1
"
"
1 exp
2 X
#
j¼1
##1ri ð2Þ ½kij ð1Þð4Þ t
j¼1
¼ 1 ½1 exp½½0:0013 þ 0:0015t4 4 exp½1½0:0013 þ 0:0015t½1 exp½½0:0013 þ 0:0015t3 ¼ 1 ½1 exp½0:0028t4 4 exp½0:0028t½1 exp½0:0028t3 ¼ 6 exp½0:0056t 8 exp½0:0084t þ 3 exp½0:0112t; ð3:140Þ
112
3 Complex Technical Systems, Reliability
ð2Þ
½R ðt; 2Þ
ð4Þ
¼
R24;2;2;2;2 ðt; 2Þ
¼1
r1 ;r2 ;r3 ;r4 ¼0 i¼1 r1 þr2 þr3 þr4 1
"
2 Y ð2Þ 1 ½Rij ðt; 2Þð4Þ j¼1
¼1 "
" # ri 4 Y 2 Y ð2Þ ð4Þ ½Rij ðt; 2Þ
1 X
1 X
4 Y
#1ri "
exp ri
2 X
r1 ;r2 ;r3 ;r4 ¼0 i¼1 r1 þr2 þr3 þr4 1
"
1 exp
2 X
j¼1
# ð2Þ ½kij ð2Þð4Þ t
j¼1
##1ri
ð2Þ ½kij ð2Þð4Þ t
j¼1
¼ 1 ½1 exp½½0:0014 þ 0:0016t4 4 exp½1½0:0014 þ 0:0016t½1 exp½½0:0014 þ 0:0016t3 ¼ 1 ½1 exp½0:003t4 4 exp½0:003t½1 exp½0:003t3 ¼ 6 exp½0:006t 8 exp½0:009t þ 3 exp½0:012t; ð3:141Þ 1 X
½Rð2Þ ðt; 3Þð4Þ ¼ R24;2;2;2;2 ðt; 3Þ ¼ 1 2 Y ð2Þ 1 ½Rij ðt; 3Þð4Þ j¼1
"
4 Y 2 Y ð2Þ ½Rij ðt; 3Þð4Þ
r1 ;r2 ;r3 ;r4 ¼0 i¼1 r1 þr2 þr3 þr4 1
"
¼1
"
1 X
4 Y
1 exp
2 X
j¼1
#1ri "
2 X ð2Þ exp ri ½kij ð3Þð4Þ t
r1 ;r2 ;r3 ;r4 ¼0 i¼1 r1 þr2 þr3 þr4 1
"
#ri
#
j¼1
##1ri
ð2Þ ½kij ð3Þð4Þ t
j¼1
¼ 1 ½1 exp½½0:0015 þ 0:0018t4 4 exp½1½0:0015 þ 0:0018t½1 exp½½0:0015 þ 0:0018t3 ¼ 1 ½1 exp½0:0033t4 4 exp½0:0033t½1 exp½0:0033t3 ¼ 6 exp½0:0066t 8 exp½0:0099t þ 3 exp½0:0132t: ð3:142Þ Considering that the system at the operation state z4 is a four-state series system composed of subsystems S1 and S2 ; after applying (1.22)–(1.23) and (3.135)– (3.138) and (3.139)–(3.142), its conditional reliability function is given by
3.6 Applications
113
½Rðt; Þð4Þ ¼ ½1; ½Rðt; 1Þð4Þ ; ½Rðt; 2Þð4Þ ; ½Rðt; 3Þð4Þ ;
t 0;
ð3:143Þ
where ½Rðt; 1Þð4Þ ¼ R2 ðt; 1Þ ¼ ½Rð1Þ ðt; 1Þð4Þ ½Rð2Þ ðt; 1Þð4Þ ¼ 12 exp½0:0088t 16 exp½0:0116t þ 6 exp½0:0144t 6 exp½0:012t þ 8 exp½0:0148t 3 exp½0:0176t;
ð3:144Þ
½Rðt; 2Þð4Þ ¼ R2 ðt; 2Þ ¼ ½Rð1Þ ðt; 2Þð4Þ ½Rð2Þ ðt; 2Þð4Þ ¼ 12 exp½0:0094t 16 exp½0:0124t þ 6 exp½0:0154t 6 exp½0:0128t þ 8 exp½0:0158t 3 exp½0:0188t;
ð3:145Þ
½Rðt; 3Þð4Þ ¼ R2 ðt; 3Þ ¼ ½Rð1Þ ðt; 3Þð4Þ ½Rð2Þ ðt; 3Þð4Þ ¼ 12 exp½0:0101t 16 exp½0:0134t þ 6 exp½0:0167t 6 exp½0:0136t þ 8 exp½0:0169t 3 exp½0:0202t:
ð3:146Þ
The expected values and standard deviations of the system conditional lifetimes in the reliability state subsets f1; 2; 3g; f2; 3g; f3g at the operation state z4 ; calculated from the results given by (3.143)–(3.146), according to (3.8)–(3.9), respectively are: l4 ð1Þ ffi 271:08;
l4 ð2Þ ffi 253:88;
l4 ð3Þ ffi 237:05;
ð3:147Þ
r4 ð1Þ ffi 163:8;
r4 ð2Þ ffi 153:35;
r4 ð3Þ ffi 142:58;
ð3:148Þ
and further, using (3.10) and (3.147), the mean values of the conditional lifetimes in the particular reliability states 1, 2, 3at the operation state z4 ; respectively are: l4 ð1Þ ffi 17:20;
l4 ð2Þ ffi 16:83;
l4 ð3Þ ffi 237:05:
ð3:149Þ
In the case when the system operation time is large enough its unconditional four-state reliability function is given by the vector Rðt; Þ ¼ ½1; Rðt; 1Þ; Rðt; 2Þ; Rðt; 3Þ;
t 0;
ð3:150Þ
where according to (3.6) and considering the exemplary system operation process transient probabilities at the operation states determined by (2.32), the vector co-ordinates are given respectively by Rðt; 1Þ ¼ p1 ½Rðt; 1Þð1Þ þ p2 ½Rðt; 1Þð2Þ þ p3 ½Rðt; 1Þð3Þ þ p4 ½Rðt; 1Þð4Þ ¼ 0:214 ½Rðt; 1Þð1Þ þ 0:038 ½Rðt; 1Þð2Þ þ 0:293 ½Rðt; 1Þð3Þ þ 0:455 ½Rðt; 1Þð4Þ
for t 0;
ð3:151Þ
114
3 Complex Technical Systems, Reliability
Fig. 3.1 The graph of the exemplary system reliability function Rðt; Þ coordinates
Rðt; 2Þ ¼ p1 ½Rðt; 2Þð1Þ þ p2 ½Rðt; 2Þð2Þ þ p3 ½Rðt; 2Þð3Þ þ p4 ½Rðt; 2Þð4Þ ¼ 0:214 ½Rðt; 2Þð1Þ þ 0:038 ½Rðt; 2Þð2Þ þ 0:293 ½Rðt; 2Þð3Þ þ 0:455 ½Rðt; 2Þð4Þ
for t 0;
ð3:152Þ
Rðt; 3Þ ¼ p1 ½Rðt; 3Þð1Þ þ p2 ½Rðt; 3Þð2Þ þ p3 ½Rðt; 3Þð3Þ þ p4 ½Rðt; 3Þð4Þ ¼ 0:214 ½Rðt; 3Þð1Þ þ 0:038 ½Rðt; 3Þð2Þ þ 0:293 ½Rðt; 3Þð3Þ þ 0:455 ½Rðt; 3Þð4Þ
for t 0;
ð3:153Þ
where coordinates ½Rðt; 1Þð1Þ ; ½Rðt; 1Þð2Þ ; ½Rðt; 1Þð3Þ ; ½Rðt; 1Þð4Þ are given by (3.107), (3.114), (3.129), (3.144), ½Rðt; 2Þð1Þ ; ½Rðt; 2Þð2Þ ; ½Rðt; 2Þð3Þ ; ½Rðt; 2Þð4Þ are given by (3.108), (3.115), (3.130), (3.145) and ½Rðt; 3Þð1Þ ; ½Rðt; 3Þð2Þ ; ½Rðt; 3Þð3Þ ; ½Rðt; 3Þð4Þ are given by (3.109), (3.116), (3.131), (3.146). The graph of the four-state exemplary system reliability function is illustrated in Fig. 3.1. The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets f1; 2; 3g; f2; 3g; f3g; calculated from the results given by (3.150)–(3.153), according to (3.7)–(3.9) and considering (2.32), (3.110), (3.117), (3.132), (3.147), respectively are: lð1Þ ¼ p1 l1 ð1Þ þ p2 l2 ð1Þ þ p3 l3 ð1Þ þ p4 l4 ð1Þ ¼ 0:214 505 þ 0:038 744:05 þ 0:293 405:56 þ 0:455 271:08 ffi 378:51;
ð3:154Þ rð1Þ ffi 286:77;
ð3:155Þ
lð2Þ ¼ p1 l1 ð2Þ þ p2 l2 ð2Þ þ p3 l3 ð2Þ þ p4 l4 ð2Þ ¼ 0:214 483:87 þ 0:038 694:44 þ 0:293 383:04 þ 0:455 253:88 ffi 357:68;
ð3:156Þ
3.6 Applications
115
rð2Þ ffi 275:18;
ð3:157Þ
lð3Þ ¼ p1 l1 ð3Þ þ p2 l2 ð3Þ þ p3 l3 ð3Þ þ p4 l4 ð3Þ ¼ 0:214 468:73 þ 0:038 651:04 þ 0:293 370:67 þ 0:455 237:05 ffi 341:51;
ð3:158Þ rð3Þ ffi 264:78:
ð3:159Þ
Further, considering (3.10) and (3.154), (3.156) and (3.158), the mean values of the system unconditional lifetimes in the particular reliability states 1, 2, 3, respectively are: lð1Þ ¼ lð1Þ lð2Þ ¼ 20:83; lð3Þ ¼ lð3Þ ¼ 341:51:
lð2Þ ¼ lð2Þ lð3Þ ¼ 16:17;
ð3:160Þ
Since the critical reliability state is r = 2, then the system risk function, according to (3.11), is given by rðtÞ ¼ 1 Rðt; 2Þ
for t 0;
ð3:161Þ
where Rðt; 2Þ is given by (3.152). Hence, by (3.12), the moment when the system risk function exceeds a permitted level, for instance d ¼ 0:05; is s ¼ r1 ðdÞ ffi 70:08:
ð3:162Þ
The graph of the risk function rðtÞ of the exemplary four-state system operating at the variable conditions is given in Fig. 3.2. The reliability characteristics of the exemplary system operating at the variable conditions predicted in this section are different from those determined in Sect. 1.4.1 for this system operating at constant conditions. This fact justifies the sensibility of considering the real systems at the variable operation conditions that is appearing out in a natural way from practice. This approach, upon the good accuracy of the systems’ operation processes identification, makes their reliability prediction more precise. Further, assuming that the system is repaired after the exceeding of its reliability critical state r = 2 and that the time of the system renovation is ignored and applying Proposition 3.4, we obtain the following results: (a) the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð357:68N; 275:18 N Þ; i.e., t 357:68N pffiffiffiffi ; t 2 ð1; 1Þ; F ðNÞ ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ 275:18 N (b) the expected value and the variance of the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2 are respectively given by
116
3 Complex Technical Systems, Reliability
Fig. 3.2 The graph of the exemplary system risk function rðtÞ
E½SN ð2Þ ffi 357:68N;
D½SN ð2Þ ffi 75724:03N;
(c) the number Nðt; 2Þ of exceeding by the system the reliability critical state r = 2 up to the moment t; t 0; for sufficiently large t, approximately has the distribution of the form 357:68ðN þ 1Þ t 357:68N t pffi pffi FNð0;1Þ ; PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 14:55 t 14:55 t N ¼ 0; 1; . . .; (d) the expected value and the variance of the number Nðt; 2Þ of exceeding by the system the reliability critical state r = 2 up to the moment t; t 0; for sufficiently large t, approximately are respectively given by Hðt; 2Þ ffi 0:0028t;
Dðt; 2Þ ffi 0:0016t:
ð3:163Þ
Assuming that the system is repaired after the exceeding of its reliability critical state r = 2 and that the time of the system renovation is non-ignored and it has the mean value l0 ð2Þ ¼ 10 and the standard deviation r0 ð2Þ ¼ 5 and applying Proposition 3.5, we obtain the following results: (a) the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð357:68N þ 10ðN 1Þ; 75724:03N þ 25ðN 1Þ; i.e., t 367:68N þ 10 ðNÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi F ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ ; t 2 ð1; 1Þ; 75749:03N 25 (b) the expected value and the variance of the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, are respectively given by
3.6 Applications
117
E½SN ð2Þ ffi 357:68N þ 10ðN 1Þ;
D½SN ð2Þ ffi 75; 724:03N þ 25ðN 1Þ;
(c) the number Nðt; 2Þ of exceeding by the system the reliability critical state r = 2 up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form PðNðt; 2Þ ¼ NÞ
367:68ðN þ 1Þ t 10 367:68N t 10 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi FNð0;1Þ ; 14:35 t þ 10 14:35 t þ 10 N ¼ 0; 1; . . .; ffi FNð0;1Þ
(d) the expected value and the variance of the number Nðt; 2Þ of exceeding by the system the reliability critical state r = 2 up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; 2Þ ffi
t þ 10 ; 367:68
Dðt; 2Þ ffi 0:0015ðt þ 10Þ;
ð3:164Þ
(e) the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffi approximately normal distribution Nð367:68N; 275:23 N Þ; i.e., ðNÞ t 367:68N pffiffiffiffi ; t 2 ð1; 1Þ; F ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ 275:23 N (f) the expected value and the variance of the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, are respectively given by E½SN ð2Þ ffi 367:68N;
D½SN ð2Þ ffi 75749:03N;
(g) the number Nðt; 2Þ of the system’s renovations up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 367:68ðN þ 1Þ t 367:68N t pffi pffi FNð0;1Þ ; PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 14:35 t 14:35 t N ¼ 0; 1; . . .; (h) the expected value and the variance of the number Nðt; 2Þ of system’s renovations up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; 2Þ ffi 0:0027t; (i)
Dðt; 2Þ ffi 0:0015t;
ð3:165Þ
the steady availability coefficient of the system at the moment t; t 0; for suffi-ciently large t, is given by Aðt; 2Þ ffi 0:97;
t 0;
118
(j)
3 Complex Technical Systems, Reliability
the steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; for sufficiently large t, is given by Aðt; s; 2Þ ffi 0:0027
Z1 Rðt; 2Þ dt;
t 0; s [ 0;
s
where Rðt; 2Þ is given by (3.152). The exemplary system reliability, renewal and availability characteristics predicted in this section are used for this system operation and reliability optimization and cost analysis performed in the appliqué part of Chap. 6.
3.6.2 Reliability, Renewal and Availability of Port Oil Piping Transportation System at Variable Operation Conditions The port oil piping transportation system reliability analysis at the constant operation conditions considered here is performed in Sect. 1.4.2, whereas its varying in time operation process analysis is performed in Sect. 2.3.2. Considering the assumptions and agreements of these two sections, we assume that its subsystems St ; t ¼ 1; 2; 3; are composed of three-state, i.e. z = 2, components ðtÞ Eij ; t ¼ 1; 2; 3; with the conditional three-state reliability functions ðtÞ
ðvÞ
ðtÞ
½Rij ðt; ÞðbÞ ¼ ½1; ½Rij ðt; 1ÞðbÞ ; ½Rij ðt; 2ÞðbÞ ;
b ¼ 1; 2; . . .; 7;
with the exponential coordinates ðtÞ
ðtÞ
½Rij ðt; 1ÞðbÞ ¼ exp½½kij ð1ÞðbÞ ;
ðtÞ
ðtÞ
½Rij ðt; 2ÞðbÞ ¼ exp½½kij ð2ÞðbÞ ;
different at various operation states zb ; b ¼ 1; 2; . . .; 7; and with the intensities of departure from the safety state subsets f1; 2g; f2g; respectively ðtÞ
½kij ð1ÞðbÞ ;
ðtÞ
½kij ð2ÞðbÞ ;
b ¼ 1; 2; . . .; 7:
In Sect. 2.3.2, it is fixed that the port oil piping transportation system reliability structure and its subsystems and components reliability depend on its changing in time operation states. The influence of the system operation states changing on the changes of the system reliability structure and its components reliability functions is as follows. At the system operation state z1 ; the system is composed of subsystem S3 illustrated in Fig. 2.10, which contains three series subsystems (k ¼ 3), each composed of 362 components (l1 ¼ 362; l2 ¼ 362; l3 ¼ 362) with the exponential reliability functions given below and is a series-’’2 out of 3’’ system (m ¼ 2).
3.6 Applications
119
Subsystem S3 consists of two pipelines of the first type and one pipeline of the second type. In each pipeline of the first type there are: • 360 pipe segments with conditional three-state reliability function co-ordinates ð3Þ
½Rij ðt; 1Þð1Þ ¼ exp½0:0059t; i ¼ 1; 2; j ¼ 1; 2; . . .; 360;
ð3Þ
½Rij ðt; 2Þð1Þ ¼ exp½0:0074t;
• two valves with conditional three-state reliability function co-ordinates ð3Þ
½Rij ðt; 1Þð1Þ ¼ exp½0:0166t; i ¼ 1; 2; j ¼ 361; 362:
ð3Þ
½Rij ðt; 2Þð1Þ ¼ exp½0:0181t;
In the pipeline of the second type there are: • 360 pipe segments with conditional three-state reliability function co-ordinates ð3Þ
½Rij ðt; 1Þð1Þ ¼ exp½0:0071t; i ¼ 3; j ¼ 1; 2; . . .; 360;
ð3Þ
½Rij ðt; 2Þð1Þ ¼ exp½0:0079t;
• two valves with conditional three-state reliability function co-ordinates ð3Þ
½Rij ðt; 1Þð1Þ ¼ exp½0:0166t; i ¼ 3; j ¼ 361; 362:
ð3Þ
½Rij ðt; 2Þð1Þ ¼ exp½0:0181t;
Thus, at the operation state z1 ; the pipeline system is identical with subsystem S3 that is a three-state series-’’2 out of 3’’ system with its structure shape parameters k ¼ 3; m ¼ 2; l1 ¼ 362; l2 ¼ 362; l3 ¼ 362; and according to (1.40)– (1.41) and (1.72)–(1.73), its conditional reliability function is given by ½Rðt; Þð1Þ ¼ ½1; ½Rðt; 1Þð1Þ ; ½Rðt; 2Þð1Þ ;
t 0;
ð3:166Þ
where ½Rðt; 1Þð1Þ ¼ R23;362;362;362 ðt; 1Þ ¼1
1 X
" #ri " #1ri 3 Y 362 362 Y Y ð3Þ ð3Þ ð1Þ ð1Þ ½Rij ðt; 1Þ 1 ½Rij ðt; 1Þ
r1 ;r2 ;r3 ¼0 i¼1 r1 þr2 þr3 1
j¼1
j¼1
¼ 1 ½1 exp½½360 0:0059 þ 2 0:0166t2 ½1 exp½½360 0:0071 þ 2 0:0166t1 2 exp½1½360 0:0059 þ 2 0:0166t
120
3 Complex Technical Systems, Reliability
½1 exp½½360 0:0059 þ 2 0:0166t1 ½1 exp½½360 0:0071 þ 2 0:0166t1 exp½½360 0:0071 þ 2 0:0166t ½1 exp½½360 0:0059 þ 2 0:0166t
ð3:167Þ
2
¼ exp½4:3144t þ 2 exp½4:7464t 2 exp½6:9036t; ½Rðt; 2Þð1Þ ¼ R23;362;362;362 ðt; 2Þ ¼1
1 X
" # ri " #1ri 3 Y 362 362 Y Y ð3Þ ð3Þ ð1Þ ð1Þ ½Rij ðt; 2Þ 1 ½Rij ðt; 2Þ
r1 ;r2 ;r3 ¼0 i¼1 r1 þr2 þr3 1
j¼1
j¼1
¼ 1 ½1 exp½½360 0:0074 þ 2 0:0181t2 ½1 exp½½360 0:0079 þ 2 0:0181t1 2 exp½1½360 0:0074 þ 2 0:0181t ½1 exp½½360 0:0074 þ 2 0:0181t1 ½1 exp½½360 0:0079 þ 2 0:0181t1 exp½½360 0:0079 þ 2 0:0181t ½1 exp½½360 0:0074 þ 2 0:0181t2 ¼ exp½5:4004t þ 2 exp½5:5804t 2 exp½8:2806t:
ð3:168Þ
The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z1 ; calculated from the results given by (3.166)–(3.168), according to (3.8)–(3.9), respectively are: l1 ð1Þ ffi 0:364; l1 ð2Þ ffi 0:302 year; ð3:169Þ r1 ð1Þ ffi 0:263; r1 ð2Þ ffi 0:219 year; ð3:170Þ and further, using (3.10) and (3.169), the mean values of the conditional lifetimes in the particular reliability states 1, 2 at the operation state z1 ; respectively are: l1 ð1Þ ffi 0:062;
l1 ð2Þ ffi 0:302 year:
ð3:171Þ
At the system operation state z2 ; the system is composed of subsystem S3 illustrated in Fig. 2.11, which contains three series subsystems (k ¼ 3), each composed of 362 components (l1 ¼ 362; l2 ¼ 362; l3 ¼ 362) with the exponential reliability functions given below and is a series–parallel system. Subsystem S3 consists of two pipelines of the first type and one pipeline of the second type with the same reliability functions as at the operation state z1 ; having the conditional reliability function co-ordinates ð3Þ
ð3Þ
½Rij ðt; uÞð2Þ ¼ ½Rij ðt; uÞð1Þ ;
u ¼ 1; 2; i ¼ 1; 2; 3; j ¼ 1; 2; . . .; 362:
3.6 Applications
121
Thus, at the operation state z2 ; the pipeline system is identical with subsystem S3 that is a three-state series–parallel system with its structure shape parameters k ¼ 3; l1 ¼ 362; l2 ¼ 362; l3 ¼ 362; and according to (1.36)–(1.37) and (1.68)– (1.69), its conditional reliability function is given by ½Rðt; Þð2Þ ¼ ½1; ½Rðt; 1Þð2Þ ; ½Rðt; 2Þð2Þ ;
t 0;
ð3:172Þ
where ð2Þ
½Rðt;1Þ
¼ R3;362;362;362 ðt;1Þ ¼ 1
" 3 Y i¼1
362 Y ð2Þ 1 ½Rij ðt;1Þð2Þ
#
j¼1
¼ 1 ½1 exp½½360 0:0059 þ 2 0:0166t2 ½1 exp½½360 0:0071 þ 2 0:0166t1 ¼ 1 ½1 exp½2:1572t2 ½1 exp½2:5892t ¼ exp½2:5892t þ 2 exp½2:1572t 2 exp½4:7464t exp½4:3144t þ exp½6:9036t; ½Rðt;2Þð2Þ ¼ R3;362;362;362 ðt;2Þ ¼ 1
ð3:173Þ
3 362 Y Y ð2Þ ½1 ½Rij ðt;2Þð2Þ i¼1
j¼1
¼ 1 ½1 exp½½360 0:0074 þ 2 0:0181t2 ½1 exp½½360 0:0079 þ 2 0:0181t1 ¼ 1 ½1 exp½2:7002t2 ½1 exp½2:8802t ¼ exp½2:8802t þ 2 exp½2:7002t 2 exp½5:5804t exp½5:4004t þ exp½8:2806t:
ð3:174Þ
The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z2 ; calculated from the results given by (3.172)–(3.174), according to (3.8)–(3.9), respectively are: ð3:175Þ l2 ð1Þ ffi 0:805; l2 ð2Þ ffi 0:665 year; r2 ð1Þ ffi 0:516;
r2 ð2Þ ffi 0:424 year;
ð3:176Þ
and further, using (3.10) and (3.175), the mean values of the conditional lifetimes in the particular reliability states 1, 2 at the operation state z2 ; respectively are: l2 ð1Þ ffi 0:140;
l2 ð2Þ ffi 0:665 year:
ð3:177Þ
At the system operation state z3 ; the pipeline system is a series composed of two series–parallel subsystems S1 and S2 illustrated in Fig. 2.12. Subsystem S1 illustrated in Fig. 2.12, contains two series pipeline subsystems (k ¼ 2), each composed of 178 components (l1 ¼ 178; l2 ¼ 178) with the
122
3 Complex Technical Systems, Reliability
exponential reliability functions identified in Chap. 5 and given below and is a series–parallel system. In each pipeline there are: • 176 pipe segments with conditional three-state reliability function co-ordinates ð1Þ
½Rij ðt; 1Þð3Þ ¼ exp½0:0062t; i ¼ 1; 2; j ¼ 1; 2; . . .; 176;
ð1Þ
½Rij ðt; 2Þð3Þ ¼ exp½0:0088t;
• two valves with conditional three-state reliability function co-ordinates ð1Þ
½Rij ðt; 1Þð3Þ ¼ exp½0:0167t; i ¼ 1; 2; j ¼ 177; 178:
ð1Þ
½Rij ðt; 2Þð3Þ ¼ exp½0:0182t;
Thus, at the operation state z3 ; subsystem S1 is a three-state series–parallel system with its structure shape parameters k ¼ 2; l1 ¼ 178; l2 ¼ 178; and according to (1.36)–(1.37) and (1.68)–(1.69), its conditional reliability function is given by ½Rð1Þ ðt; Þð3Þ ¼ ½1; ½Rð1Þ ðt; 1Þð3Þ ; ½Rð1Þ ðt; 2Þð3Þ ;
t 0;
ð3:178Þ
where ð3Þ
ð1Þ
½R ðt; 1Þ
¼ R2;178;178 ðt; 1Þ ¼ 1
2 Y
"
i¼1
178 Y ð1Þ 1 ½Rij ðt; 1Þð3Þ
#
j¼1
¼ 1 ½1 exp½½176 0:0062 þ 2 0:0167t2
½Rð1Þ ðt; 2Þð3Þ
¼ 1 ½1 exp½1:1246t2 ¼ 2 exp½1:1246t exp½2:2492t; " # 2 178 Y Y ð1Þ ð3Þ ¼ R2;178;178 ðt; 2Þ ¼ 1 1 ½Rij ðt; 2Þ i¼1
ð3:179Þ
j¼1
¼ 1 ½1 exp½½176 0:0088 þ 2 0:0182t2 ¼ 1 ½1 exp½1:5852t2 ¼ 2 exp½1:5852t exp½3:1704t:
ð3:180Þ
Subsystem S2 contains two series pipeline subsystems (k ¼ 2), each composed of 719 components (l1 ¼ 719; l2 ¼ 719) with the exponential reliability functions given below and is a series–parallel system. In each pipeline there are: • 717 pipe segments with conditional three-state reliability function co-ordinates ð2Þ
½Rij ðt; 1Þð3Þ ¼ exp½0:0062t; i ¼ 1; 2; j ¼ 1; 2; . . .; 717;
ð2Þ
½Rij ðt; 2Þð3Þ ¼ exp½0:0088t;
• two valves with conditional three-state reliability function co-ordinates
3.6 Applications
123
ð2Þ
½Rij ðt; 1Þð3Þ ¼ exp½0:0166t; i ¼ 1; 2; j ¼ 718; 719:
ð2Þ
½Rij ðt; 2Þð3Þ ¼ exp½0:0181t;
Thus, at the operation state z3 ; subsystem S2 is a three-state series–parallel system with its structure shape parameters k ¼ 2; l1 ¼ 719; l2 ¼ 719; and according to (1.36)–(1.37) and (1.68)–(1.69), its conditional reliability function is given by ½Rð2Þ ðt; Þð3Þ ¼ ½1; ½Rð2Þ ðt; 1Þð3Þ ; ½Rð2Þ ðt; 2Þð3Þ ;
t 0;
ð3:181Þ
where ð2Þ
ð3Þ
½R ðt; 1Þ
¼ R2;719;719 ðt; 1Þ ¼ 1
2 Y
"
i¼1
719 Y ð2Þ 1 ½Rij ðt; 1Þð3Þ
#
j¼1
¼ 1 ½1 exp½½717 0:0062 þ 2 0:0166t2 ¼ 1 ½1 exp½4:4786t2
½Rð2Þ ðt; 2Þð3Þ
¼ 2 exp½4:4786t exp½8:9572t; " # 2 719 Y Y ð2Þ ð3Þ ¼ R2;719;719 ðt; 2Þ ¼ 1 1 ½Rij ðt; 2Þ i¼1
ð3:182Þ
j¼1
¼ 1 ½1 exp½½717 0:0088 þ 2 0:0181t2 ¼ 1 ½1 exp½6:3458t2 ¼ 2 exp½6:3458t exp½12:6916t:
ð3:183Þ
Considering that the pipeline system at the operation state z3 is a three-state series system composed of subsystems S1 and S2 ; after applying (1.22)–(1.23) and (3.178)-(3.180) and (3.181)-(3.183), its conditional reliability function is given by ½Rðt; Þð3Þ ¼ ½1; ½Rðt; 1Þð3Þ ; ½Rðt; 2Þð3Þ ;
t 0;
ð3:184Þ
where ½Rðt; 1Þð3Þ ¼ R2 ðt; 1Þ ¼ ½Rð1Þ ðt; 1Þð3Þ ½Rð2Þ ðt; 1Þð3Þ ¼ 4 exp½5:6032t 2 exp½6:7278t 2 exp½10:0818t þ exp½11:20604t;
ð3:185Þ
½Rðt; 2Þð3Þ ¼ R2 ðt; 2Þ ¼ ½Rð1Þ ðt; 2Þð3Þ ½Rð2Þ ðt; 2Þð3Þ ¼ 4 exp½7:931t 2 exp½14:2768t 2 exp½9:5162t þ exp½15:862t:
ð3:186Þ
124
3 Complex Technical Systems, Reliability
The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z3 ; calculated from the results given by (3.184)–(3.186), according to (3.8)–(3.9), respectively are: l3 ð1Þ ffi 0:307;
l3 ð2Þ ffi 0:217 year;
ð3:187Þ
r3 ð1Þ ffi 0:222;
r3 ð2Þ ffi 0:157 year;
ð3:188Þ
and further, using (3.10) and (3.187), the mean values of the conditional lifetimes in the particular reliability states 1, 2 at the operation state z3 ; respectively are: l3 ð1Þ ffi 0:09;
l3 ð2Þ ffi 0:217 year:
ð3:189Þ
At the system operation state z4 ; the system is series and composed of two series–parallel subsystems S1 and S2 ; each containing two pipelines, and one series-‘‘2 out of 3’’ subsystem S3 containing three pipelines, with the scheme shown in Fig. 2.13. Subsystem S1 illustrated in Fig. 2.13, contains two series pipeline subsystems (k ¼ 2), each composed of 178 components (l1 ¼ 178; l2 ¼ 178), with the same reliability functions as at the operation state z3 : Thus, according to (3.178)– (3.180), subsystem S1 conditional reliability function at the operation state z4 ; is given by ½Rð1Þ ðt; Þð4Þ ¼ ½1; ½Rð1Þ ðt; 1Þð4Þ ; ½Rð1Þ ðt; 2Þð4Þ ;
t 0;
ð3:190Þ
where ½Rð1Þ ðt; 1Þð4Þ ¼ 2 exp½1:1246t exp½2:2492t;
ð3:191Þ
½Rð1Þ ðt; 2Þð4Þ ¼ 2 exp½1:5852t exp½3:1704t:
ð3:192Þ
Subsystem S2 illustrated in Fig. 2.14, contains two series pipeline subsystems (k ¼ 2), each composed of 719 components (l1 ¼ 719; l2 ¼ 719) with the same reliability functions as at the operation state z3 : Thus, according to (3.181)– (3.183), subsystem S2 conditional reliability function at the operation state z4 ; is given by ½Rð2Þ ðt; Þð4Þ ¼ ½1; ½Rð2Þ ðt; 1Þð4Þ ; ½Rð2Þ ðt; 2Þð4Þ ;
t 0;
ð3:193Þ
where ½Rð2Þ ðt; 1Þð4Þ ¼ 2 exp½4:4786t exp½8:9572t;
ð3:194Þ
½Rð2Þ ðt; 2Þð4Þ ¼ 2 exp½6:3458t exp½12:6916t:
ð3:195Þ
At the operation state z4 ; subsystem S3 illustrated in Fig. 2.13, contains three series subsystems (k ¼ 3), each composed of 362 components (l1 ¼ 362;
3.6 Applications
125
l2 ¼ 362; l3 ¼ 362) and is a series-’’2 out of 3’’ system (m ¼ 2) which means that it has the same as that at the operation state z1 reliability structure. Subsystem S3 consists of two pipelines of the first type and 1 pipeline of the second type, each composed of 362 components with the same reliability functions as at the operation state z1 : Thus, according to (3.163)–(3.165), subsystem S3 conditional reliability function at the operation state z4 ; is given by ½Rð3Þ ðt; Þð4Þ ¼ ½1; ½Rð3Þ ðt; 1Þð4Þ ; ½Rð3Þ ðt; 2Þð4Þ ;
t 0;
ð3:196Þ
where ½Rð3Þ ðt; 1Þð4Þ ¼ exp½4:3144t þ 2 exp½4:7464t 2 exp½6:9036t;
ð3:197Þ
½Rð3Þ ðt; 2Þð4Þ ¼ exp½5:4004t þ 2 exp½5:5804t 2 exp½8:2806t:
ð3:198Þ
Considering that the pipeline system at the operation state z4 is a three-state series system composed of subsystems S1 ; S2 and S3 ; after applying (1.22)–(1.23) and (3.190)–(3.192), (3.193)–(3.195) and (3.196)–(3.198), its conditional reliability function is given by ½Rðt; Þð4Þ ¼ ½1; ½Rðt; 1Þð4Þ ; ½Rðt; 2Þð4Þ ;
t 0;
ð3:199Þ
where ½Rðt; 1Þð4Þ ¼ R3 ðt; 1Þ ¼ ½Rð1Þ ðt; 1Þð4Þ ½Rð2Þ ðt; 1Þð4Þ ½Rð3Þ ðt; 1Þð4Þ ¼ 4 exp½9:9176t þ 8 exp½10:3496t 8 exp½12:5078t 2 exp½14:396t 4 exp½14:8282t þ 4 exp½16:9864t 2 exp½11:0422t 4 exp½11; 4742t þ 4 exp½13:6324t þ exp½15:5208t þ 2 exp½15:9528t 2 exp½18:111t; ð3:200Þ ½Rðt; 2Þð4Þ ¼ R3 ðt; 2Þ ¼ ½Rð1Þ ðt; 2Þð4Þ ½Rð2Þ ðt; 2Þð4Þ ½Rð3Þ ðt; 2Þð4Þ ¼ 4 exp½13:3314t þ 8 exp½13:5114t 8 exp½16:2116t 2 exp½19:6772t 4 exp½19:8572t þ 4 exp½22:5574t 2 exp½14:9166t 4 exp½15; 0966t þ 4 exp½17:7968t þ exp½21:2624t þ 2 exp½21:4424t 2 exp½24:1426t: ð3:201Þ The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z4 ; calculated from the results given by (3.199)–(3.201), according to (3.8)–(3.9), respectively are:
126
3 Complex Technical Systems, Reliability
l4 ð1Þ ffi 0:207;
l4 ð2Þ ffi 0:156 year;
ð3:202Þ
r4 ð1Þ ffi 0:137;
r4 ð2Þ ffi 0:104 year;
ð3:203Þ
and further, using (3.10) and (3.202), the mean values of the conditional lifetimes in the particular reliability states 1, 2 at the operation state z4 ; respectively are: l4 ð1Þ ffi 0:051;
l4 ð2Þ ffi 0:156 year:
ð3:204Þ
At the system operation state z5 ; the pipeline system is series and composed of two series–parallel subsystems S1 and S2 illustrated in Fig. 2.12. The system reliability structure and its components reliability functions are the same as at the operation state z3 : Thus, considering (3.184)–(3.186), its conditional reliability function is given by ½Rðt; Þð5Þ ¼ ½1; ½Rðt; 1Þð5Þ ; ½Rðt; 2Þð5Þ ;
t 0;
ð3:205Þ
where ½Rðt; 1Þð5Þ ¼ 4 exp½5:6032t 2 exp½6:7278t 2 exp½10:0818t þ exp½11:2064t;
ð3:206Þ
½Rðt; 2Þð5Þ ¼ 4 exp½7:931t 2 exp½14:2768t 2 exp½9:5162t þ exp½15:8620t:
ð3:207Þ
The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z5 ; according to (3.187)–(3.188), respectively are: l5 ð1Þ ffi 0:307;
l5 ð2Þ ffi 0:217 year;
ð3:208Þ
r5 ð1Þ ffi 0:222;
r5 ð2Þ ffi 0:157 year;
ð3:209Þ
and further, from (3.189), the mean values of the conditional lifetimes in the particular reliability states 1, 2 at the operation state z5 ; respectively are: l5 ð1Þ ffi 0:09;
l5 ð2Þ ffi 0:217 year:
ð3:210Þ
At the system operation state z6 ; the system is series and composed of two series–parallel subsystems S1 and S2 ; each containing two pipelines and one series‘‘2 out of 3’’ subsystem S3 containing three pipelines with the scheme shown in Fig. 2.13. The system reliability structure and its components reliability functions are the same as at the operation state z4 : Thus, considering (3.199)–(3.201), its conditional three-state reliability function is given by ½Rðt; Þð6Þ ¼ ½1; ½Rðt; 1Þð6Þ ; ½Rðt; 2Þð6Þ ;
t 0;
ð3:211Þ
3.6 Applications
127
where ½Rðt;1Þð6Þ ¼ 4exp½9:9176t þ 8exp½10:3496t 8exp½12:5078t 2 exp½14:396t 4 exp½14:8282t þ 4exp½16:9864t 2 exp½11:0422t 4 exp½11; 4742t þ 4 exp½13:6324t þ exp½15:5208t þ 2exp½15:9528t 2 exp½18:111t;
ð3:212Þ
½Rðt;2Þð6Þ ¼ 4exp½13:3314t þ 8exp½13:5114t 8exp½16:2116t 2 exp½19:6772t 4 exp½19:8572t þ 4exp½22:5574t 2 exp½14:9166t 4 exp½15; 0966t þ 4 exp½17:7968t þ exp½21:2624t þ 2exp½21:4424t 2 exp½24:1426t:
ð3:213Þ
The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z6 ; according to (3.202)–(3.203), respectively are: l6 ð1Þ ffi 0:207;
l6 ð2Þ ffi 0:156 year;
ð3:214Þ
r6 ð1Þ ffi 0:137;
r6 ð2Þ ffi 0:104 year;
ð3:215Þ
and further, from (3.204), the mean values of the conditional lifetimes in the particular reliability states 1, 2 at the operation state z6 ; respectively are: l6 ð1Þ ffi 0:051;
l6 ð2Þ ffi 0:156 year:
ð3:216Þ
At the system operation state z7 ; the system is composed of the subsystem S3 illustrated in Fig. 2.10. The system reliability structure and its components, reliability functions are the same as at the operation state z1 : Thus, considering (3.166)–(3.168), its conditional three-state reliability function is given by ½Rðt; Þð7Þ ¼ ½1; ½Rðt; 1Þð7Þ ; ½Rðt; 2Þð7Þ ;
t 0;
ð3:217Þ
where ½Rðt; 1Þð7Þ ¼ exp½4:3144t þ 2 exp½4:7464t 2 exp½6:9036t;
ð3:218Þ
½Rðt; 2Þð7Þ ¼ exp½5:4004t þ 2 exp½5:5804t 2 exp½8:2806t:
ð3:219Þ
The expected values and standard deviations of the pipeline system conditional lifetimes in the reliability state subsets f1; 2g; f2g at the operation state z7 ; according to (3.169)–(3.170), respectively are: l7 ð1Þ ffi 0:364;
l7 ð2Þ ffi 0:302 year;
ð3:220Þ
r7 ð1Þ ffi 0:263;
r7 ð2Þ ffi 0:219 year;
ð3:221Þ
128
3 Complex Technical Systems, Reliability
and further, from (3.171), the conditional lifetimes in the particular reliability states 1, 2 at the operation state z7 ; respectively are: l7 ð1Þ ffi 0:062;
l7 ð2Þ ffi 0:302 year:
ð3:222Þ
In the case when the operation time is large enough, the port oil transportation system unconditional reliability function is given by the vector Rðt; Þ ¼ ½1; Rðt; 1Þ; Rðt; 2Þ;
t 0;
ð3:223Þ
where according to (3.6) and considering the pipeline system operation process transient probabilities at the operation states determined by (2.38), the vector co-ordinates are given respectively by Rðt; 1Þ ¼ 0:395 ½Rðt; 1Þð1Þ þ 0:060 ½Rðt; 1Þð2Þ þ 0:003 ½Rðt; 1Þð3Þ þ 0:002 ½Rðt; 1Þð4Þ þ 0:2 ½Rðt; 1Þð5Þ þ 0:058 ½Rðt; 1Þð6Þ þ 0:282 ½Rðt; 1Þð7Þ
for t 0;
ð3:224Þ
Rðt; 2Þ ¼ 0:395 ½Rðt; 2Þð1Þ þ 0:060 ½Rðt; 2Þð2Þ þ 0:003 ½Rðt; 2Þð3Þ þ 0:002 ½Rðt; 2Þð4Þ þ 0:2 ½Rðt; 2Þð5Þ þ 0:058 ½Rðt; 2Þð6Þ þ 0:282 ½Rðt; 2Þð7Þ
for t 0;
ð3:225Þ
where ½Rðt; 1Þð1Þ ; ½Rðt; 1Þð2Þ ; ½Rðt; 1Þð3Þ ; ½Rðt; 1Þð4Þ ; ½Rðt; 1Þð5Þ ; ½Rðt; 1Þð6Þ ; ½Rðt; 1Þð7Þ are given by (3.167), (3.173), (3.185), (3.200), (3.206), (3.212), (3.218) and ½Rðt; 2Þð1Þ ; ½Rðt; 2Þð2Þ ; ½Rðt; 2Þð3Þ ; ½Rðt; 2Þð4Þ ; ½Rðt; 2Þð5Þ ; ½Rðt; 2Þð6Þ ; ð7Þ ½Rðt; 2Þ are given by (3.168), (3.174), (3.186), (3.201), (3.207), (3.213), (3.219). The graph of the three-state port oil piping transportation system reliability function is presented in Fig. 3.3. The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets f1; 2g; f2g; calculated from the above results given by (3.224)–(3.225), according to (3.7)–(3.9) and considering (2.38), (3.169), (3.175), (3.187), (3.202), (3.208), (3.214), (3.220), respectively are: lð1Þ ¼ 0:395 0:364 þ 0:060 0:805 þ 0:003 0:307 þ 0:002 0:207 þ 0:2 0:307 þ 0:058 0:207 þ 0:282 0:364 ffi 0:370 year; rð1Þ ffi 0:308 year; lð2Þ ¼ 0:395 0:302 þ 0:060 0:665 þ 0:003 0:217 þ 0:002 0:156 þ 0:2 0:217 þ 0:058 0:156 þ 0:282 0:302 ffi 0:298 year; rð2Þ ffi 0:252 year;
ð3:226Þ ð3:227Þ
ð3:228Þ ð3:229Þ
3.6 Applications
129
Fig. 3.3 The graph of the port oil piping transportation system reliability function Rðt; Þ coordinates
and further, considering (3.10) and (3.226) and (3.228), the mean values of the unconditional lifetimes in the particular reliability states 1, 2, respectively are: lð1Þ ¼ lð1Þ lð2Þ ¼ 0:072;
lð2Þ ¼ lð2Þ ¼ 0:298 year:
ð3:230Þ
Since the critical reliability state is r = 1, then according to (3.11), the system risk function is given by rðtÞ ¼ 1 Rðt; 1Þ
for t 0;
ð3:231Þ
where Rðt; 1Þ is given by (3.224). Hence, according to (3.12), the moment when the system risk function exceeds a permitted level, for instance d ¼ 0:05; is s ¼ r1 ðdÞ ffi 0:066 year:
ð3:232Þ
The graph of the port oil piping transportation system risk function r(t) is presented in Fig. 3.4. Further, assuming that the oil pipeline system is repaired after the exceeding of its critical reliability state r = 1 and that the time of the system renovation is ignored and applying Proposition 3.4, we obtain the following results: (a) the time SN ð1Þ until the Nth exceeding by the system the reliability critical state r = 1, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð0:370N; 0:308 N Þ; i.e., t 0:370N pffiffiffiffi ; t 2 ð1; 1Þ; F ðNÞ ðt; 1Þ ¼ PðSN ð1Þ\tÞ ffi FNð0;1Þ 0:308 N (b) the expected value and the variance of the time SN ð1Þ until the Nth exceeding by the system the reliability critical state r = 1 are respectively given by E½SN ð1Þ ffi 0:370N;
D½SN ð1Þ ffi 0:095N;
(c) the number Nðt; 1Þ of exceeding by the system the reliability critical state r = 1 up to the moment t; t 0; for sufficiently large t, approximately has the distribution of the form
130
3 Complex Technical Systems, Reliability
Fig. 3.4 The graph of the port oil piping transportation system risk function rðtÞ
0:370ðN þ 1Þ t 0:370N t pffi pffi ; FNð0;1Þ 0:506 t 0:506 t N ¼ 0; 1; . . .;
PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ
(d) the expected value and the variance of the number Nðt; 1Þ of exceeding by the system the reliability critical state r = 1 up to the moment t; t 0; for sufficiently large t, approximately are respectively given by Hðt; 1Þ ¼ 2:702t;
Dðt; 1Þ ¼ 1:872t:
ð3:233Þ
Assuming that the oil pipeline system is repaired after the exceeding of its reliability critical state r ¼ 1 and that the time of the system renovation is not ignored and has the mean value l0 ð1Þ ¼ 0:005 year and the standard deviation r0 ð1Þ ¼ 0:005 year and applying Proposition 3.5, we obtain the following results: (a) the time SN ð1Þ until the Nth exceeding by the system the reliability critical state r = 1, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð0:370N þ 0:005ðN 1Þ; 0:095N 0:000025ðN 1ÞÞ; i.e., t 0:375N þ 0:005 ðNÞ F ðt; 1Þ ¼ PðSN ð1Þ\tÞ ffi FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t 2 ð1; 1Þ; 0:095N 0:000025 (b) the expected value and the variance of the time SN ð1Þ until the Nth exceeding by the system the reliability critical state r = 1, for sufficiently large N, are respectively given by E½SN ð1Þ ffi 0:370N þ 0:005ðN 1Þ;
D½SN ð1Þ ffi 0:095N þ 0:000025ðN 1Þ;
(c) the number Nðt; 1Þ of exceeding by the system the reliability critical state r = 1 up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form
3.6 Applications
131
0:375ðN þ 1Þ t 0005 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ 0:503 t 0:005 0:375N t 0:005 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; FNð0;1Þ 0:503 t þ 0:005
N ¼ 0; 1; . . .;
(d) the expected value and the variance of the number Nðt; 1Þ of exceeding by the system the reliability critical state r = 1 up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; 1Þ ffi 2:667ðt þ 0:005Þ;
Dðt; 1Þ ffi 1:801ðt þ 0:005Þ;
ð3:234Þ
(e) the time SN ð1Þ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffi approximately normal distribution Nð0:375N; 0:308 N Þ; i.e., ðNÞ t 0:375NÞ pffiffiffiffi ; t 2 ð1; 1Þ; F ðt; 1Þ ¼ PðSN ð1Þ\tÞ ffi FNð0;1Þ 0:308 N (f) the expected value and the variance of the time SN ð1Þ until the Nth system’s renovation, for sufficiently large N, are respectively given by E½SN ð1Þ ffi 0:375N;
D½SN ð1Þ ffi 0:095N;
(g) the number Nðt; 1Þ of the system’s renovations up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 0:368ðN þ 1Þ t pffi PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ 0:503 t 0:368N t pffi ; N ¼ 0; 1; . . .; FNð0;1Þ 0:503 t (h) the expected value and the variance of the number Nðt; 1Þ of system’s renovations up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; 1Þ ffi 2:667t; (i)
Dðt; 1Þ ffi 1:801t;
the steady availability coefficient of the system at the moment t; t 0; for sufficiently large t, is given by Aðt; 1Þ ffi 0:987;
(j)
ð3:235Þ
t 0;
the steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; for sufficiently large t, is given by Aðt; s; 1Þ ffi 2:667
Z1 Rðt; 1Þ dt; s
t 0; s [ 0;
132
3 Complex Technical Systems, Reliability
where Rðt; 1Þ is given by (3.224). The port oil piping transportation system operating at varying in time operation conditions reliability and availability evaluation and prediction are approximate because of non-sufficiently exact input data. Although the piping system reliability structures are fixed with high accuracy, the reliability parameters of its components and its operation process parameters are not sufficiently exact because of the lack of statistical data necessary for their estimation. Therefore, the achieved results may only be considered as an illustration of the proposed methods’ possibilities of applications in the complex piping systems’ reliability and availability analysis and prediction. However, the obtained evaluation may be a very useful example in piping systems’ reliability and availability characteristics prediction, especially during the design and when planning and improving their operation processes’ safety and effectiveness. To improve the accuracy of the achieved results the statistical data should be collected for an appropriately long period of time and after that the full identification of at least the pipeline oil transportation operation process could be performed and this process, unknown parameters and main characteristics would be determined and used in its reliability, risk and availability for more precise analysis and prediction. The reliability characteristics of the port oil piping transportation system operating at the variable conditions predicted in this section are different from those determined in Sect. 1.4.2 at the system constant operation conditions. This fact suggests the necessity of considering the system at the variable operation conditions that upon the improved accuracy of the system operation process identification, makes its system reliability prediction more precise. The presented in this section reliability, renewal and availability characteristics of the port oil piping transportation system are used in its operation, reliability, availability and maintenance optimization in the appliqué part of Chap. 6. This example of the considered models practical application justifies their very good placement in the reliability practice of real complex technical systems.
3.6.3 Safety, Renewal and Availability of Maritime Ferry Technical System at Variable Operation Conditions The ferry technical system safety analysis considered here at the constant operation conditions is performed in Sect. 1.4.3, whereas its changing in time operation process analysis is performed in Sect. 2.3.3. Considering the assumptions and agreements of these sections, we assume that its subsystems St ; t ¼ 1; 2; . . .; 5; are ðtÞ composed of five-state i.e. z = 4, components, Eij ; t ¼ 1; 2; . . .; 5; having the conditional safety functions given by the vector
3.6 Applications
133
ðtÞ
ðtÞ
ðtÞ
ðtÞ
ðtÞ
½sij ðt; ÞðbÞ ¼ ½1; ½sij ðt; 1ÞðbÞ ; ½sij ðt; 2ÞðbÞ ; ½sij ðt; 3ÞðbÞ ; ½sij ðt; 4ÞðbÞ ; b ¼ 1; 2; . . .; 18; with the exponential coordinates ðtÞ
ðtÞ
½sij ðt; 2ÞðbÞ ¼ exp½½kij ð2ÞðbÞ t;
ðtÞ
ðtÞ
½sij ðt; 4ÞðbÞ ¼ exp½½kij ð4ÞðbÞ t;
½sij ðt; 1ÞðbÞ ¼ exp½½kij ð1ÞðbÞ t; ½sij ðt; 3ÞðbÞ ¼ exp½½kij ð3ÞðbÞ t;
ðtÞ
ðtÞ
ðtÞ
ð#Þ
b ¼ 1; 2; . . .; 18; different in various operation states zb ; b ¼ 1; 2; . . .; 18; and with the intensities of departure from the safety states subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g; respectively ðtÞ
ðtÞ
ðtÞ
ðtÞ
½kij ð1ÞðbÞ ; ½kij ð2ÞðbÞ ; ½kij ð3ÞðbÞ ; ½kij ð4ÞðbÞ ;
b ¼ 1; 2; . . .; 18:
In Sect. 2.3.3, it is fixed that the ferry technical system safety structure and its subsystems and components safety depend on its changing in time operation states. The influence of the system operation states changing on the changes of the system safety structure fixed in Chap. 2 and its influence on the changes of the system components exponential safety functions partly identified in Chap. 5 are given below. At the system operation state z1 ; the ferry technical system is composed of two subsystems S3 and S4 forming a series structure shown in Fig. 2.14. Subsystem S3 is a five-state series system composed of n = 3 components ð3Þ ð3Þ ð3Þ E11 ; E21 ; E31 having the exponential safety functions with coordinates: ð3Þ
ð3Þ
½s11 ðt; 1Þð1Þ ¼ exp½0:02t; ½s11 ðt; 2Þð1Þ ¼ exp½0:03t; ð3Þ
ð3Þ
½s11 ðt; 3Þð1Þ ¼ exp½0:035t; ½s11 ðt; 4Þð1Þ ¼ exp½0:04t; ð3Þ
ð3Þ
ð3Þ
ð3Þ
½s21 ðt; 1Þð1Þ ¼ exp½0:02t; ½s21 ðt; 2Þð1Þ ¼ exp½0:025t; ½s21 ðt; 3Þð1Þ ¼ exp½0:03t; ½s21 ðt; 4Þð1Þ ¼ exp½0:04t; ð3Þ
ð3Þ
ð3Þ
ð3Þ
½s31 ðt; 1Þð1Þ ¼ exp½0:033t; ½s31 ðt; 2Þð1Þ ¼ exp½0:04t; ½s31 ðt; 3Þð1Þ ¼ exp½0:045t; ½s31 ðt; 4Þð1Þ ¼ exp½0:05t: Subsystem S3 is a five-state series system that can also be considered as a fivestate parallel–series system with its structure shape parameters k ¼ 3; l1 ¼ 1; l2 ¼ 1; l3 ¼ 1: Thus, according either to the formulae (1.38)–(1.39) and (1.70)–(1.71) or to the formulae (1.22)–(1.23) and (1.58)–(1.59), after replacing in them the components’ reliability functions by their safety functions, subsystem S3 conditional safety function at the operation state z1 is given by
134
3 Complex Technical Systems, Reliability
½sð3Þ ðt; Þð1Þ ¼ ½1; ½sð3Þ ðt; 1Þð1Þ ; ½sð3Þ ðt; 2Þð1Þ ; ½sð3Þ ðt; 3Þð1Þ ; ½sð3Þ ðt; 4Þð1Þ ; t 2 h0; 1Þ;
ð3:236Þ
where ½sð3Þ ðt; uÞð1Þ ¼ s3;1;1;1 ðt; uÞ " # 3 1 3 Y Y Y ð3Þ ð3Þ ð1Þ ¼ 1 ½1 ½sij ðt; uÞ ¼ ½si1 ðt; uÞð1Þ ; i¼1
j¼1
t 2 h0; 1Þ;
i¼1
u ¼ 1; 2; 3; 4; and particularly ½sð3Þ ðt; 1Þð1Þ ¼ exp½0:02t exp½0:02t exp½0:033t ¼ exp½0:073t; ð3:237Þ ½sð3Þ ðt; 2Þð1Þ ¼ exp½0:03t exp½0:025t exp½0:04t ¼ exp½0:095t; ð3:238Þ ½sð3Þ ðt; 3Þð1Þ ¼ exp½0:035t exp½0:03t exp½0:045t ¼ exp½0:110t; ð3:239Þ ½sð3Þ ðt; 4Þð1Þ ¼ exp½0:04t exp½0:04t exp½0:05t ¼ exp½0:130t:
ð3:240Þ
ð4Þ
Subsystem S4 consists of one five-state component E11 having the exponential safety functions with the coordinates: ð4Þ
½s11 ðt; 1Þð1Þ ¼ exp½0:05t; ð4Þ
½s11 ðt; 3Þð1Þ ¼ exp½0:065t;
ð4Þ
½s11 ðt; 2Þð1Þ ¼ exp½0:06t; ð4Þ
½s11 ðt; 4Þð1Þ ¼ exp½0:07t:
Thus, at the operation state z1 ; subsystem S4 conditional safety function is ð4Þ identical with the component E11 safety function and is given by ð4Þ
½sð4Þ ðt; Þð1Þ ¼ ½1; ½sð4Þ ðt; 1Þð1Þ ; ½sð4Þ ðt; 2Þð1Þ ; ½sð4Þ ðt; 3Þð1Þ ; ½s1;1 ðt; 4Þð1Þ ; t 2 h0; 1Þ;
ð3:241Þ
where ½sð4Þ ðt; 1Þð1Þ ¼ exp½0:05t;
ð3:242Þ
½sð4Þ ðt; 2Þð1Þ ¼ exp½0:06t;
ð3:243Þ
½sð4Þ ðt; 3Þð1Þ ¼ exp½0:065t;
ð3:244Þ
½sð4Þ ðt; 4Þð1Þ ¼ exp½0:07t:
ð3:245Þ
3.6 Applications
135
Considering that the ferry technical system at the operation state z1 is a fivestate series system of subsystems S3 and S4 ; after applying (1.22)–(1.23) and (3.236)–(3.240) and (3.241)–(3.245), its conditional safety function is given by ½sðt; Þð1Þ ¼ ½1; ½sðt; 1Þð1Þ ; ½sðt; 2Þð1Þ ; ½sðt; 3Þð1Þ ; ½sðt; 4Þð1Þ ;
t 2 h0; 1Þ; ð3:246Þ
where ½sðt; uÞð1Þ ¼ s2 ðt; uÞ ¼ ½sð3Þ ðt; uÞð1Þ ½sð4Þ ðt; uÞð1Þ ;
t 2 h0; 1Þ; u ¼ 1; 2; 3; 4;
and particularly ½sðt; 1Þð1Þ ¼ exp½0:073t exp½0:05t ¼ exp½0:123t
ð3:247Þ
½sðt; 2Þð1Þ ¼ exp½0:095t exp½0:06t ¼ exp½0:155t
ð3:248Þ
½sðt; 3Þð1Þ ¼ exp½0:110t exp½0:065t ¼ exp½0:175t;
ð3:249Þ
½sðt; 4Þð1Þ ¼ exp½0:130t exp½0:07t ¼ exp½0:20t:
ð3:250Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z1 ; calculated from the results given by (3.247)–(3.250), according to (3.8)–(3.9), respectively are: l1 ð1Þ ffi 8:13;
l1 ð2Þ ffi 6:45;
l1 ð3Þ ffi 5:71;
l1 ð4Þ ffi 5:00 years; ð3:251Þ
r1 ð1Þ ffi 8:13;
r1 ð2Þ ffi 6:45;
r1 ð3Þ ffi 5:71;
r1 ð4Þ ffi 5:00 years; ð3:252Þ
and further, using (3.10) and (3.251), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z1 ; respectively are: l1 ð1Þ ffi 1:68;
l1 ð2Þ ffi 0:74;
l1 ð3Þ ffi 0:71;
l4 ð4Þ ffi 5:00 years: ð3:253Þ
At the system operation state z2 ; the ferry technical system is composed of three subsystems S1 ; S2 and S5 forming a series structure shown in Fig. 2.15. Subsystem ð1Þ S1 consists of one five-state component E11 having the exponential safety functions partly identified in Chap. 5 with coordinates: ð1Þ
½s11 ðt; 2Þð2Þ ¼ exp½0:04t;
ð1Þ
½s11 ðt; 4Þð2Þ ¼ exp½0:05t:
½s11 ðt; 1Þð2Þ ¼ exp½0:033t; ½s11 ðt; 3Þð2Þ ¼ exp½0:045t;
ð1Þ ð1Þ
Thus, at the operation state z2 ; subsystem S1 conditional safety function is ð1Þ identical with the component E11 safety function and is given by
136
3 Complex Technical Systems, Reliability
½sð1Þ ðt; Þð2Þ ¼ ½1; ½sð1Þ ðt; 1Þð2Þ ; ½sð1Þ ðt; 2Þð2Þ ; ½sð1Þ ðt; 3Þð2Þ ; ½sð1Þ ðt; 4Þð2Þ ; t 2 h0; 1Þ;
ð3:254Þ
where ½sð1Þ ðt; 1Þð2Þ ¼ exp½0:033t;
ð3:255Þ
½sð1Þ ðt; 2Þð2Þ ¼ exp½0:04t;
ð3:256Þ
½sð1Þ ðt; 3Þð2Þ ¼ exp½0:045t;
ð3:257Þ
½sð1Þ ðt; 4Þð2Þ ¼ exp½0:05t:
ð3:258Þ
Subsystem S2 is a five-state parallel–series system composed of components i ¼ 1; 2; . . .; k; j ¼ 1; 2; . . .; li ; k ¼ 7; l1 ¼ 4; l2 ¼ 2; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; l6 ¼ 1; l7 ¼ 1; having the exponential safety functions with coordinates: ð2Þ Eij ;
ð2Þ
½s1j ðt; 2Þð2Þ ¼ exp½0:04t;
ð2Þ
½s1j ðt; 4Þð2Þ ¼ exp½0:055t;
ð2Þ
½s2j ðt; 2Þð2Þ ¼ exp½0:07t;
ð2Þ
½s2j ðt; 4Þð2Þ ¼ exp½0:08t;
ð2Þ
½s31 ðt; 2Þð2Þ ¼ exp½0:07t;
ð2Þ
½s31 ðt; 4Þð2Þ ¼ exp½0:08t;
ð2Þ
½si1 ðt; 2Þð2Þ ¼ exp½0:04t;
ð2Þ
½si1 ðt; 4Þð2Þ ¼ exp½0:05t;
½s1j ðt; 1Þð2Þ ¼ exp½0:033t; ½s1j ðt; 3Þð2Þ ¼ exp½0:05t; ½s2j ðt; 1Þð2Þ ¼ exp½0:066t; ½s2j ðt; 3Þð2Þ ¼ exp½0:075t; ½s31 ðt; 1Þð2Þ ¼ exp½0:066t; ½s31 ðt; 3Þð2Þ ¼ exp½0:075t; ½si1 ðt; 1Þð2Þ ¼ exp½0:033t; ½si1 ðt; 3Þð2Þ ¼ exp½0:045t;
ð2Þ
ð2Þ
j ¼ 1; 2; 3; 4;
ð2Þ ð2Þ
j ¼ 1; 2;
ð2Þ ð2Þ ð2Þ ð2Þ
i ¼ 4; 5; 6; 7:
Subsystem S2 is a five-state parallel–series system with its structure shape parameters k ¼ 7; l1 ¼ 4; l2 ¼ 2; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; l6 ¼ 1; l7 ¼ 1: Thus, according to the formulae (1.38)–(1.39) and (1.70)–(1.71), after replacing in them the components’ reliability functions by their safety functions, subsystem S2 conditional safety function at the operational state z2 is given by ½sð2Þ ðt; Þð2Þ ¼ ½1; ½sð2Þ ðt; 1Þð2Þ ; ½sð2Þ ðt; 2Þð2Þ ; ½sð2Þ ðt; 3Þð2Þ ; ½sð2Þ ðt; 4Þð2Þ ; t 2 h0; 1Þ;
ð3:259Þ
3.6 Applications
137
where ½sð2Þ ðt; uÞð2Þ ¼ s7;4;2;1;1;1;1;1 ðt; uÞ " # li 7 Y Y ð2Þ ð2Þ 1 ½1 ½sij ðt; uÞ ; ¼ i¼1
t 2 h0; 1Þ; u ¼ 1; 2; 3; 4;
j¼1
and particularly ½sð2Þ ðt;1Þð2Þ ¼ ½6½exp½0:033t2 ½1 exp½0:033t2 þ 4½exp½0:033t3 ½1 exp½0:033t þ ½exp½0:033t4 ½1 ½1 exp½0:066t2 exp½0:066texp½0:033texp½0:033texp½0:033texp½0:033t ¼ 12exp½0:33t þ 8exp½0:429t 16exp½0:363t 3exp½0:462t; ð3:260Þ ½sð2Þ ðt; 2Þð2Þ ¼ ½6½exp½0:04t2 ½1 exp½0:04t2 þ 4½exp½0:04t3 ½1 exp½0:04t þ ½exp½0:04t4 ½1 ½1 exp½0:07t2 exp½0:07t exp½0:04t exp½0:04t exp½0:04t exp½0:04t ¼ 12 exp½0:38t þ 8 exp½0:49t þ 6 exp½0:46t 16 exp½0:42t 6 exp½0:45t 3 exp½0:53t;
ð3:261Þ
½sð2Þ ðt;3Þð2Þ ¼ ½6½exp½0:05t2 ½1 exp½0:05t2 þ 4½exp½0:05t3 ½1 exp½0:05t þ ½exp½0:05t4 ½1 ½1 exp½0:075t2 exp½0:075texp½0:045texp½0:045texp½0:045texp½0:045t ¼ 12exp½0:43t þ 8exp½0:555t þ 6exp½0:53t 16exp½0:48t 6exp½0:505t 3exp½0:605t;
ð3:262Þ
½sð2Þ ðt; 4Þð2Þ ¼ ½6½exp½0:055t2 ½1 exp½0:055t2 þ 4½exp½0:055t3 ½1 exp½0:055t þ ½exp 0:055t4 ½1 ½1 exp½0:08t2 exp½0:08t exp½0:05t exp½0:05t exp½0:05t exp½0:05t ¼ 12 exp½0:47t þ 8 exp½0:605t þ 6 exp½0:58t 16 exp½0:525t 6 exp½0:55t 3 exp½0:66t:
ð3:263Þ
Subsystem S5 a five-state series system composed of n = 3 components ð5Þ ð5Þ ð5Þ E11 ; E21 ; E31 having the exponential safety functions with coordinates:
138
3 Complex Technical Systems, Reliability ð5Þ
½s11 ðt; 2Þð2Þ ¼ exp½0:04t;
ð5Þ
½s11 ðt; 4Þð2Þ ¼ exp½0:05t;
ð5Þ
½si1 ðt; 2Þð2Þ ¼ exp½0:04t;
ð5Þ
½si1 ðt; 4Þð2Þ ¼ exp½0:055t;
½s11 ðt; 1Þð2Þ ¼ exp½0:033t; ½s11 ðt; 3Þð2Þ ¼ exp½0:045t; ½si1 ðt; 1Þð2Þ ¼ exp½0:033t; ½si1 ðt; 3Þð2Þ ¼ exp½0:05t;
ð5Þ ð5Þ ð5Þ
ð5Þ
i ¼ 2; 3:
Subsystem S5 is a five-state series system that can also be considered as a fivestate parallel–series system with its structure shape parameters k ¼ 3; l1 ¼ 1; l2 ¼ 1; l3 ¼ 1: Thus, according to either the formulae (1.38)–(1.39) and (1.70)–(1.71) or to the formulae (1.22)–(1.23) and (1.58)–(1.59), after replacing in them the components’ reliability functions by their safety functions, subsystem S5 five-state safety function at the operational state z2 is given by ½sð5Þ ðt; Þð2Þ ¼ ½1; ½sð5Þ ðt; 1Þð2Þ ; ½sð5Þ ðt; 2Þð2Þ ; ½sð5Þ ðt; 3Þð2Þ ; ½sð5Þ ðt; 4Þð2Þ ; t 2 h0; 1Þ;
ð3:264Þ
where ½sð5Þ ðt; uÞð2Þ ¼ s3;1;1;1 ðt; uÞ " # 3 1 3 Y Y Y ð5Þ ð5Þ ð2Þ 1 ½1 ½sij ðt; uÞ ¼ ½si1 ðt; uÞð2Þ ; ¼ i¼1
j¼1
t 2 h0; 1Þ;
i¼1
u ¼ 1; 2; 3; 4; and particularly ½sð5Þ ðt; 1Þð2Þ ¼ exp½0:033t exp½0:033t exp½0:033t ¼ exp½0:099t; ð3:265Þ ½sð5Þ ðt; 2Þð2Þ ¼ exp½0:04t exp½0:04t exp½0:04t ¼ exp½0:12t;
ð3:266Þ
½sð5Þ ðt; 3Þð2Þ ¼ exp½0:045t exp½0:05t exp½0:05t ¼ exp½0:145t; ð3:267Þ ½sð5Þ ðt; 4Þð2Þ ¼ exp½0:05t exp½0:055t exp½0:06t ¼ exp½0:165t: ð3:268Þ Considering that the ferry technical system at the operation state z2 is a fivestate series system of subsystems S1 ; S2 and S5 ; after applying (1.22)–(1.23) and (3.254)–(3.258), (3.259)–(3.263) and (3.264)–(3.268), its conditional safety function is given by ½sðt; Þð2Þ ¼ ½1; ½sðt; 1Þð2Þ ; ½sðt; 2Þð2Þ ; ½sðt; 3Þð2Þ ; ½sðt; 4Þð2Þ ;
t 2 h0; 1Þ; ð3:269Þ
3.6 Applications
139
where ½sðt; uÞð2Þ ¼ s3 ðt; uÞ ¼ ½sð1Þ ðt; uÞð2Þ ½sð2Þ ðt; uÞð2Þ ½sð5Þ ðt; uÞð2Þ ; t 2 h0; 1Þ; u ¼ 1; 2; 3; 4; and particularly ½sðt; 1Þð2Þ ¼ exp½0:033t½12 exp½0:33t þ 8 exp½0:429t 16 exp½0:363t 3 exp½0:462t exp½0:099t ¼ 12 exp½0:462t þ 8 exp½0:561t 16 exp½0:495t 3 exp½0:594t ð3:270Þ ½sðt; 2Þð2Þ ¼ exp½0:04t½12 exp½0:38t þ 8 exp½0:49t þ 6 exp½0:46t 16 exp½0:42t 6 exp½0:45t 3 exp½0:53t exp½0:12t ¼ 12 exp½0:54t þ 8 exp½0:65t þ 6 exp½0:62t 16 exp½0:58t 6 exp½0:61t 3 exp½0:69t;
ð3:271Þ
½sðt; 3Þð2Þ ¼ exp½0:045t½12 exp½0:43t þ 8 exp½0:555t þ 6 exp½0:53t 16 exp½0:48t 6 exp½0:505t 3 exp½0:605t exp½0:145t ¼ 12 exp½0:62t þ 8 exp½0:745t þ 6 exp½0:72t 16 exp½0:67t 6 exp½0:695t 3 exp½0:795t;
ð3:272Þ
½sðt; 4Þð2Þ ¼ exp½0:05t½12 exp½0:47t þ 8 exp½0:605t þ 6 exp½0:58t 16 exp½0:525t 6 exp½0:55t 3 exp½0:66t exp½0:165t ¼ 12 exp½0:685t þ 8 exp½0:82t þ 6 exp½0:795t 16 exp½0:74t 6 exp½0:765t 3 exp½0:875t:
ð3:273Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z2 ; calculated from the results given by (3.270)–(3.273), according to (3.8)–(3.9), respectively are: l2 ð1Þ ffi 2:86;
l2 ð2Þ ffi 2:43;
l2 ð3Þ ffi 2:14;
l2 ð4Þ ffi 1:93 years; ð3:274Þ
r2 ð1Þ ffi 2:74;
r2 ð2Þ ffi 2:35;
r2 ð3Þ ffi 2:05;
r2 ð4Þ ffi 1:85 years; ð3:275Þ
and further, using (3.10) and (3.274), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z2 ; respectively are:
140
3 Complex Technical Systems, Reliability
l2 ð1Þ ffi 0:43;
l2 ð2Þ ffi 0:29;
l2 ð3Þ ffi 0:21;
l2 ð4Þ ffi 1:93 years: ð3:276Þ
At the system operation state z3 ; the ferry technical system is composed of two subsystems S1 and S2 forming a series structure shown in Fig. 2.16. ð1Þ Subsystem S1 consists of a component E11 ; with the same safety function as at the operation state z2 : Thus, according to (3.254)–(3.258), subsystem S1 conditional function at the operation state z3 ; is given by ½sð1Þ ðt; Þð3Þ ¼ ½1; ½sð1Þ ðt; 1Þð3Þ ; ½sð1Þ ðt; 2Þð3Þ ; ½sð1Þ ðt; 3Þð3Þ ; ½sð1Þ ðt; 4Þð3Þ ; t 2 h0; 1Þ;
ð3:277Þ
where ½sð1Þ ðt; 1Þð3Þ ¼ exp½0:033t;
ð3:278Þ
½sð1Þ ðt; 2Þð3Þ ¼ exp½0:04t;
ð3:279Þ
½sð1Þ ðt; 3Þð3Þ ¼ exp½0:045t;
ð3:280Þ
½sð1Þ ðt; 4Þð3Þ ¼ exp½0:05t:
ð3:281Þ
Subsystem S2 is a five-state parallel–series system composed of components i ¼ 1; 2; . . .; k; j ¼ 1; 2; . . .; li ; k ¼ 7; l1 ¼ 4; l2 ¼ 2; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; l6 ¼ 1; l7 ¼ 1; having the exponential safety functions with coordinates: ð2Þ Eij ;
ð2Þ
½s1j ðt; 2Þð3Þ ¼ exp½0:04t;
ð2Þ
½s1j ðt; 4Þð3Þ ¼ exp½0:055t;
½s1j ðt; 1Þð3Þ ¼ exp½0:033t; ½s1j ðt; 3Þð3Þ ¼ exp½0:05t; ð2Þ ½s2j ðt; 1Þð3Þ ð2Þ ½s2j ðt; 3Þð3Þ ð2Þ ½si1 ðt; 1Þð3Þ ð2Þ ½si1 ðt; 3Þð3Þ
¼ exp½0:033t; ¼ exp½0:045t; ¼ exp½0:033t; ¼ exp½0:045t;
ð2Þ
ð2Þ
ð2Þ ½s2j ðt; 2Þð3Þ ð2Þ ½s2j ðt; 4Þð3Þ ð2Þ ½si1 ðt; 2Þð3Þ ð2Þ ½si1 ðt; 4Þð3Þ
j ¼ 1; 2; 3; 4;
¼ exp½0:04t; ¼ exp½0:05t;
j ¼ 1; 2;
¼ exp½0:04t; ¼ exp½0:05t;
i ¼ 3; 4; 5; 6; 7:
Subsystem S2 is a five-state parallel–series system with its structure shape parameters k ¼ 7; l1 ¼ 4; l2 ¼ 2; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; l6 ¼ 1; l7 ¼ 1: Thus, according to the formulae (1.38)–(1.39) and (1.70)–(1.71), after replacing in them the components’ reliability functions by their safety functions, subsystem S2 conditional safety function at the operation state z3 is given by ½sð2Þ ðt; Þð3Þ ¼ ½1; ½sð2Þ ðt; 1Þð3Þ ; ½sð2Þ ðt; 2Þð3Þ ; ½sð2Þ ðt; 3Þð3Þ ; ½sð2Þ ðt; 4Þð3Þ ; t 2 h0; 1Þ;
ð3:282Þ
3.6 Applications
141
where ½sð2Þ ðt; uÞð3Þ ¼ s7;4;2;1;1;1;1;1 ðt; uÞ " # li 7 Y Y ð2Þ ð3Þ ¼ 1 ½1 ½sij ðt; uÞ ; i¼1
t 2 h0; 1Þ; u ¼ 1; 2; 3; 4;
j¼1
and particularly ½sð2Þ ðt;1Þð3Þ ¼ ½½6exp½0:033t2 ½1 exp½0:033t2 þ 4½exp½0:033t3 ½1 exp½0:033t þ ½exp½0:033t4 ½1 ½1 exp½0:033t2 exp½0:033texp½0:033texp½0:033texp½0:033texp½0:033t ¼ 12exp½0:258t þ 8exp½0:33t þ 6exp½0:324t 16exp½0:291t 6exp½0:297t 3exp½0:363t; ð3:283Þ ½sð2Þ ðt; 2Þð3Þ ¼ ½½6 exp½0:04t2 ½1 exp½0:04t2 þ 4½exp½0:04t3 ½1 exp½0:04t þ ½exp½0:04t4 ½1 ½1 exp½0:04t2 exp½0:04t exp½0:04t exp½0:04t exp½0:04t exp½0:04t ¼ 12 exp½0:32t þ 14 exp½0:4t 22 exp½0:36t 3 exp½0:44t; ð3:284Þ ½sð2Þ ðt;3Þð3Þ ¼½½6exp½0:05t2 ½1exp½0:05t2 þ4½exp½0:05t3 ½1exp½0:05t þ½exp½0:05t4 ½1½1exp½0:045t2 exp½0:045texp½0:045t exp½0:045texp½0:045texp½0:045t ¼12exp½0:37tþ8exp½0:465tþ6exp½0:47t16exp½0:42t 6exp½0:415t3exp½0:515t;
ð3:285Þ
½sð2Þ ðt; 4Þð3Þ ¼ ½½6 exp½0:055t2 ½1 exp½0:055t2 þ 4½exp½0:055t3 ½1 exp½0:055t½exp½0:055t4 ½1 ½1 exp½0:05t2 exp½0:05t exp½0:05t exp½0:05t exp½0:05t exp½0:05t; ¼ 12 exp½0:41t þ 8 exp½0:515t þ 6 exp½0:52t 16 exp½0:465t 6 exp½0:46t 3 exp½0:57t:
ð3:286Þ
Considering that the ferry technical system at the operation state z3 is a fivestate series system of subsystems S1 and S2 ; after applying (1.22)–(1.23) and (3.277)–(3.281) and (3.282)–(3.286), its conditional five-state safety function is given by
142
3 Complex Technical Systems, Reliability
½sðt; Þð3Þ ¼ ½1; ½sðt; 1Þð3Þ ; ½sðt; 2Þð3Þ ; ½sðt; 3Þð3Þ ; ½sðt; 4Þð3Þ ;
t 2 h0; 1Þ; ð3:287Þ
where ½sðt; uÞð3Þ ¼ s2 ðt; uÞ ¼ ½sð1Þ ðt; uÞð3Þ ½sð2Þ ðt; uÞð3Þ ;
t 2 h0; 1Þ; u ¼ 1; 2; 3; 4;
and particularly ½sðt; 1Þð3Þ ¼ exp½0:033t½12 exp½0:258t þ 8 exp½0:33t þ 6 exp½0:324t 16 exp½0:291t 6 exp½0:297t 3 exp½0:363t ¼ 12 exp½0:291t þ 8 exp½0:363t þ 6 exp½0:357t 16 exp½0:324t 6 exp½0:33t 3 exp½0:396t;
ð3:288Þ
½sðt; 2Þð3Þ ¼ exp½0:04t½12 exp½0:32t þ 14 exp½0:4t 22 exp½0:36t 3 exp½0:44t ¼ 12 exp½0:36t þ 14 exp½0:44t 22 exp½0:4t 3 exp½0:48t;
ð3:289Þ
½sðt; 3Þð3Þ ¼ exp½0:045t½12 exp½0:37t þ 8 exp½0:465t þ 6 exp½0:47t 16 exp½0:42t 6 exp½0:415t 3 exp½0:515t ¼ 12 exp½0:415t þ 8 exp½0:51t þ 6 exp½0:515t 16 exp½0:465t ð3:290Þ 6 exp½0:46t 3 exp½0:56t; ½sðt; 4Þð3Þ ¼ exp½0:05t½12 exp½0:41t þ 8 exp½0:515t þ 6 exp½0:52t 16 exp½0:465t 6 exp½0:46t 3 exp½0:57t ¼ 12 exp½0:46t þ 8 exp½0:565t þ 6 exp½0:57t 16 exp½0:515t ð3:291Þ 6 exp½0:51t 3 exp½0:62t: The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z3 ; calculated from the results given by (3.288)–(3.291), according to (3.8)–(3.9), respectively are: l3 ð1Þ ffi 4:94;
l3 ð2Þ ffi 3:9;
l3 ð3Þ ffi 3:44;
r3 ð1Þ ffi 4:6;
r3 ð2Þ ffi 3:6;
r3 ð3Þ ffi 3:2;
l3 ð4Þ ffi 3:1 years; r3 ð4Þ ffi 2:9 years;
ð3:292Þ ð3:293Þ
and further, using (3.10) and (3.292), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z3 ; respectively are: l3 ð1Þ ffi 1:04;
l3 ð2Þ ffi 0:46;
l3 ð3Þ ffi 0:34;
l3 ð4Þ ffi 3:1 years:
ð3:294Þ
3.6 Applications
143
At the system operation state z4 ; the ferry technical system is composed of three subsystems S1 ; S2 and S4 forming a series structure shown in Fig. 2.17. ð1Þ Subsystem S1 consists of one component E11 ; with the same safety function as at the operation state z2 : Thus, according to (3.254)–(3.258), subsystem S1 conditional function at the operation state z4 ; is given by ½sð1Þ ðt; Þð4Þ ¼ ½1; ½sð1Þ ðt; 1Þð4Þ ; ½sð1Þ ðt; 2Þð4Þ ; ½sð1Þ ðt; 3Þð4Þ ; ½sð1Þ ðt; 4Þð4Þ ; t 2 h0; 1Þ;
ð3:295Þ
where ½sð1Þ ðt; 1Þð4Þ ¼ exp½0:033t;
ð3:296Þ
½sð1Þ ðt; 2Þð4Þ ¼ exp½0:04t;
ð3:297Þ
½sð1Þ ðt; 3Þð4Þ ¼ exp½0:045t;
ð3:298Þ
½sð1Þ ðt; 4Þð4Þ ¼ exp½0:05t:
ð3:299Þ
Subsystem S2 is a five-state parallel–series system composed of components ð2Þ Eij ; i ¼ 1; 2; . . .; k; j ¼ 1; 2; . . .; li ; k ¼ 5; l1 ¼ 4; l2 ¼ 1; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; having the exponential safety functions with coordinates: ð2Þ
½s1j ðt; 2Þð4Þ ¼ exp½0:04t;
ð2Þ
½s1j ðt; 4Þð4Þ ¼ exp½0:055t;
½s1j ðt; 1Þð4Þ ¼ exp½0:033t; ½s1j ðt; 3Þð4Þ ¼ exp½0:05t;
ð2Þ
ð2Þ
j ¼ 1; 2; 3; 4;
ð2Þ
½si1 ðt; 2Þð4Þ ¼ exp½0:04t;
ð2Þ
½si1 ðt; 4Þð4Þ
½si1 ðt; 1Þð4Þ ¼ exp½0:033t; ½si1 ðt; 3Þð4Þ ¼ exp½0:045t;
ð2Þ ð2Þ
Subsystem S2 is a five-state parallel–series system with its structure shape parameters k ¼ 5; l1 ¼ 4; l2 ¼ 1; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1: Thus, according to the formulae (1.38)–(1.39) and (1.70)–(1.71), after replacing in them the components’ reliability functions by their safety functions, the subsystem S2 conditional safety function at the operation state z4 is given by ½sð2Þ ðt; Þð4Þ ¼ ½1; ½sð2Þ ðt; 1Þð4Þ ; ½sð2Þ ðt; 2Þð4Þ ; ½sð2Þ ðt; 3Þð4Þ ; ½sð2Þ ðt; 4Þð4Þ ; t 2 h0; 1Þ;
ð3:300Þ
where ½sð2Þ ðt; uÞð4Þ ¼ s5;4;1;1;1;1 ðt; uÞ " # li 5 Y Y ð2Þ ð4Þ 1 ½1 ½sij ðt; uÞ ; ¼ i¼1
j¼1
t 2 h0; 1Þ;
u ¼ 1; 2; 3; 4;
144
3 Complex Technical Systems, Reliability
and particularly ½sð2Þ ðt; 1Þð4Þ ¼ ½6½exp½0:033t2 ½1 exp½0:033t2 þ 4½exp½0:033t3 ½1 exp½0:033t þ ½exp½0:033t4 exp½0:033t exp½0:033t exp½0:033t exp½0:033t ¼ 6 exp½0:198t 8 exp½0:231t þ 3 exp½0:264t;
ð3:301Þ
½sð2Þ ðt;2Þð4Þ ¼½6½exp½0:04t2 ½1exp½0:04t2 þ4½exp½0:04t3 ½1exp½0:04t þ½exp½0:04t4 exp½0:04texp½0:04texp½0:04texp½0:04t ¼6exp½0:24t8exp½0:28tþ3exp½0:32t; ð3:302Þ ½sð2Þ ðt; 3Þð4Þ ¼ ½6½exp½0:05t2 ½1 exp½0:05t2 þ 4½exp½0:05t3 ½1 exp½0:05t þ ½exp½0:05t4 exp½0:045t exp½0:045t exp½0:045t exp½0:045t ¼ 6 exp½0:28t 8 exp½0:33t þ 3 exp½0:38t;
ð3:303Þ
½sð2Þ ðt; 4Þð4Þ ¼ ½6½exp½0:055t2 ½1 exp½0:055t2 þ 4½exp½0:055t3 ½1 exp½0:055t þ ½exp½0:055t4 exp½0:05t exp½0:05t exp½0:05t exp½0:05t ¼ exp½0:31t 8 exp½0:365t þ 3 exp½0:42t:
ð3:304Þ
ð4Þ
Subsystem S4 consists of a component E11 having the exponential safety function with coordinates: ð4Þ
½s11 ðt; 2Þð4Þ ¼ exp½0:055t;
ð4Þ
½s11 ðt; 4Þð4Þ ¼ exp½0:065t:
½s11 ðt; 1Þð4Þ ¼ exp½0:05t; ½s11 ðt; 3Þð4Þ ¼ exp½0:06t;
ð4Þ ð4Þ
Thus, at the operation state z4 ; subsystem S4 conditional safety function is ð4Þ identical with the component E11 safety function and is given by ½sð4Þ ðt; Þð4Þ ¼ ½1; ½sð4Þ ðt; 1Þð4Þ ; ½sð4Þ ðt; 2Þð4Þ ; ½sð4Þ ðt; 3Þð4Þ ; ½sð4Þ ðt; 4Þð4Þ ; t 2 h0; 1Þ; where
ð3:305Þ
½sð4Þ ðt; 1Þð4Þ ¼ exp½0:05t;
ð3:306Þ
½sð4Þ ðt; 2Þð4Þ ¼ exp½0:055t;
ð3:307Þ
½sð4Þ ðt; 3Þð4Þ ¼ exp½0:06t;
ð3:308Þ
3.6 Applications
145
½sð4Þ ðt; 4Þð4Þ ¼ exp½0:065t:
ð3:309Þ
Considering that the ferry technical system at the operation state z4 is a fivestate series system of subsystems S1 ; S2 and S4 ; after applying (1.22)–(1.23) and (3.295)–(3.299), (3.300)–(3.304) and (3.305)–(3.309), its conditional safety function is given by ½sðt; Þð4Þ ¼ ½1; ½sðt; 1Þð4Þ ; ½sðt; 2Þð4Þ ; ½sðt; 3Þð4Þ ; ½sðt; 4Þð4Þ ;
t 2 h0; 1Þ; ð3:310Þ
where ½sðt; uÞð4Þ ¼ s3 ðt; uÞ ¼ ½sð1Þ ðt; uÞð4Þ ½sð2Þ ðt; uÞð4Þ ½sð4Þ ðt; uÞð4Þ ; u ¼ 1; 2; 3; 4;
t 2 h0; 1Þ;
and particularly ½sðt; 1Þð4Þ ¼ exp½0:033t½6 exp½0:198t 8 exp½0:231t þ 3 exp½0:264t exp½0:05t ¼ 6 exp½0:281t 8 exp½0:314t þ 3 exp½0:347t;
ð3:311Þ
½sðt; 2Þð4Þ ¼ exp½0:04t½6 exp½0:24t 8 exp½0:28t þ 3 exp½0:32t exp½0:055t ¼ 6 exp½0:335t 8 exp½0:375t þ 3 exp½0:415t;
ð3:312Þ
½sðt; 3Þð4Þ ¼ exp½0:045t½6 exp½0:28t 8 exp½0:33t þ 3 exp½0:38t exp½0:06t ¼ 6 exp½0:385t 8 exp½0:435t þ 3 exp½0:485t;
ð3:313Þ
½sðt; 4Þð4Þ ¼ exp½0:05t½exp½0:31t 8 exp½0:365t þ 3 exp½0:42t exp½0:065t ¼ 6 exp½0:425t 8 exp½0:48t þ 3 exp½0:535t:
ð3:314Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z4 ; calculated from the results given by (3.311)–(3.314), according to (3.8)–(3.9), respectively are: l4 ð1Þ ffi 4:2;
l4 ð2Þ ffi 3:80;
l4 ð3Þ ffi 3:38;
l4 ð4Þ ffi 3:05 years;
ð3:315Þ
r4 ð1Þ ffi 4:37;
r4 ð2Þ ffi 3:68;
r4 ð3Þ ffi 3:12;
r4 ð4Þ ffi 2:93 years; ð3:316Þ
146
3 Complex Technical Systems, Reliability
and further, using (3.10) and (3.315), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z4 ; respectively are: l4 ð1Þ ffi 0:72;
l4 ð2Þ ffi 0:42;
l4 ð3Þ ffi 0:33;
l4 ð4Þ ffi 3:05 years: ð3:317Þ
At the system operation state z5 ; the ferry technical system is composed of three subsystems S1 ; S2 and S4 forming a series structure shown in Fig. 2.12. The system safety structure and its components safety functions are the same as at the operation state z4 : Thus, considering (3.310)–(3.314), its conditional safety function is given by h i ½sðt; Þð5Þ ¼ 1; ½sðt; 1Þð5Þ ; ½sðt; 2Þð5Þ ; ½sðt; 3Þð5Þ ; ½sðt; 4Þð5Þ ; t 2 h0; 1Þ; ð3:318Þ where ½sðt; 1Þð5Þ ¼ 6 exp½0:281t 8 exp½0:314t þ 3 exp½0:347t;
ð3:319Þ
½sðt; 2Þð5Þ ¼ exp½0:04t½6 exp½0:24t 8 exp½0:28t þ 3 exp½0:32t exp½0:055t ¼ 6 exp½0:335t 8 exp½0:375t þ 3 exp½0:415t;
ð3:320Þ
½sðt; 3Þð5Þ ¼ exp½0:045t½6 exp½0:28t 8 exp½0:33t þ 3 exp½0:38t exp½0:06t ¼ 6 exp½0:385t 8 exp½0:435t þ 3 exp½0:485t;
ð3:321Þ
½sðt; 4Þð5Þ ¼ exp½0:05t½exp½0:31t 8 exp½0:365t þ 3 exp½0:42t exp½0:065t ¼ 6 exp½0:425t 8 exp½0:48t þ 3 exp½0:535t:
ð3:322Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z5 ; calculated from the results given by (3.319)–(3.322), according to (3.8)–(3.9), respectively are: l5 ð1Þ ffi 4:2;
l5 ð2Þ ffi 3:80;
l5 ð3Þ ffi 3:38;
l5 ð4Þ ffi 3:05 years;
ð3:323Þ
r5 ð1Þ ffi 4:37;
r5 ð2Þ ffi 3:68;
r5 ð3Þ ffi 3:12;
r5 ð4Þ ffi 2:93 years; ð3:324Þ
and further, using (3.10) and (3.323), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z5 ; respectively are: l5 ð1Þ ffi 0:72;
l5 ð2Þ ffi 0:42;
l5 ð3Þ ffi 0:33;
l5 ð4Þ ffi 3:05 years: ð3:325Þ
3.6 Applications
147
At the system operation state z6 ; the ferry technical system is composed of three subsystems S1 ; S2 and S4 forming a series structure shown in Fig. 2.18. Subsystem ð1Þ S1 consists of a component E11 ; with the same safety function as at the operation state z2 : Thus, according to (3.254)–(3.258), subsystem S1 conditional safety function at the operation state z6 is given by ½sð1Þ ðt; Þð6Þ ¼ ½1; ½sð1Þ ðt; 1Þð6Þ ; ½sð1Þ ðt; 2Þð6Þ ; ½sð1Þ ðt; 3Þð6Þ ; ½sð1Þ ðt; 4Þð6Þ ; ð3:326Þ t 2 h0; 1Þ; where ½sð1Þ ðt; 1Þð6Þ ¼ exp½0:033t;
ð3:327Þ
½sð1Þ ðt; 2Þð6Þ ¼ exp½0:04t;
ð3:328Þ
½sð1Þ ðt; 3Þð6Þ ¼ exp½0:045t;
ð3:329Þ
½sð1Þ ðt; 4Þð6Þ ¼ exp½0:05t:
ð3:330Þ
Subsystem S2 illustrated in Fig. 2.18, is a five-state parallel–series system ð2Þ composed of components Eij ; i ¼ 1; 2; . . .; k; j ¼ 1; 2; . . .; li ; k ¼ 7; l1 ¼ 4; l2 ¼ 2; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; l6 ¼ 1; l7 ¼ 1; with the same safety functions as at the operation state z3 : Thus, according to (3.282)–(3.286), subsystem S2 conditional safety function at the operation state z6 is given by h i ½sð2Þ ðt; Þð6Þ ¼ 1; ½s2Þðt; 1Þð6Þ ; ½sð2Þ ðt; 2Þð6Þ ; ½sð2Þ ðt; 3Þð6Þ ; ½sð2Þ ðt; 4Þð6Þ ; t 2 h0; 1Þ;
ð3:331Þ
where ½sð2Þ ðt; 1Þð6Þ ¼ 12 exp½0:258t þ 8 exp½0:33t þ 6 exp½0:324t 16 exp½0:291t 6 exp½0:297t 3 exp½0:363t;
ð3:332Þ
½sð2Þ ðt; 2Þð6Þ ¼ 12 exp½0:32t þ 14 exp½0:4t 22 exp½0:36t 3 exp½0:44t;
ð3:333Þ
½sð2Þ ðt; 3Þð6Þ ¼ 12 exp½0:37t þ 8 exp½0:465t þ 6 exp½0:47t 16 exp½0:42t 6 exp½0:415t 3 exp½0:515t;
ð3:334Þ
½sð2Þ ðt; 4Þð6Þ ¼ 12 exp½0:41t þ 8 exp½0:515t þ 6 exp½0:52t 16 exp½0:465t 6 exp½0:46t 3 exp½0:57t:
ð3:335Þ
148
3 Complex Technical Systems, Reliability ð4Þ
Subsystem S4 consists of a component E11 ; with the same safety functions as at the operation state z4 : Thus, according to (3.305)–(3.309), subsystem S4 conditional safety function at the operation state z6 is given by ½sð4Þ ðt; Þð6Þ ¼ ½1; ½sð4Þ ðt; 1Þð6Þ ; ½sð4Þ ðt; 2Þð6Þ ; ½sð4Þ ðt; 3Þð6Þ ; ½sð4Þ ðt; 4Þð6Þ ; t 2 h0; 1Þ;
ð3:336Þ
where ½sð4Þ ðt; 1Þð6Þ ¼ exp½0:05t;
ð3:337Þ
½sð4Þ ðt; 2Þð6Þ ¼ exp½0:055t;
ð3:338Þ
½sð4Þ ðt; 3Þð6Þ ¼ exp½0:06t;
ð3:339Þ
½sð4Þ ðt; 4Þð6Þ ¼ exp½0:065t:
ð3:340Þ
Considering that the ferry technical system at the operation state z6 is a fivestate series system of subsystems S1 ; S2 and S4 ; after applying (1.22)–(1.23) and (3.326)–(3.330), (3.331)–(3.335) and (3.336)–(3.340), its conditional safety function is given by ½sðt; Þð6Þ ¼ ½1; ½sðt; 1Þð6Þ ; ½sðt; 2Þð6Þ ; ½sðt; 3Þð6Þ ; ½sðt; 4Þð6Þ ;
t 2 h0; 1Þ; ð3:341Þ
where ½sðt; uÞð6Þ ¼ s3 ðt; uÞ ¼ ½sð1Þ ðt; uÞð6Þ ½sð2Þ ðt; uÞð6Þ ½sð4Þ ðt; uÞð6Þ ; u ¼ 1; 2; 3; 4;
t 2 h0; 1Þ;
and particularly ½sðt; 1Þð6Þ ¼ exp½0:033t½12 exp½0:258t þ 8 exp½0:33t þ 6 exp½0:324t 16 exp½0:291t 6 exp½0:297t 3 exp½0:363t½exp½0:05t ¼ 12 exp½0:341t þ 8 exp½0:413t þ 6 exp½0:407t 16 exp½0:374t ð3:342Þ 6 exp½0:38t 3 exp½0:446t; ½sðt; 2Þð6Þ ¼ exp½0:04t½12 exp½0:32t þ 14 exp½0:4t 22 exp½0:36t 3 exp½0:44t½exp½0:055t ¼ 12 exp½0:415t þ 14 exp½0:495t 22 exp½0:455t 3 exp½0:535t;
ð3:343Þ
3.6 Applications
149
½sðt; 3Þð6Þ ¼ exp½0:045t½12 exp½0:37t þ 8 exp½0:465t þ 6 exp½0:47t 16 exp½0:42t 6 exp½0:415t 3 exp½0:515t½exp½0:06t ¼ 12 exp½0:475t þ 8 exp½0:57t þ 6 exp½0:575t 16 exp½0:525t ð3:344Þ 6 exp½0:52t 3 exp½0:62t; ½sðt; 4Þð6Þ ¼ exp½0:05t½12 exp½0:41t þ 8 exp½0:515t þ 6 exp½0:52t 16 exp½0:465t 6 exp½0:46t 3 exp½0:57t½exp½0:065t ¼ 12 exp½0:525t þ 8 exp½0:63t þ 6 exp½0:635t 16 exp½0:58t 6 exp½0:575t 3 exp½0:685t: ð3:345Þ The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z6 ; calculated from the results given by (3.342)–(3.345), according to (3.8)–(3.9), respectively are: l6 ð1Þ ffi 4:01;
l6 ð2Þ ffi 3:24;
l6 ð3Þ ffi 2:88;
l6 ð4Þ ffi 2:61 years; ð3:346Þ
r6 ð1Þ ffi 3:80;
r6 ð2Þ ffi 3:10;
r6 ð3Þ ffi 2:70;
r6 ð4Þ ffi 2:47 years; ð3:347Þ
and further, using (3.10) and (3.346) the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z6 ; respectively are: l6 ð1Þ ffi 0:77;
l6 ð2Þ ffi 0:36;
l6 ð3Þ ffi 0:36;
l6 ð4Þ ffi 2:61 years
ð3:348Þ
At the system operation state z7 ; the ferry technical system is composed of three subsystems S1 ; S2 and S5 forming a series structure shown in Fig. 2.15. The system safety structure and its components safety functions are the same as at the operation state z2 : Thus, considering (3.269)–(3.273), its conditional safety function is given by ½sðt; Þð7Þ ¼ ½1; ½sðt; 1Þð7Þ ; ½sðt; 2Þð7Þ ; ½sðt; 3Þð7Þ ; ½sðt; 4Þð7Þ ;
t 2 h0; 1Þ; ð3:349Þ
where ½sðt; 1Þð7Þ ¼ 12 exp½0:462t þ 8 exp½0:561t 16 exp½0:495t 3 exp½0:594t;
ð3:350Þ
½sðt; 2Þð7Þ ¼ 12 exp½0:54t þ 8 exp½0:65t þ 6 exp½0:62t 16 exp½0:58t 6 exp½0:61t 3 exp½0:69t;
ð3:351Þ
½sðt; 3Þð7Þ ¼ 12 exp½0:62t þ 8 exp½0:745t þ 6 exp½0:72t 16 exp½0:67t 6 exp½0:695t 3 exp½0:795t;
ð3:352Þ
150
3 Complex Technical Systems, Reliability
½sðt; 4Þð7Þ ¼ 12 exp½0:685t þ 8 exp½0:82t þ 6 exp½0:795t 16 exp½0:74t 6 exp½0:765t 3 exp½0:875t:
ð3:353Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z7 ; calculated from the results given by (3.350)–(3.353), according to (3.8)–(3.9), respectively are: l7 ð1Þ ffi 2:86;
l7 ð2Þ ffi 2:43;
l7 ð3Þ ffi 2:14;
l7 ð4Þ ffi 1:93 years; ð3:354Þ
r7 ð1Þ ffi 2:74;
r7 ð2Þ ffi 2:35;
r7 ð3Þ ffi 2:05;
r7 ð4Þ ffi 1:85 years; ð3:355Þ
and further, using (3.10) and (3.354), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z7 ; respectively are: l7 ð1Þ ffi 0:43;
l7 ð2Þ ffi 0:29;
l7 ð3Þ ffi 0:21;
l7 ð4Þ ffi 1:93 years: ð3:356Þ
At the system operation state z8 ; the ferry technical system is composed of two subsystems S3 and S4 forming a series structure shown in Fig. 2.19. Subsystem S3 is a five-state series system composed of n = 2 components ð3Þ ð3Þ E11 ; E21 having the exponential safety functions identified in Chap. 5 with coordinates: ð3Þ
½s11 ðt; 1Þð8Þ ¼ exp½0:02t;
ð3Þ
½s11 ðt; 2Þð8Þ ¼ exp½0:03t;
ð3Þ
½s11 ðt; 4Þð8Þ ¼ exp½0:04t;
ð3Þ
½s21 ðt; 2Þð8Þ ¼ exp½0:04t;
ð3Þ
½s21 ðt; 4Þð8Þ ¼ exp½0:05t:
½s11 ðt; 3Þð8Þ ¼ exp½0:035t; ½s21 ðt; 1Þð8Þ ¼ exp½0:033t; ½s21 ðt; 3Þð8Þ ¼ exp½0:045t;
ð3Þ ð3Þ ð3Þ
Subsystem S3 is a five-state series system that can also be considered as a five-state parallel–series system with its structure shape parameters k ¼ 2; l1 ¼ 1; l2 ¼ 1: Thus, according to either the formulae (1.38)–(1.39) and (1.70)–(1.71) or to the formulae (1.22)–(1.23) and (1.58)–(1.59), after replacing in them the components’ reliability functions by their safety functions, the subsystem S3 conditional safety function at the operation state z8 is given by ½sð3Þ ðt; Þð8Þ ¼ ½1; ½sð3Þ ðt; 1Þð8Þ ; ½sð3Þ ðt; 2Þð8Þ ; ½sð3Þ ðt; 3Þð8Þ ; ½sð3Þ ðt; 4Þð8Þ ; t 2 h0; 1Þ;
ð3:357Þ
where ½sð3Þ ðt; uÞð8Þ ¼ s3;1;1;1 ðt; uÞ " # 2 1 2 Y Y Y ð3Þ ð3Þ ð8Þ 1 ½1 ½sij ðt; uÞ ¼ ½si1 ðt; uÞð8Þ ; ¼ i¼1
j¼1
t 2 h0; 1Þ;
i¼1
u ¼ 1; 2; 3; 4;
3.6 Applications
151
and particularly ½sð3Þ ðt; 1Þð8Þ ¼ exp½0:2t exp½0:033t ¼ exp½0:053t;
ð3:358Þ
½sð3Þ ðt; 2Þð8Þ ¼ exp½0:3t exp½0:04t ¼ exp½0:070t;
ð3:359Þ
½sð3Þ ðt; 3Þð8Þ ¼ exp½0:35t exp½0:045t ¼ exp½0:080t;
ð3:360Þ
½sð3Þ ðt; 4Þð8Þ ¼ exp½0:4t exp½0:05t ¼ exp½0:090t:
ð3:361Þ
ð4Þ
Subsystem S4 consists of a component E11 ; with the same safety function as at the operation state z1 : Thus, according to (3.241)–(3.245), subsystem S4 conditional safety function at the operation state z6 is given by ½sð4Þ ðt; Þð8Þ ¼ ½1; ½sð4Þ ðt; 1Þð8Þ ; ½sð4Þ ðt; 2Þð8Þ ; ½sð4Þ ðt; 3Þð8Þ ; ½sð4Þ ðt; 4Þð8Þ ; t 2 h0; 1Þ;
ð3:362Þ
where ½sð4Þ ðt; 1Þð8Þ ¼ exp½0:05t;
ð3:363Þ
½sð4Þ ðt; 2Þð8Þ ¼ exp½0:06t;
ð3:364Þ
½sð4Þ ðt; 3Þð8Þ ¼ exp½0:065t;
ð3:365Þ
½sð4Þ ðt; 4Þð8Þ ¼ exp½0:07t:
ð3:366Þ
Considering that the ferry technical system at the operation state z8 is a fivestate series system of subsystems S3 and S4 ; after applying (1.22)–(1.23) and (3.357)–(3.361) and (3.362)–(3.366), its conditional five-state safety function is given by ½sðt; Þð8Þ ¼ ½1; ½sðt; 1Þð8Þ ; ½sðt; 2Þð8Þ ; ½sðt; 3Þð8Þ ; ½sðt; 4Þð8Þ ;
t 2 h0; 1Þ; ð3:367Þ
where ½sðt; uÞð8Þ ¼ s2 ðt; uÞ ¼ ½sð3Þ ðt; uÞð8Þ ½sð4Þ ðt; uÞð8Þ
t 2 h0; 1Þ; u ¼ 1; 2; 3; 4;
and particularly ½sðt; 1Þð8Þ ¼ exp½0:053t exp½0:05t ¼ exp½0:103t;
ð3:368Þ
½sðt; 2Þð8Þ ¼ exp½0:070t exp½0:06t ¼ exp½0:130t;
ð3:369Þ
½sðt; 3Þð8Þ ¼ exp½0:080t exp½0:065t ¼ exp½0:145t;
ð3:370Þ
152
3 Complex Technical Systems, Reliability
½sðt; 4Þð8Þ ¼ exp½0:090t exp½0:07t ¼ exp½0:160t:
ð3:371Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z8 ; calculated from the results given by (3.68)–(3.371), according to (3.8)–(3.9), respectively are: l8 ð1Þ ffi 9:71;
l8 ð2Þ ffi 7:69;
l8 ð3Þ ffi 6:88;
l8 ð4Þ ffi 6:25 years; ð3:372Þ
r8 ð1Þ ffi 9:71;
r8 ð2Þ ffi 7:69;
r8 ð3Þ ffi 6:88;
r8 ð4Þ ffi 6:25 years; ð3:373Þ
and further, using (3.10) and (3.372), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z8 ; respectively are: l8 ð1Þ ffi 2:02;
l8 ð2Þ ffi 0:80;
l8 ð3Þ ffi 0:83;
l8 ð4Þ ffi 6:25 years: ð3:374Þ
At the system operation state z9 ; the ferry technical system is composed of two subsystems S3 and S4 forming a series structure shown in Fig. 2.19. The system safety structure and its components safety functions are the same as at the operation state z8 : Thus, considering (3.367)–(3.371), its conditional five-state safety function is given by ½sðt; Þð9Þ ¼ ½1;½ sðt; 1Þð9Þ ; ½sðt; 2Þð9Þ ; ½sðt; 3Þð9Þ ; ½sðt; 4Þð9Þ ;
t 2 h0; 1Þ; ð3:375Þ
where ½sðt; 1Þð9Þ ¼ exp½0:103t;
ð3:376Þ
½sðt; 2Þð9Þ ¼ exp½0:130t;
ð3:377Þ
½sðt; 3Þð9Þ ¼ exp½0:145t;
ð3:378Þ
½sðt; 4Þð9Þ ¼ exp½0:160t:
ð3:379Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z9 ; calculated from the results given by (3.376)–(3.379), according to (3.8)–(3.9), respectively are: l9 ð1Þ ffi 9:71;
l9 ð2Þ ffi 7:69;
l9 ð3Þ ffi 6:89;
l9 ð4Þ ffi 6:25 years; ð3:380Þ
r9 ð1Þ ffi 9:71;
r9 ð2Þ ffi 7:69;
r9 ð3Þ ffi 6:89;
r9 ð4Þ ffi 6:25 years; ð3:381Þ
and further, using (3.10) and (3.380), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z9 ; respectively are:
3.6 Applications
153
l9 ð1Þ ffi 2:02;
l9 ð2Þ ffi 0:80;
l9 ð3Þ ffi 0:64;
l9 ð4Þ ffi 6:25 years: ð3:382Þ
At the system operation state z10 ; the ferry technical system is composed of three subsystems S1 ; S2 and S5 forming a series structure shown in Fig. 2.15. The system safety structure and its components safety functions are the same as at the operation state z2 : Thus, considering (3.269)–(3.273), its conditional safety function is given by ½sðt; Þð10Þ ¼ ½1; ½sðt; 1Þð10Þ ; ½sðt; 2Þð10Þ ; ½sðt; 3Þð10Þ ; ½sðt; 4Þð10Þ ;
t 2 h0; 1Þ; ð3:383Þ
where ½sðt; 1Þð10Þ ¼ 12 exp½0:462t þ 8 exp½0:561t 16 exp½0:495t 3 exp½0:594t:
ð3:384Þ
½sðt; 2Þð10Þ ¼ 12 exp½0:54t þ 8 exp½0:65t þ 6 exp½0:62t 16 exp½0:58t 6 exp½0:61t 3 exp½0:69t;
ð3:385Þ
½sðt; 3Þð10Þ ¼ 12 exp½0:62t þ 8 exp½0:745t þ 6 exp½0:72t 16 exp½0:67t 6 exp½0:695t 3 exp½0:795t;
ð3:386Þ
½sðt; 4Þð10Þ ¼ 12 exp½0:685t þ 8 exp½0:82t þ 6 exp½0:795t 16 exp½0:74t 6 exp½0:765t 3 exp½0:875t:
ð3:387Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z10 ; calculated from the results given by (3.384)–(3.387), according to (3.8)–(3.9), respectively are: l10 ð1Þ ffi 2:86;
l10 ð2Þ ffi 2:43;
l10 ð3Þ ffi 2:14;
l10 ð4Þ ffi 1:93 years; ð3:388Þ
r10 ð1Þ ffi 2:74;
r10 ð2Þ ffi 2:35;
r10 ð3Þ ffi 2:05;
r10 ð4Þ ffi 1:85 years; ð3:389Þ
and further, using (3.10) and (3.388), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z10 ; respectively are: l10 ð1Þ ffi 0:43;
l10 ð2Þ ffi 0:29;
l10 ð3Þ ffi 0:21;
l10 ð4Þ ffi 1:93 years: ð3:390Þ
At the system operation state z11 ; the ferry technical system is composed of two subsystems S1 and S2 forming a series structure shown in Fig. 2.15.
154
3 Complex Technical Systems, Reliability ð1Þ
Subsystem S1 consists of a component E11 ; with the same safety function as at the operation state z2 : Thus, according to (3.254)–(3.258), subsystem S1 conditional safety function at the operation state z11 is given by ½sð1Þ ðt; Þð11Þ ¼ ½1; ½sð1Þ ðt; 1Þð11Þ ; ½sð1Þ ðt; 2Þð11Þ ; ½sð1Þ ðt; 3Þð11Þ ; ½sð1Þ ðt; 4Þð11Þ ; t 2 h0; 1Þ; ð3:391Þ where ½sð1Þ ðt; 1Þð11Þ ¼ exp½0:033t;
ð3:392Þ
½sð1Þ ðt; 2Þð11Þ ¼ exp½0:04t;
ð3:393Þ
½sð1Þ ðt; 3Þð11Þ ¼ exp½0:045t;
ð3:394Þ
½sð1Þ ðt; 4Þð11Þ ¼ exp½0:05t:
ð3:395Þ
Subsystem S2 is a five-state parallel–series system composed of components ð2Þ Eij ; i ¼ 1; 2; . . .; k; j ¼ 1; 2; . . .; li ; k ¼ 7; l1 ¼ 4; l2 ¼ 2; l3 ¼ 1; l4 ¼ 1; l5 ¼ 1; l6 ¼ 1; l7 ¼ 1; with the same safety functions as at the operation state z2 : Thus, according to (3.259)–(3.263), subsystem S2 conditional safety function at the operation state z11 is given by ½sð2Þ ðt; Þð11Þ ¼ ½1; ½sð2Þ ðt; 1Þð11Þ ; ½sð2Þ ðt; 2Þð11Þ ; ½sð2Þ ðt; 3Þð11Þ ; ½sð2Þ ðt; 4Þð11Þ ; t 2 h0; 1Þ;
ð3:396Þ
where ½sð2Þ ðt; 1Þð11Þ ¼ 12 exp½0:33t þ 8 exp½0:429t 16 exp½0:363t 3 exp½0:462t;
ð3:397Þ
½sð2Þ ðt; 2Þð11Þ ¼ 12 exp½0:38t þ 8 exp½0:49t þ 6 exp½0:46t 16 exp½0:42t 6 exp½0:45t 3 exp½0:53t;
ð3:398Þ
½sð2Þ ðt; 3Þð11Þ ¼ 12 exp½0:43t þ 8 exp½0:555t þ 6 exp½0:53t 16 exp½0:48t 6 exp½0:505t 3 exp½0:605t;
ð3:399Þ
½sð2Þ ðt; 4Þð11Þ ¼ 12 exp½0:47t þ 8 exp½0:605t þ 6 exp½0:58t 16 exp½0:525t 6 exp½0:55t 3 exp½0:66t:
ð3:400Þ
Considering that the ferry technical system at the operation state z11 is a fivestate series system of subsystems S1 and S2 ; after applying (1.22)–(1.23) and (3.391)–(3.395) and (3.396)–(3.400) its conditional safety function is given by
3.6 Applications
155
½sðt; Þð11Þ ¼ ½1; ½sðt; 1Þð11Þ ; ½sðt; 2Þð11Þ ; ½sðt; 3Þð11Þ ; ½sðt; 4Þð11Þ ;
t 2 h0; 1Þ; ð3:401Þ
where ½sðt; uÞð11Þ ¼ s2 ðt; uÞ ¼ ½sð1Þ ðt; uÞð11Þ ½sð2Þ ðt; uÞð11Þ ;
t 2 h0; 1Þ; u ¼ 1; 2; 3; 4;
and particularly ½sðt;1Þð11Þ ¼ exp½0:033t½12exp½0:33t þ 8exp½0:429t 16exp½0:363t 3exp½0:462t ¼ 12exp½0:363t þ 8exp½0:462t 16exp½0:396t 3exp½0:495t; ð3:402Þ ½sðt; 2Þð11Þ ¼ exp½0:04t½12 exp½0:38t þ 8 exp½0:49t þ 6 exp½0:46t 16 exp½0:42t 6 exp½0:45t 3 exp½0:53t ¼ 12 exp½0:42t þ 8 exp½0:53t þ 6 exp½0:5t 16 exp½0:46t 6 exp½0:49t 3 exp½0:57t;
ð3:403Þ
½sðt; 3Þð11Þ ¼ exp½0:045t½12 exp½0:43t þ 8 exp½0:555t þ 6 exp½0:53t 16 exp½0:48t 6 exp½0:505t 3 exp½0:605t ¼ 12 exp½0:475t þ 8 exp½0:6t þ 6 exp½0:575t 16 exp½0:525t 6 exp½0:55t 3 exp½0:65t;
ð3:404Þ
½sðt; 4Þð11Þ ¼ exp½0:05t½12 exp½0:47t þ 8 exp½0:605t þ 6 exp½0:58t 16 exp½0:525t 6 exp½0:55t 3 exp½0:66t ¼ 12 exp½0:52t þ 8 exp½0:655t þ 6 exp½0:63t 16 exp½0:575t ð3:405Þ 6 exp½0:6t 3 exp½0:71t: The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z11 ; calculated from the results given by (3.402)–(3.405), according to (3.8)–(3.9), respectively are: l11 ð1Þ ffi 3:91;
l11 ð2Þ ffi 3:37;
l11 ð3Þ ffi 3:07;
l11 ð4Þ ffi 2:76 years; ð3:406Þ
r11 ð1Þ ffi 3:64;
r11 ð2Þ ffi 3:16;
r11 ð3Þ ffi 2:82;
r11 ð4Þ ffi 2:57 years; ð3:407Þ
and further, using (3.10) and (3.406), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z11 ; respectively are:
156
3 Complex Technical Systems, Reliability
l11 ð1Þ ffi 0:54;
l11 ð2Þ ffi 0:34;
l11 ð3Þ ffi 0:27;
l11 ð4Þ ffi 2:76 years: ð3:408Þ
At the system operation state z12 ; i.e. at the leaving state, the ferry technical system is composed of three subsystems S1 ; S2 and S4 forming a series structure shown in Fig. 2.17. The system safety structure and its components safety functions are the same as at the operation state z4 : Thus, considering (3.310)–(3.314), its conditional safety function is given by ½sðt; Þð12Þ ¼ ½1; ½sðt; 1Þð12Þ ; ½sðt; 2Þð12Þ ; ½sðt; 3Þð12Þ ; ½sðt; 4Þð12Þ ;
t 2 h0; 1Þ; ð3:409Þ
where ½sðt; 1Þð12Þ ¼ 6 exp½0:281t 8 exp½0:314t þ 3 exp½0:347t;
ð3:410Þ
½sðt; 2Þð12Þ ¼ 6 exp½0:335t 8 exp½0:375t þ 3 exp½0:415t;
ð3:411Þ
½sðt; 3Þð12Þ ¼ 6 exp½0:385t 8 exp½0:435t þ 3 exp½0:485t;
ð3:412Þ
½sðt; 4Þð12Þ ¼ 6 exp½0:425t 8 exp½0:48t þ 3 exp½0:535t:
ð3:413Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z12 ; calculated from the results given by (3.410)–(3.413), according to (3.8)–(3.9), respectively are: l12 ð1Þ ffi 4:2;
l12 ð2Þ ffi 3:80;
l12 ð3Þ ffi 3:38;
l12 ð4Þ ffi 3:05 years; ð3:414Þ
r12 ð1Þ ffi 4:37;
r12 ð2Þ ffi 3:68;
r12 ð3Þ ffi 3:12;
r12 ð4Þ ffi 2:93 years; ð3:415Þ
and further, using (3.10) and (3.414), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z12 ; respectively are: l12 ð1Þ ffi 0:72;
l12 ð2Þ ffi 0:42;
l12 ð3Þ ffi 0:33;
l12 ð4Þ ffi 3:05 years: ð3:416Þ
At the system operation state z13 ; the ferry technical system is composed of three subsystems S1 ; S2 and S4 forming a series structure shown in Fig. 2.17. The system safety structure and its components safety functions are the same as at the operation state z4 : Thus, considering (3.310)–(3.314), its conditional safety function is given by h i ½sðt; Þð13Þ ¼ 1; ½sðt; 1Þð13Þ ; ½sðt; 2Þð13Þ ; ½sðt; 3Þð13Þ ; ½sðt; 4Þð13Þ ; t 2 h0; 1Þ; ð3:417Þ
3.6 Applications
157
where ½sðt; 1Þð13Þ ¼ 6 exp½0:281t 8 exp½0:314t þ 3 exp½0:347t;
ð3:418Þ
½sðt; 2Þð13Þ ¼ 6 exp½0:335t 8 exp½0:375t þ 3 exp½0:415t;
ð3:419Þ
½sðt; 3Þð13Þ ¼ 6 exp½0:385t 8 exp½0:435t þ 3 exp½0:485t;
ð3:420Þ
½sðt; 4Þð13Þ ¼ 6 exp½0:425t 8 exp½0:48t þ 3 exp½0:535t:
ð3:421Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z13 ; calculated from the results given by (3.418)–(3.421), according to (3.8)–(3.9), respectively are: l13 ð1Þ ffi 4:2;
l13 ð2Þ ffi 3:80;
r13 ð1Þ ffi 4:37;
l13 ð3Þ ffi 3:38;
r13 ð2Þ ffi 3:68;
l13 ð4Þ ffi 3:05 years; ð3:422Þ
r13 ð3Þ ffi 3:12;
r13 ð4Þ ffi 2:93 years; ð3:423Þ
and further, using (3.10) and (3.422), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z13 ; respectively are: l13 ð1Þ ffi 0:72;
l13 ð2Þ ffi 0:42;
l13 ð3Þ ffi 0:33;
l13 ð4Þ ffi 3:05 years: ð3:424Þ
At the system operation state z14 ; the ferry technical system is composed of three subsystems S1 ; S2 and S4 forming a series structure shown in Fig. 2.17. The system safety structure and its components safety functions are the same as at the operation state z4 : Thus, considering (3.310)–(3.314), its conditional safety function is given by h i ½sðt; Þð14Þ ¼ 1; ½sðt; 1Þð14Þ ; ½sðt; 2Þð14Þ ; ½sðt; 3Þð14Þ ; ½sðt; 4Þð14Þ ; t 2 h0; 1Þ; ð3:425Þ where ½sðt; 1Þð14Þ ¼ 6 exp½0:281t 8 exp½0:314t þ 3 exp½0:347t;
ð3:426Þ
½sðt; 2Þð14Þ ¼ 6 exp½0:335t 8 exp½0:375t þ 3 exp½0:415t;
ð3:427Þ
½sðt; 3Þð14Þ ¼ 6 exp½0:385t 8 exp½0:435t þ 3 exp½0:485t;
ð3:428Þ
½sðt; 4Þð14Þ ¼ 6 exp½0:425t 8 exp½0:48t þ 3 exp½0:535t:
ð3:429Þ
158
3 Complex Technical Systems, Reliability
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z14 ; calculated from the results given by (3.426)–(3.429), according to (3.8)–(3.9), respectively are: l14 ð1Þ ffi 4:2;
l14 ð2Þ ffi 3:80;
r14 ð1Þ ffi 4:37;
l14 ð3Þ ffi 3:38;
r14 ð2Þ ffi 3:68;
l14 ð4Þ ffi 3:05 years; ð3:430Þ
r14 ð3Þ ffi 3:12;
r14 ð4Þ ffi 2:93 years; ð3:431Þ
and further, using (3.10) and (3.430), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z14 ; respectively are: l14 ð1Þ ffi 0:72;
l14 ð2Þ ffi 0:42;
l14 ð3Þ ffi 0:33;
l14 ð4Þ ffi 3:05 years: ð3:432Þ
At the system operation states z15 ; the ferry technical system is composed of two subsystems S1 and S2 forming a series structure shown in Fig. 2.16. The system safety structure and its components safety functions are the same as at the operation state z3 : Thus, considering (3.287)–(3.291), its conditional safety function is given by h i ½sðt; Þð15Þ ¼ 1; ½sðt; 1Þð15Þ ; ½sðt; 2Þð15Þ ; ½sðt; 3Þð15Þ ; ½sðt; 4Þð15Þ ; t 2 h0; 1Þ; ð3:433Þ where ½sðt; 1Þð15Þ ¼ 12 exp½0:291t þ 8 exp½0:363t þ 6 exp½0:357t 16 exp½0:324t 6 exp½0:33t 3 exp½0:396t;
ð3:434Þ
½sðt; 2Þð15Þ ¼ 12 exp½0:36t þ 14 exp½0:44t 22 exp½0:4t 3 exp½0:48t;
ð3:435Þ
½sðt; 3Þð15Þ ¼ 12 exp½0:415t þ 8 exp½0:51t þ 6 exp½0:515t 16 exp½0:465t 6 exp½0:46t 3 exp½0:56t;
ð3:436Þ
½sðt; 4Þð15Þ ¼ 12 exp½0:46t þ 8 exp½0:565t þ 6 exp½0:57t 16 exp½0:515t 6 exp½0:51t 3 exp½0:62t:
ð3:437Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z15 ; calculated from the results given by (3.434)–(3.437), according to (3.8)–(3.9), respectively are:
3.6 Applications
159
l15 ð1Þ ffi 4:94;
l15 ð2Þ ffi 3:9;
l15 ð3Þ ffi 3:44;
r15 ð1Þ ffi 4:6;
r15 ð2Þ ffi 3:6;
r15 ð3Þ ffi 3:2;
l15 ð4Þ ffi 3:1 years; ð3:438Þ r15 ð4Þ ffi 2:9 years;
ð3:439Þ
and further, using (3.10) and (3.438), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z15 ; respectively are: l15 ð1Þ ffi 1:04;
l15 ð2Þ ffi 0:46;
l15 ð3Þ ffi 0:34;
l15 ð4Þ ffi 3:1 years: ð3:440Þ
At the system operation state z16 ; the ferry technical system is composed of two subsystems S1 and S2 forming a series structure shown in Fig. 2.16. The system safety structure and its components safety functions are the same as at the operation state z11 : Thus, considering (3.401)–(3.405), its conditional safety function is given by h i ½sðt; Þð16Þ ¼ 1; ½sðt; 1Þð16Þ ; ½sðt; 2Þð16Þ ; ½sðt; 3Þð16Þ ; ½sðt; 4Þð16Þ ; t 2 h0; 1Þ; ð3:441Þ where ½sðt; 1Þ16 ¼ 12 exp½0:363t þ 8 exp½0:462t 16 exp½0:396t 3 exp½0:495t;
ð3:442Þ
½sðt; 2Þ16 ¼ 12 exp½0:42t þ 8 exp½0:53t þ 6 exp½0:5t 16 exp½0:46t 6 exp½0:49t 3 exp½0:57t;
ð3:443Þ
½sðt; 3Þ16 ¼ 12 exp½0:475t þ 8 exp½0:6t þ 6 exp½0:575t 16 exp½0:525t 6 exp½0:55t 3 exp½0:65t;
ð3:444Þ
½sðt; 4Þ16 ¼ 12 exp½0:52t þ 8 exp½0:655t þ 6 exp½0:63t 16 exp½0:575t 6 exp½0:6t 3 exp½0:71t:
ð3:445Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z16 ; calculated from the results given by (3.442)–(3.445), according to (3.8)–(3.9), respectively are: l16 ð1Þ ffi 3:91;
l16 ð2Þ ffi 3:37;
l16 ð3Þ ffi 3:07;
l16 ð4Þ ffi 2:76 years; ð3:446Þ
r16 ð1Þ ffi 3:64;
r16 ð2Þ ffi 3:16;
r16 ð3Þ ffi 2:82;
r16 ð4Þ ffi 2:57 years; ð3:447Þ
and further, using (3.10) and (3.446), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z16 ; respectively are:
160
3 Complex Technical Systems, Reliability
l16 ð1Þ ffi 0:54;
l16 ð2Þ ffi 0:34;
l16 ð3Þ ffi 0:27;
l16 ð4Þ ffi 2:76 years: ð3:448Þ
At the system operation state z17 ; the ferry technical system is composed of three subsystems S1 ; S2 and S5 forming a series structure shown in Fig. 2.15. The system safety structure and its components safety functions are the same as at the operation state z2 : Thus, considering (3.269)–(3.273), its conditional safety function is given by h i ½sðt; Þð17Þ ¼ 1; ½sðt; 1Þð17Þ ; ½sðt; 2Þð17Þ ; ½sðt; 3Þð17Þ ; ½sðt; 4Þð17Þ ; t 0; ð3:449Þ where ½sðt; 1Þð17Þ ¼ 12 exp½0:462t þ 8 exp½0:561t 16 exp½0:495t 3 exp½0:594t:
ð3:450Þ
½sðt; 2Þð17Þ ¼ 12 exp½0:54t þ 8 exp½0:65t þ 6 exp½0:62t 16 exp½0:58t 6 exp½0:61t 3 exp½0:69t;
ð3:451Þ
½sðt; 3Þð17Þ ¼ 12 exp½0:62t þ 8 exp½0:745t þ 6 exp½0:72t 16 exp½0:67t 6 exp½0:695t 3 exp½0:795t;
ð3:452Þ
½sðt; 4Þð17Þ ¼ 12 exp½0:685t þ 8 exp½0:82t þ 6 exp½0:795t 16 exp½0:74t 6 exp½0:765t 3 exp½0:875t:
ð3:453Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z17 ; calculated from the results given by (3.450)–(3.453), according to (3.8)–(3.9), respectively are: l17 ð1Þ ffi 2:86;
l17 ð2Þ ffi 2:43;
l17 ð3Þ ffi 2:14;
l17 ð4Þ ffi 1:93 years; ð3:454Þ
r17 ð1Þ ffi 2:74;
r17 ð2Þ ffi 2:35;
r17 ð3Þ ffi 2:05;
r17 ð4Þ ffi 1:85 years; ð3:455Þ
and further, using (3.10) and (3.454), the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z17 ; respectively are: l17 ð1Þ ffi 0:43;
l17 ð2Þ ffi 0:29;
l17 ð3Þ ffi 0:21;
l17 ð4Þ ffi 1:93 years: ð3:456Þ
At the system operation state z18 ; the ferry technical system is composed of two subsystems S3 and S4 forming a series structure shown in Fig. 2.14. The system
3.6 Applications
161
safety structure and its components safety functions are the same as at the operation state z1 : Thus, considering (3.246)–(3.250), its conditional safety function is given by h i ½sðt; Þð18Þ ¼ 1; ½sðt; 1Þð18Þ ; ½sðt; 2Þð18Þ ; ½sðt; 3Þð18Þ ; ½sðt; 4Þð18Þ ; t 2 h0; 1Þ; ð3:457Þ where ½sðt; 1Þð18Þ ¼ exp½0:123t
ð3:458Þ
½sðt; 2Þð18Þ ¼ exp½0:155t
ð3:459Þ
½sðt; 3Þð18Þ ¼ exp½0:175t;
ð3:460Þ
½sðt; 4Þð18Þ ¼ exp½0:20t:
ð3:461Þ
The expected values and standard deviations of the ferry technical system conditional lifetimes in the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g at the operation state z18 ; calculated from the results given by (3.458)–(3.461), according to (3.8)–(3.9), respectively are: l18 ð1Þ ffi 8:13;
l18 ð2Þ ffi 6:45;
l18 ð3Þ ffi 5:71;
l18 ð4Þ ffi 5:00 years; ð3:462Þ
r18 ð1Þ ffi 8:13;
r18 ð2Þ ffi 6:45;
r18 ð3Þ ffi 5:71;
r18 ð4Þ ffi 5:00 years; ð3:463Þ
and further, using (3.10) and (3.462) the mean values of the ferry technical system conditional lifetimes in the particular safety states 1, 2, 3, 4 at the operation state z18 ; respectively are: l18 ð1Þ ffi 1:68;
l18 ð2Þ ffi 0:74;
l18 ð3Þ ffi 0:71;
l18 ð4Þ ffi 5:00 years: ð3:464Þ
In the case when the ferry technical system operation time is large enough, its unconditional safety function is given by the vector sðt; Þ ¼ ½1; sðt; 1Þ; sðt; 2Þ; sðt; 3Þ; sðt; 4Þ;
t 0;
ð3:465Þ
where according to (3.101) and considering the ferry technical system operation process transient probabilities at the operation states determined by (2.48), the vector coordinates are given respectively by
162
3 Complex Technical Systems, Reliability
Fig. 3.5 The graph of the ferry technical system safety function sðt; Þ coordinates
sðt; 1Þ ¼ 0:038 ½sðt; 1Þð1Þ þ0:002 ½sðt; 1Þð2Þ þ0:026 ½sðt; 1Þð3Þ þ 0:036 ½sðt; 1Þð4Þ þ0:363 ½sðt; 1Þð5Þ þ0:026 ½sðt; 1Þð6Þ þ 0:005 ½sðt; 1Þð7Þ þ0:016 ½sðt; 1Þð8Þ þ0:037 ½sðt; 1Þð9Þ þ 0:002 ½sðt; 1Þð10Þ þ0:003 ½sðt; 1Þð11Þ þ0:016 ½sðt; 1Þð12Þ þ 0:351 ½sðt; 1Þð13Þ þ0:034 ½sðt; 1Þð14Þ þ0:024 ½sðt; 1Þð15Þ þ 0:003 ½sðt; 1Þð16Þ þ0:005 ½sðt; 1Þð17Þ þ0:013 ½sðt; 1Þð18Þ ;
ð3:466Þ
sðt; 2Þ ¼ 0:038 ½sðt; 2Þð1Þ þ0:002 ½sðt; 2Þð2Þ þ0:026 ½sðt; 2Þð3Þ þ 0:036 ½sðt; 2Þð4Þ þ0:363 ½sðt; 2Þð5Þ þ0:026 ½sðt; 2Þð6Þ þ 0:005 ½sðt; 2Þð7Þ þ0:016 ½sðt; 2Þð8Þ þ0:037 ½sðt; 2Þð9Þ þ 0:002 ½sðt; 2Þð10Þ þ0:003 ½sðt; 2Þð11Þ þ0:016 ½sðt; 2Þð12Þ þ 0:351 ½sðt; 2Þð13Þ þ0:034 ½sðt; 2Þð14Þ þ0:024 ½sðt; 2Þð15Þ h ið17Þ þ 0:003 ½sðt; 2Þð16Þ þ0:005 sðt; 2Þ þ0:013 ½sðt; 2Þð18Þ ; ð3:467Þ sðt; 3Þ ¼ 0:038 ½sðt; 3Þð1Þ þ0:002 ½sðt; 3Þð2Þ þ0:026 ½sðt; 3Þð3Þ þ 0:036 ½sðt; 3Þð4Þ þ0:363 ½sðt; 3Þð5Þ þ0:026 ½sðt; 3Þð6Þ þ 0:005 ½sðt; 3Þð7Þ þ0:016 ½sðt; 3Þð8Þ þ0:037 ½sðt; 3Þð9Þ þ 0:002 ½sðt; 3Þð10Þ þ0:003 ½sðt; 3Þð11Þ þ0:016 ½sðt; 3Þð12Þ þ 0:351 ½sðt; 3Þð13Þ þ0:034 ½sðt; 3Þð14Þ þ0:024 ½sðt; 3Þð15Þ þ 0:003 ½sðt; 3Þð16Þ þ0:005 ½sðt; 3Þð17Þ þ0:013 ½sðt; 3Þð18Þ ; ð3:468Þ
3.6 Applications
163
sðt; 4Þ ¼ 0:038 ½sðt; 4Þð1Þ þ0:002 ½sðt; 4Þð2Þ þ0:026 ½sðt; 4Þð3Þ þ 0:036 ½sðt; 4Þð4Þ þ0:363 ½sðt; 4Þð5Þ þ0:026 ½sðt; 4Þð6Þ þ 0:005 ½sðt; 4Þð7Þ þ0:016 ½sðt; 4Þð8Þ þ0:037 ½sðt; 4Þð9Þ þ 0:002 ½sðt; 4Þð10Þ þ0:003 ½sðt; 4Þð11Þ þ0:016 ½sðt; 4Þð12Þ þ 0:351 ½sðt; 4Þð13Þ þ0:034 ½sðt; 4Þð14Þ þ0:024 ½sðt; 4Þð15Þ þ 0:003 ½sðt; 4Þð16Þ þ0:005 ½sðt; 4Þð17Þ þ0:013 ½sðt; 4Þð18Þ ; ð3:469Þ where ½sðt; uÞðbÞ ; u ¼ 1; 2; 3; 4; b ¼ 1; 2; . . .; 18; are given by (3.247)–(3.250), (3.270)–(3.273), (3.288)–(3.291), (3.311)–(3.314), (3.318)–(3.322), (3.342)– (3.345), (3.350)–(3.353), (3.368)–(3.371), (3.376)–(3.379), (3.384)-(3.387), (3.402)–(3.405), (3.410)–(3.413), (3.418)–(3.421), (3.426)–(3.429), (3.434)– (3.437), (3.442)–(3.445), (3.450)–(3.453), (3.458)–(3.461). The graph of the five-state ferry technical system safety function is shown in Fig. 3.5. The expected values and the standard deviations of the ferry unconditional lifetimes in the safety state subsets, f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g; calculated from the results given by (3.466)–(3.469), according to (3.7)–(3.9), and considering (2.48), (3.251), (3.274), (3.292), (3.315), (3.323), (3.346), (3.354), (3.372), (3.380), (3.388), (3.406), (3.414), (3.422), (3.430), (3.438), (3.446), (3.454), (3.462), respectively are: lð1Þ ffi 0:038 8:13 þ 0:002 2:86 þ 0:026 4:94 þ 0:036 4:2 þ 0:363 4:2 þ 0:026 4:01 þ 0:005 2:86 þ 0:016 9:71 þ 0:037 9:71 þ 0:002 2:86 þ 0:003 3:91 þ 0:016 4:2 þ 0:351 4:2 þ 0:034 4:2 þ 0:024 4:94 þ 0:003 3:91 þ 0:005 2:86 þ 0:013 8:1 ffi 4:70 years; rð1Þ ffi 4:3 years;
ð3:470Þ ð3:471Þ
lð2Þ ffi 0:038 6:45 þ 0:002 2:43 þ 0:026 3:90 þ 0:036 3:80 þ 0:363 3:80 þ 0:026 3:24 þ 0:005 2:43 þ 0:016 7:69 þ 0:037 7:69 þ 0:002 2:43 þ 0:003 3:37 þ 0:016 3:80 þ 0:351 3:80 þ 0:034 3:80 þ 0:024 3:90 ð3:472Þ þ 0:003 3:37 þ 0:005 2:43 þ 0:013 6:45 ffi 4:11 years; rð2Þ ffi 3:64 years;
ð3:473Þ
lð3Þ ffi 0:037 5:71 þ 0:002 2:14 þ 0:026 3:44 þ 0:036 3:38 þ 0:363 3:38 þ 0:026 2:88 þ 0:005 2:14 þ 0:016 6:89 þ 0:037 6:89 þ 0:002 2:14 þ 0:003 3:07 þ 0:016 3:38 þ 0:351 3:38 þ 0:034 3:38 þ 0:024 3:44 þ 0:003 3:07 þ 0:005 2:14 þ 0:013 5:71 ffi 3:66 years;
ð3:474Þ
164
3 Complex Technical Systems, Reliability
Fig. 3.6 The graph of the ferry technical system risk function rðtÞ
rð3Þ ffi 3:34 years;
ð3:475Þ
lð4Þ ffi 0:038 5:00 þ 0:002 1:93 þ 0:026 3:1 þ 0:036 3:05 þ 0:363 3:05 þ 0:026 2:61 þ 0:005 1:93 þ 0:016 6:06 þ 0:037 6:25 þ 0:002 1:93 þ 0:003 2:76 þ 0:016 3:05 þ 0:351 3:05 þ 0:034 3:05 þ 0:024 3:10 þ 0:003 2:76 þ 0:005 1:93 þ 0:013 5:00 ffi 3:29 years; ð3:476Þ rð2Þ ffi 2:89 years;
ð3:477Þ
And further, considering (3.10) and (3.470), (3.472), (3.474) and (3.476), the unconditional lifetimes in the particular safety states 1, 2, 3, 4 respectively are: lð1Þ ¼ lð1Þ lð2Þ ¼ 0:59;
lð2Þ ¼ lð2Þ lð3Þ ¼ 0:77 year;
lð3Þ ¼ lð3Þ lð4Þ ¼ 0:45;
lð4Þ ¼ lð4Þ ¼ 2:29 years:
ð3:478Þ ð3:479Þ
Since the critical safety state is r ¼ 2; the system risk function, according to (3.11), is given by rðtÞ ¼ 1 sðt; 2Þ; ð3:480Þ where sðt; 2Þ is given by (3.467). The graph of the ferry technical system risk function rðtÞ is presented in Fig. 3.6. From (3.480), according to (3.12), the moment when the system risk function exceeds a permitted level, for instance d ¼ 0:05; is s ¼ r1 ðdÞ ffi 0:21 year:
ð3:481Þ
The safety characteristics of the ferry technical system operating at the variable conditions predicted in this section are different from those determined in Sect. 1.4.3 at the system constant operation conditions. This fact suggests the necessity of considering the system at the variable operation conditions, which
3.6 Applications
165
upon the improved accuracy of the system operation process identification, makes its system safety prediction more precise. Using the results of the maritime ferry technical system safety prediction given by (3.472)–(3.473) and the results of the classical renewal theory presented in Sect. 3.4, we may predict the renewal and availability characteristics of this technical system in the case when it is repairable and its time of renovation is either ignored or non-ignored. First, assuming that the maritime ferry technical system is repaired after the exceeding of its safety critical state r ¼ 2 and that the time of the system renovation is ignored and applying Proposition 3.4, we obtain the following results: (a) the time SN ð2Þ until the Nth exceeding by the system the safety critical state r ¼ 2; for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð4:11N; 3:64 N Þ; i.e., t 4:11N pffiffiffiffi ; t 2 ð1; 1Þ; F ðNÞ ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ 3:64 N (b) the expected value and the variance of the time SN ð2Þ until the Nth exceeding by the system the safety critical state r ¼ 2 are respectively given by E½SN ð2Þ ffi 4:11N;
D½SN ð2Þ ffi 13:250N;
(c) the number Nðt; 2Þ of exceeding by the system the safety critical state r ¼ 2 up to the moment t; t 0; for sufficiently large t, approximately has the distribution of the form 4:11ðN þ 1Þ t 4:11N t pffi pffi ; FNð0;1Þ PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 1:795 t 1:795 t N ¼ 0; 1; . . .; (d) the expected value and the variance of the number Nðt; 2Þ of exceeding by the system the safety critical state r ¼ 2 up to the moment t; t 0; for sufficiently large t; approximately are respectively given by Hðt; 2Þ ¼ 0:243t;
Dðt; 2Þ ¼ 0:191t:
Assuming that the maritime ferry technical system is repaired after the exceeding its safety critical state r ¼ 2 and that the time of the system renovation is non-ignored and it has the mean value l0 ð2Þ ¼ 0:019 year and the standard deviation r0 ð2Þ ¼ 0:0095 year and applying Proposition 3.5, we obtain the following results: (a) the time SN ð2Þ until the Nth exceeding by the system the safety critical state r ¼ 2; for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð4:11N þ 0:019ðN 1Þ; 13:250N 0:0000903ðN 1ÞÞ; i.e.,
166
3 Complex Technical Systems, Reliability
F
ðNÞ
t 4:129N þ 0:019 ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 13:25009N 0:0000903 t 2 ð1; 1Þ;
(b) the expected value and the variance of the time SN ð2Þ until the Nth exceeding by the system the safety critical state r ¼ 2; for sufficiently large N, are respectively given by E½SN ð2Þ ffi 4:11N þ 0:019ðN 1Þ;
D½SN ð2Þ ffi 13:25N þ 0:0000903ðN 1Þ;
(c) the number Nðt; 2Þ of exceeding by the system the safety critical state r ¼ 2 up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 4:129ðN þ 1Þ t 0:019 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 1:791 t þ 0:019 4:129N t 0:019 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; N ¼ 0; 1; . . .; FNð0;1Þ 1:791 t þ 0:019 (d) the expected value and the variance of the number Nðt; 2Þ of exceeding by the system the safety critical state r ¼ 2 up to the moment t; t 0; for sufficiently large t, are respectively given by Hðt; 2Þ ffi 0:242ðt þ 0:019Þ;
Dðt; 2Þ ffi 0:188ðt þ 0:019Þ;
(e) the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffi approximately normal distribution Nð4:129N; 3:640 N Þ; i.e., ðNÞ t 4:129N pffiffiffiffi ; t 2 ð1; 1Þ; F ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ 3:640 N (f) the expected value and the variance of the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, are respectively given by E½SN ð2Þ ffi 4:129N;
D½SN ð2Þ ffi 13:25009N;
(g) the number Nðt; 2Þ of the system’s renovations up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 4:129ðN þ 1Þ t 4:129N t pffi pffi ; FNð0;1Þ PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 1:791 t 1:791 t N ¼ 0; 1; . . .; (h) the expected value and the variance of the number Nðt; 2Þ of system’s renovations up to the moment t; t 0; for sufficiently large t, are respectively given by
3.6 Applications
167
Hðt; 2Þ ffi 0:242t; (i)
Dðt; 2Þ ffi 0:188t;
the steady availability coefficient of the system at the moment t; t 0; for sufficiently large t, is given by Aðt; 2Þ ffi 0:995;
(j)
t 0;
the steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; for sufficiently large t, is given by Aðt; s; 2Þ ffi 0:242
Z1 sðt; 2Þ dt;
t 0; s [ 0;
s
where sðt; 2Þ is given by (7.378). The ferry technical system operating at the varying in time operation conditions safety and risk evaluation and prediction is approximate because of non-sufficiently exact input data. Although the ferry operation process parameters are sufficiently exact because of the collected very good and sufficient statistical data necessary for their estimation, the ferry technical system safety structures are fixed on a very high level of generality and without high detailed accuracy. The input safety parameters of the ferry technical system components also are not sufficiently accurate because of the lack exact statistical data necessary for their estimation. Therefore, the achieved results may only be considered as an illustration of the proposed methods, possibilities of applications in the ferry technical system safety and risk analysis and prediction. However, the obtained evaluation may be a very useful example in technical systems of ships operating at restricted and open sea waters, safety and risk characteristics prediction, especially during the design and when planning and improving their operation processes, safety and effectiveness. To improve the quality of the achieved results it is supposed that the statistical data should be collected for the additional period of time and after that the full identification of at least the ferry technical system operation process would be performed and this process, unknown parameters and main characteristics could be determined and used in its safety and risk for more precise analysis and prediction. The final characteristics of the ferry technical system safety and risk are used in the appliqué part of Chap. 6 for its operation and safety optimization.
3.7 Summary In this chapter the final results coming from the general analytical models of complex technical multistate systems, reliability, availability and safety are based on the results of the previous chapters of the book. The material given in this chapter delivers the procedures and algorithms that allow to find the main and
168
3 Complex Technical Systems, Reliability
practically important reliability and safety and renewal and availability characteristics of the complex technical systems at the variable operation condition. The reliability characteristics of the exemplary system and the port oil transportation system and the safety characteristics of the ferry technical system operating at the variable conditions predicted in this chapter are different from those determined in Chap. 1 for these systems operating at constant conditions. This fact justifies the sensibility of considering complex technical systems at the variable operation conditions, that is appearing out in a natural way from practice. This approach, upon the good accuracy of systems’ operation processes and their components reliability and safety parameters identification, makes their reliability, safety, renewal and availability characteristics prediction more precise. The results of this chapter, together with the results of their applications, and the linear programming are used in Chap. 6 for complex technical systems, reliability, availability and safety improvement, optimization and operation cost analysis.
References 1. Amari SV, Misra RB (1997) Comment on: Dynamic reliability analysis of coherent multistate systems. IEEE Trans Reliab 46:460–461 2. Aven T (1985) Reliability evaluation of multistate systems with multistate components. IEEE Trans Reliab 34:473–479 3. Aven T (1993) On performance measures for multistate monotone systems. Reliab Eng Syst Saf 41:259–266 4. Aven T, Jensen U (1999) Stochastic models in reliability. Springer, New York 5. Barlow RE, Wu AS (1978) Coherent systems with multi-state components. Math Oper Res 4:275–281 6. Brunelle RD, Kapur KC (1999) Review and classification of reliability measures for multistate and continuum models. IEEE Trans 31:1117–1180 7. Ferreira F, Pacheco A (2007) Comparison of level-crossing times for Markov and semiMarkov processes. Stat Probab Lett 77(2):151–157 8. Glynn PW, Haas PJ (2006) Laws of large numbers and functional central limit theorems for generalized semi-Markov processes. Stoch Model 22(2):201–231 9. Grabski F (2002) Semi-Markov models of systems reliability and operations analysis. System Research Institute, Polish Academy of Science (in Polish) 10. Guze S, Kołowrocki K (2008) Reliability analysis of multi-state ageing consecutive ‘‘k out of n: F’’ systems. Int J Mater Struct Reliab 6(1):47–60 11. Guze S, Kołowrocki K, Soszyn´ska J (2008) Modeling environment and infrastructure influence on reliability and operation processes of port transportation systems. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 2(1):179–188 12. Habibullah MS, Lumanpauw E, Kolowrocki K, Soszynska J, Ming NG (2009) A computational tool for general model of operation processes in industrial systems. Operation processes. Electron J Reliab Risk Anal Theory Appl 2(4):181–191 13. Huang J, Zuo MJ, Wu Y (2000) Generalized multi-state k-out-of-n:G systems. IEEE Trans Reliab 49:105–111 14. Hudson JC, Kapur KC (1982) Reliability theory for multistate systems with multistate components. Microelectron Reliab 22:1–7
References
169
15. Hudson JC, Kapur KC (1983) Reliability analysis of multistate systems with multistate components. Trans Inst Ind Eng 15:127–135 16. Hudson J, Kapur K (1985) Reliability bounds for multistate systems with multistate components. Oper Res 33:735–744 17. Kołowrocki K (1998) On applications of asymptotic reliability functions to the reliability and risk evaluation of pipelines. Int J Press Vessel Pip 75:545–558 18. Kołowrocki K (2003) An asymptotic approach to reliability evaluation of large multi-state systems with applications to piping transportation systems. Int J Press Vessel Pip 80:59–73 19. Kołowrocki K (2004) Reliability of large systems. Elsevier, Amsterdam 20. Kołowrocki K (2006) Reliability and risk evaluation of complex systems in their operation processes. Int J Mater Struct Reliab 4(2):129–147 21. Kołowrocki K (2007) Reliability modelling of complex systems—part 1. Int J Gnedenko e-Forum Reliab Theory Appl 2(3–4):116–127 22. Kołowrocki K (2007) Reliability modelling of complex systems—part 2. Int J Gnedenko e-Forum Reliab Theory Appl 2(3–4):128–139 23. Kołowrocki K (2008) Reliability and risk analysis of multi-state systems with degrading components. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 2(2):205–216 24. Kołowrocki K (2008) Reliability of large systems. In: Encyclopedia of quantitative risk analysis and assessment, vol 4. Wiley, New York, pp 1466–1471 25. Kołowrocki K, Soszyn´ska J (2006) Reliability and availability of complex systems. Qual Reliab Eng Int 22(1):79–99 26. Kołowrocki K, Soszyn´ska J (2008) A general model of industrial systems operation processes related to their environment and infrastructure. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 2(2):223–226 27. Kolowrocki K, Soszynska J (2009) Modeling environment and infrastructure influence on reliability and operation process of port oil transportation system. Electron J Reliab Risk Anal Theory Appl 2(3):131–142 28. Kolowrocki K, Soszynska J (2009) Safety and risk evaluation of Stena Baltica ferry in variable operation conditions. Electron J Reliab Risk Anal Theory Appl 2(4):168–180 29. Kołowrocki K, Soszyn´ska J (2009) Statistical identification and prediction of the port oil pipeline system’s operation process and its reliability and risk evaluation. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 3(2):241–250 30. Kołowrocki K, Soszyn´ska J (2009) Methods and algorithms for evaluating unknown parameters of components reliability of complex technical systems. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 4(2):223–230 31. Kolowrocki K, Soszynska J (2010) Reliability, availability and safety of complex technical systems: modelling – identification – prediction – optimization. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 4(1):133–158 32. Kolowrocki K, Soszynska J (2010) Reliability modeling of a port oil transportation system’s operation processes. Int J Perform Eng 6(1):77–87 33. Kołowrocki K, Soszyn´ska J (2010) Safety and risk evaluation of a ferry technical system. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 4(1):159–172 34. Kołowrocki K, Soszyn´ska J, Judzin´ski M, Dziula P (2007) On multi-state safety analysis in shipping. Int J Reliab Qual Saf Eng Syst Reliab Saf 146:547–567 35. Kołowrocki K, Soszyn´ska J, Xie M, Kien M, Salahudin M (2008) Safety and reliability of complex industrial systems and process. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 4(2):227–234 36. Kołowrocki K, Soszyn´ska J (2011) On safety analysis of complex technical maritime transportation system. J Risk Reliab 225(1):1–10 37. Kossow A, Preuss W (1995) Reliability of linear consecutively-connected systems with multistate components. IEEE Trans Reliab 44:518–522 38. Limnios N, Oprisan G (2005) Semi-Markov processes and reliability. Birkhauser, Boston 39. Lisnianski A, Levitin G (2003) Multi-state system reliability assessment, optimisation and applications. World Scientific Publishing Co. Pte. Ltd., Singapore
170
3 Complex Technical Systems, Reliability
40. Meng F (1993) Component-relevancy and characterisation in multi-state systems. IEEE Trans Reliab 42:478–483 41. Mercier S (2008) Numerical bounds for semi-Markovian quantities and application to reliability. Methodol Comput Appl Probab 10(2):179–198 42. Natvig B (1982) Two suggestions of how to define a multi-state coherent system. Adv Appl Probab 14:434–455 43. Natvig B (1984) Multi-state coherent systems. In: Encyclopaedia of statistical sciences. Wiley, New York 44. Natvig B, Streller A (1984) The steady-state behaviour of multistate monotone systems. J Appl Probab 21:826–835 45. Ohio F, Nishida T (1984) On multistate coherent systems. IEEE Trans Reliab 33:284–287 46. Soszyn´ska J (2004) Reliability of large series system in variable operation conditions. In: Jt Proc 17. Gdynia Maritime University Press, Gdynia, pp 36–43 47. Soszyn´ska J (2004) Reliability of large parallel systems in variable operation conditions. In Fac of Navig Res Work 16. Gdynia, pp 168–180 48. Soszyn´ska J (2006) Reliability of large series–parallel system in variable operation conditions. Int J Autom Comput 3(2):199–206 49. Soszyn´ska J (2006) Reliability evaluation of a port oil transportation system in variable operation conditions. Int J Press Vessel Pip 83(4):304–310 50. Soszyn´ska J (2006) Safety analysis of multistate systems in variable operations conditions (in Polish). Diagnostyka 3(39):25–34 51. Soszyn´ska J (2007) Systems reliability analysis in variable operation conditions. PhD thesis, Gdynia Maritime University-System Research Institute Warsaw (in Polish) 52. Soszyn´ska J (2008) Asymptotic approach to reliability evaluation of large ‘‘m out of l’’-series system in variable operation conditions. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 2(2):323–346 53. Soszyn´ska J (2010) Reliability and risk evaluation of a port oil pipeline transportation system in variable operation conditions. Int J Press Vessel Pip 87(2–3):81–87 54. Soszyn´ska J, Kołowrocki K, Blokus-Roszkowska A, Guze S (2010) Prediction of complex technical systems operation processes. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 4(2):79–510 55. Xue J (1985) On multi-state system analysis. IEEE Trans Reliab 34:329–337 56. Xue J, Yang K (1995) Dynamic reliability analysis of coherent multi-state systems. IEEE Trans Reliab 4(44):683–688 57. Xue J, Yang K (1995) Symmetric relations in multi-state systems. IEEE Trans Reliab 4(44):689–693 58. Yu K, Koren I, Guo Y (1994) Generalised multistate monotone coherent systems. IEEE Trans Reliab 43:242–250
Chapter 4
Complex Technical System Operation Processes Identification
4.1 Introduction The general model of complex technical systems operation processes is proposed in Chap. 2. The reliability and safety models of various multistate complex technical systems are considered in Chap. 1. Consequently, the general joint models linking these system reliability models with the model of their operation processes, allowing us for the reliability and safety analysis of complex technical systems at the variable operation conditions, are constructed in Chap. 3. To be able to apply these general models practically in the evaluation and prediction of the reliability and safety of real complex technical systems it is necessary to have the statistical methods concerned with determining the unknown parameters of the proposed models [1–4, 6, 7, 10, 11, 13, 14, 17, 25]. Particularly, concerning the system operation process, the probabilities of the system operation process stay at the operation states at the initial moment, the probabilities of the system operation process transitions between the system operation states and the distributions of the conditional sojourn times of the system operation process at the particular operation states should be identified [5, 10–13, 15, 16]. It is also necessary to have the methods of testing the hypotheses concerned with the conditional sojourn times of the system operation process at the operation states and the procedures of testing the uniformity of their realizations coming from different sets of empirical data [13, 18].
4.2 Identification of Complex Technical Systems Operation Processes We assume, as in Chaps. 2 and 3, that a system during its operation at the fixed moment t, t 2 h0; þ1i; may be at one of m, v 2 N; different operations states zb ; b ¼ 1; 2; . . .; m: Next, we mark by ZðtÞ; t 2 h0; þ1i; the system operation K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8_4, Springer-Verlag London Limited 2011
171
172
4 Complex Technical System Operation Processes Identification
process, which is a function of a continuous variable t, taking discrete values in the set fz1 ; z2 ; . . .; zv g of the system operation states. We assume a semi-Markov model [8, 9, 13, 17, 19–22, 24] of the system operation process ZðtÞ and we mark by hbl its random conditional sojourn times at the operation states zb, when its next operation state is zl ; b; l ¼ 1; 2; . . .; v; b 6¼ l: Under these assumption, the operation process may be described by the vector ½pb ð0Þ1m of probabilities of the system operation process stay at the particular operations states at the initial moment t ¼ 0; the matrix ½pbl ðtÞmm of the probabilities of the system operation process transitions between the operation states and the matrix ½Hbl ðtÞmm of the distribution functions of the conditional sojourn times hbl of the system operation process at the operation states or equivalently by the matrix ½hbl ðtÞmm of the density functions of the conditional sojourn times hbl , b; l ¼ 1; 2; . . .; v; b 6¼ l; of the system operation process at the operation states. All these parameters of the complex technical system operation process are unknown and before their use for the prognosis of this process their characteristics have to be estimated on the basis of statistical data coming from practice.
4.2.1 Defining Unknown Parameters of System Operation Process and Data Collection To make the estimation of the unknown parameters of the system operations process, the experiment delivering the necessary statistical data should be precisely planned. First, before the experiment, we should perform the following preliminary steps: (i) to analyze the system operation process; (ii) to fix or to define the system operation process following general parameters: • the number of the operation states of the system operation process m, • the operation states of the system operation process z1 ; z2 ; . . .; zm ; (iii) to fix the possible transitions between the system operation states; (iv) to fix the set of the unknown parameters of the system operation process semi-Markov model. Next, to estimate the unknown parameters of the system operation process, based on the experiment, we should collect the necessary statistical data performing the following steps: (i)
to fix and collect the following statistical data necessary for evaluating the probabilities pb ð0Þ of the system operation process stay at the operation states at the initial moment t ¼ 0: • the duration time of the experiment H, • the number of the investigated (observed) realizations of the system operation process nð0Þ,
4.2 Identification of Complex Technical Systems Operation Processes
173
• the vector of the realizations nb ð0Þ; b ¼ 1; 2; . . .; m; of the number of stays of the operation process respectively at the operation states z1, z2, …, zm, at the initial moments t ¼ 0 of all n(0) observed realizations of the system operation process ½nb ð0Þ ¼ ½n1 ð0Þ; n2 ð0Þ; . . .; nm ð0Þ; where n1 ð0Þ þ n2 ð0Þ þ nm ð0Þ ¼ nð0Þ; (ii) to fix and collect the following statistical data necessary for evaluating the probabilities pbl of the system operation process transitions between the system operation states: • the matrix of the realization of the numbers nbl, b, l =1, 2, …, v, b = l, of the transitions of the system operation process from the operation state zb into the operation state zl at all observed realizations of the system operation process 2 3 n11 n12 . . .n1m 6 n n . . .n 7 2m 7 6 21 22 ½nbl ¼ 6 7; 4 ... 5 nm1 nm2 . . .nmm where nbb ¼ 0 for b ¼ 1; 2; . . .; v; • the vector of the realizations of the numbers nb, b = 1, 2, …, v, of departures of the system operation process from the operation states zb (the sums of the numbers of the bth rows of the matrix ½nbl ) ½nb ¼ ½n1 ; n2 ; . . .; nm ; where n1 ¼ n11 þ n12 þ þ n1m ; n2 ¼ n21 þ n22 þ þ n2m ; ... nm ¼ nm1 þ nm2 þ þ nmm ;
(iii) to fix and collect the following statistical data necessary for evaluating the unknown parameters of the distributions Hbl ðtÞ of the conditional sojourn times hbl of the system operation process at the particular operation states:
174
4 Complex Technical System Operation Processes Identification
• the numbers nbl, b, l = 1, 2, …, v, b 6¼ l, of realizations of the conditional sojourn times hbl ; b, l = 1, 2, …, v, b 6¼ l, of the system operation process at the operation state zb when the next transition is to the operation state zl during the observation time H, • the realizations hkbl ; k = 1, 2, …, nbl, of the conditional sojourn times hbl of the system operation process at the operation state zb when the next transition is to the operation state zl during the observation time H for each b, l = 1, 2, … , v, b 6¼ l.
4.2.2 Estimating Basic Parameters of System Operation Process After collecting the statistical data, it is possible to estimate the unknown parameters of the system operation process performing the following steps: (i)
to determine the vector ½pð0Þ ¼ ½p1 ð0Þ; p2 ð0Þ; . . .; pm ð0Þ;
ð4:1Þ
of the realizations of the probabilities pb ð0Þ, b ¼ 1; 2; . . .; m; of the system operation process stay at the operation states at the initial moment t ¼ 0, according to the formula pb ð0Þ ¼
nb ð0Þ nð0Þ
for b ¼ 1; 2; . . .; m;
where nð0Þ ¼
m X
nb ð0Þ;
ð4:2Þ
ð4:3Þ
b¼1
is the number of realizations of the system operation process starting at the initial moment t = 0; (ii) to determine the matrix 2 3 p11 p12 . . . p1v 6p p ... p 7 2v 7 6 21 22 ½pbl ¼ 6 ð4:4Þ 7; 4 5 ... pv1 pv2 . . . pvv of realizations of the probabilities pbl, b; l ¼ 1; 2; . . .; m; of system operation process transitions from the operation state zb to the operation state zl according to the formula n pbl ¼ bl for b; l ¼ 1; 2; . . .; m; b 6¼ l; pbb ¼ 0 for b ¼ 1; 2; . . .; m; nb ð4:5Þ
4.2 Identification of Complex Technical Systems Operation Processes
175
where nb ¼
m X
nbl ;
b ¼ 1; 2; . . .; m;
ð4:6Þ
b6¼l
is the realization of the total number of the system operation process departures from the operation state zb during the experiment time H:
4.2.3 Estimating Parameters of Distributions of System Conditional Sojourn Times at Operation States Prior to estimating the parameters of the distributions of the conditional sojourn times of the system operation process at the particular operation states, we have to determine the following empirical characteristics of the realizations of the conditional sojourn time of the system operation process at the particular operation states: • the realizations of the empirical mean values hbl of the conditional sojourn times hbl of the system operation process at the operation state zb when the next transition is to the operation state zl, according to the formula nbl X h ¼ 1 hk ; bl nbl k¼1 bl
b; l ¼ 1; 2; . . .; m;
b 6¼ l;
ð4:7Þ
• the number rbl of the disjoint intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; rbl which include the realizations hkbl , k ¼ 1; 2; . . .; nbl ; of the conditional sojourn times hbl at the operation state zb when the next transition is to the operation state zl, according to the formula pffiffiffiffiffiffi rbl ffi nbl ; • the length dbl of the intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; rbl , according to the formula dbl ¼
bl R ; rbl 1
where bl ¼ max hkbl min hkbl ; R 1 k nbl
1 k nbl
• the ends ablj ; bblj ; of the intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; rbl , according to the formulae
176
4 Complex Technical System Operation Processes Identification
a1bl ¼ maxf min hkbl 1 k nbl
bblj ¼ a1bl þ jdbl ; j1 ablj ¼ bbl ;
dbl ; 0g; 2
j ¼ 1; 2; . . .; rbl ;
j ¼ 2; 3. . .; rbl ;
in such a way that I1 [ I2 [ . . . [ Irbl ¼ ha1bl ; brblbl Þ and Ii \ Ij ¼ ;
for all i 6¼ j;
i; j 2 f1; 2; . . .; rbl g;
• the number nblj of the realizations hkbl in the intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; rbl , according to the formula nblj ¼ #fk : hkbl 2 Ij ; k 2 f1; 2; . . .; nbl gg;
j ¼ 1; 2; . . .; rbl ;
where rbl X
nblj ¼ nbl ;
j¼1
whereas the symbol # means the number of elements of the set. To estimate the parameters of the distributions of the conditional sojourn times of the system operation process at the particular operation states distinguished in Chap. 2, we proceed respectively in the following way: • for the uniform distribution with the density function given by (2.5), the estimates of the unknown parameters are: xbl ¼ a1bl
ybl ¼ xbl þ rbl dbl ;
ð4:8Þ
• for the triangular distribution with the density function given by (2.6), the estimates of the unknown parameters are: xbl ¼ a1bl ;
ybl ¼ xbl þ rbl dbl ;
zbl ¼ hbl ;
ð4:9Þ
• for the double trapezium distribution with the density function given by (2.7), the estimates of the unknown parameters are: xbl ¼ a1bl ;
ybl ¼ xbl þ rbl dbl ;
qbl ¼
n1bl ; nbl dbl
wbl ¼
nrblbl ; nbl dbl
zbl ¼ hbl ; ð4:10Þ
• for the quasi-trapezium distribution with the density function given by (2.8), the estimates of the unknown parameters are:
4.2 Identification of Complex Technical Systems Operation Processes
xbl ¼ a1bl ; ybl ¼ xbl þ rbl dbl ; qbl ¼
177
n1bl nrbl ; wbl ¼ bl ; z1bl ¼ h1bl ; z2bl ¼ h2bl ; nbl dbl nbl dbl ð4:11Þ
where nðmeÞ
X h1 ¼ 1 hk ; bl nðmeÞ k¼1 bl
h2bl ¼
nbl X 1 hk ; nbl nðmeÞ k¼n þ1 bl
nðmeÞ
ðmeÞ
nbl þ 1 ¼ ; 2
ð4:12Þ
and ½x denotes the entire part of x; • for the exponential distribution with the density function given by (2.9), the estimates of the unknown parameters are: xbl ¼ a1bl ;
1 abl ¼ ; hbl xbl
ð4:13Þ
• for the Weibull’s distribution with the density function given by (2.10), the estimates of the unknown parameters are (the expressions for estimates of parameters abl and bbl are not explicit): Pnbl k nbl nbl k¼1 lnðhbl xbl Þ bbl þ 1 ; ð4:14Þ ; a ¼ xbl ¼ abl ; abl ¼ Pn P bl nbl k bbl k bbl bl lnðhkbl xbl Þ k¼1 ðhbl Þ k¼1 ðhbl Þ • for the chimney distribution with the density function given by (2.11), the estimates of the unknown parameters are: xbl ¼ a1bl ;
ybl ¼ xbl þ rbl dbl ;
ð4:15Þ
and moreover, if nbl ¼ maxf nblj g
_
ð4:16Þ
1 j rbl
and i, i 2 f1; 2; . . .; rbl g; is the number of the interval including the largest number of realizations i.e. such as that _
nibl ¼ nbl ;
ð4:17Þ
then: • for i ¼ 1 either z1bl ¼ xbl þ ði 1Þ dbl ;
z2bl ¼ xbl þ idbl ;
niþ1 þ þ nrblbl Dbl ¼ bl ; nbl
Abl ¼ 0;
Cbl ¼
nibl ; nbl
ð4:18Þ
178
4 Complex Technical System Operation Processes Identification
while iþ1 ¼0 nbl
iþ1 nbl 6¼ 0
or
nibl 3; iþ1 nbl
and
ð4:19Þ
or z1bl ¼ xbl þ ði 1Þdbl ; Cbl ¼
z2bl ¼ xbl þ ði þ 1Þdbl ;
iþ1 nibl þ nbl ; nbl
Dbl ¼
Abl ¼ 0;
ð4:20Þ
iþ2 nbl þ þ nrblbl ; nbl
ð4:21Þ
while iþ1 nbl 6¼ 0
nibl \3; iþ1 nbl
and
ð4:22Þ
• for i ¼ 2; 3; . . .; rbl 1 either z1bl ¼ xbl þ ði 1Þdbl ;
z2bl ¼ xbl þ idbl ;
Dbl ¼
i1 n1bl þ þ nbl ; nbl
Abl ¼
nibl ; nbl ð4:23Þ
Cbl ¼
iþ1 nbl þ þ nrblbl ; nbl
ð4:24Þ
while i1 ¼0 nbl
i1 or nbl 6¼ 0
and
nibl 3 i1 nbl
ð4:25Þ
iþ1 nbl ¼0
iþ1 or nbl 6¼ 0
and
nibl 3; iþ1 nbl
ð4:26Þ
and while
or z1bl ¼ xbl þ ði 1Þdbl ; Cbl ¼
z2bl ¼ xbl þ ði þ 1Þd; iþ1 nibl þ nbl ; nbl
Dbl ¼
Abl ¼
i1 n1bl þ þ nbl ; nbl
iþ2 nbl þ þ nrblbl ; nbl
ð4:27Þ ð4:28Þ
while i1 i1 nbl ¼ 0 or nbl 6¼ 0
and
nibl 3 i1 nbl
ð4:29Þ
4.2 Identification of Complex Technical Systems Operation Processes
179
and while iþ1 6¼ 0 nbl
nibl \3; iþ1 nbl
and
ð4:30Þ
or z1bl ¼ xbl þ ði 2Þdbl ; z2bl ¼ xbl þ idbl ; Abl ¼
Dbl ¼
i2 n1bl þ þ nbl ni1 þ nibl ; Cbl ¼ bl ; nbl nbl ð4:31Þ
iþ1 nbl þ þ nrblbl ; nbl
ð4:32Þ
while i1 6¼ 0 nbl
nibl \3 i1 nbl
and
ð4:33Þ
and while iþ1 ¼0 nbl
iþ1 or nbl 6¼ 0
and
nibl 3; iþ1 nbl
ð4:34Þ
or z1bl ¼ xbl þ ði 2Þdbl ; Cbl ¼
z2bl ¼ xbl þ ði þ 1Þdbl ;
iþ1 i1 nbl þ nibl þ nbl ; nbl
Dbl ¼
Abl ¼
i2 n1bl þ þ nbl ; ð4:35Þ nbl
iþ2 nbl þ þ nrblbl ; nbl
ð4:36Þ
while i1 nbl 6¼ 0
and
nibl \3 i1 nbl
ð4:37Þ
iþ1 6¼ 0 nbl
and
nibl \3; iþ1 nbl
ð4:38Þ
and while
• for i ¼ rbl either z1bl ¼ xbl þ ði 1Þdbl ;
z2bl ¼ xbl þ idbl ;
Abl ¼
i1 n1bl þ þ nbl ; nbl
ð4:39Þ
180
4 Complex Technical System Operation Processes Identification
Cbl ¼
nibl ; nbl
Dbl ¼ 0;
ð4:40Þ
while i1 i1 ¼ 0 or nbl 6¼ 0 and nbl
nibl 3; i1 nbl
ð4:41Þ
or z1bl ¼ xbl þ ði 2Þdbl ;
z2bl ¼ xbl þ idbl ; Cbl ¼
i1 nbl þ nibl ; nbl
Abl ¼
i2 n1bl þ þ nbl ; nbl
Dbl ¼ 0;
ð4:42Þ ð4:43Þ
while i1 6¼ 0 nbl
and
nibl \3: i1 nbl
ð4:44Þ
4.2.4 Identification of Distribution Functions of System Conditional Sojourn Times at Operation States To formulate and then to verify the non-parametric hypothesis concerning the form of the distribution of the system operation process conditional sojourn time hbl at the operation state zb when the next transition is to the operation state zl, on the basis of at least 30 of its realizations hkbl ; k ¼ 1; 2; . . .; nbl ; it is due to proceed according to the following (Fig. 4.1) scheme: • to construct and plot the realization of the histogram of the system operation process conditional sojourn time hbl at the operation state zb, defined by the following formula j
n hnbl ðtÞ ¼ bl nbl
for t 2 I j ;
ð4:45Þ
n ðtÞ, comparing it with the graphs • to analyze the realization of the histogram h bl of the density functions hbl ðtÞ of the previously distinguished distributions in Chap. 2, to select one of them and to formulate the null hypothesis H0 , concerning the unknown form of the distribution of the conditional sojourn time hbl in the following form: H0 : The system operation process conditional sojourn time hbl at the operation state zb when the next transition is to the operation state zl, has the distribution with the density function hbl ðtÞ;
4.2 Identification of Complex Technical Systems Operation Processes Fig. 4.1 The graph of the realization of the histogram of the system operation process conditional sojourn time hbl at the operation state zb
181
h nbl (t )
1
0
a bl1
abl2 = bbl1
bbl2
a blrbl
bblrbl
t
• to join each of the intervals Ij that has the number nblj of realizations less than 4 either with the neighbor interval Ijþ1 or with the neighbor interval Ij1 in such a way that the number of realizations in all the intervals is not less than 4; • to fix a new number of intervals rbl ; • to determine new intervals Ij ¼ h ablj ; bblj Þ;
j ¼ 1; 2; ::; rbl ;
blj of realizations in new intervals Ij ; j ¼ 1; 2; ::; r bl ; • to fix the number n • to calculate the hypothetical probabilities so that the variable hbl takes values from the interval Ij ; under the assumption that the hypothesis H0 is true, i.e. the probabilities pj ¼ Pðhbl 2 Ij Þ ¼ Pð ablj hbl \ bblj Þ ¼ Hbl ð bblj Þ Hbl ðablj Þ;
j ¼ 1; 2; . . .; rbl ; ð4:46Þ
where Hbl ðbblj Þ and Hbl ð ablj Þ are the values of the distribution function Hbl ðtÞ of the random variable hbl corresponding to the density function hbl ðtÞ assumed in the null hypothesis H0 ; • to calculate the realization of the v2 (chi-square)-Pearson’s statistics Unbl , according to the formula r bl X ð nblj nbl pj Þ2 unbl ¼ ; ð4:47Þ nbl pj j¼1 • to assume the significance level a ða ¼ 0:01; a ¼ 0:02; a ¼ 0:05 or a ¼ 0:10Þ of the test; • to fix the number rbl l 1 of degrees of freedom, substituting for l for the distinguished in Chap. 2 distributions respectively the following values: l ¼ 0 for the uniform, triangular, double trapezium, quasi-trapezium and chimney distributions, l ¼ 1 for the exponential distribution and l ¼ 2 for the Weibull’s distribution; • to read from the tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a and the number of degrees of freedom r bl l 1 such that the following equality holds
182
4 Complex Technical System Operation Processes Identification
Fig. 4.2 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
f
x2
(t )
1- α
0
α
Critical domain
uα
PðUnbl [ ua Þ ¼ a;
t
ð4:48Þ
and then to determine the critical domain in the form of the interval (Fig. 4.2) ðua ; þ1Þ and the acceptance domain in the form of the interval h0; ua i; • to compare the obtained value unbl of the realization of the statistics Unbl with at read from the tables, critical value ua of the chi-square random variable and to decide on the previously formulated null hypothesis H0 in the following way: if the value unbl does not belong to the critical domain, i.e. when unbl ua ; then we do not reject the hypothesis H0 , if the value unbl belongs to the critical domain, i.e. when unbl [ ua ; then we reject the hypothesis H0 :
4.2.5 Testing Uniformity of Statistical Data of Complex Technical System Operation Processes The statistical data needed in Sect. 4.2.4 for estimating the unknown parameters of the complex technical system operation process very often come from different experiments of the same operation process and are collected into separate data sets. Before combining them into one set of data in order to perform the unknown parameters, evaluation with the methods and procedures described in Sect. 4.2.4, we have to carryout the uniformity testing of these statistical data sets. 4.2.5.1 Procedure of System Operation Process Data Collection To carryout the uniformity testing of the statistical data collected in two separate data sets coming from the same system operation process realizations in two different experiments, we should collect the necessary statistical data performing the following steps: (i)
to fix two independent experiments of the system operation process data collection and their following basic parameters: • the duration times of the experiments H1 , H2 , • the system operation processes, single realizations, • the number of the investigated (observed) realizations of the system operation process n1 ð0Þ, n2 ð0Þ;
4.2 Identification of Complex Technical Systems Operation Processes
183
(ii) to fix and collect the following statistical data concerned with the empirical distributions of the conditional sojourn times h1bl and h2bl , b; l 2 f1; 2; . . .; mg; b 6¼ l; of the system operation process at the particular operation states, respectively in the first and second experiments: • the number of realizations n1bl ;
b; l 2 f1; 2; . . .; mg;
b 6¼ l;
of the sojourn time h1bl , b; l 2 f1; 2; . . .; mg; in the first experiment, • the sample of non-decreasing ordered realizations h1k bl ;
k ¼ 1; 2; . . .; n1bl ;
b 6¼ l;
ð4:49Þ
of the sojourn time h1bl ; b; l 2 f1; 2; . . .; mg; in the first experiment, • the number of realizations n2bl ;
b; l 2 f1; 2; . . .; mg;
b 6¼ l;
of the sojourn time h2bl ; b; l 2 f1; 2; . . .; mg; in the second experiment, • the sample of non-decreasing ordered realizations h2k bl ;
k ¼ 1; 2; . . .; n2bl ;
b 6¼ l;
ð4:50Þ
of the sojourn time h2bl ; b; l 2 f1; 2; . . .; mg; in the second experiment.
4.2.5.2 Procedure of Testing Uniformity of Distributions of System Operation Process Conditional Sojourn Times at Operation States We consider test k based on the Kolmogorov-Smirnov theorem [18] that can be used for testing whether two independent samples of realizations of the conditional sojourn time hbl , b; l 2 f1; 2; . . .; mg; b 6¼ l; at the particular operation states of the system operation process are drawn from the population with the same distribution. We assume that we have defined in the previous section two independent samples of non-decreasing ordered realizations (4.49) and (4.50) of the sojourn times h1bl and h2bl , b; l 2 f1; 2; . . .; mg; b 6¼ l; coming from two different experiments, respectively composed of n1bl and n2bl realizations and we define their corresponding empirical distribution functions Hbl1 ðtÞ ¼
1 1 #fk : h1k bl \t; k 2 f1; 2; . . .; nbl gg; t 0; b; l 2 f1; 2; . . .; mg; b 6¼ l; n1bl ð4:51Þ
184
4 Complex Technical System Operation Processes Identification
and Hbl2 ðtÞ ¼
1 2 #fk : h2k bl \t; k 2 f1; 2; . . .; nbl gg; t 0; b; l 2 f1; 2; . . .; mg; b 6¼ l: n2bl ð4:52Þ
Then, according to the Kolmogorov-Smirnov theorem [18], the sequence of distribution functions given by the equation k ð4:53Þ Qn1 n2 ðkÞ ¼ P Dn1 n2 \ pffiffiffi n defined for k [ 0; where n1 n2 ; n1 þ n2
ð4:54Þ
1 H ðtÞ H 2 ðtÞ;
ð4:55Þ
n1 ¼ n1bl ;
n2 ¼ n2bl ;
Dn1 n2 ¼
max
and
1\t\þ1
n¼
bl
bl
is convergent, as n ! 1; to the limit distribution function QðkÞ ¼
þ1 X
ð1Þk e2k
2 2
k
;
k [ 0:
ð4:56Þ
k¼1
The distribution function QðkÞ given by (4.56) is called k distribution and its table of values are available. The convergence of the sequence Qn1 n2 ðkÞ to the k distribution QðkÞ means that for sufficiently large n1 and n2 we may use the following approximate formula Qn1 n2 ðkÞ ffi QðkÞ:
ð4:57Þ
Hence, it follows that if we define the statistic pffiffiffi Un ¼ Dn1 n2 n;
ð4:58Þ
where Dn1 n2 is defined by (4.55), then by (4.53) and (4.57), we have pffiffiffi k PðUn \kÞ ¼ PðDn1 n2 n\kÞ ¼ P Dn1 n2 \ pffiffiffi ¼ Qn1 n2 ðkÞ ffi QðkÞ for k [ 0: n ð4:59Þ This result means that in order to formulate and then verify the hypothesis that the two independent samples of the realizations of the system operation process conditional sojourn times h1bl and h2bl , b; l 2 f1; 2; . . .; mg; b 6¼ l; at the operation state zb when the next transition is to the operation state zl are coming from the population with the same distribution, it is necessary to proceed according to the following scheme:
4.2 Identification of Complex Technical Systems Operation Processes
185
• to fix the number of realizations n1bl and n2bl in the samples, • to collect the realizations (4.49) and (4.50) of the conditional sojourn times h1bl and h2bl of the system operation process in the samples, • to find the realization of the empirical distribution functions Hbl1 ðtÞ and Hbl2 ðtÞ defined by (4.51) and (4.52) respectively, in the following forms: 8 > n11 bl > > ¼ 0; t h11 > bl 1 > n > bl > > > n12 > > 12 > h11 > bl1 ; bl \t hbl > > n > bl > > > > n13 > 13 bl > h12 > 1 ; bl \t hbl > n > < : bl : : ; ð4:60Þ Hbl1 ðtÞ ¼ n1k 1k1 1k > bl > ; h \t h > bl bl 1 > n > > > : bl : : > > > 1n1 > > nbl bl 1n1 1 1n1 > > hbl bl \t hbl bl > 1 ; > > nbl > > > > 1n1 þ1 > > nbl bl 1n1bl > > : 1 ¼ 1; t hbl nbl 8 21 nbl > > > ¼ 0; t h21 > bl 2 > n > bl > > > n22 > > 22 bl > ; h21 > bl \t hbl 2 > > n > bl > > > n23 > 23 bl > > h22 > n2 ; bl \t hbl > < bl : : : Hbl2 ðtÞ ¼ n2k ; ð4:61Þ 2k1 bl > hbl \t h2k > 2 ; bl > n bl > > > : : : > > > 2n2 > > nbl bl 2n2 1 2n1 > > ; hbl bl \t hbl bl > > 2 > nbl > > > > 2n2 þ1 > > nbl bl 2n2 > > : 2 ¼ 1; t hbl bl nbl where n11 bl ¼ 0;
1 n1bl þ1
nbl
¼ n1bl ;
ð4:62Þ
and 1j 1k n1k bl ¼ #fj : hbl \hbl ;
j 2 f1; 2; . . .; n1bl gg;
k ¼ 2; 3; . . .; n1bl ;
ð4:63Þ
186
4 Complex Technical System Operation Processes Identification
Fig. 4.3 The graphical interpretation of the critical domain and the acceptance domain for the two-sample Smirnov-Kolmogorov test
f (t )
1−
Critical domain
t
0
0
is the number of the sojourn time h1bl realizations less than its realization h1k bl ; k ¼ 2; 3; . . .; n1bl ; and respectively n21 bl ¼ 0;
2 n2bl þ1
nbl
¼ n2bl ;
ð4:64Þ
and 2j 2k n2k bl ¼ #fj : hbl \hbl ;
j 2 f1; 2; . . .; n2bl gg;
k ¼ 2; 3; . . .; n2bl ;
ð4:65Þ
is the number of the sojourn time h2bl realizations less than its realization h2k bl ; k ¼ 2; 3; . . .; n2bl ; • to calculate the realization of the statistic un defined by (4.58) according to the formula pffiffiffi ð4:66Þ un ¼ dn1bl n2bl n; where dn1bl n2bl ¼ maxfdn11 n2 ; dn21 n2 g; bl bl
bl bl
ð4:67Þ
1k 2 dn11 n2 ¼ maxfHbl1 ðh1k bl Þ Hbl ðhbl Þ ;
k 2 f1; 2; . . .; n1bl gg;
ð4:68Þ
2k 2 dn21 n2 ¼ maxfHbl1 ðh2k bl Þ Hbl ðhbl Þ ;
k 2 f1; 2; . . .; n2bl gg;
ð4:69Þ
bl bl
bl bl
n¼
n1bl n2bl ; þ n2bl
n1bl
ð4:70Þ
• to formulate the null hypothesis H0 in the following form: H0 : The samples of realizations (4.49) and (4.50) come from the populations with the same distributions, • to fix the significance level a ða ¼ 0:01; a ¼ 0:02; a ¼ 0:05 or a ¼ 0:10Þ of the test, • to read from the tables of k distribution, corresponding to 1 a; the value k0 such that the following equality holds PðUn \k0 Þ ¼ Qðk0 Þ ¼ 1 a;
ð4:71Þ
4.2 Identification of Complex Technical Systems Operation Processes
187
• to determine the critical domain in the form of the interval (Fig. 4.3) ðk0 ; þ1Þ and the acceptance domain in the form of the interval ð0; k0 i; • to compare the obtained value un of the realization of the statistics Un with that read from the tables critical value k0 , • to decide on the previously formulated null hypothesis H0 in the following way: • if the value un does not belong to the critical domain, i.e. when un k0 ; then we do not reject the hypothesis H0, if the value un belongs to the critical domain, i.e. when un [ k0 ; then we reject the hypothesis H0. In the case when the null hypothesis H0 is not rejected we may join the statistical data from the considered two separate sets into one new set of data and if there are no other sets of statistical data including the realizations of the sojourn time hbl ; we proceed with the data of this new set in the way described in Sects. 4.2.1–4.2.4. Conversely, if there are other sets of statistical data including the realizations of the sojourn time hbl ; we select the next one and perform the procedure of this section for data from this set and data from the previously formed new set. We continue this procedure up to the moment when the store of the statistical data sets including the realizations of the sojourn time hbl ; is exhausted.
4.3 Applications 4.3.1 Statistical Identification of Exemplary System Operation Process The operation process of the considered exemplary system is described in Sect. 2.3.1, where the number of system operation process states m ¼ 4 is fixed and the operation states zb, b ¼ 1; 2; 3; 4 are arbitrarily defined. Moreover, it is fixed that there are possible transitions between all system operation states.
4.3.1.1 Defining Parameters and Data Collection of Exemplary System Operation Process The unknown parameters of the system operation process semi-markov model are: • the initial probabilities pb ð0Þ, b ¼ 1; 2; 3; 4; of the exemplary system operation process stay at the particular states zb at the moment t = 0, • the probabilities pbl , b; l ¼ 1; 2; 3; 4; b 6¼ l; of the exemplary system operation process transitions from the operation state zb into the operation state zl, • the distributions of the exemplary system operation process conditional sojourn times hbl , b; l ¼ 1; 2; 3; 4; b 6¼ l; at the particular operation states and their mean values Mbl ¼ E½hbl ; b; l ¼ 1; 2; 3; 4; b 6¼ l. To identify all these parameters of the exemplary system operation process the statistical data about this process are needed. As the considered system is an
188
4 Complex Technical System Operation Processes Identification
exemplary one, we do not have the statistical data collected that are needed for estimating the parameters. Therefore, only the values of the unknown parameters of the exemplary system operation process we need in our considerations are arbitrarily fixed in the two following sections.
4.3.1.2 Evaluating Basic Parameters of Exemplary System Operation Process As we do not have the statistical data collected that are needed for estimating the unknown basic parameters of the exemplary system operation process, we assume their values arbitrarily as in Sect. 2.3.1, where the matrix ½pbl of the probabilities pbl , b; l ¼ 1; 2; 3; 4; of the system operation process transitions from the operation state zb into the operation state zl is arbitrarily defined.
4.3.1.3 Evaluating Parameters of Distributions of Exemplary System Conditional Sojourn Times at Operation States We do not have the realizations of the conditional sojourn times hbl , b; l ¼ 1; 2; 3; 4; of the exemplary system operation process at the particular operation states that could allow to identify their distributions. Therefore, we assume arbitrarily their mean values Mbl ¼ E½hbl , b; l ¼ 1; 2; 3; 4; that are given in Sect. 2.3.1.
4.3.1.4 Identification of Distribution Functions of Exemplary System Conditional Sojourn Times at Operation States As we do not have the realizations of the conditional sojourn times hbl , b; l ¼ 1; 2; 3; 4; of the exemplary system operation process at the particular operation states, it is not possible to identify their distributions.
4.3.2 Statistical Identification of Port Oil Piping Transportation System Operation Process On the basis of the expert opinions concerning the operation process of the considered port oil pipeline transportation system, in Sect. 2.3.2, the number of the pipeline system operation process states m ¼ 7 is fixed and the operation states zb, b ¼ 1; 2; . . .; 7; are defined. Moreover, it is fixed that there are possible transitions between all system operation states.
4.3 Applications
189
4.3.2.1 Defining Parameters and Data Collection of Port Oil Piping Transportation System Operation Process The unknown parameters of the system operation process semi-Markov model are: • the initial probabilities pb ð0Þ, b ¼ 1; 2; . . .; 7; of the pipeline system operation process stay at the particular states zb at the moment t = 0, • the probabilities pbl , b; l ¼ 1; 2; . . .; 7; b 6¼ l; of the pipeline system operation process transitions from the operation state zb into the operation state zl, • the distributions of the pipeline system conditional sojourn times hbl , b; l ¼ 1; 2; . . .; 7; b 6¼ l; at the particular operation states and their mean values Mbl ¼ E½hbl ; b; l ¼ 1; 2; . . .; 7; b; l ¼ 1; 2; . . .; 7; b 6¼ l: To identify all these parameters of the pipeline system operation process the statistical data about this process is needed. The statistical data collected by the system operators necessary for evaluating the initial transient probabilities of the piping system operation process at the particular states are: • the pipeline system operation process observation/experiment time H ¼ 329 days ¼ 47 weeks; • the number of pipeline system operation process realizations nð0Þ ¼ 41; • the vector of realizations n1 ð0Þ ¼ 14; n2 ð0Þ ¼ 2; n3 ð0Þ ¼ 0; n4 ð0Þ ¼ 0; n5 ð0Þ ¼ 9; n6 ð0Þ ¼ 8; n7 ð0Þ ¼ 8; of the number of pipeline system operation processes stay at the particular operation states zb at the initial moment t = 0 ½nb ð0Þ ¼ ½14; 2; 0; 0; 9; 8; 8: The collected statistical data necessary for evaluating the probabilities of transitions of the pipeline system operation process between the operation states are: • the matrix of realizations n11 n21 n31 n41 n51 n61 n71
¼ 0; ¼ 1; ¼ 1; ¼ 0; ¼ 21; ¼ 2; ¼ 17;
n12 n22 n32 n42 n52 n62 n72
¼ 1; ¼ 0; ¼ 0; ¼ 0; ¼ 1; ¼ 0; ¼ 2;
n13 n23 n33 n43 n53 n63 n73
¼ 1; ¼ 0; ¼ 0; ¼ 0; ¼ 0; ¼ 0; ¼ 0;
n14 n24 n34 n44 n54 n64 n74
¼ 0; ¼ 0; ¼ 0; ¼ 0; ¼ 1; ¼ 0; ¼ 0;
n15 n25 n35 n45 n55 n65 n75
¼ 24; ¼ 0; ¼ 0; ¼ 0; ¼ 0; ¼ 14; ¼ 7;
n16 n26 n36 n46 n56 n66 n76
¼ 5; ¼ 0; ¼ 0; ¼ 0; ¼ 10; ¼ 0; ¼ 6;
n17 n27 n37 n47 n57 n67 n77
¼ 14; ¼ 4; ¼ 0; ¼ 1; ¼ 10; ¼ 5; ¼ 0;
190
4 Complex Technical System Operation Processes Identification
of the number of pipeline system operation process transitions from the state zb into the state zl during the experiment time H ¼ 329 days 3 2 0 1 1 0 24 5 14 6 1 0 0 0 0 0 4 7 7 6 6 1 0 0 0 0 0 0 7 7 6 ½nbl ¼ 6 0 1 7 7; 6 0 0 0 0 0 6 21 1 0 1 0 10 10 7 7 6 4 2 0 0 0 14 0 5 5 17 2 0 0 7 6 0 • the vector of realizations n1 ¼ n11 þ n12 þ n13 þ n14 þ n15 þ n16 þ n17 ¼ 45; n2 ¼ n21 þ n22 þ n23 þ n24 þ n25 þ n26 þ n27 ¼ 5; n3 ¼ n31 þ n32 þ n33 þ n34 þ n35 þ n36 þ n37 n4 ¼ n41 þ n42 þ n43 þ n44 þ n45 þ n46 þ n47 n5 ¼ n51 þ n52 þ n53 þ n54 þ n55 þ n56 þ n57 n6 ¼ n61 þ n62 þ n63 þ n64 þ n65 þ n66 þ n67
¼ 1; ¼ 1; ¼ 43; ¼ 21;
n7 ¼ n71 þ n72 þ n73 þ n74 þ n75 þ n76 þ n77 ¼ 32; (the sums of the number of matrix ½nbl rows) of the total numbers of the pipeline system operation process transitions from the operation state zb during the experiment time H ¼ 329 days ½nb ¼ ½45; 5; 1; 1; 43; 21; 32: The collected statistical data necessary for evaluating the unknown parameters of the distributions of the conditional sojourn times of the port oil piping transportation system operation process at the particular operation states are: • the numbers nbl ; b; l ¼ 1; 2; . . .; 7; of the realizations hkbl of the conditional sojourn times hbl of the port oil piping transportation system operation process at the operation state zb when the next transition is to the operation state zl; • the realizations hkbl , k = 1,2, …, nbl ; of the conditional sojourn times hbl of the port oil piping transportation system operation process at the operation state zb when the next transition is to the operation state zl. For instance, the statistical data for the sojourn time h15 collected by the experts operating the piping system are as follows: • the number of realizations n15 ¼ 24, • the realizations: h115 ¼ 930, h215 ¼ 3840, h315 ¼ 1290, h415 ¼ 480, h515 ¼ 5575, 11 12 h615 ¼ 4680, h715 ¼ 4350, h815 ¼ 2100, h915 ¼ 840, h10 15 ¼ 2460, h15 ¼ 1560, h15 ¼ 14 15 16 17 18 19 1020, h13 15 ¼ 1860, h15 ¼ 960, h15 ¼ 930, h15 ¼ 910, h15 ¼ 480, h15 ¼ 410, h15 ¼ 20 21 22 23 24 960, h15 ¼ 480, h15 ¼ 1440, h15 ¼ 4710, h15 ¼ 540, h15 ¼ 5180.
4.3 Applications
191
4.3.2.2 Evaluating Basic Parameters of Port Oil Piping Transportation System Operation Process On the basis of the statistical data from Sect. 4.3.2.1, using respectively the formulae (4.1–4.3) and (4.4–4.6) given in Sect. 4.2.2, it is possible to evaluate the following unknown basic parameters of the port oil piping transportation system operation process: • the vector ½pð0Þ ¼ ½0:34; 0:05; 0; 0; 0:23; 0:19; 0:19 of the initial probabilities pb ð0Þ, b ¼ 1; 2; . . . 7; of the pipeline system operation process stay at the particular states zb at the t = 0, • the matrix 2 3 0 0:022 0:022 0 0:534 0:111 0:311 6 0:2 0 0 0 0 0 0:8 7 6 7 6 7 6 1 0 0 0 0 0 0 7 6 7 0 0 0 0 0 1 7 ½pbl ¼ 6 6 0 7 6 0:488 0:023 0 0:023 0 0:233 0:233 7 6 7 6 7 4 0:095 0 0 0 0:667 0 0:238 5 0:531 0:062 0 0 0:219 0:188 0 of the probabilities pbl , b; l ¼ 1; 2; . . .; 7; of transitions of the pipeline system operation process from the operation state zb into the operation state zl. The values of some probabilities existing in the vector ½pð0Þ and in the matrix ½pbl , besides of those standing on the main diagonal, equal to zero do not mean that the events they are concerned with, cannot appear. They are evaluated on the basis of real statistical data and their values may change and become more precise if the duration of the experiment is longer.
4.3.2.3 Evaluating Parameters of Distributions of Port Oil Piping Transportation System Conditional Sojourn Times at Operation States On the basis of the statistical data partly presented in Sect. 4.3.2.1, using the procedure and the formulae given in Sect. 4.2.3, it is possible to determine the empirical parameters of the conditional sojourn times of the pipeline system operation process at the particular operation states. To illustrate the application of this procedure and these formulae, we perform it for h15 , that is one of the conditional sojourn times having the most populous set of realizations listed at the end of Sect. 4.3.2.1.
192
4 Complex Technical System Operation Processes Identification
The results for the conditional sojourn time h15 are: • the realization h15 of the defined by (4.7) mean value of the conditional sojourn time h15 of the pipeline system operation process at the operation state z1 when the next transition is to the operation state z5 24 1 X h15 ¼ hk ¼ 1999:4; 24 k¼1 15 j j ; b15 Þ; j ¼ 1; 2; . . .; r15 ; that • the number r15 of the disjoint intervals Ij ¼ ha15 k include the realizations h15 , k ¼ 1; 2; . . .; 24; of the conditional sojourn times h15 at the operation state z1 when the next transition is to the operation state z5 pffiffiffiffiffi r15 ffi 24 ffi 5; j j ; b15 Þ; j ¼ 1; 2; . . .; 5; which after • the length d15 of the intervals Ij ¼ ha15 considering
15 ¼ max hk15 min hk15 ¼ 5575 410 ¼ 5165; R 1 k 24
1 k 24
is d15 ¼
15 5165 R ¼ ¼ 1291; r15 1 4
j j j j • the ends a15 ; b15 , of the intervals Ij ¼ ha15 ; b15 Þ; j ¼ 1; 2; . . .; 5, which after considering
min hk15
1 k 24
d15 1291 ¼ 410 ¼ 235:5; 2 2
are a115 ¼ maxf235:5; 0g ¼ 0;
b115 ¼ a115 þ 1291 ¼ 0 þ 1291 ¼ 1291;
a215 ¼ b115 ¼ 1291;
b215 ¼ a115 þ 2 1291 ¼ 0 þ 2582 ¼ 2582;
a315 ¼ b215 ¼ 2582;
b315 ¼ a115 þ 3 1291 ¼ 0 þ 3873 ¼ 3873;
a415 ¼ b315 ¼ 3873;
b415 ¼ a115 þ 4 1291 ¼ 0 þ 5164 ¼ 5164;
a515 ¼ b415 ¼ 5164;
b515 ¼ a115 þ 5 1291 ¼ 0 þ 6455 ¼ 6455;
j j j • the number n15 of the realizations hk15 in particular intervals Ij ¼ ha15 ; b15 Þ; j ¼ 1; 2; . . .; 5,
n115 ¼ 13;
n215 ¼ 5;
n315 ¼ 1;
n415 ¼ 4;
n515 ¼ 1:
4.3 Applications
193
4.3.2.4 Identification of Distribution Functions of Port Oil Piping Transportation System Conditional Sojourn Times at Operation States Using the procedure given in Sect. 4.2.4 and the statistical data from Sect. 4.3.2.1 and the results from Sect. 4.3.2.3, we may verify the hypotheses on the distributions of the pipeline system operation process conditional sojourn times hbl , b; l ¼ 1; 2; . . .; 7; b 6¼ l; at the particular operation states. To do this, we need a sufficient number of realizations of these variables [2, 4, 23, 26, 27], namely, the sets of their realizations should contain at least 30 realizations coming from the experiment. This condition is not satisfied for the statistical data we have in disposal and that are presented in Sect. 4.3.2.1. However, to make the procedure familiar to the reader, we perform it for the conditional sojourn time h15 , the one having the most numerous set of realizations and preliminarily analyzed in Sect. 4.3.2.3. The realization h15 ðtÞ of the histogram of the pipeline system operation process conditional sojourn time h15 , defined by (4.45), is presented in Table 4.1 and illustrated in Fig. 4.4. After analyzing and comparing the realization h15 ðtÞ of the histogram with the graphs of the density functions hbl ðtÞ of the previously distinguished distributions in Chap. 2, we formulate the null hypothesis H0 in the following form: H0 : The pipeline system operation process conditional sojourn time h15 at the operation state z1 when the next transition is to the operation state z5, has the chimney distribution with the density function defined by (2.11) of the form 8 0; t\x15 > > > A > 15 > > x15 t z115 > 1 x > > z 15 > 15 > < C 15 ; z115 t z215 h15 ðtÞ ¼ ð4:72Þ z215 z115 > > > > > D15 > > ; z215 t y15 > 2 > y z > > 15 : 15 0; t [ y15 : Since, according to (4.16–4.17), we have j n15 ¼ maxf n15 g ¼ 13 and
_
1j5
_
n115 ¼ n15 ¼ 13;
Table 4.1 The realization of the histogram of the pipeline system operation process conditional sojourn time h15 Histogram of the conditional sojourn time h15 j j Ij ¼ ha15 ; b15 Þ
0–1291
1291–2582
2582–3873
3873–5164
5164–6455
j n15
13
5
1
4
1
j =n15 h15 ðtÞ ¼ n15
13/24
5/24
1/24
4/24
1/24
194
4 Complex Technical System Operation Processes Identification
Fig. 4.4 The graph of the histogram of the pipeline system operation process conditional sojourn time h15
0,6 0,5 0,4 0,3 0,2 0,1 0 0-1291
1291-2582
2582-3873
3873-5164
5164-6455
then i ¼ 1. Moreover, by (4.22), we get n215 ¼ 5 6¼ 0
n115 13 ¼ \3: 5 n215
and
Therefore, we estimate the unknown parameters of the density function of the hypothetical chimney distribution using the formulae (4.15), (4.20) and (4.21) and obtain the following results x15 ¼ a115 ¼ 0;
y15 ¼ x15 þ r15 d15 ¼ 0 þ 5 1291 ¼ 6455;
z115 ¼ x15 þ ð1 1Þd15 ¼ 0; A15 ¼ 0; C15 ¼
z215 ¼ x15 þ ð1 þ 1Þd15 ¼ 0 þ 2 1291 ¼ 2582;
n115 þ n215 13 þ 5 ¼ ¼ 0:75; 24 n15
D15 ¼
n315 þ n415 þ n515 1 þ 4 þ 1 ¼ ¼ 0:25: 24 n15
Substituting the above results into (4.72), we get completely defined the hypothetical density function in the form 8 0; t\0 > > > 0:75 > > > < ; 0 t\2582 h15 ðtÞ ¼ 2582 0 0:25 > > > ; 2582 t\6455 > > 6455 2582 > : 0; t 6455 8 0; > > > < 0:000290473; ¼ > 0:000064549; > > : 1;
t\0 0 t\2582 2582 t\6455 t 6455:
ð4:73Þ
Hence the hypothetical distribution function H15 ðtÞ of the conditional sojourn time h15 , after taking the integral of the hypothetical density function h15 ðtÞ given by (4.73), takes the following form
4.3 Applications
H15 ðtÞ ¼
Zt 0
195
8 0; > > < 0:000290473t; h15 ðtÞdt ¼ > 0:000064549t þ 0:583336205; > : 1;
t\0 0 t\2582 2582 t\6455 t 6455: ð4:74Þ
Next, we join the intervals defined in the realization of the histogram h15 ðtÞ that j have the number n15 ; of realizations less than 4 into new intervals and we perform the following steps: • fix the new number of intervals r 15 ¼ 3, • determine the new intervals I1 ¼ h0; 1291Þ;
I2 ¼ h1291; 2582Þ;
I3 ¼ h2582; 6455Þ;
• fix the number of realizations in the new intervals n115 ¼ 13;
n215 ¼ 5;
n315 ¼ 6;
• calculate, using (4.46), the hypothetical probabilities that the variable h15 takes values from the new intervals p1 ¼ Pðh15 2 I1 Þ ¼ Pð0 h15 \1291Þ ¼ H15 ð1291Þ H15 ð0Þ ¼ 0:375 0 ¼ 0:375; p2 ¼ Pðh15 2 I2 Þ ¼ Pð1291 h15 \2582Þ ¼ H15 ð2582Þ H15 ð1291Þ ¼ 0:75 0:375 ¼ 0:375;
p3 ¼ Pðh15 2 I3 Þ ¼ Pð2582 h15 \6455Þ ¼ H15 ð6455Þ H15 ð2582Þ ¼ 1 0:75 ¼ 0:25; • calculate, using (4.47), the realization of the v2 (chi-square)-Pearson’s statistics u24 ¼
j 3 X ðn15 n15 pj Þ2 ð13 24 0:375Þ2 ð5 24 0:375Þ2 ð6 24 0:25Þ2 þ þ ¼ n15 pj 24 0:375 24 0:375 24 0:25 j¼1
ffi 1:78 þ 1:78 þ 0 ¼ 3:56; • assume the significance level a ¼ 0:05, • fix the number of degrees of freedom r15 l 1 ¼ 3 0 1 ¼ 2; • read from the tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a ¼ 0:05 and the number of degrees of freedom r15 l ¼ 1 ¼ 2; such that, according to (4.48), the following equality holds
196
4 Complex Technical System Operation Processes Identification
Fig. 4.5 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
f 2 (t ) x
1 −α
0
α u24 = 3.56
Critical domain
uα = 5.99
t
PðU24 [ ua Þ ¼ a ¼ 0:05; this amounts to ua ¼ 5:99 and we determine the critical domain in the form of the interval ð5:99; þ1Þand the acceptance domain in the form of the interval (Fig. 4.5) h0; 5:99i • we compare the obtained value u24 ¼ 3:56 of the realization of the statistics U24 with that read from the tables, critical value ua ¼ 5:99 of the chi-square random variable and since the value u24 ¼ 3:56 does not belong to the critical domain, i.e. u24 ¼ 3:56 ua ¼ 5:99; we do not reject the hypothesis H0. After getting such a result, in the case that have enough statistical data, we may assume that the sojourn time h15 has the chimney distribution with the density function given by (4.73). Otherwise, if the null hypothesis H0 is rejected, we should select other density function from the distinguished distributions and repeat the procedure of testing. If we decide to accept the density function h15 ðtÞ of the conditional sojourn time h15 given by (4.73), then either applying the general formula (2.12) for mean value or applying the particular formula (2.11) for chimney distribution, it is possible to find its mean value evaluation: M15 ¼
Z1
1 th15 ðtÞdt ¼ ½A15 ðx15 þ z115 Þ þ C15 ðz115 þ z215 Þ þ D15 ðz215 þ y15 Þ 2
0
1 ¼ ½0ð0 þ 0Þ þ 0:75ð0 þ 2582Þ þ 0:25ð2582 þ 6455Þ ffi 2098: 2 Because of the lack of sufficient number of realizations of the piping system operation process conditional sojourn times at the operation states, it is not possible to identify statistically their distributions. In those cases of not identified distributions, using formula (4.7), it is possible to find the approximate empirical values of the mean values Mbl ¼ E½hbl of the conditional sojourn times at the particular operation states that are as follow:
4.3 Applications
M12 ¼ 1920; M21 ¼ 9960; M54 ¼ 300; M67 ¼ 438;
197
M13 ¼ 480; M27 ¼ 810; M56 ¼ 436:3; M71 ¼ 874:1;
M15 ¼ 1999:4; M16 ¼ 1250; M17 ¼ 1129:6; M31 ¼ 575; M47 ¼ 380; M51 ¼ 874:7; M52 ¼ 480; M57 ¼ 1042:5; M61 ¼ 325; M65 ¼ 510:7; M72 ¼ 510; M75 ¼ 2585:7; M76 ¼ 2380:
These approximate values are used in Sect. 2.3.2 in the evaluation and prediction of the characteristics of the port oil piping transportation system operation process. As there are no realizations of the port oil piping transportation system operation process conditional sojourn times at the operation states h14 ; h23 ; h24 ; h25 ; h26 ; h32 ; h34 ; h35 ; h36 ; h37 ; h41 ; h42 ; h43 ; h45 ; h46 ; h53 ; h62 ; h63 ; h64 ; h73 ; h74 ; it is impossible to estimate their empirical conditional mean values M14 ;
M23 ; M24 ; M42 ; M43 ; M45 ;
M25 ; M46 ;
M26 ; M53 ;
M32 ; M62 ;
M34 ; M63 ;
M35 ; M64 ;
M36 ; M73 ;
M37 ; M74 ;
M41 ;
4.3.3 Statistical Identification of Maritime Ferry Technical System Operation Process On the basis of expert opinion concerning the operation process of the considered maritime ferry technical system, in Sect. 2.3.3, the number of system operation process states m ¼ 18 is fixed and the operation states zb ; b ¼ 1; 2; . . .; 18; are defined. Moreover, it is fixed that there are possible only transitions between the neighboring system operation states, i.e., from the operation states zb to the operation states zbþ1 , b ¼ 1; 2; . . .; 17; and from the operation state z18 to the operation state z1.
4.3.3.1 Defining Parameters and Data Collection of Ferry Technical System Operation Process The unknown parameters of the system operation process semi-Markov model are: • the initial probabilities pb ð0Þ, b ¼ 1; 2; . . .; 18; of the ferry technical system operation process stay at the particular states zb at the moment t = 0, • the probabilities pb bþ1 , b ¼ 1; 2; . . .; 17; and p181 of the ferry technical system operation process transitions from the operation state zb into the operation state zbþ1 and from the operation state z18 into the operation state z1 ;
198
4 Complex Technical System Operation Processes Identification
• the distributions of the ferry technical system operation process conditional sojourn times hb bþ1 , b ¼ 1; 2; . . .; 17; and h181 at the particular operation states and their mean values Mb bþ1 ¼ E½hb bþ1 , b ¼ 1; 2; . . .; 17; and M181 ¼ E½h181 . To identify all these parameters of the ferry technical system operation process the statistical data about this process is needed. The statistical data collected during the ferry spring voyages are partly given below. The collected spring statistical data necessary to evaluate the probabilities of straying of the ferry technical system operation process at the particular states at the initial moment t ¼ 0, are: • the ferry technical system operation process observation/experiment time H ¼ 42 days; • the number the ferry operation process realizations nð0Þ ¼ 42; • the vector of realizations n1 ð0Þ ¼ 42; n2 ð0Þ ¼ 0; . . .; n18 ð0Þ ¼ 0; of the number of ferry technical system operation process transients at the particular operation states zb at the initial moment t = 0 ½nb ð0Þ ¼ ½42; 0; . . . ; 0: The collected statistical data necessary to evaluate the probabilities of transitions of the ferry technical system operation process between the operation states are: • the matrix of realizations n11 ¼ 0; n12 ¼ 42; n13 ¼ 0; . . .; n117 ¼ 0; n118 ¼ 0; n21 ¼ 0; n22 ¼ 0; n23 ¼ 42; . . .; n217 ¼ 0; n218 ¼ 0; n171 ¼ 0; n172 ¼ 0; n173 ¼ 0; . . .; n1717 ¼ 0; n1718 ¼ 42; n181 ¼ 42; n182 ¼ 0; n183 ¼ 0; . . .; n1817 ¼ 0; n1818 ¼ 0; of the number of ferry technical system operation process transitions from the state zb into the state zl during the experiment time H ¼ 42 days 2 3 0 42 0 . . .0 0 6 7 6 0 0 42 . . . 0 0 7 6 7 7; ... ½nbl ¼ 6 6 7 6 7 4 0 0 0 . . .0 42 5 42 0 0 . . . 0 0
4.3 Applications
199
• the matrix of realizations n1 ¼ n11 þ n12 þ þ n118 ¼ 42; n2 ¼ n21 þ n22 þ þ n218 ¼ 42; ... n17 ¼ n171 þ n172 þ þ n1718 ¼ 42; n18 ¼ n181 þ n182 þ þ n1818 ¼ 42; (the sums of the number of matrix ½nbl rows) of the total number of ferry technical system operation process transitions from the operation state zb during the experiment time H ¼ 42 days ½nb ¼ ½42; 42; . . .; 42: The statistical data necessary collected by experts to evaluate the unknown parameters of the distributions of the conditional sojourn times of the ferry technical system operation process at the particular operation states are: • the numbers nb bþ1 ; b ¼ 1; 2; . . .; 17; of the realizations hkb bþ1 of the conditional sojourn times hb bþ1 of the ferry technical system operation process at the operation state zb when the next transition is to the operation state zbþ1 and the number n181 of the realizations hk181 of the conditional sojourn times h181 of the ferry technical system operation process at the operation state z18 when the next transition is to the operation state z1; • the realizations hkb bþ1 , k = 1,2, …, nb bþ1 ; b ¼ 1; 2; . . .; 17; of the conditional sojourn times hb bþ1 of the ferry technical system operation process at the operation state zb when the next transition is to the operation state zbþ1 and the realizations hk181 ; k = 1,2, …, n181 ; of the conditional sojourn times h181 of the ferry technical system operation process at the operation state z18 when the next transition is to the operation state z1. The exemplary statistical data collected by experts for two of the conditional sojourn times, namely for h12 and h23 , are presented below. The statistical data for the sojourn time h12 collected by the experts operating the ferry technical system are as follows: • the number of realizations n12 ¼ 42; • the realizations: 55, 52, 47, 75, 60, 60, 62, 43, 50, 61, 65, 63, 45, 45, 40, 20, 33, 50, 43, 15, 45, 57, 97, 68, 58, 35, 45, 75, 72, 62, 37, 44, 46, 78, 59, 65, 53, 25, 55, 84, 71, 67. The statistical data for the sojourn time h23 collected by the experts operating the ferry technical system are as follows: • the number of realizations n23 ¼ 42; • the realizations: 4, 3, 3, 2, 2, 2, 2, 3, 3, 4, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 4, 3, 3, 2, 3, 6, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2.
200
4 Complex Technical System Operation Processes Identification
4.3.3.2 Evaluating Basic Parameters of Ferry Technical System Operation Process On the basis of the statistical data from Sect. 4.3.3.1, using the formulae (4.1–4.3) and respectively (4.4–4.6) given in Sect. 4.2.2, it is possible to evaluate the following basic parameters of the ferry technical system operation process: • the vector ½pð0Þ ¼ ½1; 0; 0; . . .; 0; of the initial probabilities pb ð0Þ, b ¼ 1; 2; . . .; 18; of the ferry technical system operation process stay at the particular operation states zb at the moment t = 0, • the matrix 2 3 0 1 0 ... 0 0 6 7 60 0 1 ... 0 07 6 7 7; ... ½pbl ¼ 6 6 7 6 7 40 0 0 ... 0 15 1 0 0 ... 0 0 of the probabilities pbl , b; l ¼ 1; 2; . . .; 18; of the ferry technical system operation process transitions from the operation state zb into the operation state zl.
4.3.3.3 Evaluating Parameters of Distributions of Ferry Technical System Conditional Sojourn Times at Operation States On the basis of the statistical data partly presented in Sect. 4.3.3.1, using the formulae given in Sect. 4.2.3, it is possible to determine the empirical parameters of the conditional sojourn times of the ferry technical system operation process at the particular operation states. To illustrate this procedure and the formulae application, we perform it for two of the conditional sojourn times, namely h12 and h23 : The results for the ferry technical system operation process conditional sojourn time h12 are: • the realization h12 of the defined by (4.7) mean value of the conditional sojourn times h12 of the ferry technical system operation process at the operation state z1 when the next transition is to the operation state z2 42 1 X hk ¼ 54:3; h12 ¼ 42 k¼1 12
ð4:75Þ
4.3 Applications
201
j j • the number r12 of disjoint intervals Ij ¼ ha12 ; b12 Þ; j ¼ 1; 2; . . .; r12 , that include the realizations hk12 , k ¼ 1; 2; . . .; 42; of the conditional sojourn times h12 at the operation state z1 when the next transition is to the operation state z2 pffiffiffiffiffiffi pffiffiffiffiffi r12 ¼ n12 ¼ 42 ffi 6; j j ; b12 Þ; j ¼ 1; 2; . . .; 6; which after • the length d12 of the intervals Ij ¼ ha12 considering
12 ¼ max hk12 min hk12 ¼ 97 15 ¼ 82; R 1 j 42
1 j 42
is d12 ¼
R 82 ¼ ffi 16; r12 1 5
j j j j • the ends a12 ; b12 ; of the intervals Ij ¼ ha12 ; b12 Þ; j ¼ 1; 2; . . .; 6; which after considering
min hk12
1 k 42
d12 16 ¼ 15 ¼ 15 8 ¼ 7; 2 2
are a112 ¼ maxf7; 0g ¼ 7;
b112 ¼ a112 þ d ¼ 7 þ 16 ¼ 23;
a212 ¼ b112 ¼ 23;
b212 ¼ a112 þ 2 16 ¼ 7 þ 32 ¼ 39;
a312 ¼ b212 ¼ 39;
b312 ¼ a112 þ 3 16 ¼ 7 þ 48 ¼ 55;
a412 ¼ b312 ¼ 55;
b312 ¼ a112 þ 3 16 ¼ 7 þ 48 ¼ 55;
a512 ¼ b412 ¼ 71;
b512 ¼ a112 þ 5 16 ¼ 7 þ 80 ¼ 87;
a612 ¼ b512 ¼ 87;
b612 ¼ a112 þ 6 16 ¼ 7 þ 96 ¼ 103;
j j j • the number n12 of the realizations hk12 in particular intervals Ij ¼ ha12 ; b12 Þ; j ¼ 1; 2; . . .; 6;
n112 ¼ 2;
n212 ¼ 4;
n312 ¼ 14;
n412 ¼ 15;
n512 ¼ 6;
n612 ¼ 1:
The results for the ferry technical system operation process conditional sojourn time h23 are: • the realization of the mean value h23 of the conditional sojourn times h23 of the ferry technical system operation process at the operation state z2 when the next transition is to the operation state z3 42 X h23 ¼ 1 hk ¼ 2:6; 42 k¼1 23
b; l ¼ 1; 2; . . .; 18;
b 6¼ l;
ð4:76Þ
202
4 Complex Technical System Operation Processes Identification
j j • the number r23 of the disjoint intervals Ij ¼ ha23 ; b23 Þ, j ¼ 1; 2; . . .; r23 , which k include the realizations h23 ; k ¼ 1; 2; . . .; 42; of the conditional sojourn times h23 at the operation state z2 when the next transition is to the operation state z3 pffiffiffiffiffi r23 ¼ 42 ffi 6; j j • the length d23 of the intervals Ij ¼ ha23 ; b23 Þ, j ¼ 1; 2; . . .; 6; which after considering
23 ¼ max hk12 min hk23 ¼ 6 2 ¼ 4; R l k 42
l k 42
is d23 ¼
23 4 R ¼ ¼ 0:8; r23 1 5
j j j j • the ends a23 ; b23 ; of the intervals Ij ¼ ha23 ; b23 Þ, j ¼ 1; 2; . . .; 6; which after considering
min hk23
1 k 42
d23 0:8 ¼2 ¼ 2 0:4 ¼ 1:6; 2 2
are a123 ¼ maxf1:6; 0g ¼ 1:6;
b123 ¼ a123 þ 0:8 ¼ 1:6 þ 0:8 ¼ 2:4;
a223 ¼ b123 ¼ 2:4;
b223 ¼ a123 þ 2 0:8 ¼ 1:6 þ 1:6 ¼ 3:2;
a323 ¼ b223 ¼ 3:2;
b323 ¼ a123 þ 3 0:8 ¼ 1:6 þ 2:4 ¼ 4:0;
a423 ¼ b323 ¼ 4:0;
b423 ¼ a123 þ 4 0:8 ¼ 1:6 þ 3:2 ¼ 4:8;
a523 ¼ b423 ¼ 4:8;
b523 ¼ a523 þ 5 0:8 ¼ 1:6 þ 4:0 ¼ 5:6;
a623 ¼ b523 ¼ 5:6;
b623 ¼ a623 þ 6 0:8 ¼ 1:6 þ 4:8 ¼ 6:4;
j j j • the number n23 of the realizations hk23 in particular intervals Ij ¼ ha23 ; b23 Þ; j ¼ 1; 2; . . .; 6;
n123 ¼ 24;
n223 ¼ 14;
n323 ¼ 0;
n423 ¼ 3;
n523 ¼ 0;
n623 ¼ 1:
4.3.3.4 Identification of Distribution Functions of Ferry Technical System Conditional Sojourn Times at Operation States Using the procedure given in Sect. 4.2.4 and data from Sect. 4.3.3.1 and the results from Sect. 4.3.3.3, we may verify the hypotheses on the distributions of the ferry technical system conditional sojourn times hbl , b; l ¼ 1; 2; . . .; 18; b 6¼ l; in the particular operation states. To do this, we need a sufficient number of realizations of these variables [2, 4, 23, 26–27], namely the sets of their realizations should
4.3 Applications
203
Table 4.2 The realization of the histogram of the ferry technical system operation process conditional sojourn time h12 Histogram of the conditional sojourn time h12 Ij ¼ hablj ; bblj Þ
7–23
23–39
39–55
55–71
71–87
j n12 12 ðtÞ ¼ n j =n12 h 12
2
4
14
15
6
1
2/42
4/42
14/42
15/42
6/42
1/42
Fig. 4.6 The histogram of the ferry technical system operation process conditional sojourn time h12
87–103
0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 7-23
23-39
39-55
55-71
71-87
87-103
contain at least 30 realizations coming from the experiment. This condition is satisfied for the statistical data we have at disposal and that are partly presented in Sect. 4.3.3.1. To make the procedure familiar to the reader, we perform it for the conditional sojourn times h12 and h23 preliminarily analyzed in Sect. 4.3.3.3. The realization h12 ðtÞ defined by (4.45) of the we get histogram of the ferry technical system operation process conditional sojourn time h12 is presented in Table 4.2 and illustrated in Fig. 4.6. After analyzing and comparing the realization h12 ðtÞ of the histogram with the graphs of the density functions hbl ðtÞ of the previously distinguished distributions in Chap. 2, we formulate the null hypothesis H0 in the following form: H0 : The ferry technical system operation time conditional sojourn time h12 at the operation state z1 when the next transition is to the operation state z2, has the triangular distribution with the density function defined by (2.6) of the form 8 0; t\x12 > > > > 2 t x12 > > ; x12 t z12 < y12 x12 z12 x12 h12 ðtÞ ¼ ð4:77Þ 2 y12 t > > > ; z12 t y12 > > > : y12 x12 y12 z12 0; t [ y12 : We estimate the unknown parameters of the density function of the hypothetical triangular distribution using the formulae (4.9) and the evaluation (4.75) and we obtain the following results x12 ¼ a1 ¼ 7; y12 ¼ x12 þ r12 d12 ¼ 7 þ 6:16 ¼ 103; z12 ¼ h12 ¼ 54:3: 12
Substituting the above results into (4.77), we completely define the hypothetical density function in the form
204
4 Complex Technical System Operation Processes Identification
8 0; t\7 > > > > 2 t 7 > > < ; 7 t\54:3 h12 ðtÞ ¼ 103 7 54:3 7 2 103 t > > > ; 54:3 t\103 > > 103 7 103 54:3 > : 0; t 103 8 0; t\7 > > > < 0:000440451t 0:003083157; 7 t\54:3 ¼ > 0:000427789t þ 0:044062286; 54:3 t\103 > > : 0; t 103:
ð4:78Þ
Hence, the hypothetical distribution function H12 ðtÞ of the conditional sojourn time h12 , after taking the integral of the hypothetical density function h12 ðtÞ given by (4.78), takes the following form Rt H12 ðtÞ ¼ h12 ðtÞdt 0
8 0; > > > > < 0:0002202255t2 0:003083157t þ 0:0107910495; ¼ > 0:0002138945t2 þ 0:044062286t 1:2692080266 > > > : 1;
t\7 7 t\54:3 54:3 t\103 t 103: ð4:79Þ
Next, we join the intervals defined in the realization of the histogram h12 ðtÞ j which have the numbers n12 ; of realizations less than 4 into new intervals and we perform the following steps: • fix the new number of intervals r 12 ¼ 4, • determine the new intervals I1 ¼ h7; 39Þ; I2 ¼ h39; 55Þ; I3 ¼ h55; 71Þ;
I4 ¼ h71; 103Þ;
• fix the number of realizations in the new intervals n112 ¼ 6;
n212 ¼ 14;
n312 ¼ 15;
n412 ¼ 7;
• calculate, using (4.46), the hypothetical probabilities that the variable h12 takes values from the new intervals p1 ¼ Pðh12 2 I1 Þ ¼ Pð7 h12 \39Þ ¼ H12 ð39Þ H12 ð7Þ ¼ 0:2255 0 ¼ 0:2255;
p2 ¼ Pðh12 2 I2 Þ ¼ Pð39 h12 \55Þ ¼ H12 ð55Þ H12 ð39Þ ¼ 0:5072 0:2255 ¼ 0:2817;
4.3 Applications Fig. 4.7 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
205
f
x2
(t )
1− α
0
u42 = 3.26
α
Critical domain
uα = 7.81
t
p3 ¼ Pðh12 2 I3 Þ ¼ Pð55 h12 \71Þ ¼ H12 ð71Þ H12 ð55Þ ¼ 0:7810 0:5072 ¼ 0:2738
p4 ¼ Pðh12 2 I4 Þ ¼ Pð71 h12 \103Þ ¼ H12 ð103Þ H12 ð71Þ ¼ 1 0:7810 ¼ 0:2190; • calculate, using (4.47), the realization of the v2 (chi-square)-Pearson’s statistics u42 ¼
j 4 X ðn12 n12 pj Þ2 ð6 42 0:2255Þ2 ð14 42 0:2817Þ2 þ ¼ n12 pj 42 0:2255 42 0:2817 j¼1
þ
ð15 42 0:2738Þ2 ð7 42 0:2190Þ2 þ 42 0:2738 42 0:2190
ffi 1:27 þ 0:39 þ 1:07 þ 0:53 ¼ 3:26; • assume the significance level a ¼ 0:05, • fix the number of degrees of freedom r12 l 1 ¼ 4 0 1 ¼ 3; • read from the tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a ¼ 0:05 and the number of degrees of freedom r 12 l ¼ 1 ¼ 3; such that, according to (4.49), the following equality holds PðU42 [ ua Þ ¼ a ¼ 0:05 that amounts ua ¼ 7:81 and determine the critical domain in the form of the interval ð7:81; þ1Þ and the acceptance domain in the form of the interval (Fig. 4.7) h0; 7:81i • we compare the obtained value u42 ¼ 3:26 of the realization of the statistics U42 with the read from the tables, critical value ua ¼ 7:81 of the chi-square
206
4 Complex Technical System Operation Processes Identification
Table 4.3 The realization of the histogram of the ferry technical system operation process conditional sojourn time h23 Histogram of the conditional sojourn time h23 Ij ¼ hablj ; bblj Þ
1.6–2.4
2.4–3.2
3.2–4.0
4.0–4.8
4.8–5.6
5.6–6.4
j n23 23 ðtÞ ¼ n j =n23 h 23
24
14
0
3
0
1
24/42
14/42
0/42
3/42
0/42
1/42
Fig. 4.8 The histogram of the ferry technical system operation process conditional sojourn time h23
0,6 0,5 0,4 0,3 0,2 0,1 0 1,6-2,4
2,4-3,2
3,2-4,0
4,0-4,8
4,8-5,6
5,6-6,4
random variable and since the value u42 ¼ 3:26 does not belong to the critical domain, i.e. u42 ¼ 3:26 ua ¼ 7:81; we do not reject the hypothesis H0. The realization h23 ðtÞ of the histogram defined by (4.45) of the ferry technical system operation process conditional sojourn time h23 is presented in Table 4.3 and illustrated in Fig. 4.8. After analyzing and comparing the realization h23 ðtÞ of the histogram with the graphs of the density functions hbl ðtÞ of the previously distinguished distributions in Chap. 2, we formulate the null hypothesis H0 in the following form: H0 : The ferry technical system operation process conditional sojourn time h23 at the operation state z2 when the next transition is to the operation state z3, has the exponential distribution with the density function defined by (2.9) of the form h23 ðtÞ ¼
0; t\x23 a23 exp½a23 ðt x23 Þ; t x23 ;
ð4:80Þ
We estimate the unknown parameters of the density function of the hypothetical exponential distribution using the formulae (4.13) and the evaluation (4.76) and obtain the following results
4.3 Applications
207
1 1 a23 ¼ ¼ ¼1 2:6 1:6 h23 x23
x23 ¼ a123 ¼ 1:6;
Substituting the above results into (4.80), we completely define the hypothetical density function in the form 0; t\1:6 0; t\1:6 ð4:81Þ ¼ h23 ðtÞ ¼ exp½t þ 1:6; t 1:6: 1 exp½1ðt 1:6Þ: t 1:6 Hence, the hypothetical distribution function H23 ðtÞ of the conditional sojourn time h23 , after taking the integral of the hypothetical density function h23 ðtÞ given by (4.81), takes the following form H23 ðtÞ ¼
Zt
h23 ðtÞdt ¼
0; 1 exp½t þ 1:6;
t\1:6 t 1:6:
ð4:82Þ
0
Next, we join the intervals defined in the realization of the histogram h23 ðtÞ j which have the numbers n23 ; of realizations less than 4 into new intervals and we perform the following steps: • fix the new number of intervals r 23 ¼ 3, • determine the new intervals I1 ¼ h1:6; 2:4Þ;
I2 ¼ h2:4; 3:2Þ;
I3 ¼ h3:2; 1Þ;
• fix the numbers of realizations in the new intervals n123 ¼ 24;
n223 ¼ 14;
n323 ¼ 4;
• calculate, using (4.46), the hypothetical probabilities that the variable h23 takes values from the new intervals p1 ¼ Pðh23 2 I1 Þ ¼ Pð1:6 h23 \2:4Þ ¼ H23 ð2:4Þ H23 ð1:6Þ ¼ ð1 exp½2:4 þ 1:6Þ ð1 exp½1:6 þ 1:6Þ ¼ 1 exp½0:8 ¼ 1 0:4493 ¼ 0:5507; p2 ¼ Pðh23 2 I2 Þ ¼ Pð2:4 h23 \3:2Þ ¼ H23 ð3:2Þ H23 ð2:4Þ ¼ ð1 exp½3:2 þ 1:6Þ ð1 exp½2:4 þ 1:6Þ ¼ exp½1:6 þ exp½0:8 ¼ 0:2019 þ 0:4493 ¼ 0:2474; p3 ¼ Pðh23 2 I3 Þ ¼ Pð2:4 h23 \1Þ ¼ 1 ðp1 þ p2 Þ ¼ 1 ð0:5507 þ 0:2474Þ ¼ 0:2019; • calculate, using (4.47), the realization of the v2 (chi-square)-Pearson’s statistics
208
4 Complex Technical System Operation Processes Identification
Fig. 4.9 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
f
x2
(t )
1− α
0
u42 ¼
α
Critical domain
u 42 = 3.65 uα = 3.84
t
j 3 X ðn23 n23 pj Þ2 ð24 42 0:5507Þ2 ð14 42 0:2474Þ2 ¼ þ n23 pj 42 0:5507 42 0:2474 j¼1
þ
ð4 42 0:2019Þ2 ffi 0:03 þ 1:25 þ 2:37 ¼ 3:65; 42 0:2019
• assume the significance level a ¼ 0:05, • fix the number of degrees of freedom r23 l 1 ¼ 3 1 1 ¼ 1; • read from the tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a ¼ 0:05 and the number of degrees of freedom r23 l ¼ 1 ¼ 1; such that, according to (4.49), the following equality holds PðU42 [ ua Þ ¼ a ¼ 0:05 this amounts to ua ¼ 3:84 and we determine the critical domain in the form of the interval (Fig. 4.9) ð3:84; þ1Þ and the acceptance domain in the form of the interval h0; 3:84i • we compare the obtained value u42 ¼ 3:65 of the realization of the statistics U42 with that read from the table, critical value ua ¼ 3:84 of the chi-square random variable and since the value u42 ¼ 3:65 does not belong to the critical domain, i.e. u42 ¼ 3:65 ua ¼ 3:84; we do not reject the hypothesis H0. Proceeding afterwards in an analogous way as in the case of the conditional sojourn times h12 and h23 , we identify the density functions of the remaining sojourn times. Their forms are given in Chap. 2. For the identified distributions, by application of either the general formulae for the mean value given by (2.12) or the particular formulae (2.13–2.19), the mean values Mb bþ1 ¼ E½hb bþ1 ; b ¼ 1; 2; . . .; 17; and M181 ¼ E½h181 of the ferry technical system operation process conditional sojourn times at the particular operation states are determined in
4.3 Applications
209
Sect. 2.3.3 and applied for the evaluation and prediction of the characteristics of the ferry technical system operation process.
4.3.4 Testing Uniformity of Statistical Data of Ferry Technical System Operation Process The ferry technical system operation process is described in Sect. 2.3.3. Its semiMarkov model unknown parameters are evaluated in Sect. 4.3.3 on the basis of the statistical data coming from its spring realizations. Additionally, we have at disposal the statistical data coming from this winter operation process realizations given below in this section. Thus, we may consider the results of two different experiments of the statistical data collection from the ferry technical system operation process, one performed in spring and another in winter. From these experiments two sets of the realizations of the ferry technical system operation process conditional sojourn times at particular operation states hb bþ1 , b ¼ 1; 2; . . .; 17; and h181 are collected. Joining these two sets of data into one new set of data containing all data from the two experiments and using it for the evaluation of the ferry technical system operation process may lead to a much better estimation of these parameters. But, before the joining of these data into one new set it is necessary to be sure that they are the realizations of the same operation process. Otherwise, the evaluations may be very far from these parameters, real values. Testing the uniformity of the statistical data of these two separate sets before joining them into one set of data may give the reason for doing this.
4.3.4.1 Ferry Technical System Operation Process Data Collection The statistical data of the spring experiment on the ferry technical system operation process given in Sect. 4.3.3.1 are characterized by the following parameters: • the ferry technical system operation process experiment time H1 = 42 days, • the number of ferry technical system operation process realizations n1 ð0Þ ¼ 42; 1k • the number of realizations h1k b bþ1 , b ¼ 1; 2; . . .; 17; and h181 of the ferry technical system operation process conditional sojourn times at the particular operation states
n1bl ¼ 42: k The realizations h1k b bþ1 ¼ hb bþ1 , b ¼ 1; 2; . . .; 17; k ¼ 1; 2; . . .; 42; of the ferry technical system operation process conditional sojourn times h1b bþ1 for
210
4 Complex Technical System Operation Processes Identification
k b ¼ 1; 2; . . .; 17; and the realizations h1k 181 ¼ h181 , k ¼ 1; 2; . . .; 42; of the technical system operation process conditional sojourn time h1181 ; are partly given in Sect. 4.3.3.1.
The winter experiment on the ferry technical system operation process is characterized by the following parameters: • the ferry technical system operation process experiment time H2 = 40 days, • the number of ferry technical system operation process realizations n2 ð0Þ ¼ 40; 2k • the number of realizations h2k b bþ1 , b ¼ 1; 2; . . .; 17; and h181 of the ferry technical system operation process conditional sojourn times at particular operation states
n2bl ¼ 40: The realizations h2k b bþ1 , b ¼ 1; 2; . . .; 17; k ¼ 1; 2; . . .; 40; of the ferry technical system operation process conditional sojourn time h2b bþ1 for b ¼ 1; 2; . . .; 17; and the realizations h2k 181 , k ¼ 1; 2; . . .; 40; of the ferry technical system operation process conditional sojourn time h2181 ; are collected by experts and partly presented in Sect. 4.3.4.2.
4.3.4.2 Spring and Winter Statistical Data of Ferry Technical System Operation Process Uniformity Analysis We use the two-sample Smirnov-Kolmogorov test described in Sect. 4.2.5.2 to verify the hypotheses that the spring and the winter data sets consisting of the ferry technical system operation process conditional sojourn times at the particular operation states come from the population with the same distribution. The procedure of testing the uniformity of data sets collected by the experts in the spring and in winter is illustrated on the example of the spring realizations h1k 12 , 2k k ¼ 1; 2; . . .; 42; and the winter realizations h12 , k ¼ 1; 2; . . .; 40; of the ferry technical system operation process conditional sojourn time h112 and respectively the ferry technical system operation process conditional sojourn time h212 at the operation state z1 while the next operation state is z2. For the spring data, the ordered sample of realizations h1k 12 presented in Sect. 4.3.3.1 is: 15, 20, 25, 33, 35, 37, 40, 43, 43, 44, 45, 45, 45, 45, 46, 47, 50, 50, 52, 53, 55, 55, 57, 58, 59, 60, 60, 61, 62, 62, 63, 65, 65, 67, 68, 71, 72, 75, 75, 78, 84, 97, and after applying (4.60–4.62), the conditional sojourn time h112 has the empirical distribution function of the form
4.3 Applications
211
8 0 > > > > > 1=42; > > > > > 2=42; > > > > > 3=42; > > > > 4=42 > > > > > 5=42; > > > > > 6=42; > > > > > 8=42; > > > > > 9=42; > > > > > 13=42; > > > > 14=42; > > > > > 15=42; > > > > > 17=42; > > > > > 18=42; > > > > > 19=42; > > > > > < 21=42; 1 H12 ðtÞ ¼ 22=42; > > > 23=42; > > > > > 24=42; > > > > > 26=42; > > > > > 27=42; > > > > 29=42; > > > > > > 30=42; > > > > 32=42; > > > > > 33=42; > > > > > 34=42; > > > > > 35=42; > > > > > 36=42; > > > > 38=42; > > > > > 39=42; > > > > > 40=42; > > > > > 41=42; > > : 1;
t 15; 15\t 20; 20\t 25; 25\t 33; 33\t 35; 35\t 37; 37\t 40; 40\t 43; 43\t 44; 44\t 45; 45\t 46; 46\t 47; 47\t 50; 50\t 52; 52\t 53; 53\t 55; 55\t 57; 57\t 58; 58\t 59; 59\t 60; 60\t 61; 61\t 62; 62\t 63; 63\t 65; 65\t 67; 67\t 68; 68\t 71; 71\t 72; 72\t 75; 75\t 78; 78\t 84; 84\t 97; t [ 97;
For the winter data collected by the experts, the ordered sample of realizations is: 12, 15, 15, 18, 19, 20, 25, 33, 33, 34, 36, 37, 37, 37, 40, 41, 44, 46, 48, 48, 50, 53, 55, 57, 59, 60, 60, 61, 62, 63, 65, 65, 65, 67, 69, 75, 75, 75, 80, 90, and after applying (4.63–4.65), the conditional sojourn time h212 has the empirical distribution function of the form h2k 12
212
4 Complex Technical System Operation Processes Identification
8 0 > > > > > 1=40; > > > > > 3=40; > > > > > 4=40; > > > > > 5=40; > > > > 6=40; > > > > > 7=40; > > > > > 9=40; > > > > > 10=40; > > > > > 11=40; > > > > > 14=40; > > > > > 15=40; > > > > > 16=40; > > > > > 17=40; > > > > > < 18=40; 2 H12 ðtÞ ¼ 20=40; > > > 21=40; > > > > > 22=40; > > > > > 23=40; > > > > > 24=40; > > > > > 25=40; > > > > > 27=40; > > > > > 28=40; > > > > > 29=40; > > > > 30=40; > > > > > 34=40; > > > > > 35=40; > > > > > 36=40; > > > > > 38=40; > > > > > 39=40; > > : 1;
t 12; 12\t 15; 15\t 18; 18\t 19; 19\t 20; 20\t 25; 25\t 33; 33\t 34; 34\t 36; 36\t 37; 37\t 40; 40\t 41; 41\t 44; 44\t 46; 46\t 48; 48\t 50; 50\t 53; 53\t 55; 55\t 57; 57\t 59; 59\t 60; 60\t 61; 61\t 62; 62\t 63; 63\t 65; 65\t 67; 67\t 69; 69\t 75; 75\t 80; 80\t 90; t [ 90:
Consequently, the null hypothesis is as follows: H0 : The winter and spring sets of realizations of the ferry technical system operation process conditional sojourn times h112 and h212 come from populations with the same distribution. 1 2 Using the empirical distributions H12 ðtÞ and H12 ðtÞ, we form a common Table 4.4 composed of all their values. In the Table 4.4, the values tk assume all 2k realizations h1k 12 ; k ¼ 1; 2; . . .; 42; and h12 ; k ¼ 1; 2; . . .; 40; of the conditional
4.3 Applications
213
sojourn times h112 and h212 i.e. they represent all discontinuity points of the 1 2 empirical distribution functions H12 ðtÞ and H12 ðtÞ where they have jumps in their 1 2 values H12 ðtk Þ and H12 ðtk Þ respectively. Further, according to (4.68–4.70), from Table 4.4, we get 1 2 ðtk Þ H12 ðtk Þ ffi 0:209; d42 40 ¼ maxH12 tk
and according to (4.70) n¼
42 40 ¼ 20:49: 42 þ 40
Thus, considering (4.66), the realization of the statistics Un defined by (4.59), is pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi un ¼ d42 40 n ¼ 0:209 20:49 ffi 0:946: To verify the hypothesis H0, we use the two-sample Smirnov-Kolmogorov test k at the significance level a ¼ 0:05: From the table of the k distribution for 1 a ¼ 1 0:05 ¼ 0:95, we find the critical value k0 such that according to (4.71) the condition PðUn \k0 Þ ¼ Qðk0 Þ ¼ 1 0:05 ¼ 0:95 is satisfied. This value amounts to k0 ffi 1:36: Since un ffi 0:946\k0 ¼ 1:36; we do not have arguments to reject the null hypothesis H0. Proceeding in an analogous way with the statistical data collected at the remaining operation states, we may test positively the uniformity of the spring sets of the realizations of the ferry technical system operation process conditional sojourn times h1b bþ1 , b ¼ 2; 3; . . .; 17; and h1181 and respectively the winter sets of the realizations of the ferry technical system operation process conditional sojourn times h2b bþ1 , b ¼ 2; 3; . . .; 17; and h2181 . After that, we may join the statistical data collected in spring and winter and create new statistical data sets of realizations of the ferry technical system operation process conditional sojourn times hb bþ1 , b ¼ 1; 2; . . .; 17; and h181 with the following new statistical data: • the ferry technical system operation process experiment time H ¼ 82 days; • the number of ferry technical system operation process realizations nð0Þ ¼ 82;
214
4 Complex Technical System Operation Processes Identification
Table 4.4 The values and 2k t ¼ h1k 12 _ h12 differences of the distribution k 1 2 ðtk Þ and H12 ðtk Þ 12 functions H12 15 at all their discontinuity 18 points 19 20 25 33 34 35 36 37 40 41 43 44 45 46 47 48 50 52 53 55 57 58 59 60 61 62 63 65 67 68 69 71 72 75 78 80 84 90 97 [97
1 H12 ðtk Þ
2 H12 ðtk Þ
1 H ðtk Þ H 2 ðtk Þ 12 12
0 0 1/42 1/42 1/42 2/42 3/42 4/42 4/42 5/42 5/42 6/42 8/42 8/42 9/42 13/42 14/42 15/42 17/42 17/42 18/42 19/42 21/42 22/42 23/42 24/42 26/42 27/42 29/42 30/42 32/42 33/42 34/42 35/42 35/42 36/42 38/42 39/42 40/42 40/42 41/42 41/42 1
0 1/40 3/40 4/40 5/40 6/40 7/40 9/40 10/40 10/40 11/40 14/40 15/40 16/40 16/40 17/40 17/40 18/40 18/40 20/40 21/40 21/40 22/40 23/40 24/40 24/40 25/40 24/40 28/40 29/40 30/40 34/40 35/40 35/40 36/40 36/40 36/40 38/40 38/40 39/40 39/40 1 1
0 0.025 0.051 0.076 0.101 0.102 0.104 0.129 0.155 0.131 0.156 0.207 0.185 0.209 0.186 0.115 0.092 0.093 0.045 0.095 0.096 0.073 0.05 0.051 0.052 0.029 0.006 0.043 0.009 0.011 0.012 0.064 0.065 0.042 0.067 0.043 0.005 0.021 0.002 0.023 0.001 0.024 0
4.3 Applications
215
• the number of realizations hkb bþ1 , b ¼ 1; 2; . . .; 17; and hk181 of the ferry technical system operation process conditional sojourn times hb bþ1 , b ¼ 1; 2; . . .; 17; and h181 at the particular operation states nb bþ1 ¼ 82; n181 ¼ 82; • the realizations hkb bþ1 , b ¼ 1; 2; . . .; 17; and hk181 , k ¼ 1; 2; . . .; 82; of the ferry technical system operation process conditional sojourn times hb bþ1 , b ¼ 1; 2; . . .; 17; and h181 : After these join the statistical data of two experiments, we may go to the ferry technical system operation process identification and proceeding accordingly to the procedures proposed in Sects. 4.2.1–4.2.4, we may improve the accuracy of the results obtained in Sect. 4.3.3.
4.4 Summary The proposed statistical methods of identification of the unknown parameters of the system operation processes allow us for the identification of the models discussed in Chaps. 3 and 6 and also their practical applications in the evaluation, prediction and optimization of reliability, availability and safety of real complex technical systems. This possibility is illustrated in the appliqué sections of these chapters, where on the basis of these methods and the statistical data presented in this chapter, the identification and then the evaluation, prediction and optimization of reliability, availability and safety of the exemplary system, the port oil pipeline system and the ferry technical system are done.
References 1. Barbu V, Limnios N (2006) Empirical estimation for discrete-time semi-Markov processes with applications in reliability. J Nonparametric Stat 18(7–8):7–8 483-498 2. Collet J (1996) Some remarks on rare-event approximation. IEEE Trans Reliab 45:106–108 3. Gamiz ML, Roman Y (2008) Non-parametric estimation of the availability in a general repairable. Reliab Eng Syst Saf 93(8):1188–1196 4. Giudici P, Figini S (2009) Applied data mining for business and industry. Wiley, London 5. Habibullah MS, Lumanpauw E, Kolowrocki K, Soszynska J, Ming NG (2009) A computational tool for general model of operation processes in industrial systems operation processes. Electron J Reliab Risk Anal: Theory Appl 2(4):181–191 6. Helvacioglu S, Insel M (2008) Expert system applications in marine technologies. Ocean Eng 35(11–12):1067–1074 7. Hryniewicz O (1995) Lifetime tests for imprecise data and fuzzy reliability requirements. In: Onisawa T, Kacprzyk J (eds) Reliability and safety analyses under fuzziness. Physica Verlag, Heidelberg, pp 169–182 8. Kołowrocki K (2004) Reliability of large systems. Elsevier, Amsterdam
216
4 Complex Technical System Operation Processes Identification
9. Kołowrocki K, Soszyn´ska J (2008) A general model of industrial systems operation processes related to their environment and infrastructure. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 2(2):223–226 10. Kolowrocki K, Soszynska J (2009) Modeling environment and infrastructure influence on reliability and operation process of port oil transportation system. Electron J Reliab Risk Anal: Theory Appl 2(3):131–142 11. Kolowrocki K, Soszynska J (2009) Safety and risk evaluation of Stena Baltica ferry in variable operation conditions. Electron J Reliab Risk Anal: Theory Appl 2(4):68–180 12. Kołowrocki K, Soszyn´ska J (2009) Statistical identification and prediction of the port oil pipeline system’s operation process and its reliability and risk evaluation. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 3(2):241–250 13. Kołowrocki K, Soszyn´ska J (2009) Methods and algorithms for evaluating unknown parameters of operation processes of complex technical systems. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 3(1–2):211–222 14. Kołowrocki K, Soszyn´ska J (2009) Methods and algorithms for evaluating unknown parameters of components reliability of complex technical systems. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(2):223–230 15. Kołowrocki K, Soszyn´ska J (2009) Statistical identification and prediction of the port oil pipeline system’s operation process and its reliability and risk evaluation. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(2):241–250 16. Kolowrocki K, Soszynska J (2010) Reliability modeling of a port oil transportation system’s operation processes. Int J of Perform Eng 6(1):77–87 17. Kolowrocki K, Soszynska J (2010) Reliability, availability and safety of complex technical systems: modelling–identification–prediction–optimization. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(1):133–158 18. Kolowrocki K, Soszynska J (2010) Testing uniformity of statistical data sets coming from complex systems operation processes. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(1):123–132 19. Limnios N, Oprisan G (2005) Semi-Markov processes and reliability. Birkhauser, Boston 20. Limnios N, Ouhbi B, Sadek A (2005) Empirical estimator of stationary distribution for semiMarkov processes. Commun Stat-Theory Methods 34(4):987–995 12 21. Macci C (2008) Large deviations for empirical estimators of the stationary distribution of a semi-Markov process with finite state space. Commun Stat-Theory Methods 37(19):3077– 3089 22. Mercier S (2008) Numerical bounds for semi-Markovian quantities and application to reliability. Methodol Comput Appl Probab 10(2):179–198 23. Rice JA (2007) Mathematical statistics and data analysis. Duxbury. Thomson Brooks/Cole. University of California, Berkeley 24. Soszyn´ska J (2007) Systems reliability analysis in variable operation conditions. PhD thesis, Gdynia Maritime University-System Research Institute Warsaw (in Polish) 25. Soszyn´ska J, Kołowrocki K, Blokus-Roszkowska A, Guze S (2010) Identification of complex technical system components safety models. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(2):399–496 26. Vercellis S (2009) Data mining and optimization for decision making. Wiley, Indianapolis 27. Wilson AG, Graves TL, Hamada MS et al (2006) Advances in data combination, analysis and collection for system reliability assessment. Stat Sci 21(4):514–531
Chapter 5
Complex Technical System Components Reliability and Safety Identification
5.1 Introduction The general joint models linking the multistate systems’ reliability and safety models considered in Chap. 1 with the models of their operation processes proposed in Chap. 2, giving the possibility of the reliability and safety analysis of complex technical systems at the variable operating conditions, are constructed in Chap. 3. To be able to apply these general models practically in the evaluation and prediction of the reliability and safety of real complex technical systems it is necessary to elaborate the statistical methods concerned with determining unknown parameters of the proposed models [2–6, 8–13, 19–22, 24]. Particularly, concerning the multistate systems’ safety and reliability models, the unknown parameters of the conditional multistate reliability and safety functions of the system components at the various operation states should be identified [1, 7, 14–17, 25–28]. It is also necessary to have the methods of testing the hypotheses concerned with the conditional multistate reliability functions of the system components at the system various operation states.
5.2 Identification of Complex Technical System Components Reliability and Safety Models We assume, as in Sects. 3.2 and 3.5, that the changes of the system operation process Z(t) have an influence on the system components’ reliability and safety characteristics. We mark the conditional multistate reliability function of the system component while the system is at the operation state zb ; b ¼ 1; 2; . . .; m; by the vector
K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8_5, Springer-Verlag London Limited 2011
217
218
5 Complex Technical System Components Reliability
½Rðt; ÞðbÞ ¼ ½1; ½Rðt; 1ÞðbÞ ; . . .; ½Rðt; zÞðbÞ ;
ð5:1Þ
where ½Rðt; uÞðbÞ ¼ P T ðbÞ ðuÞ [ tjZðtÞ ¼ zb
for t 2 h0; 1Þ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v;
ð5:2Þ
is the conditional reliability function standing the probability that the conditional lifetime T ðbÞ ðuÞ of the system component at the reliability states subset fu; u þ 1; . . .; zg is greater than t, while the system operation process Z(t) is in the operation state zb, b ¼ 1; 2; . . .; m: Further, we assume that the coordinates (5.2) of the vector of the conditional multistate reliability function (5.1) are exponential reliability functions of the form h i ½Rðt; uÞðbÞ ¼ exp ½kðuÞðbÞ t for t 2 h0; 1Þ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v: ð5:3Þ Similarly, we mark the conditional multistate safety function of the system component while the system is at the operation state zb ; b ¼ 1; 2; . . .; m; by the vector h i ½sðt; ÞðbÞ ¼ 1; ½sðt; 1ÞðbÞ ; . . .; ½sðt; zÞðbÞ ; ð5:4Þ where ½sðt; uÞðbÞ ¼ PðT ðbÞ ðuÞ [ tjZðtÞ ¼ zb Þ u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v;
for t 2 h0; 1Þ;
ð5:5Þ
is the conditional safety function standing the probability that the conditional lifetime T ðbÞ ðuÞ of the system component in the safety state subset fu; u þ 1; . . .; zg is greater than t, while the system operation process Z(t) is at the operation state zb ; b ¼ 1; 2; . . .; m: Further, we assume that the coordinates (5.5) of the vector of the conditional multistate safety function (5.4) are exponential safety functions of the form h i ½sðt; uÞðbÞ ¼ exp ½kðuÞðbÞ t for t 2 h0; 1Þ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v: ð5:6Þ The above assumptions mean that the density function of the system component conditional lifetime T ðbÞ ðuÞ in the reliability or safety state subset fu; u þ 1; . . .; zg ; u ¼ 1; 2; . . .; z; at the operation state zb ; b ¼ 1; 2; . . .; m; is exponential of the form ½f ðt; uÞðbÞ ¼ ½kðuÞðbÞ exp½½kðuÞðbÞ t
for t 2 h0; 1Þ;
ð5:7Þ
5.2 Identification of Complex Technical System Components Fig. 5.1 The graph of the conditional density function ½f ðt; uÞðbÞ
219
[ f (t , u )] ( b ) [ (u )] ( b )
0
t
where ½kðuÞðbÞ ; ½kðuÞðbÞ 0; is an unknown intensity of the system component departure from this subset of the reliability or safety states. The exemplary graph of the conditional density function ½f ðt; uÞðbÞ defined by (5.7) is illustrated in Fig. 5.1. The assumptions on the system components’ reliability and safety models expressed respectively in the formulae (5.1–5.3) and (5.4–5.6) are identical. Thus, the methods of estimating the unknown intensities of departures from the reliability state subsets or respectively from the safety state subsets, existing in the formula (5.7) for the density function, are the same. Therefore, in this section we present these methods in detail mainly for the system components’ reliability models and after that we apply them for the identification of the system components’ reliability models and the system components’ safety models as well.
5.2.1 Defining Unknown Parameters of System Components Reliability and Safety Models and Data Collection First, before fixing the subsystems and components of the system that are intended to be observed at the various operation states, we should analyze the system operation process and to fix or to define its following general parameters: • the number of the operation states of the system operation process m, • the operation states of the system operation process z1 ; z2 ; . . .; zv : Next, we should perform the following steps: (i) (ii)
fix the subsystems of the system operating at the particular operation states; fix, describe and mark the components of the subsystems operating at the particular operation states.
220
5 Complex Technical System Components Reliability
To make the estimation of the unknown parameters of the system components conditional multistate reliability or safety functions the experiment delivering the necessary statistical data should be precisely planned. Before the experiment, we should perform the following preliminary steps: (i) (ii)
analyze the processes of reliability or safety states changing of all system components at the particular operation states; fix or define its following general parameters:
• the number of the reliability or safety states of the system components z þ 1, • the reliability or safety states of the system components 0; 1; . . .; z; (iii) fix the possible transitions between the system components reliability or safety states; (iv) fix the set of the unknown parameters of the system components reliability models.
5.2.1.1 Collecting Data Coming from Components Reliability and Safety States Changing Processes To estimate the unknown parameters of the system components’ multistate reliability or safety models, during the experiment, we should collect the necessary statistical data performing the following steps: (i) (ii)
fix the experiment kind subjected to the defined below Cases 1–6; fix and to collect, in Cases 1–6, the following statistical data necessary to evaluate the component unknown intensities of departure from the reliability or safety state subsets:
• the number of experiment posts, • the experiments duration times, • the number of the observed realizations of the component lifetimes up to the first departure from the reliability or safety state subsets, • the realizations of the component lifetimes up to the first departure from the reliability or safety state subsets. The fixed kinds of experiments and the collected statistical data sets are described below. They are only defined in the reliability aspect as their definitions in the safety aspect are analogous. Case 1 The estimation of the component intensity of departure from the reliability state subset on the basis of the realizations of the component lifetimes up to the first departure from the reliability state subset on several experimental posts—Completed investigations, the same observation time on all experimental posts.
5.2 Identification of Complex Technical System Components
t1( b ) (1)
1 2
t 2( b ) (1)
t 3( b ) (1)
3
221
ti(b) (1)
i
t n( b( b) ) (1)
n (b ) 0
(b )
t
Fig. 5.2 The scheme of the realizations of the component lifetimes up to the first departure from the reliability state subset on n(b) observational posts (completed investigations, the same observation time on all experimental posts)
We assume that during the time sðbÞ ; sðbÞ [ 0; we have been observing the realizations of the component lifetime T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the operation state zb ; b ¼ 1; 2; . . .; m, on nðbÞ identical experimental posts. We assume that at the beginning of the experiment all components are new identical components straying at the best reliability state z and during the fixed observation time sðbÞ all components have left the reliability state subset f1; 2; . . .; zg, i.e. all observed components reached the worst reliability state 0 (Fig. 5.2). It means that the number mðbÞ ðuÞ of components that have left the reliability state subset fu; u þ 1; . . .; zg, u ¼ 1; 2; . . .; z; is equal to nðbÞ , i.e. mðbÞ ðuÞ ¼ nðbÞ ; u ¼ 1; 2; . . .; z. We mark by ðbÞ
AðbÞ ðuÞ ¼ fti ðuÞ : i ¼ 1; 2; . . .; mðbÞ ðuÞg; u ¼ 1; 2; . . .; z; ðbÞ
the set of the moments ti ðuÞ; i ¼ 1; 2; . . .; mðbÞ ðuÞ; u ¼ 1; 2; . . .; z; of departures from the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the component on the ith observational post, i.e. the realizations of the component lifetimes ðbÞ Ti ðuÞ to the first departure from the reliability state subset, that are the independent random variables with the exponential distribution defined by the density function (5.1). Case 2 The estimation of the component intensity of departure from the reliability state subset on the basis of the realizations of the component lifetimes up to the first departure from the reliability state subset on several experimental posts— Non-completed investigations, the same observation time on all experimental posts.
222
5 Complex Technical System Components Reliability
t1( b ) (1)
1 2
t 2( b ) (1)
ti(b ) (1)
i
t m( b( )b ) (1)
m (b )
t m( b( )b ) +1 (1)
t n( b( b) ) (1)
n (b ) (b )
0
t
Fig. 5.3 The scheme of the realizations of the component lifetimes up to the first departure from the reliability state subset on n(b) observational posts (non-completed investigations, the same observation time on all experimental posts)
We assume that during the time sðbÞ ; sðbÞ [ 0; we have been observing the realizations of the component lifetimes T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the operation state zb ; b ¼ 1; 2; . . .; m; on nðbÞ identical experimental posts. We assume that at the beginning of the experiment all components are new identical components straying at the best reliability state z and that during the fixed observation time sðbÞ not all components have left the reliability state subset f1; 2; . . .; zg; i.e. mðbÞ ; mðbÞ \nðbÞ ; observed components reached the worst reliability state 0 (Fig. 5.3). It means that the number mðbÞ ðuÞ of components that have left the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is less or equal to nðbÞ ; i:e: ðmðbÞ ðuÞ nðbÞ ; u ¼ 1; 2; . . .; z: We mark by ðbÞ
AðbÞ ðuÞ ¼ fti ðuÞ : i ¼ 1; 2; . . .; mðbÞ ðuÞg; u ¼ 1; 2; . . .; z; ðbÞ
the set of the moments ti ðuÞ; i ¼ 1; 2; . . .; mðbÞ ðuÞ; u ¼ 1; 2; . . .; z; of departures from the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the component on the ith observational post, i.e. the realizations of the component lifetimes ðbÞ Ti ðuÞ to the first departure from the reliability state subset, that are the independent random variables with the exponential distribution defined by the density function (5.1). Case 3 The estimation of the component intensity of departure from the reliability state subset on the basis of the realizations of the component lifetimes up to the first departure from the reliability state subset on several experimental posts— Non-completed investigations, different observation times on particular experimental posts. We assume that we have been observing the realizations of the component lifetimes T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at
5.2 Identification of Complex Technical System Components
1
(b) 1
t1(b) (1)
2
223
(b) 2
t2(b) (1)
t m( b( )b ) (1)
(b)
m
τ
n (b )
(b) m ( b )+1
(b ) m( b )
t m( b( )b ) 1 (1) +
(b) n(b)
t n( b( b) ) (1)
0
t
Fig. 5.4 The scheme of the realizations of the component lifetimes up to the first departure from the reliability state subset on n(b) observational posts (non-completed investigations, different observation times on all experimental posts)
the operation state zb ; b ¼ 1; 2; . . .; m, on nðbÞ identical experimental posts. We assume that the observation times on particular experimental posts are different ðbÞ ðbÞ and we mark by si ; si [ 0; i ¼ 1; 2; . . .; nðbÞ , the observation time respectively on the ith experimental post. We assume that at the beginning of the experiment all components are new identical components straying at the best reliability state z ðbÞ and that during the fixed observation times si not all components have left the reliability state subset f1; 2; . . .; zg, i.e. mðbÞ ; mðbÞ \nðbÞ , observed components reached the worst reliability state 0 (Fig. 5.4). It means that the number mðbÞ ðuÞ of components that have left the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is less or equal to nðbÞ ; i:e: mðbÞ ðuÞ nðbÞ ; u ¼ 1; 2; . . .; z. We mark by ðbÞ
AðbÞ ðuÞ ¼ fti ðuÞ : i ¼ 1; 2; . . .; mðbÞ ðuÞg; u ¼ 1; 2; . . .; z; ðbÞ
the set of the moments ti ðuÞ; i ¼ 1; 2; . . .; mðbÞ ðuÞ; u ¼ 1; 2; . . .; z; of departures from the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the component on the ith observational post, i.e. the realizations of the component lifetimes ðbÞ Ti ðuÞ to the first departure from the reliability state subset, that are the independent random variables with the exponential distribution defined by the density function (5.1). Case 4 The estimation of the component intensity of departure from the reliability state subset on the basis of the realizations of the component simple renewal flow (stream) on one experimental post. We assume that during the time sðbÞ ; sðbÞ [ 0; we have been observing the realizations of the component lifetime T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the operation state zb ; b ¼ 1; 2; . . .; m, on one
224
5 Complex Technical System Components Reliability
0 t1( b ) (1)
t 2( b ) (1)
ti(b ) (1)
t m( b()b ) +1 (1)
(b )
t
Fig. 5.5 The scheme of the realizations of the component simple renewal flow (stream) on one experimental post
experimental post. We assume that at the moment when the component is leaving the reliability state subset f1; 2; . . .; zg, i.e. the observed component reached the worst reliability state 0, it is replaced at once by the same new component straying at the reliability state z (Fig. 5.5). It means that at the beginning all components are new identical components straying at the best reliability state z. We assume that during the fixed observation time m(b) components have left the reliability state subset f1; 2; . . .; zg; i:e: mðbÞ observed components reached the worst reliability state 0. It means that the number mðbÞ ðuÞ of components that have left the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is equal either to m(b) or to mðbÞ þ 1, i.e. mðbÞ ðuÞ ¼ mðbÞ or mðbÞ ðuÞ ¼ mðbÞ þ 1, u ¼ 1; 2; . . .; z. We mark by ðbÞ
AðbÞ ðuÞ ¼ fti ðuÞ : i ¼ 1; 2; . . .; mðbÞ ðuÞg; u ¼ 1; 2; . . .; z; ðbÞ
the set of the moments ti ðuÞ; i ¼ 1; 2; . . .; mðbÞ ðuÞ; u ¼ 1; 2; . . .; z; of departures from the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the component ðbÞ on the ith observational post, i.e. the realizations of the component lifetime Ti ðuÞ to the first departure from the reliability state subset, that are the independent random variables with the exponential distribution defined by the density function (5.1). Case 5 The estimation of the component intensity of departure from the reliability state subset on the basis of the realizations of the component simple renewal flows (streams) on several experimental posts—The same observation time on all experimental posts. ðbÞ
We assume that during the time si ; sðbÞ [ 0; we have been observing the realizations of the component lifetime T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the operation state zb ; b ¼ 1; 2; . . .; m, on n(b) experimental posts. We assume that, at each observation post, at the moment when the component is leaving the reliability state subset f1; 2; . . .; zg, i.e. the observed component reached the worst reliability state 0, it is replaced at once by the same new component straying at the reliability state z (Fig. 5.6). It means that, at each experiment post, at the beginning all components are new identical components straying at the best reliability state z. We assume that, at the jth, j ¼ 1; 2; . . .; nðbÞ ðbÞ experimental post, during the fixed observation time mj components have left the ðbÞ
reliability state subset f1; 2; . . .; zg; i:e: mj
observed components reached the ðbÞ
worst reliability state 0. It means that the number mj ðuÞ of components that have
5.2 Identification of Complex Technical System Components 1 2
[t 2( b) (1)](1)
[t1(b ) (1)](1)
[t1(b ) (1)]( 2 )
n
m1
+1
[t ( b()b ) (1)]( 2) m2 +1
[t i( b ) (1)]( j )
[t 2( b ) (1)]( j )
[t1(b ) (1)]( j )
[t ( b()b ) (1)](1)
[ti(b ) (1)]( 2)
[t 2(b ) (1)]( 2 )
j
[ti(b ) (1)](1)
225
[t ( b()b ) (1)]( j ) m j +1
(b )
0 [t (b) (1)]( n (b) ) 1
[t2(b) (1)]( n
(b) )
[ti(b ) (1)]( n
(b) )
[ t ( b()b ) m
+1 n (b )
(1)] ( n
(b ) )
(b )
t
Fig. 5.6 The scheme of the realizations of the component simple renewal flows (streams) on several experimental posts (the same observation time on all experimental posts)
left the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is equal either to ðbÞ ðbÞ ðbÞ ðbÞ ðbÞ ðbÞ mj or to mj þ 1, i.e. mj ðuÞ ¼ mj or mj ðuÞ ¼ mj þ 1; u ¼ 1; 2; . . .; z We mark by ðbÞ
ðbÞ
ðbÞ
Aj ðuÞ ¼ f½ti ðuÞðjÞ : i ¼ 1; 2; . . .; mj ðuÞg; ðbÞ
u ¼ 1; 2; . . .; z; j ¼ 1; 2; . . .; nðbÞ ; ðbÞ
the sets of the times ½ti ðuÞðjÞ ; i ¼ 1; 2; . . .; mj ðuÞ, to the components departures from the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the jth, j ¼ 1; 2; . . .; nðbÞ , experimental post, i.e. the realizations of the component lifetimes ðbÞ Ti ðuÞ to the first departure from the reliability state subset, that are random variable with the exponential distribution defined by the density function (5.1). Case 6 The estimation of the component intensity of departure from the reliability state subset on the basis of the realizations of the component simple renewal flows (streams) on several experimental posts—Different observation times on experimental posts. We assume that we have been observing the realizations of the component lifetime T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the operation state zb ; b ¼ 1; 2; . . .; m; on nðbÞ experimental posts. We assume that the observation times on particular experimental posts are different and we mark ðbÞ ðbÞ by sj ; sj [ 0; i ¼ 1; 2; . . .; nðbÞ ; the observation time respectively on the ith experimental post. We assume that, at each observation post, at the moment when the component is leaving the reliability state subset f1; 2; . . .; zg, i.e. the observed component reached the worst reliability state 0, it is replaced at once by the same new component straying at the reliability state z (Fig. 5.7). It means that, at each experiment post, at the beginning all components are new identical components straying at the best reliability state z. We assume that, at the jth, j ¼ 1; 2; . . .; nðbÞ , experimental post, during the fixed observation time m(b) j components have left the ðbÞ reliability state subset f1; 2; . . .; zg, i.e. mj observed components reached the
226
5 Complex Technical System Components Reliability
1
... (b)
[t1 (1)](1)
2
.. .
...
(b )
[t 2 (1)](1)
(b) 2
[t ( b()b )+ (1)]( 2 )
(b)
[t 2 (1)]( 2 )
m2
1
...
[t1( b ) (1)]n
(b )
(b)
[t ( b()b ) (1)]( j )
[t ( b()b ) (1)]( j )
( b)
(b )
[t i (1)]( j )
[t 2 (1)]( j )
n (b )
... (b)
(b )
(b )
[t1 (1)] ( j )
...
[t 2( b ) (1)]n
(b)
[t i( b ) (1)]n
(b) 1
+
1
... (b )
[t1 (1)] ( 2 )
j
.. .
[t m( b()b ) 1 (1)]( 2 )
(b )
[t i (1)](1)
mj
...
(b ) j
m j +1
(b)
[t ( b()b ) (1)]n m n( b )
[t m(b()b ) 1 (1)]n n
n( b )
+ (b )
Fig. 5.7 The scheme of the realizations of the component simple renewal flows (streams) on several experimental posts (different observation times on experimental posts)
ðbÞ
worst reliability state 0. It means that the number mj ðuÞ of components that have left the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is equal either to ðbÞ ðbÞ ðbÞ ðbÞ ðbÞ ðbÞ mj or to mj þ 1, i.e. mj ðuÞ ¼ mj or mj ðuÞ ¼ mj þ 1; u ¼ 1; 2; . . .; z. We mark by h iðjÞ ðbÞ ðbÞ ðbÞ ti ðuÞ : i ¼ 1; 2; . . .; mj ðuÞ ; Aj ðuÞ ¼ u ¼ 1; 2; . . .; z; j ¼ 1; 2; . . .; nðbÞ ; h iðjÞ ðbÞ ðbÞ the sets of the times ti ðuÞ ; i ¼ 1; 2; . . .; mj ðuÞ; to the components departures from the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z, at the jth, j ¼ 1; 2; . . .; nðbÞ , experimental post, i.e. the realizations of the component lifetimes ðbÞ Ti ðuÞ to the first departure from the reliability state subset, that are random variable with the exponential distribution defined by the density function (5.1). To make clear the way of data collection, it has to be expressed that the realizations of the component lifetimes to the first departure from the reliability state subset f1; 2; . . .; zg, i.e. the times up to the moment when the component reached the worst reliability state 0 presented in Figs. 5.2–5.7 include the realizations of the component lifetimes to the first departure from the reliability state ðbÞ subsets fu; u þ 1; . . .; zg; u ¼ 2; 3; . . .; z:, For instance, the realization t1 ð1Þ preðbÞ
ðbÞ
ðbÞ
sented in Fig. 5.2 includes the realizations t1 ðzÞ; t1 ðz 1Þ; . . .; t1 ð2Þ, which because of ageing, fulfill the following condition ðbÞ
ðbÞ
ðbÞ
ðbÞ
t1 ðzÞ t1 ðz 1Þ t1 ð2Þ t1 ð1Þ illustrated in Fig. 5.8.
5.2 Identification of Complex Technical System Components
227
u z z-1
2 1 0
t1( b) ( z )
0
t1( b ) ( z − 1)
t1(b ) (2)
t 1( b ) (1 )
t
Fig. 5.8 The relationship between the realizations of the component lifetimes in the reliability state subsets
5.2.1.2 Collecting Reliability and Safety Data Coming from Experts In this case, as we do not have statistical data coming from the system components’ reliability or safety states’ changing processes, we try to get approximate data on components’ reliability and safety from experts. To make the estimation of the unknown parameters of the system components’ multistate reliability or safety models on the basis of the expert opinions the approximate values ½^ lðuÞðbÞ ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m;
of the mean values ½lðuÞðbÞ ¼ E½T ðbÞ ðuÞ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m;
of the system component lifetimes T ðbÞ ðuÞ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m; in the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; while the system is operating at the operation state zb ; b ¼ 1; 2; . . .; m should be fixed.
5.2.2 Estimating Parameters of Conditional Multistate Exponential Reliability and Safety Functions of System Components Prior to the estimation of the system components unknown intensities of departures from the reliability or safety states subsets, we have to decide whether we have statistical data coming from components reliability or safety states changing processes or from experts and after that we have to select one of two procedures of estimating presented below, respectively in Sects. 5.2.2.1 and 5.2.2.2.
228
5 Complex Technical System Components Reliability
5.2.2.1 Evaluating System Components Intensities of Departures from Reliability and Safety State Subsets on Basis of Data Coming from Components Reliability and Safety States Changing Processes On the basis of statistical data described in Sect. 5.2.1.1, we can find the estimate ½^kðuÞðbÞ of the system component unknown intensity ½kðuÞðbÞ of departure from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z: The formulae for all kinds of experiments, considered in Sect. 5.2.1.1 are presented below. Case 1 The maximum likelihood evaluation of the unknown component intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is nðbÞ ½^ kðuÞðbÞ ¼ PnðbÞ ðbÞ ; i¼1 ti ðuÞ
u ¼ 1; 2; . . .; z:
ð5:8Þ
Case 2 The maximum likelihood evaluation of the unknown component intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is ½^kðuÞðbÞ ¼ PmðbÞ ðuÞ i¼1
mðbÞ ðuÞ ðbÞ
ti ðuÞ þ sðbÞ ½nðbÞ mðbÞ ðuÞ
;
u ¼ 1; 2; . . .; z:
ð5:9Þ
Assuming the observation time sðbÞ as the moment of departure from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the components that have not left this reliability or safety state subset we get so-called pessimistic evaluation of the intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the form ½^kðuÞðbÞ ¼ PmðbÞ ðuÞ i¼1
nðbÞ ðbÞ
ti ðuÞ þ sðbÞ ½nðbÞ mðbÞ ðuÞ
;
u ¼ 1; 2; . . .; z:
ð5:10Þ
Case 3 The maximum likelihood evaluation of the unknown component intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is ½^kðuÞðbÞ ¼ PmðbÞ ðuÞ i¼1
mðbÞ ðuÞ ; P ðbÞ ðbÞ ðbÞ ti ðuÞ þ ni¼mðbÞ ðuÞþ1 si ðbÞ
u ¼ 1; 2; . . .; z:
ð5:11Þ
Assuming the observation times si ; i ¼ mðbÞ ðuÞ; mðbÞ ðuÞ þ 1; . . .; nðbÞ ; as the moment of departure from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the components that have not left this reliability or
5.2 Identification of Complex Technical System Components
229
safety state subset we get a so-called pessimistic evaluation of the intensity of departure kðbÞ ðuÞ from the reliability or safety state subset of the form ½^kðuÞðbÞ ¼ PmðbÞ ðuÞ i¼1
nðbÞ ; P ðbÞ ðbÞ ðbÞ ti ðuÞ þ ni¼mðbÞ ðuÞþ1 si
u ¼ 1; 2; . . .; z:
ð5:12Þ
Case 4 The maximum likelihood evaluation of the unknown component intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is mðbÞ ðuÞ ; ½^kðuÞðbÞ ¼ PmðbÞ ðuÞ ðbÞ ðbÞ ðuÞ t ðuÞ þ d i i¼1
u ¼ 1; 2; . . .; z;
ð5:13Þ
where dðbÞ ðuÞ ¼
8 > <
s
> :
0;
ðbÞ
mðbÞ PðuÞ i¼1
ðbÞ
ti ð1Þ; if mðbÞ ðuÞ ¼ mðbÞ
ð5:14Þ
if mðbÞ ðuÞ ¼ mðbÞ þ 1; u ¼ 1; 2; . . .; z:
In the case when mðbÞ ðuÞ ¼ mðbÞ ; u ¼ 1; 2; . . .; z; after assuming the observation time sðbÞ as the moment of departure from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the last component that has not left this reliability or safety state subset, we get a so-called pessimistic evaluation of the intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the form mðbÞ þ 1 ; ½^kðuÞðbÞ ¼ PmðbÞ ðuÞ ðbÞ ti ðuÞ þ d ðbÞ ðuÞ i¼1
u ¼ 1; 2; . . .; z:
ð5:15Þ
Case 5 The maximum likelihood evaluation of the unknown component intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is either ½^kðuÞðbÞ ¼
PnðbÞ PnðbÞ
j¼1
where
PmðbÞ ðuÞ h j
i¼1
ðbÞ
mj ðuÞ ; i j P ðbÞ ðbÞ ðbÞ ti ðuÞ þ nj¼1 dj ðuÞ j¼1
u ¼ 1; 2; . . .; z;
ð5:16Þ
230
5 Complex Technical System Components Reliability
ðbÞ dj ðuÞ
¼
8 > > <
ðbÞ
ðbÞ
s
> > : 0;
mj ðuÞh
P
i¼1
ðbÞ
ti ð1Þ
iðjÞ
ðbÞ
ðbÞ
ðbÞ
ðbÞ
; if mj ðuÞ ¼ mj
if mj ðuÞ ¼ mj þ 1; u ¼ 1; 2; . . .; z; ð5:17Þ
for j ¼ 1; 2; . . .; nðbÞ : ðbÞ ðbÞ In the case if there exist j, j 2 f1; 2; . . .; nðbÞ g; such that mj ðuÞ ¼ mj ; u ¼ 1; 2; . . .; z; assuming the observation time s(b) as the moment of departures from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the last components on all experimental posts that have not left this reliability or safety state subset we get a so-called pessimistic evaluation of the intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the form PnðbÞ
ðbÞ
mj þ nðbÞ ; h i j P ðbÞ PnðbÞ PmðbÞ ðbÞ ðbÞ n j ðuÞ ti ðuÞ þ j¼1 dj ðuÞ i¼1 j¼1 j¼1
½^kðuÞðbÞ ¼
u ¼ 1; 2; . . .; z:
ð5:18Þ
Case 6 The maximum likelihood evaluation of the unknown component intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; is either ½^kðuÞ
ðbÞ
PnðbÞ
¼
ðbÞ
mj ðuÞ ; h ðbÞ PnðbÞ Pmj ðuÞ ðbÞ i j PnðbÞ ðbÞ ti ðuÞ þ j¼1 dj ðuÞ j¼1 i¼1 j¼1
u ¼ 1; 2; . . .; z;
ð5:19Þ
where
ðbÞ dj ðuÞ
¼
8 > > <
ðbÞ
ðbÞ sj
> > : 0;
mj ðuÞh
P
i¼1
ðbÞ
ti ð1Þ
iðjÞ
ðbÞ
ðbÞ
ðbÞ
ðbÞ
; if mj ðuÞ ¼ mj
if mj ðuÞ ¼ mj þ 1; u ¼ 1; 2; . . .; z; ð5:20Þ
for j ¼ 1; 2; . . .; nðbÞ : ðbÞ
ðbÞ
In the case when there exist j, j 2 f1; 2; . . .; nðbÞ g; such that mj ðuÞ ¼ mj ; u ¼ ðbÞ
1; 2; . . .; z; assuming the observation times sj ; j ¼ 1; 2; . . .; nðbÞ ; as the moments of departures from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the last components on experimental posts that have not left this reliability or safety state subset, we get a so-called pessimistic evaluation of the intensity of departure ½kðuÞðbÞ from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; of the form
5.2 Identification of Complex Technical System Components
ðbÞ
½^kðuÞ
PnðbÞ
¼
231
ðbÞ
mj þ nðbÞ ; h i j P ðbÞ PnðbÞ PmðbÞ ðbÞ n ðbÞ j ðuÞ d ti ðuÞ þ j¼1 j ðuÞ j¼1 i¼1 j¼1
u ¼ 1; 2; . . .; z:
ð5:21Þ
5.2.2.2 Evaluating System Components Intensities of Departure from Reliability and Safety State Subsets on the Basis of Data Coming from Experts On the basis of the approximate values ½^ lðuÞðbÞ ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m;
of the mean values ½lðuÞðbÞ ¼ E½TðuÞðbÞ ;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m;
of the system component lifetimes ½TðuÞðbÞ ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m; in the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; while the system is operating at the operation state zb ; b ¼ 1; 2; . . .; m coming from experts and described in Sect. 5.2.1.2, we want to estimate the values ½^kðuÞðbÞ of the components unknown intensities ½kðuÞðbÞ of departure from the reliability or safety state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; while the system is operating at the operation state zb ; b ¼ 1; 2; . . .; m The formula for all system components is given by the following approximate equation ½kðuÞðbÞ ffi ½^kðuÞðbÞ ¼
1 ½^ lðuÞðbÞ
;
u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m:
ð5:22Þ
5.2.3 Identification of Conditional Multistate Exponential Reliability and Safety Functions of System Components Prior to the identification of the system components conditional multistate exponential reliability or safety functions, we have to decide whether we have statistical data coming from components reliability states changing processes or from experts and after that we have to select one of two procedures of estimation presented below, in Sects. 5.2.3.1 and 5.2.3.2, respectively.
232
5 Complex Technical System Components Reliability
5.2.3.1 Identifying System Components Conditional Multistate Exponential Reliability and Safety Functions on the Basis of Data Coming from Components Reliability and Safety States Changing Processes Having the evaluation of the unknown intensity ½kðuÞðbÞ of the component departure from the reliability or safety state subset, performed according to one of the formulae given in Sect. 5.2.2.1 on the basis of the realizations of the system component lifetime T ðbÞ ðuÞ in the reliability or safety state subset fu; u þ 1; . . .; zg at the operation state zb, described in Sect. 5.2.1.1, we may formulate and then verify the non-parametric hypothesis concerning the exponential form of the coordinates ½Rðt; uÞðbÞ ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v; of the vector of its conditional multistate reliability function ½Rðt; ÞðbÞ or the coordinates ½sðt; uÞðbÞ ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; v; of the vector of its conditional multistate safety function ½sðt; ÞðbÞ : The procedures for the identification of the system components’ reliability functions and safety functions are analogous. Therefore, we present the way of the identification of the reliability functions only. To perform this task in the reliability aspect, it is due to act according to the scheme below: • to formulate the null hypothesis H0, concerned with the form of the component multistate reliability ½Rðt; ÞðbÞ in the following form: H0: The conditional multistate reliability function of the system component ½Rðt; ÞðbÞ ¼ ½1; ½Rðt; 1ÞðbÞ ; . . .; ½Rðt; zÞðbÞ ;
ð5:23Þ
has the exponential reliability function coordinates of the forms ½Rðt; uÞðbÞ ¼ exp½½kðuÞðbÞ t for t 2 h0; 1Þ;
u ¼ 1; 2; . . .; z;
ð5:24Þ
ðbÞ ; b ¼ 1; 2; . . .; mof observed components’ realizations of • to fix the number n the system component conditional lifetime T ðbÞ ð1Þ; b ¼ 1; 2; . . .; m in the reliability state subset f1; 2; . . .; zg; i.e., the number of observed components that reached the worst reliability state 0, according to the formula
nðbÞ ¼
8 ðbÞ n > > < mðbÞ þ 1
in Cases 1; 2; 3 in Case 4
ðbÞ
nP ðbÞ > > : mj þ nðbÞ
in Cases 5; 6;
j¼1
ðbÞ ðuÞ; u ¼ 1; 2; . . .; z; b ¼ 1; 2; . . .; m; of realizations of the • to fix the number m system component conditional lifetime T ðbÞ ðuÞ; b ¼ 1; 2; . . .; m, in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z;
5.2 Identification of Complex Technical System Components
233
8 ðbÞ > in Cases 1; 2; 3; 4 < m ðuÞ ðbÞ nðbÞ P ðuÞ ¼ m ðbÞ > mj ðuÞ in Cases 5; 6; :¼ j¼1
ðbÞ
ðbÞ
ðbÞ
• to fix the realizations t1 ðuÞ; t2 ðuÞ; . . .; tm ðbÞ ðuÞ ðuÞ; u ¼ 1; 2; . . .; z; of the system component conditional lifetime T ðbÞ ðuÞ; b ¼ 1; 2; . . .; m; in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; ðbÞ ðbÞ ðbÞ • to determine the number r ðbÞ ðuÞ of disjoint intervals Ij ðuÞ ¼ hxj ðuÞ; yj ðuÞÞ; ðbÞ
ðbÞ
ðbÞ
j ¼ 1; 2; . . .; rðbÞ ðuÞ, that include the realizations t2 ðuÞ; t1 ðuÞ; . . .; tm ðbÞ ðuÞ ðuÞ; u ¼ 1; 2; . . .; z; of the system component conditional lifetime T ðbÞ ðuÞ in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; according to the formula qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbÞ ðuÞ; r ðbÞ ðuÞ ffi m ðbÞ
ðbÞ
ðbÞ
• to determine the length d ðbÞ ðuÞ of the intervals Ij ðuÞ ¼ hxj ðuÞ; yj ðuÞÞ, j ¼ 1; 2; . . .; r ðbÞ ðuÞ, according to the formula
d ðbÞ ðuÞ ¼
ðbÞ ðuÞ R ; 1
r ðbÞ ðuÞ
where ðbÞ ðuÞ ¼ R
n
max
1im ðbÞ ðuÞ
ðbÞ
o ðbÞ ti ðuÞ
min
n
1 i mðbÞ ðuÞ
ðbÞ
o ðbÞ ti ðuÞ ;
ðbÞ
ðbÞ
ðbÞ
• to determine the ends xj ðuÞ; yj ðuÞ; of the intervals Ij ðuÞ ¼ hxj ðuÞ; yj ðuÞÞ; j ¼ 1; 2; . . .; rðbÞ ðuÞ, according to the formulae n o d ðbÞ ðuÞ ðbÞ ðbÞ min ti ðuÞ ; 0 x1 ðuÞ ¼ max 2 1im ðbÞ ðuÞ ðbÞ
ðbÞ
yj ðuÞ ¼ x1 ðuÞ þ jd ðbÞ ðuÞ; ðbÞ
bÞ
xj ðuÞ ¼ yj1 ðuÞ;
j ¼ 1; 2; . . .; r ðbÞ ðuÞ;
j ¼ 2; 3; . . .; r ðbÞ ðuÞ;
in a way such that ðbÞ
ðbÞ
ðbÞ
ðbÞ
ðbÞ
I1 ðuÞ [ I2 ðuÞ [ . . . [ IrðbÞ ðuÞ ðuÞ ¼ hx1 ðuÞ; yrðbÞ ðuÞ ðuÞÞ;
234
5 Complex Technical System Components Reliability
and ðbÞ
ðbÞ
Ii ðuÞ \ Ij ðuÞ ¼ £
n o i; j 2 1; 2; . . .; rðbÞ ðuÞ
for all i 6¼ j; ðbÞ
ðbÞ
ðbÞ
and the ends of the interval IrðbÞ ðuÞþ1 ðuÞ ¼ hxr ðbÞ ðuÞþ1 ðuÞ; yrðbÞ þ1 ðuÞÞ; including the remaining unknown realizations, assuming ðbÞ
ðbÞ
ðbÞ
xrðbÞ ðuÞþ1 ðuÞ ¼ yr ðbÞ ðuÞ;
yr ðbÞ þ1 ðuÞ ¼ þ1; ðbÞ
• to determine the number of realizations nj ðuÞ in particular intervals ðbÞ
Ij ðuÞ; j ¼ 1; 2; . . .; r ðbÞ ðuÞ þ 1;according to the formula n n oo ðbÞ ðbÞ ðbÞ ðbÞ ðuÞ ; j ¼ 1; 2; . . .;r ðbÞ ðuÞ; nj ðuÞ ¼ # i : ti ðuÞ 2 Ij ðuÞ; i 2 1; 2; . . .; m ðbÞ
ðbÞ m ðbÞ ðuÞ nrðbÞ ðuÞþ1 ðuÞ ¼ n where r ðbÞX ðuÞþ1
ðbÞ
nj ðuÞ ¼ nðbÞ ;
j¼1
whereas the symbol # means the number of elements of a set; • to construct the realization of the histogram of the conditional system component lifetime T ðbÞ ðuÞ; b ¼ 1; 2; . . .; m; in the reliability state subset fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; at the system operation state zb ; b ¼ 1; 2; . . .; m; ðbÞ
f ðbÞ ðt; uÞ ¼ nðbÞ
nj ðuÞ nðbÞ
ðbÞ
for t 2 Ij ðuÞ;
ðbÞ
• to join each of the intervals Ij ðuÞ; j ¼ 1; 2; . . .; r ðbÞ ðuÞ that has the number ðbÞ
ðbÞ
nj ðuÞ of realizations \4 either with the neighbor interval Ijþ1 ðuÞ or with the ðbÞ
neighbor interval Ij1 ðuÞ; in such a way that the number of realizations in all ðbÞ
intervals are not \4, and to join the interval IrðbÞ ðuÞþ1 ðuÞ including unknown realizations with its neighboring new interval; • to fix a new number of intervals r ðbÞ ðuÞ; • to determine new intervals
5.2 Identification of Complex Technical System Components ðbÞ IjðbÞ ðuÞ ¼ hxðbÞ yj ðuÞÞ; j ðuÞ;
235
j ¼ 1; 2; . . .; r ðbÞ ðuÞ;
ðbÞ ðbÞ j ðuÞ of realizations in new intervals Ij ðuÞ; j ¼ • to fix the number n 1; 2; . . .; rðbÞ ðuÞ; • to calculate the hypothetical probabilities so that the lifetime T ðbÞ ðuÞ takes ðbÞ values from the interval Ij ðuÞ; under the assumption that the hypothesis H0 is true, i.e. the probabilities
ðbÞ ðbÞ ðbÞ ðbÞ pj ðuÞ ¼ P T ðbÞ ðuÞ 2 Ij ðuÞ ¼ P xj ðuÞ T ðbÞ ðuÞ\yj ðuÞ h iðbÞ h iðbÞ ðbÞ ðbÞ ¼ Rðxj ðuÞ; uÞ Rðyj ðuÞ; uÞ ; j ¼ 1; 2; . . .; r ðbÞ ðuÞ; ðbÞ
ð5:25Þ
ðbÞ
where ½Rðxj ðuÞ; uÞðbÞ and ½Rðyj ðuÞ; uÞðbÞ are the values of the coordinate reliability function ½Rðt; uÞðbÞ of the multistate reliability function defined by (5.23– 5.24) in the null hypothesis H0 ; • to calculate the realization un of the v2 (chi-square)-Pearson’s statistics Un , according to the formula un ¼
rðbÞ ðuÞ X
ð nj ðuÞ nðbÞ pj ðuÞÞ2
j¼1
nðbÞ pj ðuÞ
ðbÞ
ðbÞ
ðbÞ
;
ð5:26Þ
• to assume the significance level a ða ¼ 0:01; a ¼ 0:02; a ¼ 0:05 or a ¼ 0:10Þ of the test; • to fix the number rðbÞ ðuÞ l 1 of degrees of freedom, substituting l = 1; • to read from the Tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a and the number of degrees of freedom r ðuÞ l 1 such that the following equality holds PðUn [ ua Þ ¼ a;
ð5:27Þ
and then to determine the critical domain in the form of the interval ðua ; þ1Þ and the acceptance domain in the form of the interval h0; ua i (see: Fig. 5.9); • to compare the obtained value un of the realization of the statistics Un with that read from the Tables’ critical value ua of the chi-square random variable and to verify the previously formulated null hypothesis H0 in the following way: if the value un does not belong to the critical domain, i.e. when un ua ; we do not reject the hypothesis H0, if the value un belongs to the critical domain, i.e. when un [ ua ; we reject the hypothesis H0.
236
5 Complex Technical System Components Reliability
f
x2
(t )
1−
0
Critical domain
u
t
Fig. 5.9 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
5.2.3.2 Identifying System Components Conditional Multistate Exponential Reliability and Safety Functions on Basis of Data Coming from Experts In this case, as we do not have statistical data from the system components’ reliability or safety states changing processes, we arbitrarily assume the conditional multistate exponential reliability or safety functions with the approximately estimated intensities of departures from the reliability or safety states according to the formula (5.22) given in Sect. 5.2.2.2.
5.3 Applications 5.3.1 Statistical Identification of Exemplary System Components Reliability The considered exemplary system reliability structure changing at the various operation states and its components and their unknown reliability parameters are described in Sects. 2.3.1 and 3.6.1.
5.3.1.1 Defining Parameters of Exemplary System Components Reliability Models and Data Collection At all the system operation process states zb ; b ¼ 1; 2; 3; 4; defined in Sect. 2.3.1, we distinguish the following four reliability states 0, 1, 2, 3, (z = 3) of the system and its components, defined in Sect. 1.4.1. Moreover, we fix the possible transitions between the components’ reliability states only from better to worse. From the above, the subsystems St ; t ¼ 1; 2; are composed of four-state, i.e. ðtÞ z = 3, components Eij ; t ¼ 1; 2; with the conditional four-state reliability functions given by the vector
5.3 Applications
237
h h iðbÞ iðbÞ h iðbÞ h iðbÞ ðtÞ ðvÞ ðtÞ ðtÞ ; Rij ðt; Þ ¼ 1 ; Rij ðt; 1Þ ; Rij ðt; 2Þ ; Rij ðt; 3Þ
b ¼ 1; 2; 3; 4; ð5:28Þ
with the exponential co-ordinates h h h iðbÞ iðbÞ h iðbÞ iðbÞ ðtÞ ðtÞ ðtÞ ðtÞ Rij ðt; 1Þ ¼ exp kij ð1Þ ; Rij ðt; 2Þ ¼ exp kij ð2Þ ; h iðbÞ h iðbÞ ðtÞ ðtÞ ; ð5:29Þ Rij ðt; 3Þ ¼ exp kij ð3Þ different at various operation states zb ; b ¼ 1; 2; 3; 4; and with the intensities of departure from the reliability state subsets f1; 2; 3g; f2; 3g; f3g; respectively h
ðtÞ
kij ð1Þ
iðbÞ h iðbÞ h iðbÞ ðtÞ ðtÞ ; kij ð1Þ ; kij ð3Þ ; b ¼ 1; 2; 3; 4:
5.3.1.2 Collecting Data Coming from Exemplary System Components Reliability States Changing Processes We arbitrarily suppose that we have at disposal data collected from the exemplary system components’ reliability states changing processes due to Case 2 described in Sect. 5.2.1.1. Namely, we have at disposal the following data for particular ðtÞ components Eij ; t ¼ 1; 2; of the exemplary system: ðbÞ
• the number of identical experiment posts nðbÞ ¼ nij , ðbÞ
• the observation times sðbÞ ¼ sij ; ðbÞ
• the numbers mðbÞ ðuÞ ¼ mij ðuÞ of components that have left the reliability state subsets fu; u þ 1; . . .; 3g; u ¼ 1; 2; 3; ðbÞ ðbÞ • the sets AðbÞ ðuÞ ¼ Aij ðuÞ ¼ fti ðuÞ : i ¼ 1; 2; . . .; mðbÞ ðuÞg of realizations ðbÞ
ðbÞ
ðbÞ
ti ðuÞ ¼ tij ðuÞ of the component lifetimes Tij ðuÞ in the reliability state subsets fu; u þ 1; . . .; 3g; u ¼ 1; 2; 3; at the operation state zb ; b ¼ 1; 2; 3; 4. ð1Þ
For instance, we suppose that the collected data for the component E11 of the subsystem S1 at the operation state z1 are as follows: nð1Þ ¼ 40;
sð1Þ ¼ 2600; mð1Þ ð1Þ ¼ 32;
Að1Þ ð1Þ ¼ f30; 44; 209; 240; 263; 265; 280; 285; 288; 289; 289; 302; 307; 350; 381; 400; 430; 441; 452; 490; 490; 790; 837; 852; 856; 869; 1176; 1191; 1253; 1697; 1700; 2454g; ð5:30Þ
238
5 Complex Technical System Components Reliability
u 3
2
1
0
t1(1) (3) = 20 t1(1) ( 2 ) = t1(1) (1) = 30
0
t
u 3
2
1
0
t 2(1) (3) = 27 t 2(1) ( 2) = 37 t2(1) (1) = 44
0
ð1Þ
ð1Þ
t ð1Þ
ð1Þ
Fig. 5.10 The realizations of the components E11 lifetimes T11 ð1Þ; T11 ð2Þ and T11 ð3Þ in the reliability state subsets f1; 2; 3g; f2; 3g; and f3g
nð1Þ ¼ 40;
sð1Þ ¼ 2600; mð1Þ ð2Þ ¼ 32;
Að1Þ ð2Þ ¼ f30; 37; 37; 60; 63; 65; 69; 69; 80; 85; 88; 302; 307; 350; 352; 381; 400; 430; 441; 462; 470; 490; 637; 652; 656; 669; 776; 891; 1053; 1597; 1600; 2254g ð5:31Þ nð1Þ ¼ 40;
sð1Þ ¼ 2600; mð1Þ ð3Þ ¼ 32;
Að1Þ ð3Þ ¼ f20; 27; 37; 60; 63; 65; 69; 69; 80; 85; 88; 302; 307; 350; 352; 381; 400; 430; 441; 462; 470; 490; 637; 652; 656; 669; 776; 891; 1053; 1597; 1600; 2054g: ð5:32Þ ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
The first realizations t1 ð1Þ ¼ 30; t1 ð2Þ ¼ 30; t1 ð3Þ ¼ 20 and the second ð1Þ
ð1Þ
realizations t2 ð1Þ ¼ 44; t2 ð2Þ ¼ 37; t2 ð3Þ ¼ 27 of the component E11 lifetimes ð1Þ T11 ð1Þ;
ð1Þ T11 ð2Þ
taken from the
ð1Þ and T11 ð3Þ in the reliability state subsets f1; 2; 3g; f2; 3g; and ð1Þ ð1Þ ð1Þ sets A1 ð1Þ; A1 ð2Þ and A1 ð3Þ and are presented in Fig. 5.10.
f3g
5.3 Applications
239
5.3.1.3 Evaluating Exemplary System Components Intensities of Departures from Reliability State Subsets on Basis of Data Coming from Components Reliability States Changing Processes As by the arbitrary assumption, there are data collected from the exemplary system components’ reliability states changing processes, their reliability models identification using the methods of Sect. 5.2.2.1 is possible. To identify the intensities of departures from the reliability state subsets, we can use statistical data included in Sect. 5.3.1.2 and formula (5.9) in order to find the ðtÞ ðtÞ ðtÞ approximate values ½^ kij ð1ÞðbÞ ; ½^ kij ð2ÞðbÞ and ½^ kij ð3ÞðbÞ of the subsystems St ; t ¼ ðtÞ
ðtÞ
ðtÞ
1; 2; components unknown intensities ½kij ð1ÞðbÞ ; ½kij ð2ÞðbÞ and ½kij ð3ÞðbÞ of departure respectively from the reliability states subsets f1; 2; 3g; f2; 3g; f3g, while the system is operating at the operation state zb ; b ¼ 1; 2; 3; 4; and we can use the formula (5.10) to get their pessimistic evaluations. ð1Þ ð1Þ To illustrate this procedure, we find the evaluations ½^k11 ð1Þð1Þ ; ½^k11 ð2Þð1Þ ð1Þ ð1Þ ð1Þ ð1Þ and ½^k ð3Þð1Þ of the intensities ½k ð1Þð1Þ ; ½k ð2Þð1Þ and ½k ð3Þð1Þ of depar11
11
11
11
tures respectively from the reliability state subsets f1; 2; 3g; f2; 3g; and f3g of the ð1Þ component E11 of the subsystem S1 , while the system is operating in the operation state z1 : We proceed as follows: • from data (5.30), we have nð1Þ ¼ 40; ð1Þ mX ð1Þ
sð1Þ ¼ 2600; mð1Þ ð1Þ ¼ 32;
ð1Þ
t1 ð1Þ ¼ 30 þ 44 þ þ 1700 þ 2454 ¼ 20200;
i¼1 ð1Þ
ð1Þ
then, according to (5.9), the evaluations ½^ k11 ð1Þð1Þ of the intensity ½k11 ð1Þð1Þ of departure from the reliability state subset f1; 2; 3g is h
^ k11 ð1Þ
ið1Þ
¼ Pmð1Þ
i¼1
¼
mð1Þ ð1Þ ð1Þ ð1Þ ti ð1Þ
þ sð1Þ ½nð1Þ mð1Þ ð1Þ
32 ffi 0:0008 20200 þ 2600½40 32
ð5:33Þ
and according to (5.10), its pessimistic evaluation is ½^k11 ð1Þð1Þ ¼ Pmð1Þ ð1Þ i¼1
ffi 0:0010:
nð1Þ ð1Þ ti ð1Þ
þ
sð1Þ ½nð1Þ
mð1Þ ð1Þ
¼
40 20200 þ 2600½40 32
240
5 Complex Technical System Components Reliability
• from data (5.31), we have
nð1Þ ¼ 40; ð1Þ mX ð2Þ
sð1Þ ¼ 2600; mð1Þ ð2Þ ¼ 32;
ð1Þ
t1 ð2Þ ¼ 30 þ 37 þ þ 1600 þ 2254 ¼ 15853;
i¼1 ð1Þ
ð1Þ
then, according to (5.9), the evaluations ½^ k11 ð2Þð1Þ of the intensity ½k11 ð2Þð1Þ of departure from the reliability state subset f2; 3g is h
^ k11 ð2Þ
ið1Þ
¼ Pmð1Þ ð2Þ i¼1
¼
mð1Þ ð2Þ ð1Þ ti ð2Þ
þ sð1Þ ½nð1Þ mð1Þ ð2Þ
32 ffi 0:0009 15853 þ 2600½40 32
ð5:34Þ
and according to (5.10), its pessimistic evaluation is ½^k11 ð2Þð1Þ ¼ Pmð1Þ ð2Þ i¼1
nð1Þ ð1Þ ti ð2Þ
þ
sð1Þ ½nð1Þ
mð1Þ ð2Þ
¼
40 15853 þ 2600½40 32
ffi 0:0011:
• from data (5.32), we have
nð1Þ ¼ 40; ð1Þ mX ð3Þ
sð1Þ ¼ 2600; mð1Þ ð3Þ ¼ 32;
ð1Þ
t1 ð3Þ ¼ 20 þ 27 þ þ 1600 þ 2054 ¼ 15633;
i¼1 ð1Þ
ð1Þ
then, according to (5.9), the evaluations ½^ k11 ð3Þð1Þ of the intensity ½k11 ð3Þð1Þ of departure from the reliability state subset f3g is h
^ k11 ð3Þ
ið1Þ
¼ Pmð1Þ ð3Þ
mð1Þ ð3Þ ð1Þ
ti ð3Þ þ sð1Þ ½nð1Þ mð1Þ ð3Þ 32 ¼ ffi 0:0009 15633 þ 2600½40 32 i¼1
and according to (5.10), its pessimistic evaluation is
ð5:35Þ
5.3 Applications
h
^k11 ð3Þ
ið1Þ
241
¼ Pmð1Þ ð3Þ i¼1
mð1Þ ð3Þ ð1Þ ti ð3Þ
þ
sð1Þ ½nð1Þ
mð1Þ ð3Þ
¼
40 15633 þ 2600½40 32
ffi 0:0011: ð1Þ
The evaluations of the intensities of departure ½k21 ðuÞð1Þ ; u ¼ 1; 2; 3; of the ð1Þ
component E21 are the same as that determined by (5.33–5.35) as this component ð1Þ
is identical with the component E11 : In this way, we may obtain the evaluations of the unknown intensities of departure for all the remaining system components. Substituting the evaluations of the intensities of departures respectively into the formulae (5.29), we get the exponential coordinates of the exemplary system components’ reliability functions (5.28) which after testing, partly presented in Sect. 5.3.1.3, are used in Sect. 3.6.1 of Chap. 3 for evaluation and prediction of this system reliability.
5.3.1.4 Identifying Exemplary System Components Conditional Multistate Exponential Reliability Functions on the Basis of Data Coming from System Components Reliability States Changing Processes As by the arbitrary assumption, there are data collected from the system components reliability states changing processes, it is possible to verify the hypotheses on the exponential forms of the system components’ conditional reliability functions. To this end, we use the procedure given in Sect. 5.2.3.1. Applying this procedure and using the statistical data from Sect. 5.3.1.2 and the results from Sect. 5.3.1.3, we may verify the hypotheses on the conditional exponential four-state exemplary system components’ reliability functions ðtÞ ½Rij ðt; ÞðbÞ ; t ¼ 1; 2; b ¼ 1; 2; 3; 4; at the particular operation states zb ; b ¼ 1; 2; 3; 4: To do this, we need a sufficient number of realizations of the system components lifetime in the reliability state subsets [2, 4, 18, 23, 24], namely the sets of their realizations should contain at least 30 realizations coming from the experiment. This condition is satisfied for the statistical data we have at disposal which are partly presented in Sect. 5.3.1.2. To make the procedure familiar to the reader, we perform it for the conditional reliability function of the subsystem S1 ð1Þ component E11 at the system operation state z1. Considering the evaluated values of the unknown intensities of the component ð1Þ E11 departure from the reliability state subsets given by (5.33–5.35), we formulate the null hypothesis H0 and the alternative hypothesis HA, concerned with the form of its multistate reliability ½Rðt; Þð1Þ in the following form: ð1Þ H0: The conditional multistate reliability function of the system component E11 at the operation state z1
242
5 Complex Technical System Components Reliability
h
ð1Þ
R11 ðt; Þ
ið1Þ
h ið1Þ h ið1Þ h ið1Þ ð1Þ ð1Þ ð1Þ ; ¼ 1; R11 ðt; 1Þ ; R11 ðt; 2Þ ; R11 ðt; 3Þ
has the exponential reliability function coordinates of the form h ið1Þ h ið1Þ ð1Þ ð1Þ R11 ðt; 1Þ ¼ exp½0:0008t; R11 ðt; 2Þ ¼ exp½0:0009t; ð1Þ
½R11 ðt; 3Þð1Þ ¼ exp½0:0009t for t 2 h0; 1Þ:
ð5:36Þ
To verify the hypothesis concerning the exponential form of the coordinate 1Þð1Þ defined by (5.36), it is due to act according to the scheme below:
ð1Þ ½R11 ðt;
• we fix the number of observed components and the number of realizations of the ð1Þ exemplary system component conditional lifetime T11 ð1Þ in the reliability state subset {1, 2, 3} which according to (5.30) are nð1Þ ¼ nð1Þ ¼ 40; m ð1Þ ¼ mð1Þ ð1Þ ¼ 32; ð1Þ
• we fix the realizations ti ð1Þ; i ¼ 1; 2; . . .; 32; of the exemplary system comð1Þ
ponent conditional lifetime T11 ð1Þ in the reliability state subset f1; 2; 3g that are given by (5.30), ð1Þ ð1Þ • we determine the number r ð1Þ ð1Þ of the disjoint intervals Ij ð1Þ ¼ hxj ð1Þ; ð1Þ
yj ð1ÞÞ j ¼ 1; 2; . . .; r ð1Þ ð1Þ,
which
include
the
realizations
ð1Þ
ti ð1Þ; i ¼
ð1Þ
1; 2; . . .; 32; of the system component conditional lifetime T11 ð1Þ in the reliability state subset f1; 2; 3g rð1Þ ð1Þ ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ð1Þ ð1Þ ¼ 32 ffi 6; m ð1Þ
ð1Þ
ð1Þ
• we determine the length d ð1Þ ð1Þ of the intervals Ij ð1Þ ¼ hxj ð1Þ; yj ð1ÞÞ; j ¼ 1; 2; . . .; 6; which after considering
ð1Þ ð1Þ ¼ max ftið1Þ ð1Þg min ftið1Þ ð1Þg ¼ 2454 30 ¼ 2424; R 1 i 32
1 i 32
is dð1Þ ð1Þ ¼
ð1Þ
2424 ffi 485; 61
ð1Þ
ð1Þ
ð1Þ
• we determine the ends xj ð1Þ; yj ð1Þ; of the intervals Ij ð1Þ ¼ hxj ð1Þ; ð1Þ
yj ð1ÞÞ; j ¼ 1; 2; . . .; 6; which after considering
5.3 Applications
243
ð1Þ
min fti ð1Þg
1 i 32
d ð1Þ ¼ 30 242:5 ¼ 212:5; 2
are ð1Þ
ð1Þ
ð1Þ
x1 ð1Þ ¼ maxf212:5; 0g ¼ 0; y1 ð1Þ ¼ x1 ð1Þ þ 1d ð1Þ ð1Þ ¼ 0 þ 1 485 ¼ 485; ð1Þ
ð1Þ
x2 ð1Þ ¼ y1 ð1Þ ¼ 485; ð1Þ
ð1Þ
x3 ð1Þ ¼ y2 ð1Þ ¼ 970;
ð1Þ
ð1Þ
y2 ð1Þ ¼ x1 ð1Þ þ 2dð1Þ ð1Þ ¼ 0 þ 2 485 ¼ 970; ð1Þ
ð1Þ
y3 ð1Þ ¼ x1 ð1Þ þ 3dð1Þ ð1Þ ¼ 0 þ 3 485 ¼ 1455;
ð1Þ
ð1Þ
y4 ð1Þ ¼ x1 ð1Þ þ 4dð1Þ ð1Þ ¼ 0 þ 4 485 ¼ 1940;
ð1Þ
ð1Þ
y5 ð1Þ ¼ x1 ð1Þ þ 5dð1Þ ð1Þ ¼ 0 þ 5 485 ¼ 2425;
ð1Þ
ð1Þ
y6 ð1Þ ¼ x1 ð1Þ þ 6dð1Þ ð1Þ ¼ 0 þ 6 485 ¼ 2910;
x4 ð1Þ ¼ y3 ð1Þ ¼ 1455; x5 ð1Þ ¼ y4 ð1Þ ¼ 1940; x6 ð1Þ ¼ y5 ð1Þ ¼ 2425;
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
and the ends of the interval I7 ð1Þ ¼ hx7 ð1Þ; y7 ð1ÞÞ, including the remaining unknown realizations ð1Þ
ð1Þ
x7 ð1Þ ¼ y6 ð1Þ ¼ 2910;
ð1Þ
y7 ð1Þ ¼ þ1; ð1Þ
• we determine the numbers of realizations nj ð1Þ in particular intervals ð1Þ
ð1Þ
ð1Þ
Ij ð1Þ ¼ hxj ð1Þ; yj ð1ÞÞ; j ¼ 1; 2; . . .; 7; ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
n1 ð1Þ ¼ 19; n2 ð1Þ ¼ 7; n3 ð1Þ ¼ 3; n4 ð1Þ ¼ 2; n5 ð1Þ ¼ 0; ð1Þ
ð1Þ
n6 ð1Þ ¼ 1; n7 ð1Þ ¼ 40 32 ¼ 8; • we find the realization f40 ðt; 1Þ of the histogram of the exemplary system ð1Þ component conditional lifetime T11 ð1Þ which is presented in Table 5.1, • we join the intervals defined in the realization of the histogram f40 ðt; 1Þ; which ð1Þ have the numbers nj ð1Þ; of realizations \4 into new intervals and the interval ð1Þ
I7 ð1Þwith its new neighboring interval, • we fix the new number of intervals r ð1Þ ð1Þ ¼ 3; • we determine the new intervals I ð1Þ ð1Þ ¼ h0; 485Þ; I ð1Þ ð1Þ ¼ h485; 970Þ; I ð1Þ ð1Þ ¼ h970; þ1Þ; 1 2 3 • we fix the number of realizations in the new intervals ð1Þ
ð1Þ
ð1Þ
n1 ð1Þ ¼ 19; n2 ð1Þ ¼ 7; ~ n3 ð1Þ ¼ 14;
244
5 Complex Technical System Components Reliability ð1Þ
Table 5.1 The realization of the histogram of the exemplary system conditional lifetime T11 ð1Þ Histogram of the conditional lifetime ð1Þ
Ij ð1Þ ð1Þ ð1Þ ¼ hxj ð1Þ; yj ð1ÞÞ ð1Þ nj ð1Þ
f40 ðt; 1Þ ¼
ð1Þ T11 ð1Þ
0– 485
485– 970
970– 1455
1455– 1940
1940– 2425
2425– 2910
2910– þ1
19
7
3
2
0
1
8
3/40
2/40
0/40
1/40
8/40
19/40 7/40
ð1Þ nj ð1Þ=nð1Þ
• we calculate, using (5.25), the hypothetical probabilities that the conditional ð1Þ lifetime T11 ð1Þ takes values from the new intervals ð1Þ ð1Þ ð1Þ ð1Þ p1 ð1Þ ¼ PðT11 ð1Þ 2 I1 ð1ÞÞ ¼ Pð0 T11 ð1Þ\485Þ ð1Þ
ð1Þ
¼ ½R11 ð0; 1Þð1Þ ½R11 ð485; 1Þð1Þ ¼ exp½0:0008 0 exp½0:0008 485Þ ¼ 1 exp½0:388 ¼ 1 0:6784 ¼ 0:3216; ð1Þ ð1Þ ð1Þ ð1Þ p2 ð1Þ ¼ PðT11 ð1Þ 2 I2 ð1ÞÞ ¼ Pð485 T11 ð1Þ\970Þ ð1Þ
ð1Þ
¼ ½R11 ð485; 1Þð1Þ ½R11 ð970; 1Þð1Þ ¼ exp½0:0008 485 exp½0:0008 970Þ ¼ exp½0:388 exp½0:776 ¼ 0:6784 0:4602 ¼ 0:2182; ð1Þ ð1Þ ð1Þ ð1Þ p3 ð1Þ ¼ PðT11 ð1Þ 2 I3 ð1ÞÞ ¼ Pð970 T11 ð1Þh1Þ ð1Þ
ð1Þ
¼ 1 ½p1 ð1Þ þ p2 ð1Þ ¼ 1 ½0:3216 þ 0:2182 ¼ 0:4602; • we calculate, using (5.26), the realization of the v2 (chi-square)-Pearson’s statistics u40 ¼
ð1Þ ð1Þ 3 X ð nj ð1Þ nð1Þ pj ð1ÞÞ2 ð1Þ
j¼1
þ
nð1Þ pj ð1Þ
¼
ð19 40 0:3216Þ2 40 0:3216
ð7 40 0:2182Þ2 ð14 40 0:4602Þ2 þ 40 0:2182 40 0:4602
ffi 2:93 þ 0:34 þ 1:06 ¼ 4:33; • we assume the significance level a = 0.01, • we fix the number of degrees of freedom rð1Þ ð1Þ l 1 ¼ 3 1 1 ¼ 1;
5.3 Applications
245
Fig. 5.11 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
f 2 (t ( x
1−
0
Critical domain
u40 = 4.33 u = 6.63
t
• we read from the Tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a ¼ 0:01 and the number of degrees of freedom r ð1Þ ð1Þ l ¼ 1 ¼ 1; such that, according to (5.27), the following equality holds PðU40 [ ua Þ ¼ a ¼ 0:01 which amounts to ua ¼ 6:63 and determine the critical domain in the form of the interval ð6:63; þ1Þ and the acceptance domain in the form of the interval h0; 6:63i (Fig. 5.11), • we compare the obtained value u40 ¼ 4:33 of the realization of the statistics U40 read from the Tables of the chi-square distribution critical value ua ¼ 6:63 and since the value u40 ¼ 4:33 does not belong to the critical domain, i.e. u40 ¼ 4:33 ua ¼ 6:63; then we do not reject the hypothesis H0 in its part concerned with the coordinate ð1Þ ½R11 ðt; 1Þð1Þ . To verify the hypothesis concerning the exponential form of the coordinate ð1Þ ½R11 ðt;
2Þð1Þ defined by (5.36), it is due to act according to the scheme below:
• we fix the number of observed components and the number of realizations of the ð1Þ exemplary system component conditional lifetime T11 ð2Þ in the reliability state subset {2, 3} which according to (5.31) are
nð1Þ ¼ nð1Þ ¼ 40;
ð1Þ ð2Þ ¼ mð1Þ ð2Þ ¼ 32; m
ð1Þ
• we fix the realizations ti ð2Þ; i ¼ 1; 2; . . .; 32; of the exemplary system comð1Þ
ponent conditional lifetime T11 ð2Þ in the reliability state subset f2; 3g that are given by (5.31),
246
5 Complex Technical System Components Reliability ð1Þ
ð1Þ
• we determine the number rð1Þ ð2Þ of the disjoint intervals Ij ð2Þ ¼ hxj ð2Þ; ð1Þ
ð1Þ
yj ð2ÞÞ; j ¼ 1; 2; . . .; rð1Þ ð2Þ; which include the realizations ti ð2Þ; i ¼ ð1Þ
1; 2; . . .; 32; of the system component conditional lifetime T11 ð2Þ in the reliability states subsets f2; 3g rð1Þ ð2Þ ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ð1Þ ð2Þ ¼ 32 ffi 6; m ð1Þ
ð1Þ
ð1Þ
• we determine the length d ð1Þ ð2Þ of the intervals Ij ð2Þ ¼ hxj ð2Þ; yj ð2ÞÞ; j ¼ 1; 2; . . .; 6; which after considering
ð1Þ ð2Þ ¼ max R
1 i 32
n o n o ð1Þ ð1Þ ti ð2Þ min ti ð2Þ ¼ 2254 30 ¼ 2224; 1 i 32
is d ð1Þ ð2Þ ¼
• we ð1Þ
determine
the
2224 ffi 445; 61
ð1Þ
ð1Þ
xj ð2Þ; yj ð2Þ;
ends
of
the
intervals
ð1Þ
Ij ð2Þ ¼
ð1Þ
hxj ð2Þ; yj ð2ÞÞ; j ¼ 1; 2; . . .; 6; which after considering n o d ð1Þ ð2Þ ð1Þ ti ð2Þ ¼ 30 222:5 ¼ 192:5; 1 i 32 2 min
are ð1Þ
ð1Þ
ð1Þ
x1 ð2Þ ¼ maxf192:5; 0g ¼ 0; y1 ð2Þ ¼ x1 ð2Þ þ 1d ð1Þ ð2Þ ¼ 0 þ 1 445 ¼ 445 ð1Þ
ð1Þ
x2 ð2Þ ¼ y1 ð2Þ ¼ 445; ð1Þ
ð1Þ
x3 ð2Þ ¼ y2 ð2Þ ¼ 890;
ð1Þ
ð1Þ
y2 ð2Þ ¼ x1 ð2Þ þ 2dð1Þ ð2Þ ¼ 0 þ 2 445 ¼ 890; ð1Þ
ð1Þ
y3 ð2Þ ¼ x1 ð2Þ þ 3dð1Þ ð2Þ ¼ 0 þ 3 445 ¼ 1335;
ð1Þ
ð1Þ
y4 ð2Þ ¼ x1 ð2Þ þ 4dð1Þ ð2Þ ¼ 0 þ 4 445 ¼ 1780;
ð1Þ
ð1Þ
y5 ð2Þ ¼ x1 ð2Þ þ 5dð1Þ ð2Þ ¼ 0 þ 5 445 ¼ 2225;
ð1Þ
ð1Þ
y6 ð2Þ ¼ x1 ð2Þ þ 6dð1Þ ð2Þ ¼ 0 þ 6 445 ¼ 2670;
x4 ð2Þ ¼ y3 ð2Þ ¼ 1335; x5 ð2Þ ¼ y4 ð2Þ ¼ 1780; x6 ð2Þ ¼ y5 ð2Þ ¼ 2225;
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
5.3 Applications
247 ð1Þ
Table 5.2 The realization of the histogram of the exemplary system conditional lifetime T11 ð2Þ ð1Þ
Histogram of the conditional lifetime T11 ð2Þ ð1Þ
Ij ð2Þ ð1Þ
ð1Þ
0– 445
445– 890
890– 1335
1335– 1780
1780– 2225
2225– 2670
2670– þ1
19
8
2
2
0
1
8
19/40
8/40
2/40
2/40
0/40
1/40
8/40
¼ hxj ð2Þ; yj ð2ÞÞ ð1Þ
nj ð2Þ f40 ðt; 2Þ ¼
ð1Þ nj ð2Þ=nð1Þ
ð1Þ
ð1Þ
ð1Þ
and the ends of the interval I7 ð2Þ ¼ hx7 ð2Þ; y7 ð2ÞÞ; including the remaining unknown realizations ð1Þ
ð1Þ
x7 ð2Þ ¼ y6 ð2Þ ¼ 2670;
ð1Þ
y7 ð2Þ ¼ þ1; ð1Þ
• we determine the numbers of realizations mj ð2Þ in particular intervals ð1Þ
ð1Þ
ð1Þ
Ij ð2Þ ¼ hxj ð2Þ; yj ð2ÞÞ; j ¼ 1; 2; . . .; 7; ð1Þ
ð1Þ
ð1Þ
ð1Þ
n1 ð2Þ ¼ 19; n2 ð2Þ ¼ 8; n3 ð2Þ ¼ 2; ð1Þ ð1Þ n6 ð2Þ ¼ 1; n7 ð2Þ ¼ 40 32 ¼ 8;
n4 ð2Þ ¼ 2;
ð1Þ
n5 ð2Þ ¼ 0;
• we find the realization f40 ðt; 2Þ of the histogram of the exemplary system ð1Þ component conditional lifetime T11 ð2Þ presented in Table 5.2, • we join the intervals defined in the realization of the histogram f40 ðt; 2Þ that have ð1Þ ð1Þ the numbers nj ð2Þ; of realizations\4 into new intervals and the interval I7 ð2Þ with its new neighboring interval, • we fix the new number of intervals r ð1Þ ð2Þ ¼ 3; • we determine the new intervals I1ð1Þ ð2Þ ¼ h0; 445Þ;
I2ð1Þ ð2Þ ¼ h445; 890Þ;
I3ð1Þ ð2Þ ¼ h890; þ1Þ;
• we fix the number of realizations in the new intervals ð1Þ
n1 ð2Þ ¼ 19;
ð1Þ
n2 ð2Þ ¼ 8;
ð1Þ
n3 ð2Þ ¼ 13;
• we calculate, using (5.25), the hypothetical probabilities that the conditional ð1Þ lifetime T11 ð2Þ takes values from the new intervals
248
5 Complex Technical System Components Reliability ð1Þ ð1Þ ð1Þ ð1Þ p1 ð2Þ ¼ PðT11 ð2Þ 2 I1 ð2ÞÞ ¼ Pð0 T11 ð2Þ\445Þ ð1Þ
ð1Þ
¼ ½R11 ð0; 2Þð1Þ ½R11 ð445; 2Þð1Þ ¼ exp½0:0009 0 exp½0:0009 445Þ ¼ 1 exp½0:4005 ¼ 1 0:6700 ¼ 0:3300; ð1Þ
ð1Þ
ð1Þ
ð1Þ
p2 ð2Þ ¼ PðT11 ð2Þ 2 I2 ð2ÞÞ ¼ Pð445T11 ð2Þ\890Þ ð1Þ
ð1Þ
¼ ½R11 ð445;2Þð1Þ ½R11 ð890;2Þð1Þ ¼ exp½0:0009 445 exp½0:0009 890Þ ¼ exp½0:4005 exp½0:8010 ¼ 0:6700 0:4489 ¼ 0:2211; ð1Þ ð1Þ ð1Þ ð1Þ p3 ð2Þ ¼ PðT11 ð2Þ 2 I3 ð2ÞÞ ¼ Pð890 T11 ð2Þh0; 1Þ ð1Þ
ð1Þ
¼ 1 ½p1 ð2Þ þ p2 ð2Þ ¼ 1 ½0:3300 þ 0:2211 ¼ 0:4489; • we calculate, using (5.26), the realization of the v2 (chi-square)-Pearson’s statistics
u40 ¼
2 ð1Þ ð1Þ 3 nj ð2Þ nð1Þ pj ð2Þ X ð1Þ
j¼1
þ
nð1Þ pj ð2Þ
¼
ð19 40 0:3300Þ2 ð8 40 0:2211Þ2 þ 40 0:3300 40 0:2211
ð13 40 0:4489Þ2 ffi 2:55 þ 0:08 þ 1:37 ¼ 4:00; 40 0:4489
• we assume the significance level a = 0.01, • we fix the number of degrees of freedom rð1Þ ð2Þ l 1 ¼ 3 1 1 ¼ 1; • we read from the Tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a = 0.01 and the number of degrees of freedom r ð1Þ ð2Þ l ¼ 1 ¼ 1; such that, according to (5.27), the following equality holds PðU40 [ ua Þ ¼ a ¼ 0:01 which amounts to ua ¼ 6:63 and determine the critical domain in the form of the interval ð6:63; þ1Þ and the acceptance domain in the form of the interval h0; 6:63i; (Fig. 5.12),
5.3 Applications
249
Fig. 5.12 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
f 2 (t ( x
1−
0
Critical domain
u40 = 4.00
u = 6.63
t
• we compare the obtained value u40 ¼ 4:00 of the realization of the statistics U40 read from the Tables of the chi-square distribution critical value ua = 6.63 and since the value u40 ¼ 4:00 does not belong to the critical domain, i.e. u40 ¼ 4:00 ua ¼ 6:63; we do not reject the hypothesis H0 in its part concerned with the coordinate ð1Þ ½R11 ðt; 2Þð1Þ . To verify the hypothesis concerning the exponential form of the coordinate ð1Þ ½R11 ðt; 3Þð1Þ , defined by (5.36), it is due to act according to the scheme below: • we fix the number of observed components and the number of realizations of the ð1Þ system component conditional lifetime T11 ð3Þ in the reliability state subset {3} which according to (5.33) are ð1Þ ð3Þ ¼ mð1Þ ð3Þ ¼ 32; nð1Þ ¼ nð1Þ ¼ 40; m ð1Þ
• we fix the realizations ti ð3Þ; i ¼ 1; 2; . . .; 32; of the exemplary system comð1Þ
ponent conditional lifetime T11 ð3Þ in the reliability state subset {3} which are given by (5.33), ð1Þ ð1Þ • we determine the number rð1Þ ð3Þ of the disjoint intervals Ij ð3Þ ¼ hxj ð3Þ; ð1Þ
ð1Þ
yj ð3ÞÞ; j ¼ 1; 2; . . .; rð1Þ ð3Þ; which include the realizations ti ð3Þ; i ¼ ð1Þ
1; 2; . . .; 32; of the system component conditional lifetime T11 ð3Þ in the reliability state subset {3} rð1Þ ð3Þ ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ð1Þ ð3Þ ¼ 32 ffi 6; m ð1Þ
ð1Þ
ð1Þ
• we determine the length d ð1Þ ð3Þ of the intervals Ij ð3Þ ¼ hxj ð3Þ; yj ð3ÞÞ; j ¼ 1; 2; . . .; 6; which after considering
250
5 Complex Technical System Components Reliability
n o ð1Þ ð1Þ ti ð3Þ min fti ð3Þg ¼ 2054 20 ¼ 2034;
ð1Þ ¼ max R
1 i 32
1 i 32
is d ð1Þ ð3Þ ¼ ð1Þ
2034 ffi 407; 61
ð1Þ
ð1Þ
ð1Þ
• we determine the ends xj ð3Þ; yj ð3Þ; of the intervals Ij ð3Þ ¼ hxj ð3Þ; ð1Þ
yj ð3ÞÞ; j ¼ 1; 2; . . .; 6; which after considering ð1Þ
min fti ð3Þg
1 i 32
d ð1Þ ð3Þ ¼ 20 203:5 ¼ 183:5; 2
are ð1Þ
x1 ð3Þ ¼ maxf183:5; 0g ¼ 0; ð1Þ ð1Þ y1 ð3Þ ¼ x1 ð3Þ þ 1dð1Þ ð3Þ ¼ 0 þ 1 407 ¼ 407; ð1Þ
ð1Þ
x2 ð3Þ ¼ y1 ð3Þ ¼ 407; ð1Þ
ð1Þ
x3 ð3Þ ¼ y2 ð3Þ ¼ 814;
ð1Þ
ð1Þ
y2 ð3Þ ¼ x1 ð3Þ þ 2dð1Þ ð3Þ ¼ 0 þ 2 407 ¼ 814; ð1Þ
ð1Þ
y3 ð3Þ ¼ x1 ð3Þ þ 3dð1Þ ð3Þ ¼ 0 þ 3 407 ¼ 1221;
ð1Þ
ð1Þ
y4 ð3Þ ¼ x1 ð3Þ þ 4dð1Þ ð3Þ ¼ 0 þ 4 407 ¼ 1628;
ð1Þ
ð1Þ
y5 ð3Þ ¼ x1 ð3Þ þ 5dð1Þ ð3Þ ¼ 0 þ 5 407 ¼ 2035;
ð1Þ
ð1Þ
y6 ð3Þ ¼ x1 ð3Þ þ 6dð1Þ ð3Þ ¼ 0 þ 6 407 ¼ 2442;
x4 ð3Þ ¼ y3 ð3Þ ¼ 1221; x5 ð3Þ ¼ y4 ð3Þ ¼ 1628; x6 ð3Þ ¼ y5 ð3Þ ¼ 2035;
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
and the ends of the interval I7 ð3Þ ¼ hx7 ð3Þ; y7 ð3ÞÞ, including the remaining unknown realizations ð1Þ
ð1Þ
ð1Þ
x7 ð3Þ ¼ y6 ð3Þ ¼ 2442; y7 ð3Þ ¼ þ1; ð1Þ
ð1Þ
• we determine the number of realizations nj ð3Þ in particular intervals Ij ð3Þ ¼ ð1Þ
ð1Þ
hxj ð3Þ; yj ð3ÞÞ; j ¼ 1; 2; . . .; 6; according ð1Þ
ð1Þ
n1 ð3Þ ¼ 17; n2 ð3Þ ¼ 10; ð1Þ ð1Þ n6 ð3Þ ¼ 1; n7 ð3Þ ¼ 8;
ð1Þ
n3 ð3Þ ¼ 2;
ð1Þ
n4 ð3Þ ¼ 2;
ð1Þ
n5 ð3Þ ¼ 0;
• we find the realization of the histogram of the exemplary system component ð1Þ conditional lifetimes T11 ð3Þ as presented in Table 5.3,
5.3 Applications
251 ð1Þ
Table 5.3 The realization of the histogram of the exemplary system conditional lifetime T11 ð3Þ ð1Þ
Histogram of the conditional lifetime T11 ð3Þ ð1Þ
Ij ð3Þ ð1Þ ð1Þ ¼ hxj ð3Þ; yj ð3ÞÞ ð1Þ nj ð3Þ
f40 ðt; 3Þ ¼
0– 407
407– 814
814– 1221
1221– 1628
1628– 2035
2035– 2442
2442– þ1
17
10
2
2
0
1
8
17/40
10/40
2/40
2/40
0/40
1/40
8/40
ð1Þ nj ð3Þ=nð1Þ
• we join the intervals defined in the realization of the histogram f40 ðt; 3Þ that have ð1Þ ð1Þ the numbers nj ð3Þ; of realizations\4 into new intervals and the interval I7 ð2Þ with its new neighboring interval, • we fix the new number of intervals r ð1Þ ð3Þ ¼ 3; • we determine the new intervals I1ð1Þ ð3Þ ¼ h0; 407Þ;
I2ð1Þ ð3Þ ¼ h407; 814Þ;
I3ð1Þ ð3Þ ¼ h814; þ1Þ;
• we fix the number of realizations in the new intervals ð1Þ
ð1Þ
ð1Þ
n2 ð3Þ ¼ 10; n3 ð3Þ ¼ 13; n1 ð3Þ ¼ 17; • we calculate, using (5.25), the hypothetical probabilities that the conditional ð1Þ lifetime T11 ð3Þ takes values from the new intervals ð1Þ ð1Þ ð1Þ ð1Þ p1 ð3Þ ¼ PðT11 ð3Þ 2 I1 ð3ÞÞ ¼ Pð0 T11 ð3Þ\407Þ ð1Þ
ð1Þ
¼ ½R11 ð0; 3Þð1Þ ½R11 ð407; 3Þð1Þ ¼ exp½0:0009 0 exp½0:0009 407Þ ¼ 1 exp½0:3663 ¼ 1 0:6933 ¼ 0:3067; ð1Þ ð1Þ ð1Þ ð1Þ p2 ð3Þ ¼ PðT11 ð3Þ 2 I2 ð3ÞÞ ¼ Pð407T11 ð3Þ\814Þ ð1Þ
ð1Þ
¼ ½R11 ð407;3Þð1Þ ½R11 ð814;3Þð1Þ ¼ exp½0:0009 407 exp½0:0009 814Þ ¼ exp½0:3663 exp½0:7326 ¼ 0:6933 0:4807 ¼ 0:2126; ð1Þ ð1Þ ð1Þ ð1Þ p3 ð3Þ ¼ PðT11 ð3Þ 2 I3 ð3ÞÞ ¼ Pð814 T11 ð3Þh1Þ ð1Þ
ð1Þ
¼ 1 ½p1 ð3Þ þ p2 ð3Þ ¼ 1 ½0:3067 þ 0:2126 ¼ 0:4807;
252
5 Complex Technical System Components Reliability
• we calculate, using (5.26), the realization of the v2 (chi-square)-Pearson’s statistics
u40 ¼
ð1Þ ð1Þ 3 X ðnj ð3Þ nð1Þ pj ð3ÞÞ2 j¼1
þ
ð1Þ nð1Þ pj ð3Þ 2
¼
ð17 40 0:3067Þ2 ð10 40 0:2126Þ2 þ 40 0:3067 40 0:2126
ð13 40 0:4807 ffi 1:83 þ 0:26 þ 2:02 ¼ 4:11; 40 0:4807
• we assume the significance level a ¼ 0:01, • we fix the number of degrees of freedom rð1Þ ð3Þ l 1 ¼ 3 1 1 ¼ 1; • we read from the Tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a ¼ 0:01 and the number of degrees of freedom r ð1Þ ð3Þ l ¼ 1 ¼ 1; such that, according to (5.27), the following equality holds PðU40 [ ua Þ ¼ a ¼ 0:01 which amounts to ua ¼ 6:63 and determine the critical domain in the form of the interval ð6:63; þ1Þand the acceptance domain in the form of the interval ð0; 6:63Þ(Fig. 5.13),
• we compare the obtained value u40 ¼ 4:11 of the realization of the statistics U40 read from the Tables of the chi-square distribution critical value ua ¼ 6:63 and since the value u40 ¼ 4:11 does not belong to the critical domain, i.e. u40 ¼ 4:11 ua ¼ 6:63; we do not reject the hypothesis H0 in its part concerned with the coordinate h ið1Þ ð1Þ R11 ðt; 3Þ . In conclusion, we accept in full the hypothesis H0 that the reliability function of ð1Þ the component E11 at the operation state z1 is exponential with the coordinates given by (5.36). For the remaining cases, proceeding afterwards in an analogous way as in the h ið1Þ ð1Þ ð1Þ case of the conditional reliability function R11 ðt; Þ of the component E11 at the operation state z1, we can get’all the results of testing the hypotheses on the forms of the reliability functions defined by (5.28–5.29) given in Sect. 3.6.1 and used for the evaluation and prognosis of the reliability characteristics of the exemplary system operating at the variable operation conditions.
5.3 Applications
253
Fig. 5.13 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
f 2 (t ) x
1−
Critical domain
u40 = 4.11 u = 6.63
0
t
5.3.2 Statistical Identification of Port Oil Piping Transportation System Components Reliability The considered port oil piping transportation system reliability structure changing at the various operation states and its components and their unknown reliability parameters are described in Sects. 2.3.2 and 3.6.2.
5.3.2.1 Defining Parameters of Port Oil Piping Transportation System Components Reliability Models and Data Collection At all the system operation states zb ; b ¼ 1; 2; . . .; 7;, defined in Sect. 2.3.2, we distinguish the following three reliability states 0, 1, 2 (z = 2) of the system and its components, defined in Sect. 1.4.2. Moreover, we fix the possible transitions between the components reliability states only from better to worse. From the above, the port oil pipeline transportation subsystems St ; t ¼ 1; 2; 3; ðtÞ are composed of three-state, i.e. z = 2, components Eij ; t ¼ 1; 2; 3; having the conditional three-state reliability functions ðtÞ
ðtÞ
ðtÞ
b ¼ 1; 2; . . .; 7;
ð5:37Þ
ðtÞ
ðtÞ
ð5:38Þ
½Rij ðt; ÞðbÞ ¼ ½1; ½Rij ðt; 1ÞðbÞ ; ½Rij ðt; 2ÞðbÞ ; with the exponential coordinates ðtÞ
ðtÞ
½Rij ðt; 1ÞðbÞ ¼ exp½½kij ð1ÞðbÞ ; ½Rij ðt; 2ÞðbÞ ¼ exp½½kij ð2ÞðbÞ ;
and with the intensities of departure from the reliability state subsets f1; 2g; f2g; respectively ðtÞ
ðtÞ
½kij ð1ÞðbÞ ; ½kij ð2ÞðbÞ ;
b ¼ 1; 2; . . .; 7;
different at the particular operation states zb ; b ¼ 1; 2; . . .; 7; where i ¼ 1; 2 and j ¼ 1; 2; . . .; 178 for t ¼ 1; i ¼ 1; 2 and j ¼ 1; 2; . . .; 719 for t ¼ 2; i ¼ 1; 2; 3 and j ¼ 1; 2; . . .; 362 for t = 3.
254
5 Complex Technical System Components Reliability ð1Þ
Table 5.4 The approximate mean values ½^ lij ðuÞðbÞ of the piping subsystem components’ ð1Þ
conditional lifetimes ½Tij ðuÞðbÞ at the operation states zb ð1Þ
Component of subsystem S1
Mean value ½^ lij ðuÞðbÞ (in years)
Operation state zb
Reliability state subsetfu; u þ 1; . . .; 2g
Eij ; i ¼ 1; 2; j ¼ 1; 2; . . .; 176
Eij i ¼ 1; 2; j ¼ 177; 178
{1,2} {2} {1,2} {2} {1,2} {2} {1,2} {2}
161 114 161 114 161 114 161 114
60 55 60 55 60 55 60 55
z1 z2 z3 z4 z5 z6
ð1Þ
ð1Þ
z7
5.3.2.2 Collecting Data on Port Oil Piping Transportation System Components Reliability Coming from Experts We have the approximate realizations ðtÞ
½^ lij ðuÞðbÞ ;
t ¼ 1; 2; 3; u ¼ 1; 2; b ¼ 1; 2; . . .; 7;
of the mean values ðtÞ
ðtÞ
½lij ðuÞðbÞ ¼ E½½Tij ðuÞðbÞ ;
t ¼ 1; 2; 3; u ¼ 1; 2; b ¼ 1; 2; . . .; 7;
ðtÞ
of the conditional lifetimes ½Tij ðuÞðbÞ ; t ¼ 1; 2; 3; u ¼ 1; 2; b ¼ 1; 2; . . .; 7; at the reliability state subsets fu; u þ 1; . . .; 2g; u ¼ 1; 2; of the components ðtÞ Eij ; t ¼ 1; 2; 3; of the oil pipeline subsystems St ; t ¼ 1; 2; 3; at the particular operation states zb ; b ¼ 1; 2; . . .; 7 estimated on the basis of expert opinion. In Table 5.4, as an example of data coming from experts, there is given the approximate realizations of the conditional lifetime at the reliability state subsets of the components of the oil pipeline subsystems S1 :
5.3.2.3 Evaluating Port Oil Piping Transportation System Components Intensities of Departures from Reliability State Subsets on the Basis of Data Coming from Experts To evaluate approximately the parameters of multistate reliability functions of the port oil pipeline system components the statistical data from experts presented in
5.3 Applications
255
Sect. 5.3.2.2 can be used. The collected statistical data partly presented in Table 5.4 and the resulting from (5.22) formula ðtÞ
½^kij ðuÞðbÞ ¼
1 ; ðtÞ ^ ½kij ðuÞðbÞ
t ¼ 1; 2; 3; u ¼ 1; 2; b ¼ 1; 2; . . .; 7;
ð5:39Þ
ðtÞ application yield the approximate values ½^ kij ðuÞðbÞ of subsystems St ; t ¼ 1; 2; 3; ðtÞ
components’ unknown intensities ½kij ðuÞðbÞ of departure from the reliability state subsets, {1, 2}, {2}, while the system is operating at the operation state zb ; b ¼ 1; 2; . . .; 7. For instance, substituting into (5.39) the values of the mean lifetimes h ið3Þ ð1Þ ^ ij ð1Þ ¼ 161; l
h ið3Þ ð1Þ ^ij ð2Þ ¼ 114; l
taken from the fifth and sixth rows of the third column of Table 5.4, we obtain the approximate evaluations of the unknown intensities of departure of components ð1Þ Eij ; i ¼ 1; 2 j ¼ 1; 2; . . .; 176, of the subsystem from the reliability state subsetsf1; 2g; f2g while the piping system is operating at the operation state z3 that respectively amount to h ið3Þ ^kð1Þ ð1Þ ¼h ij h ið3Þ ^kð1Þ ð2Þ ¼h ij
1 ð1Þ ^ ij ð1Þ l
1 ð1Þ
^ ij ð2Þ l
ið3Þ ¼
1 ffi 0:0062; 161
ið3Þ ¼
1 ffi 0:0088; 114
i ¼ 1; 2; j ¼ 1; 2; . . .; 176: ð5:40Þ
Similarly, substituting into (5.39) the values of the mean lifetimes ð1Þ
½^ lij ð1Þð3Þ ¼ 60;
ð1Þ
½^ lij ð2Þð3Þ ¼ 55;
taken from the fifth and sixth rows of the fourth column of Table 5.4, we obtain the approximate evaluations of the unknown intensities of departure of compoð1Þ nents Eij ; i ¼ 3; j ¼ 177; 178; of subsystem S1 from the reliability state subsets f1; 2g; f2g while the piping system is operating at the operation state z3 that respectively amount to h ið3Þ ^kð1Þ ð1Þ ¼h ij h
^kð1Þ ð2Þ ij
ið3Þ
¼h
1 ðtÞ
^ij ð1Þ l 1 ðtÞ
^ij ð2Þ l
ið3Þ ¼ ið3Þ
¼
1 ffi 0:0167; 610 1 ffi 0:0182; 55
ð5:41Þ i ¼ 3; j ¼ 177; 178:
256
5 Complex Technical System Components Reliability
The evaluations of unknown intensities of departure from the reliability state subsets {1, 2}, {2} of all components of the port oil piping transportation system operating at the various operation states, can be obtained in the same way. Substituting the above and the remaining evaluations of the intensities of departures respectively into the formulae (5.37–5.38), we get the exponential coordinates of the port oil piping transportation system components three-state reliability functions which after arbitrary acceptance in Sect. 5.3.2.4 are used in Sect. 3.6.2 for the evaluation and prediction of the system reliability characteristics. 5.3.2.4 Identifying Port Oil Piping Transportation System Components Conditional Multistate Exponential Reliability Functions on the Basis of Data Coming from Experts As there are no data collected from the port oil piping transportation system components’ reliability states changing processes, it is not possible to verify the hypotheses on the exponential forms of the port oil piping transportation system components’ conditional reliability functions. We arbitrarily assume that these reliability functions are exponential and using the results of the previous section and the relationships (5.37–5.38) given in Sect. 5.3.2.1 we fix their forms. For instance, using the evaluations (5.40–5.41) of the previous section and the formulae (5.37–5.38), we conclude that at the system operation state, the subsystem S1 consists of two identical pipelines, each composed of 176 pipe segments with the conditional reliability functions coordinates h ið3Þ h ið3Þ ð1Þ ð1Þ Rij ðt; 1Þ ¼ exp½0:0062t; Rij ðt; 2Þ ¼ exp½0:0088t; i ¼ 1; 2; j ¼ 1; 2; . . .; 176 and two valves with the conditional reliability functions co-ordinates h ið3Þ h ið3Þ ð1Þ ð1Þ Rij ðt; 1Þ ¼ exp½0:0167t; Rij ðt; 2Þ ¼ exp½0:0182t; i ¼ 1; 2; j ¼ 177; 178: The exponential coordinates of the port oil piping transportation system all components three-state reliability functions arbitrarily fixed in this way are given and used in Sect. 3.6.2 for the evaluation and prediction of the system reliability characteristics.
5.3.3 Statistical Identification of Maritime Ferry Technical System Components Safety The considered ferry technical system safety structure changing at the various operation states and its components and their unknown safety parameters are described in Sects. 2.3.3 and 3.6.3.
5.3 Applications
257
5.3.3.1 Defining Parameters of Ferry Technical System Components Safety Models and Data Collection At all the system operation states zb ; b ¼ 1; 2. . .; 18; defined in Sect. 2.3.3, we distinguish the following five safety states 0; 1; 2; 3; 4ðz ¼ 4Þ of the system and its components, defined in Sect. 1.4.3. Moreover, we fix the possible transitions between the components safety states only from better to worse. From the above, the ferry technical subsystems St ; t ¼ 1; 2. . .; 5 are composed ðtÞ of five-state, i.e. z = 4, components Eij ; t ¼ 1; 2; . . .; 5 having the conditional five-state safety functions h iðbÞ h iðbÞ h iðbÞ h iðbÞ h iðbÞ ðtÞ ðtÞ ðtÞ ðtÞ ðtÞ ¼ 1; sij ðt; 1Þ ; sij ðt; 2Þ ; sij ðt; 3Þ ; sij ðt; 4Þ ; sij ðt; Þ b ¼ 1; 2; . . .; 18; ð5:42Þ with the exponential coordinates h h h iðbÞ iðbÞ h iðbÞ iðbÞ ðtÞ ðtÞ ðtÞ ðtÞ sij ðt; 1Þ ¼ exp kij ð1Þ ; sij ðt; 2Þ ¼ exp kij ð2Þ ; h
ðtÞ
sij ðt; 3Þ
iðbÞ
h h iðbÞ h iðbÞ iðbÞ ðtÞ ðtÞ ðtÞ ; sij ðt; 4Þ ¼ exp kij ð4Þ ; ¼ exp kij ð3Þ ð5:43Þ
and with the intensities of departure from the safety states subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g; respectively h
ðtÞ
kij ð1Þ
iðbÞ h
iðbÞ h iðbÞ h iðbÞ ðtÞ ðtÞ ðtÞ kij ð2Þ ; kij ð3Þ ; kij ð4Þ ;
b ¼ 1; 2; . . .; 18;
different at the particular operation states zb ; b ¼ 1; 2; . . .; 18; where i = 1 and j = 1 for t = 1, i = 1 and j = 1, 2, 3, 4, i = 2 and j = 1, 2, i = 3, 4, 5, 6, 7 and j = 1 for t = 2, i = 1, 2, 3, 4, 5 and j = 1 for t = 3, i = 1, 2 and j = 1 for t = 4, i = 1, 2, 3 and j = 1 for t = 5.
5.3.3.2 Collecting Data on Ferry Technical System Components Safety Models Coming from Experts We have the approximate realizations ðtÞ
½^ lij ðuÞðbÞ ; of the mean values
t ¼ 1; 2; 3; 4; 5; u ¼ 1; 2; 3; 4; b ¼ 1; 2; . . .; 18;
258
5 Complex Technical System Components Reliability ðtÞ
ðtÞ
½lij ðuÞðbÞ ¼ E½½Tij ðuÞðbÞ
t ¼ 1; 2; 3; 4; 5; u ¼ 1; 2; 3; 4; b ¼ 1; 2; . . .; 18; ðtÞ
of the conditional lifetimes ½Tij ðuÞðbÞ ; t ¼ 1; 2; 3; 4; 5; u ¼ 1; 2; 3; 4; b ¼ 1; 2; . . .; 18; in the safety state subsets fu; u þ 1; . . .; 4g; u ¼ 1; 2; 3; 4; of the ðtÞ components Eij of the ferry technical subsystems Sb ; t ¼ 1; 2; 3; 4; 5; at the particular operation states zb ; b ¼ 1; 2; . . .; 18; estimated on the basis of expert opinions. In the Table 5.5, as an example of data coming from experts, there are given the approximate realizations of the conditional lifetime at the safety state subsets of the components of the ferry technical subsystems S1.
5.3.3.3 Evaluating Ferry Technical System Components Intensities of Departures from Safety State Subsets on the Basis of Data Coming from Experts To evaluate approximately the parameters of multistate safety functions of the ferry technical system components the statistical data coming from experts partly presented in Sect. 5.3.3.2 can be used. The statistical data collected in Table 5.5 and the resulting from (5.22) formula 1
ðtÞ
½^kij ðuÞðbÞ ¼
ðtÞ ½^ lij ðuÞðbÞ
;
t ¼ 1; 2; 3; 4; 5; u ¼ 1; 2; 3; 4;
b ¼ 1; 2; . . .; 18; ð5:44Þ
ðtÞ
application yield the approximate values ½^ kij ðuÞðbÞ of subsystems St ; t ¼ 1; 2; 3; ðtÞ
components unknown intensities ½kij ðuÞðbÞ of departure from the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g; while the system is operating at the operation state zb ; b ¼ 1; 2; . . .; 18: For instance, substituting into (5.44) the values the mean lifetimes ð1Þ
ð1Þ
ð1Þ
ð1Þ
lij ð2Þð2Þ ¼ 25; ½^ lij ð3Þð2Þ ¼ 22; ½^ lij ð4Þð2Þ ¼ 20; ½^ lij ð1Þð2Þ ¼ 30; ½^ taken from the fourth, fifth, sixth and seventh rows of the third column of Table 5.5, we obtain the approximate evaluations of the unknown intensities of ð1Þ departure of component E11 of subsystem S1 from the safety state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g; while the ferry technical system is operating at the operation state z2 that respectively amount to h ið2Þ ^kð1Þ ð1Þ ¼ h ij
1 ð1Þ
^ij ð1Þ l
ið2Þ ¼
h ið2Þ 1 ð1Þ ffi 0:033; ^ kij ð2Þ ¼ h 30
1 ð1Þ
^ij ð2Þ l
ið2Þ ¼
1 ffi 0:040; 25
5.3 Applications
259 ð1Þ
Table 5.5 The approximate mean values ½^ l11 ðuÞðbÞ of the ferry subsystem components’ conð1Þ ditional lifetimes ½T11 ðuÞðbÞ at the operation states zb ð1Þ Component of subsystem S1 E11 ð1Þ Operation state zb Safety state subsetfu; u þ 1; . . .; 4g Mean value ½^ l ð1ÞðbÞ (in years) 11
z1 z2
z3
z4
z5
z6
z7
z8 z9 z10
z11
z12
{1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4}
30 25 22 20 30 25 22 20 30 25 22 20 30 25 22 20 30 25 22 20 30 25 22 20
{1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4}
30 25 22 20 30 25 22 20 30 25 22 20 (continued)
260
5 Complex Technical System Components Reliability
Table 5.5 (continued) Component of subsystem S1
ð1Þ
E11
Operation state zb
Safety state subsetfu; u þ 1; . . .; 4g
Mean value ½^ l11 ð1ÞðbÞ (in years)
z13
{1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4} {1,2,3,4} {2,3,4} {3,4} {4}
30 25 22 20 30 25 22 20 30 25 22 20 30 25 22 20 30 25 22 20
z14
z15
z16
z17
ð1Þ
z18
h
^kð1Þ ð3Þ ij
ið2Þ
¼h
1 ð1Þ
^ij ð3Þ l
ið2Þ ¼
h ið2Þ 1 ð1Þ ffi 0:045; ^ kij ð4Þ ¼ h 22
1 ð1Þ
^ij ð4Þ l
ið2Þ ¼
1 ffi 0:050: 20 ð5:45Þ
The evaluations of all unknown intensities of departure from the reliability state subsets f1; 2; 3; 4g; f2; 3; 4g; f3; 4g; f4g; of components of the ferry technical system operating at various operation states, can be obtained in the same way. Substituting the above evaluations of the intensities of departures respectively into the formulae (5.42–5.43), we get the exponential coordinates of the ferry technical system components’ five-state safety functions which after arbitrary acceptance in Sect. 5.3.3.4 are used in Sect. 3.6.2 for the evaluation and prediction of the system safety characteristics. 5.3.3.4 Identifying Ferry Technical System Components Conditional Multistate Exponential Safety Functions on the Basis of Data Coming from Experts As there are no data collected from the ferry technical system components safety states changing processes, it is not possible to verify the hypotheses on the
5.3 Applications
261
exponential forms of this system components’ conditional safety functions. We arbitrarily assume that these safety functions are exponential and using the results of the previous section and the relationships (5.42–5.43) given in Sect. 5.3.3.1 we fix their forms. For instance, using the evaluations (5.45) of the previous section and (5.42– 5.43), we conclude that at the system operation state z2, subsystem S1 consists of a ð1Þ component E11 with the conditional multistate safety function co-ordinates ð1Þ
½s11 ðt; 2Þð2Þ ¼ exp½0:040t;
ð1Þ
½s11 ðt; 4Þð2Þ ¼ exp½0:05t:
½s11 ðt; 1Þð2Þ ¼ exp½0:033t; ½s11 ðt; 3Þð2Þ ¼ exp½0:045t;
ð1Þ ð1Þ
The exponential coordinates of the ferry technical system components safety functions arbitrarily fixed in this way are used in Sect. 3.6.3 for the evaluation and prediction of the system safety characteristics.
5.4 Summary The proposed statistical methods of identification of the unknown parameters of the system reliability and safety of the components of complex technical systems allow for the models discussed in Chap. 3 and their unknown parameters identification and practical applications in evaluation and prediction of the reliability, availability and safety of real complex technical systems. The methods and the results of these models’ identification allow for the optimization of the reliability, availability and safety of the complex technical systems presented in Chap. 6.
References 1. Aven T (1985) Reliability evaluation of multistate systems with multistate components. IEEE Trans Reliab 34:473–479 2. Collet J (1996) Some remarks on rare-event approximation. IEEE Trans Reliab 45:106–108 3. Gamiz ML, Roman Y (2008) Non-parametric estimation of the availability in a general repairable. Reliab Eng Sys Saf 93(8):1188–1196 4. Giudici P, Figini S (2009) Applied data mining for business and industry. Wiley, Chichester 5. Helvacioglu S, Insel M (2008) Expert system applications in marine technologies. Ocean Eng 35(11–12):1067–1074 6. Hryniewicz O (1995) Lifetime tests for imprecise data and fuzzy reliability requirements. In: Onisawa T, Kacprzyk J (eds) Reliability and safety analyses under fuzziness. Physica Verlag, Heidelberg, pp 169–182 7. Kołowrocki K (2004) Reliability of large systems. Elsevier, Amsterdam 8. Kołowrocki K (2007) Reliability modelling of complex systems–Part 1. Int J Gnedenko eForum Reliab Theory Appl 2(3–4):116–127 9. Kołowrocki K (2007) Reliability modelling of complex systems–Part 2. Int J Gnedenko eForum Reliab Theory Appl 2(3–4):128–139
262
5 Complex Technical System Components Reliability
10. Kolowrocki K, Soszynska J (2009) Modeling environment and infrastructure influence on reliability and operation process of port oil transportation system. Electron J Reliab Risk Anal: Theory Appl 2(3):131–142 11. Kolowrocki K, Soszynska J (2009) Safety and risk evaluation of Stena Baltica ferry in variable operation conditions. Electron J Reliab Risk Anal Theory Appl 2(4):168–180 12. Kołowrocki K, Soszyn´ska J (2009) Methods and algorithms for evaluating unknown parameters of components reliability of complex technical systems Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(2):223–230 13. Kolowrocki K, Soszynska J (2010) Reliability, availability and safety of complex technical systems modelling identification prediction optimization Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(1):133–158 14. Kołowrocki K, Soszyn´ska J (2010) Safety and risk evaluation of a ferry technical system Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(1):159–172 15. Kołowrocki K, Soszyn´ska J (2011) On safety analysis of complex technical maritime transportation system. J Risk Reliab 225(1):1–10 16. Kołowrocki K, Soszyn´ska J, Judzin´ski M, Dziula P (2007) On multi-state safety analysis in shipping. Int J Reliab Qual Saf Eng Sys Reliab Saf 14(6):547–567 17. Lisnianski A, Levitin G (2003) Multi-state system reliability. Assessment, optimisation and applications. World Scientific Publishing Co. Pte. Ltd, London 18. Rice JA (2007) Mathematical statistics and data analysis. Duxbury. Thomson Brooks/Cole, University of California, Berkeley 19. Soszyn´ska J (2006) Reliability evaluation of a port oil transportation system in variable operation conditions. Int J Press Vessel Pip 83(4):304–310 20. Soszyn´ska J (2007) Systems reliability analysis in variable operation conditions. PhD Thesis, Gdynia Maritime University-System Research Institute Warsaw (in Polish) 21. Soszyn´ska J (2010) Reliability and risk evaluation of a port oil pipeline transportation system in variable operation conditions. Int J Press Vessel Pip 87(2–3):81–87 22. Soszyn´ska J, Kołowrocki K, Blokus-Roszkowska A, Guze S (2010) Identification of complex technical system components safety models Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(2):399–496 23. Vercellis S (2009) Data mining and optimization for decision making. Wiley, Hoboken 24. Wilson AG, Graves TL, Hamada MS et’al (2006) Advances in data combination, analysis and collection for system reliability assessment. Stat Sci 21(4):514–531 25. Xue J (1985) On multi-state system analysis. IEEE Trans Reliab 34:329–337 26. Xue J, Yang K (1995) Dynamic reliability analysis of coherent multi-state systems. IEEE Trans Reliab 4(44):683–688 27. Xue J, Yang K (1995) Symmetric relations in multi-state systems. IEEE Trans Reliab 4(44):689–693 28. Yu K, Koren I, Guo Y (1994) Generalized multistate monotone coherent systems. IEEE Trans Reliab 43:242–250
Chapter 6
Complex Technical Systems Operation, Reliability, Availability, Safety and Cost Optimization
6.1 Introduction Complex technical systems’ reliability and safety improvement and decreasing their operation costs and risk are of great value in the industrial practice [1, 4, 6–9, 15, 16]. In everyday practice, the tools that could be applied to improve the reliability and safety characteristics of the considered multistate non-repairable systems are needed [3, 5]. Tools are needed for allowing to find the distributions, the expected values and the variances of the optimal times until the successive exceeding of the reliability or the safety critical state and the distributions, the expected values and the variances of the optimal number of exceedings of the reliability or the safety critical state at a fixed moment of the operation time for these systems when they are repairable and when their renovation is ignored, and to compare the values of these characteristics before and after their operation processes’ optimization [2, 3, 5, 10–12, 14]. Tools can also be applied to find the distribution functions, the expected values and the variances of the optimal times until the successive renovations and distribution functions, the expected values and variances of the optimal times until the successive exceeding of the reliability or safety critical state and distributions, the expected values and variances of the optimal number of renovations up to a fixed moment of the operation time and distributions, the expected values and variances of the optimal number of exceedings of the reliability or safety critical state up to a fixed moment of the operation time and steady optimal availability coefficients and optimal availability coefficients in a fixed operation time interval for these systems when they are repairable and the time of their renovation is non-ignored and to compare the values of these characteristics before and after their operation processes’ optimization [2, 3, 5, 10–12, 14]. These tools may be applied to perform the effective cost analysis [13] of complex technical systems at the variable operating conditions and its application to the evaluation the cost before and after the system operation process optimization [3, 5]. The tools based on the methods of corrective and K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8_6, Springer-Verlag London Limited 2011
263
264
6 Complex Technical Systems Operation, Reliability, Availability, Safety
preventive maintenance analysis [17] may be applied to fix the optimal maintenance policy maximizing the availability and minimizing the renovation cost of the complex technical systems at the variable operating conditions [3, 5].
6.2 Optimization of Operation, Reliability and Safety and Cost Analysis of Complex Technical Systems 6.2.1 Optimal Transient Probabilities of Complex Technical System Operation Process at Operation States Considering Eq. 3.6, it is natural to assume that the system operation process has significant influence on the system reliability. This influence is also clearly expressed in Eq. 3.7 for the mean values of the system unconditional lifetimes in the reliability state subsets. From the linear Eq. 3.7, we can see that the mean value of the system unconditional lifetime lðuÞ; u ¼ 1; 2; . . .; z, is determined by the limit values of transient probabilities pb ; b ¼ 1; 2; . . .; m; of the system operation process at the operation states given by (2.22) and the mean values lb ðuÞ, b ¼ 1; 2; . . .; m; u ¼ 1; 2; . . .; z; of the system conditional lifetimes in the reliability state subsets fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; given by (3.8). Therefore, the system lifetime optimization approach based on the linear programming [2, 3, 5] can be proposed. Namely, we may look for the corresponding optimal values p_ b ; b ¼ 1; 2; . . .; m; of the transient probabilities pb ; b ¼ 1; 2; . . .; m; of the system operation process at the operation states to maximize the mean value lðuÞ of the unconditional system lifetimes in the reliability state subsets fu; u þ 1; . . .; zg; u ¼ 1; 2; . . .; z; under the assumption that the mean values lb ðuÞ; b ¼ 1; 2; . . .; m; u ¼ 1; 2; . . .; z; of the system conditional lifetimes in the reliability state subsets are fixed. As a special and practically important case of the above formulated system lifetime optimization problem, if r; r ¼ 1; 2; . . .; z; is a system critical reliability state, we may look for the optimal values p_ b ; b ¼ 1; 2; . . .; m; of the transient probabilities pb ; b ¼ 1; 2; . . .; m; of the system operation process at the system operation states to maximize the mean value lðrÞ of the unconditional system lifetime in the reliability state subset fr; r þ 1; . . .; zg; r ¼ 1; 2; . . .; z; under the assumption that the mean values lb ðrÞ; b ¼ 1; 2; . . .; m; r ¼ 1; 2; . . .; z; of the system conditional lifetimes in this reliability state subset are fixed. More exactly, we may formulate the optimization problem as a linear programming model with the objective function of the following form lðrÞ ¼
m X
pb lb ðrÞ
b¼1
for a fixed r 2 f1; 2; . . .; zg and with the following bound constraints
ð6:1Þ
6.2 Optimization of Operation, Reliability and Safety and Cost Analysis ^
_
pb pb pb ;
b ¼ 1; 2; . . .; m;
m X
pb ¼ 1;
265
ð6:2Þ ð6:3Þ
b¼1
where lb ðrÞ; lb ðrÞ 0;
b ¼ 1; 2; . . .; m;
ð6:4Þ
are fixed mean values of the system conditional lifetimes in the reliability state subset fr; r þ 1; . . .; zg and ^
^
pb ; 0 pb 1
_
_
^
_
pb ; 0 pb 1; pb pb ;
and
b ¼ 1; 2; . . .; m;
ð6:5Þ
are lower and upper bounds of the unknown transient probabilities pb ; b ¼ 1; 2; . . .; m; respectively. Now, we can obtain the optimal solution of the formulated by (6.1–6.5) the linear programming problem, i.e. we can find the optimal values p_ b of the transient probabilities pb, b ¼ 1; 2; . . .; m; which maximize the objective function given by (6.1). First, we arrange the system conditional lifetime mean values lb ðrÞ; b ¼ 1; 2; . . .; m; in non-increasing order lb1 ðrÞ lb2 ðrÞ lbm ðrÞ; where bi 2 f1; 2; . . .; mg for i ¼ 1; 2; . . .; m: Next, we substitute ^
^
_
_
xi ¼ pbi ; xi ¼ pbi ; xi ¼ pbi
for i ¼ 1; 2; . . .; m
ð6:6Þ
and we maximize with respect to xi ; i ¼ 1; 2; . . .; m; the linear form (6.1) which after this transformation takes the form lðrÞ ¼
m X
xi lbi ðrÞ
ð6:7Þ
i¼1
for a fixed r 2 f1; 2; . . .; zg with the following bound constraints ^
_
xi xi x i ; m X
i ¼ 1; 2; . . .; m; xi ¼ 1;
ð6:8Þ ð6:9Þ
i¼1
where lbi ðrÞ; lbi ðrÞ 0;
i ¼ 1; 2; . . .; m;
are fixed mean values of the system conditional lifetimes in the reliability state subset fr; r þ 1; . . .; zg arranged in non-increasing order and
266
6 Complex Technical Systems Operation, Reliability, Availability, Safety ^
^
x i ; 0 xi 1
_
_
^
_
xi ; 0 x i 1; x i xi ;
and
i ¼ 1; 2; . . .; m;
ð6:10Þ
are lower and upper bounds of the unknown probabilities xi ; i ¼ 1; 2; . . .; m; respectively. To find the optimal values of xi ; i ¼ 1; 2; . . .; m; we define m X ^ ^ ^ x¼ xi ; ^y ¼ 1 x ð6:11Þ i¼1
and ^0
_0
x ¼ 0; x ¼ 0 and
^I
x ¼
I I X X ^ _I _ xi ; x ¼ xi i¼1
for I ¼ 1; 2; . . .; m:
ð6:12Þ
i¼1
Next, we find the largest value I 2 f0; 1; . . .; mg such that _I
^I
x x \^y
ð6:13Þ
and we fix the optimal solution that maximizes (6.7) in the following way: (i)
if I ¼ 0; the optimal solution is ^
x_ 1 ¼ ^y þ x 1
^
x_ i ¼ x i
and
for i ¼ 2; 3; . . .; m;
ð6:14Þ
(ii) if 0\I\m; the optimal solution is _I
_
x_ i ¼ xi
^I
^
for i ¼ 1; 2; . . .; I; x_ Iþ1 ¼ ^y x þ x þ xIþ1 ^
x_ i ¼ x i
for i ¼ I þ 2; I þ 3; . . .; m;
and ð6:15Þ
(iii) if I ¼ m; the optimal solution is _
x_ i ¼ x i
for i ¼ 1; 2; . . .; m:
ð6:16Þ
Finally, after making the inverse substitution to (6.6), we get the optimal limit transient probabilities p_ bi ¼ x_ i
for i ¼ 1; 2; . . .; m;
ð6:17Þ
that maximize the system mean lifetime in the reliability state subset fr; r þ 1; . . .; zg; defined by the linear form (6.1), giving its maximum value in the following form _ lðrÞ ¼
m X b¼1
for a fixed r 2 f1; 2; . . .; zg.
p_ b lb ðrÞ
ð6:18Þ
6.2 Optimization of Operation, Reliability and Safety and Cost Analysis
267
6.2.2 Optimal Reliability and Safety Characteristics of Complex Technical Systems _ From expression (6.18) for the maximum mean value lðrÞ of the system unconditional lifetime in the reliability state subset fr; r þ 1; . . .; zg; replacing in it the critical reliability state r by the reliability state u, u ¼ 1; 2; . . .; z; we obtain the corresponding optimal solutions for the mean values of the system unconditional lifetimes in the reliability state subsets fu; u þ 1; . . .; zg of the form _ lðuÞ ¼
m X
p_ b lb ðuÞ
for u ¼ 1; 2; . . .; z:
ð6:19Þ
b¼1
Further, according to (3.5–3.6), the corresponding optimal unconditional multistate reliability function of the system is the vector _ Þ ¼ ½1; Rðt; _ 1Þ; . . .; Rðt; _ zÞ; Rðt;
ð6:20Þ
with the coordinates given by _ uÞ ffi Rðt;
v X
p_ b ½Rðt; uÞðbÞ
for t 0;
u ¼ 1; 2; . . .; z:
ð6:21Þ
b¼1
By applying (3.9), the corresponding optimal values of the variances of the system unconditional lifetimes in the system reliability state subsets are 2
r_ ðuÞ ¼ 2
Z1
2 _ uÞdt ½lðuÞ _ t Rðt; ;
u ¼ 1; 2; . . .; z;
ð6:22Þ
0
_ uÞ is given by (6.21). _ where lðuÞ is given by (6.19) and Rðt; And, by (3.10), the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are _ ðuÞ ¼ lðuÞ _ _ þ 1Þ; l lðu
_ ðzÞ ¼ lðzÞ: _ u ¼ 1; . . .; z 1; l
ð6:23Þ
Moreover, considering (3.11) and (3.12), the corresponding optimal system risk function and the optimal moment when the risk exceeds a permitted level d, respectively are given by _ rÞ; r_ ðtÞ ¼ 1 Rðt;
t 0;
ð6:24Þ
and s_ ¼ r_ 1 ðdÞ;
ð6:25Þ
_ rÞ is given by (6.21) for u ¼ r and r_ 1 ðtÞ; if it exists, is the inverse where Rðt; function of the optimal risk function r_ ðtÞ:
268
6 Complex Technical Systems Operation, Reliability, Availability, Safety
All the considered reliability characteristics and presented results in this section may be transferred to the complex technical systems safety analysis and formulated by simply replacing their reliability function Rðt; Þ and its optimal form _ Þ respectively by their safety function sðt; Þ and its optimal form s_ ðt; Þ: Rðt;
6.2.3 Optimal Renewal and Availability Characteristics of Complex Technical Systems _ _ The direct replacing l(r) by lðrÞ and rðrÞ by rðrÞ in the expressions for the repairable systems renewal and availability characteristics pointed in Propositions 3.4 and 3.5, we get their corresponding optimal values pointed out in the following two propositions [3, 5]. Proposition 6.1 If components of the multistate repairable system with ignored time of renovation have the exponential reliability functions at the operation states zb ; b ¼ 1; 2; . . .; v; given by (3.13–3.14) or respectively by (3.15–3.16) and the system reliability critical state is r; r 2 f1; 2; . . .; zg; then after the system operation process optimization: (a) the optimal time S_ N ðrÞ until the Nth exceeding by the system, the reliability critical state r, for sufficiently large N , has approximately normal distribution pffiffiffiffi _ _ NðN lðrÞ; N rðrÞÞ, i.e., _ t N lðrÞ F_ ðNÞ ðt; rÞ ¼ PðS_ N ðrÞ\tÞ ffi FNð0;1Þ pffiffiffiffi ; _ N rðrÞ t 2 ð1; 1Þ;
r 2 f1; 2; . . .; zg;
ð6:26Þ
(b) the expected value and the variance of the optimal time S_ N ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, are respectively given by _ E½S_ N ðrÞ ¼ N lðrÞ;
D½S_ N ðrÞ ¼ N r_ 2 ðrÞ;
r 2 f1; 2; . . .; zg;
ð6:27Þ
_ rÞ of exceedings by the system the reliability critical (c) the optimal number Nðt; state r up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 1 1 0 0 _ _ ðN þ 1ÞlðrÞ tC tC BN lðrÞ _ rÞ ¼ NÞ ffi FNð0;1Þ B qffiffiffiffiffiffi A FNð0;1Þ @ qffiffiffiffiffiffiA; PðNðt; @ t t _ _ rðrÞ rðrÞ _ _ lðrÞ lðrÞ N ¼ 0; 1; . . .;
r 2 f1; 2; . . .; zg;
ð6:28Þ
(d) the expected value and the variance of the optimal number Nðt; rÞ of exceeding by the system the reliability critical state r up to the moment t; t 0; for sufficiently large t, are respectively given by
6.2 Optimization of Operation, Reliability and Safety and Cost Analysis
_ rÞ ¼ Hðt;
t ; _ lðrÞ
_ rÞ ¼ Dðt;
t 3
_ ½lðrÞ
2 _ ½rðrÞ ;
r 2 f1; 2; . . .; zg;
269
ð6:29Þ
_ _ where lðrÞ and rðrÞ are given by (6.19 and 6.22) for u ¼ r: Proposition 6.2 If components of the multistate repairable system with nonignored time of renovation at the operational states zb ; b ¼ 1; 2; . . .; m; have the exponential reliability functions at the operation states zb ; b ¼ 1; 2; . . .; v; given by (3.13–3.14) or respectively by (3.15–3.16), the system reliability critical state is r, r 2 f1; 2; . . .; zg; and the successive times of system’s renovations are independent and have an identical distribution function with the expected value lo ðrÞ and the variance r2o ðrÞ; then after the system operation process optimization: (a) the optimal time S_ N ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ NðN lðrÞ þ ðN 1Þlo ðrÞÞ; N r_ 2 ðrÞ þ ðN 1Þr2o ðrÞÞ, i.e., _ ðNÞ
F
S_ N ðrÞ\tÞ ffi FNð0;1Þ ðt; rÞ ¼ Pð t 2 ð1; 1Þ;
! _ t NðlðrÞ þ lo ðrÞÞ þ lo ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Nðr_ 2 ðrÞ þ r2o ðrÞÞ r2o ðrÞ
r 2 f1; 2; . . .; zg;
ð6:30Þ
(b) the expected value and the variance of the optimal time S_ N ðrÞ until the Nth exceeding by the system the reliability critical state r, for sufficiently large N, are respectively given by _ þ ðN 1Þlo ðrÞ; D½ S_ N ðrÞ ffi N r_ 2 ðrÞ þ ðN 1Þr2o ðrÞ; E½S_ N ðrÞ ffi N lðrÞ r 2 f1; 2; . . .; zg;
ð6:31Þ
_ rÞ of exceedings by the system the reliability critical (c) the optimal number Nðt; state r up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 1 0 C B BðN þ 1ÞðlðrÞ _ þ lo ðrÞÞ t l0 ðrÞC _ B PðNðt; rÞ ¼ NÞ ffi FNð0;1Þ B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C A @ t þ l0 ðrÞ ðr_ 2 ðrÞ þ r2o ðrÞÞ _ lðrÞ þ l0 ðrÞ 1 0 C B B NðlðrÞ _ þ lo ðrÞÞ t l0 ðrÞ C C; s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi FNð0;1Þ B C B A @ t þ l0 ðrÞ 2 2 ðr_ ðrÞ þ ro ðrÞÞ _ lðrÞ þ l0 ðrÞ N ¼ 0; 1; . . .;
r 2 f1; 2; . . .; zg;
ð6:32Þ
270
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ rÞ of (d) the expected value and the variance of the optimal number Nðt; exceedings by the system the reliability critical state r up to the moment t; t 0; for sufficiently large t, are respectively given by _ rÞ ffi Hðt;
t þ l0 ðrÞ t þ l0 ðrÞ _ rÞ ffi ; Dðt; ðr_ 2 ðrÞ þ r2o ðrÞÞ; _ lðrÞ þ lo ðrÞ _ ðlðrÞ þ lo ðrÞÞ3 r 2 f1; 2; . . .; zg;
ð6:33Þ
_ SN ðrÞ until the Nth system’s renovation, for sufficiently (e) the optimal time _ large N, has approximately normal distribution NðNðlðrÞ þ lo ðrÞÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nðr_ 2 ðrÞ þ r2o ðrÞÞÞ, i.e., ! _ þ lo ðrÞÞ t NðlðrÞ _ _ ðNÞ F ðt;rÞ ¼ PðSN ðrÞ\tÞ ffi FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t 2 ð1;1Þ; Nðr_ 2 ðrÞ þ r2o ðrÞÞ r 2 f1; 2;...;zg;
ð6:34Þ
_ (f) the expected value and the variance of the optimal time SN ðrÞ until the Nth system’s renovation, for sufficiently large N, are respectively given by _ _ _ þ lo ðrÞÞ; D½ SN ðrÞ ffi Nðr_ 2 ðrÞ þ r2o ðrÞÞ; E½SN ðrÞ ffi NðlðrÞ r 2 f1; 2; . . .; zg;
ð6:35Þ
_ rÞ of system’s renovations up to the moment t; t 0; (g) the optimal number Nðt; for sufficiently large t, has approximately distribution of the form 0 1 B ðN þ 1ÞðlðrÞ _ þ lo ðrÞÞ t C _ rÞ ¼ NÞ ffi F B C PðNðt; Nð0;1Þ @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA t 2 2 _ ðr ðrÞ þ ro ðrÞÞ _ lðrÞ þ l0 ðrÞ 0 1 B C _ NðlðrÞ þ lo ðrÞÞ t ffiC FNð0;1Þ B @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A; t ðr_ 2 ðrÞ þ r2o ðrÞÞ _ lðrÞ þ l0 ðrÞ N ¼ 0; 1; . . .;
r 2 f1; 2; . . .; zg;
ð6:36Þ
_ rÞ of system’s (h) the expected value and the variance of the optimal number Nðt; renovations up to the moment t; t 0; for sufficiently large t, are respectively given by _ rÞ ffi Hðt;
t t _ rÞ ffi ðr_ 2 ðrÞ þ r2o ðrÞÞ; ; Dðt; _ lðrÞ þ lo ðrÞ _ ðlðrÞ þ lo ðrÞÞ3 r 2 f1; 2; . . .; zg;
ð6:37Þ
6.2 Optimization of Operation, Reliability and Safety and Cost Analysis
(i)
the optimal steady availability coefficient of the system at the moment t; t 0; for sufficiently large t, is given by _ rÞ ffi Aðt;
(j)
271
_ lðrÞ ; _ lðrÞ þ lo ðrÞ
t 0;
r 2 f1; 2; . . .; zg;
ð6:38Þ
the optimal steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; for sufficiently large t, is given by _ s; rÞ ffi Aðt;
1 _ lðrÞ þ lo ðrÞ
Z1
_ rÞdt; Rðt;
t 0;
s [ 0;
r 2 f1; 2; . . .; zg;
s
ð6:39Þ _ rÞ is given by the formula (6.21) and lðrÞ _ _ where Rðt; and rðrÞ are given by (6.19 and 6.22) for u ¼ r:
6.2.4 Optimal Sojourn Times of Complex Technical System Operation Processes at Operation States and Operation Strategy Replacing in (2.22) the limit transient probabilities pb of the system operation process at the operation states by their optimal values p_ b found in Sect. 6.2.1 and the mean values Mb of the unconditional sojourn times at the operation states by _ b maximizing the mean value of the their corresponding unknown optimal values M system lifetime in the reliability states subset fr; r þ 1; . . .; zg defined by (6.1), we get the system of equations _b pb M ; p_ b ¼ Pv _ l¼1 pl Ml
b ¼ 1; 2; . . .; v:
ð6:40Þ
After simple transformations the above system takes the form _ 1 þ p_ 1 p2 M _ 2 þ þ p_ 1 pm M _m¼0 ðp_ 1 1Þp1 M _ 1 þ ðp_ 2 1Þp2 M _ 2 þ þ p_ 2 pm M _m¼0 p_ 2 p1 M .. . _ _ _ m ¼ 0; p_ m p1 M1 þ p_ m p2 M2 þ þ ðp_ m 1Þpm M
ð6:41Þ
_ b are unknown variables that we want to find, p_ b are optimal transient where M probabilities determined by (6.17) and pb are steady probabilities determined by (2.23). Since the system of equations (6.41) is homogeneous and it can be proved that the determinant of its main matrix is equal to zero, then it has nonzero solutions and moreover, these solutions are ambiguous. Thus, if we fix some of the optimal values
272
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ b of the mean values Mb of the unconditional sojourn times at the operation states, M for instance by arbitrarily fixing one or a few of them, we may find the values of the remaining ones and in this way get the solution of this equation. _ bl of the Having this solution, it is also possible to look for the optimal values M mean values Mbl of the conditional sojourn times at the operation states using the following system of equations v X
_ b; _ bl ¼ M pbl M
b ¼ 1; 2; . . .; v;
ð6:42Þ
l¼1
_ b and Mbl by M _ bl ; where pbl are known obtained from (2.21) by replacing Mb by M probabilities of the system operation process transitions between the operation states zb i zl, b; l ¼ 1; 2; . . .; v; b 6¼ l; defined by (2.22). Another very useful and much easier to be applied in practice tool that can help in planning the operation processes of complex technical systems are the system operation process optimal mean values of the total system operation process sojourn times ^hb at the particular operation states zb ; b ¼ 1; 2; . . .; v; during the fixed system operation time h; which can be obtained by replacing in the formula (2.24) the transient probabilities pb at the operation states zb by their optimal values p_b and resulting in the following expression _ ^ E½ hb ¼ p_ b h;
b ¼ 1; 2; . . .; v:
ð6:43Þ
_ b of the mean values of the unconditional The knowledge of the optimal values M _ sojourn times and the optimal values Mbl of the mean values of the conditional _ ^hb of the total sojourn times at the operation states and the optimal mean values E½ sojourn times at the particular operation states during the fixed system operation time may be the basis for changing the complex technical systems’ operation processes in order to ensure that these systems operates more reliably and safely. Their knowledge may also be useful in these systems’ operation cost analysis.
6.2.5 Cost Analysis of Complex Technical System Operations We consider the complex technical multistate system consisted of n components and we assume that the operation costs of its single basic components at the operation state zb, b ¼ 1; 2; . . .; v; during the system operation time h; h 0; amount to ci ðh; bÞ;
i ¼ 1; 2; . . .; n:
First, we suppose that the system is non-repairable, i.e. the system during the operation has not exceeded the critical reliability state r. In this case, the total cost of the non-repairable system during the operation time h; h 0; is given by
6.2 Optimization of Operation, Reliability and Safety and Cost Analysis
CðhÞ ffi
m X
pb
b¼1
n X
ci ðh; bÞ;
h 0;
273
ð6:44Þ
i¼1
where pb ; b ¼ 1; 2; . . .; m; are transient probabilities defined by (2.22). Further, we additionally assume that the system is repairable after exceeding the critical reliability state r, its renovation time is ignored and the cost of the system singular renovation is cig : Then, the total operation cost of the repairable system to ignoring its renovation time during the operation time h; h 0; amounts to Cig ðhÞ ffi
m X
pb
n X
ci ðh; bÞ þ cig Hðh; rÞ;
h 0;
ð6:45Þ
i¼1
b¼1
where pb ; b ¼ 1; 2; . . .; m; are transient probabilities defined by (2.22) and Hðh; rÞ is the mean value of the number of exceedings of the critical reliability state r by the system operating at the variable conditions during the operation time h defined by (3.58). Now, we assume that the system is repairable after exceeding the critical reliability state r and its renewal time is non-ignored and have a distribution function with mean value l0 ðrÞand standard deviation r0 ðrÞ and the cost of the system singular renovation is cnig : Then, the total operation cost of the repairable system while not ignoring its renovation time during the operation time h; h 0; amounts to Cnig ðhÞ ffi
m X b¼1
pb
n X
rÞ; ci ðh; bÞ þ cnig Hðh;
h 0;
ð6:46Þ
i¼1
rÞ where pb ; b ¼ 1; 2; . . .; m; are transient probabilities defined by (2.22) and Hðh; is the mean value of the number of renovations of the system operating at the variable conditions during the operation time h defined by (3.92). rÞ for the The particular expressions for the mean values Hðh; rÞ and Hðh; repairable systems with ignored and non-ignored renovation times existing in the formulae (6.45) and (6.46), respectively defined by (3.58) and (3.92), are determined in Chap. 3 for typical multistate repairable systems, i.e. for multistate series, parallel, ‘‘m out of n’’, consecutive ‘‘m out of n: F’’, series-parallel, parallel-series, series‘‘m out of k’’, ‘‘mi out of li’’-series, series-consecutive ‘‘m out of k: F’’ and consecutive ‘‘mi out of li: F’’-series systems operating at the variable operating conditions. After the optimization of the system operation process, the system operation total costs given by (6.44–6.46) assume their optimal values. The total optimal cost of the non-repairable system during the operation time h; h 0; after its operation process optimization is given by
274
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ CðhÞ ¼
m X
p_ b
b¼1
n X
ci ðh; bÞ;
h 0;
ð6:47Þ
i¼1
where p_ b ; b ¼ 1; 2; . . .; m; are optimal transient probabilities defined by (6.17). The optimal total operation cost of the repairable system ignoring its renovation time during the operation time h; h 0; after its operation process optimization amounts to C_ ig ðhÞ ffi
m X
p_ b
n X
_ ci ðh; bÞ þ cig Hðh; rÞ;
h 0;
ð6:48Þ
i¼1
b¼1
where p_ b ; b ¼ 1; 2; . . .; m; are optimal transient probabilities defined by (6.17) and _ Hðh; rÞ is the mean value of the optimal number of exceedings of the critical reliability state r by the system operating at the variable conditions during the operation time h defined by (6.29). The total optimal operation cost of the repairable system while non-ignoring its renovation time during the operation time h; h 0; after its operation process optimization amounts to C_ nig ðhÞ ffi
m X b¼1
p_ b
n X
_ rÞ; ci ðh; bÞ þ cnig Hðh;
h 0;
ð6:49Þ
i¼1
where p_ b ; b ¼ 1; 2; . . .; m; are optimal transient probabilities defined by (6.17) and _ Hðh; rÞ is the mean value of the optimal number of renovations of the system operating at the variable operating conditions during the operation time h defined by (6.37). _ _ The particular expressions for the optimal mean values Hðh; rÞ and Hðh; rÞ for the repairable systems with ignored and non-ignored renovation times existing in the formulae (6.48) and (6.49), respectively defined by (6.29) and (6.37), may be obtained by replacing the transient probabilities pb by their optimal values p_ b in the rÞ defined by (3.58) and (3.92) which are expressions for Hðh; rÞ and Hðh; determined in Chap. 3 for typical multistate repaired systems, i.e. for multistate series, parallel, ‘‘m out of n’’, consecutive ‘‘m out of n: F’’, series-parallel, parallelseries, series-‘‘m out of k’’, ‘‘mi out of li’’-series, series-consecutive ‘‘m out of k: F’’ and consecutive ‘‘mi out of li: F’’-series systems operating at variable operating conditions. The application of the formulae (6.44–6.46) and (6.47–6.49) allow us to compare the costs of the non-repairable and repairable systems with ignored and non-ignored times of renovations operating at variable operating conditions before and after the optimization of their operation processes.
6.3 Corrective and Preventive Maintenance Policy Optimization of Complex
275
6.3 Corrective and Preventive Maintenance Policy Optimization of Complex Technical Systems 6.3.1 Corrective and Preventive Maintenance Policy Maximizing Availability of Complex Technical System We consider the maintenance policy applied to the complex technical multistate system operating at the variable conditions which includes both corrective renovation taken upon the system exceeding the critical reliability state r and preventive renovation taken periodically with the period of time g; g 0: As before, we denote by TðrÞ; r ¼ 1; 2; . . .; z; the system lifetime in the reliability states subset fr; r þ 1; . . .; zg and by l0 ðrÞ the mean value of the system corrective renovation time upon the system exceeding the critical reliability state r. Additionally, we mark by l1 ðrÞ the mean value of the system preventive renovation time upon the system correction after each period of time g independently of the system exceeding the critical reliability state r. In both cases, after renovation, the system becomes ‘‘as good as new’’, i.e. it is in the best reliability state z. Under these assumptions, the system renovation can be performed either at the moment t; t\g; or at the moment g and the system mean time between the renovations is given by lU ðg; rÞ ¼ PfTðrÞ ¼ t \ t\gg t þ PfTðrÞ ¼ t \ t gg g. Hence lU ðg; rÞ ¼
¼
¼
¼
Zg
tf ðt; rÞdt þ PfTðrÞ gg g
0 Zþ1
Zþ1
tf ðt; rÞdt
0
g
Zþ1
Zþ1
Rðt; rÞdt
0
g
Zg
Zþ1
0
Rðt; rÞdt þ
tf ðt; rÞdt þ Rðg; rÞg
tf ðt; rÞdt þ Rðg; rÞg
Rðt; rÞdt
g
Zþ1
tf ðt; rÞdt þ Rðg; rÞg;
g
where Rðt; rÞ ¼ PfTðrÞ [ tg;
t 2 h0; þ1Þ;
is the rth coordinate of the system reliability function Rðt; Þ ¼ ½Rðt; 1Þ; Rðt; 2Þ; . . .; Rðt; zÞ;
t 2 h0; þ1Þ;
276
6 Complex Technical Systems Operation, Reliability, Availability, Safety
and f ðt; r Þ ¼
d 0 Rðt; rÞ ¼ R ðt; rÞ dt
is its corresponding density function. From the above, after integrating by parts, we get Zg Zþ1 Zþ1 þ1 lU ðg; rÞ ¼ Rðt; rÞdt þ Rðt; rÞdt þ tRðt; rÞjg Rðt; rÞdt þ Rðg; rÞg 0
g
g
and finally lU ðg; rÞ ¼
Zg Rðt; rÞdt:
ð6:50Þ
0
As the system renovation can be performed either at the moment t; t\g; with the mean value of the system corrective renovation time l0 ðrÞ or at the moment g with the mean value of the system preventive renovation time l1 ðrÞ, the system mean renovation time is given by lD ðg; rÞ ¼ PfTðrÞ ¼ t \ t\gg l0 ðrÞ þ PfTðrÞ ¼ t \ t gg l1 ðrÞ ¼ ½1 Rðg; rÞl0 ðrÞ þ Rðg; rÞl1 ðrÞ:
ð6:51Þ
After applying the results (6.50) and (6.51), we get the formula for the availability coefficient of the system Rg lU ðg; rÞ 0 Rðt; rÞdt R ; ¼ g Aðg; rÞ ffi lU ðg; rÞ þ lD ðg; rÞ 0 Rðt; rÞdt þ ½1 Rðg; rÞl0 ðrÞ þ Rðg; rÞl1 ðrÞ g 0:
ð6:52Þ
Our objective now is that of finding the optimal value g_ of the preventive maintenance period g that maximizes Aðg; rÞ given by (6.52). To solve this problem, we find the derivative of Aðg; rÞ with respect to g that takes the form Rg dAðg; rÞ Rðg; rÞ½ 0 Rðt; rÞdt þ ½1 Rðg; rÞl0 ðrÞ þ Rðg; rÞl1 ðrÞ ¼ Rg dg ½ 0 Rðt; rÞdt þ ½1 Rðg; rÞl0 ðrÞ þ Rðg; rÞl1 ðrÞ2 Rg Rðt; rÞdt½Rðg; rÞ R0 ðg; rÞl0 ðrÞ þ R0 ðg; rÞl1 ðrÞ 0R g ½ 0 Rðt; rÞdt þ ½1 Rðg; rÞl0 ðrÞ þ Rðg; rÞl1 ðrÞ2 and further we get
6.3 Corrective and Preventive Maintenance Policy Optimization of Complex
277
dAðg; rÞ dg Rg R0 ðg; rÞ½l1 ðrÞ l0 ðrÞ 0 Rðt; rÞdt þ Rðg; rÞl0 ðrÞ þ R2 ðg; rÞ½l1 ðrÞ l0 ðrÞ ¼ : Rg ½ 0 Rðt; rÞdt þ ½1 Rðg; rÞl0 ðrÞ þ Rðg; rÞl1 rÞ2 ð6:53Þ Since Rðg; rÞ is a non-increasing function of g, R0 ðg; rÞ 0 and from (6.53) it follows that in the case when l0 ðrÞ l1 ðrÞ the inequality dAðg; rÞ [0 dg
for g 0
holds, which means that Aðg; rÞ is an increasing function of g independently of the form of Rðg; rÞ. Thus, there is no optimal value g_ of the preventive maintenance period g that maximizes Aðg; rÞ given by (6.52). If l0 ðrÞ [ l1 ðrÞ and either dAðg; rÞ [0 dg
for g 0
or dAðg; rÞ \0 for g 0 dg then also there is no optimal value g_ of the preventive maintenance period g that maximizes Aðg; rÞ given by (6.52). Finally, the optimal value g_ of the preventive maintenance period g that maximizes Aðg; rÞ given by (6.51) can only be found when l0 ðrÞ [ l1 ðrÞ and there exists the solution g 0 of the equality dAðg; rÞ ¼0 dg Thus, after joining (6.53) and (6.54), we get the equation
ð6:54Þ
278
6 Complex Technical Systems Operation, Reliability, Availability, Safety
0
R ðg; rÞ½l0 ðrÞ l1 ðrÞ
Zg
Rðt; rÞdt R2 ðg; rÞ½l0 ðrÞ l1 ðrÞ ¼ Rðg; rÞl0 ðrÞ
0
ð6:55Þ from which the optimal value g_ of g can be determined. From (6.55), using simple arithmetical operations, we can obtain a slightly simpler equation Zg R 0 ðg; rÞ l0 ðrÞ Rðt; rÞdt þ Rðg; rÞ ¼ ð6:56Þ Rðg; rÞ l0 ðrÞ l1 ðrÞ 0
from which the optimal value g_ of g can be determined. Introducing the rate of departure from the system reliability states set fr; r þ 1; . . .; zg kðg; rÞ ¼
R0 ðg; rÞ Rðg; rÞ
ð6:57Þ
from (6.56), we obtain the following expression Zg kðg; rÞ 0
Rðt; rÞdt þ Rðg; rÞ ¼
l0 ðrÞ : l0 ðrÞ l1 ðrÞ
ð6:58Þ
Substituting either Rðg; rÞ into Eq. 6.56 or Rðg; rÞ and kðg; rÞ into Eq. 6.58 we may find the optimal value g_ of g. Later substituting this value into the expression (6.52) _ rÞ of the availability coefficient Aðg; rÞ of the we may obtain the maximum value Aðg; system. Unfortunately, as a rule, it is not possible to find the explicit solution of Eqs. 6.56 and 6.58 and they have to be found by the approximate methods. One of the approximate methods suggested to optimize the complex technical system’s corrective and preventive maintenance policy maximizing its availability is the method of secants which we apply by performing the following procedure: • we check the relationship between the mean value of the system corrective renovation time l0 ðrÞ and the mean value of the system preventive renovation time l1 ðrÞ; • if the inequality l0 ðrÞ l1 ðrÞ holds, then there is no optimal value g_ of the preventive maintenance period g which maximizes the availability coefficient of the system Aðg; rÞ given by (6.52) which is an increasing function of g and we determine its values for the arbitrarily fixed values of the preventive maintenance period g1 ; g2 ; . . .; gj ;
6.3 Corrective and Preventive Maintenance Policy Optimization of Complex
279
• if the inequality is l0 ðrÞ [ l1 ðrÞ we are looking for the optimal value g_ of the preventive maintenance period g which maximizes the availability coefficient of the system Aðg; rÞ given by (6.52) by determining, if it exists, its approximate value from Eq. 6.58 by applying the method of secants in the interval ha; bi as follows: • we define the function f ðgÞ ¼ kðg; rÞ
Zg
Rðt; rÞdt þ Rðg; rÞ
0
l0 ðrÞ l0 ðrÞ l1 ðrÞ
for g 0;
where kðg; rÞ is given by (6.57), • we define the interval ha; bi assuming a ¼ 0 and finding b such that f ðbÞ [ 0; • we use the recurrent formula g0 ¼ a; gkþ1 ¼ gk
f ðgk Þ ðb gk Þ f ðbÞ f ðgk Þ
for k ¼ 0; 1; . . .; K;
where K is such that f ðgKþ1 Þ\e and e is the measure of the method of secants accuracy, • we fix the optimal value g_ of the preventive maintenance period of time g maximizing the system availability assuming g_ ¼ gKþ1 ; _ rÞ of the availability coefficient Aðg; rÞ and determine the maximum value Aðg; of the system substituting this optimal value g_ for g in expression (6.52) in the case when the recurrent procedure is convergent. Otherwise, there is no optimal value g_ of the preventive maintenance period g that maximizes the availability coefficient of the system and we find its value for the arbitrarily fixed values of the preventive maintenance period of time g1 ; g2 ; . . .; gj :
6.3.2 Corrective and Preventive Maintenance Policy Minimizing Cost of Renovation of Complex Technical Systems We consider the maintenance policy applied to complex technical multistate systems operating in variable conditions which includes both a corrective
280
6 Complex Technical Systems Operation, Reliability, Availability, Safety
renovation taken upon the system exceeding the critical reliability state r and a preventive renovation taken at the system’s fixed age 1; 1 [ 0: As in Sect. 6.3.1, we denote by TðrÞ; r ¼ 1; 2; . . .; z; the system lifetime in the reliability states subset fr; r þ 1; . . .; zg and by Rðt; rÞ ¼ PfTðrÞ [ tg;
t 2 h0; þ1Þ;
the rth coordinate of the system reliability function Rðt; Þ ¼ ½Rðt; 1Þ; Rðt; 2Þ; . . .; Rðt; zÞ;
t 2 h0; þ1Þ;
and by f ðt; rÞ ¼
d 0 Rðt; rÞ ¼ R ðt; rÞ; dt
the density function corresponding to the coordinate Rðt; rÞ: Moreover, we denote by c0 ðrÞ the mean value of the cost of the system corrective renovation upon the system exceeding the critical reliability state r and by c1 ðrÞ the mean value of the cost of the system preventive renovation upon the system correction after reaching by the system the age 1; independently of the system exceeding the critical reliability state r. We assume that the cost c0 ðrÞ is greater than the cost c1 ðrÞ and that the system intensity of departure kðt; rÞ from the system reliability states fr; r þ 1; . . .; zg is non decreasing, which are natural assumptions. Under these assumptions, the system renovation can be performed in one cycle up to the first renovation either at the moment t; t\1; because of exceeding by the system the critical reliability state r with the probability PfTðrÞ 1g ¼ 1 Rð1; rÞ or at the moment 1 because of reaching by the system the age 1 with the probability PfTðrÞ [ 1g ¼ Rð1; rÞ: Therefore, the expected value of the length of the time of one cycle up to the first renovation is given by LC ð1; rÞ ¼ Lð1; rÞ ½1 Rð1; rÞ þ 1 Rð1; rÞ;
ð6:59Þ
where Lð1; rÞ ¼
Z1
f ðt; rÞ t dt ¼ 1 Rð1; rÞ
0
0
t
R ðt; rÞ dt 1 Rð1; rÞ
0
1Rð1; rÞ þ ¼
Z1
R1
Rð1; rÞdt
0
1 Rð1; rÞ
ð6:60Þ
6.3 Corrective and Preventive Maintenance Policy Optimization of Complex
281
is the conditional expected value of the length of the time up to exceeding by the system the critical reliability state r before the system reaches the age 1. From the assumptions, it also follows that the expected value of the cost of the system renovation during one cycle is CC ð1; rÞ ¼ c0 ðrÞ ½1 Rð1; rÞ þ c1 ðrÞ Rð1; rÞ:
ð6:61Þ
Now, using (6.59) and (6.61), we may define the expected cost of the system renovation per unit time as Cð1; rÞ ¼
CC ð1; rÞ c0 ðrÞ½1 Rð1; rÞ þ c1 ðrÞRð1; rÞ ¼ : LC ð1; rÞ Lð1; rÞ½1 Rð1; rÞ þ 1 Rð1; rÞ
ð6:62Þ
The above formula relates the renovation age 1 to the expected renovation cost per unit time. Taking the derivative of the function Cð1; rÞ given by (6.62) with respect to 1 we get dCð1; rÞ ½c0 ðrÞ c1 ðrÞf ð1; rÞ½Lð1; rÞ½1 Rð1; rÞ þ 1 Rð1; rÞ ¼ d1 ½Lð1; rÞ½1 Rð1; rÞ þ 1Rð1; rÞ2
½c0 ðrÞ ½c0 ðrÞ c1 ðrÞRð1; rÞ½1f ð1; rÞ þ Lð1; rÞf ð1; rÞ þ Rð1; rÞ 1 f ð1; rÞ ½Lð1; rÞ½1 Rð1; rÞ þ 1 Rð1; rÞ2
and then dCð1; rÞ ½c0 ðrÞ c1 ðrÞ½1 f ð1; rÞRð1; rÞ þ R2 ð1; rÞ c0 ðrÞRð1; rÞ c1 ðrÞf ð1; rÞLð1; rÞ ¼ d1 ½Lð1; rÞ½1 Rð1; rÞ þ 1 Rð1; rÞ2
ð6:63Þ Hence, the equality dCð1; rÞ ¼0 d1 holds if and only if R2 ð1; rÞ þ 1 f ð1; rÞRð1; rÞ ¼
c0 ðrÞRð1; rÞ þ c1 ðrÞf ð1; rÞLð1; rÞ c0 ðrÞ c1 ðrÞ
ð6:64Þ
Thus, finally, we get 0
c0 ðrÞRð1; rÞ c1 ðrÞR ð1; rÞLð1; rÞ ; R ð1; rÞ 1R ð1; rÞRð1; rÞ ¼ c0 ðrÞ c1 ðrÞ 2
0
ð6:65Þ
where Lð1; rÞ is given by (6.60). 0 Substituting Rð1; rÞ and R ð1; rÞ into Eq. 6.65 we receive the expression from which we may find the optimal value 1_ of the system age 1 at which the system preventive maintenance minimizing the cost of the system renovation is
282
6 Complex Technical Systems Operation, Reliability, Availability, Safety
performed. Afterwards, substituting this value into expression (6.62), we may obtain the minimum value Cð_1; rÞ of the expected cost Cð1; rÞof the system renovation per unit time. Unfortunately, very often, it is not possible to find the explicit solution of Eq. (6.65) and it has to be solved by approximate methods. One of the approximate methods suggested to optimize the complex technical system corrective and preventive maintenance policy minimizing its cost of renovation is the method of secants that we apply by performing the following procedure: • we check the relationship between the mean value of the cost of the system corrective renovation c0 ðrÞ and the mean value of the cost of the system preventive renovation c1 ðrÞ; • if the inequality c0 ðrÞ c1 ðrÞ holds, then there is no optimal value 1_ of the system age 1 which minimizes the system renovation cost per unit time Cð1; rÞgiven by (6.62) and we determine its values for the arbitrarily fixed values of the system age at which the system preventive maintenance is performed 11 ; 12 ; . . .; 1j ; • if the inequality c0 ðrÞ [ c1 ðrÞ we are looking for the optimal value 1_ of the system age 1 which minimizes the system renovation cost per unit time Cð1; rÞgiven by (6.62) by determining, if it exists, its approximate value from Eq. 6.65 by applying the method of secants in the interval ha; bi as follows: • we define the function 0
0
wð1Þ ¼ R2 ð1; rÞ 1R ð1; rÞRð1; rÞ
c0 ðrÞRð1; rÞ c1 ðrÞR ð1; rÞLð1; rÞ for c0 ðrÞ c1 ðrÞ 1 [ 0;
where Lð1; rÞ is given by (6.60), • we define the interval ha; bi assuming a ¼ 0 and finding b such that wðbÞ [ 0; • we use the recurrent formula 10 ¼ a; 1kþ1 ¼ 1k
wðgk Þ ðb 1k Þ for k ¼ 0; 1; . . .; K; wðbÞ wðgk Þ
where K is such that wð1Kþ1 Þ\e and e is the measure of the method of secants accuracy,
6.3 Corrective and Preventive Maintenance Policy Optimization of Complex
283
• we fix the optimal value 1_ of the system age 1 at which the system preventive maintenance minimizing the cost of the system renovation is performed assuming 1_ ¼ 1Kþ1 ; and we determine the minimum value Cð_1; rÞ of the expected cost Cð1; rÞ of the system renovation per unit time substituting this optimal value 1_ for 1 in to expression (6.62) in the case when the recurrent procedure is convergent. Otherwise, there is no optimal value 1_ of the system age 1 which minimizes the system renovation cost and we find its values for the arbitrarily fixed values of the system age at which the system preventive maintenance is performed 11 , 12 , . . .; 1j .
6.4 Applications 6.4.1 Optimization of Operation, Reliability and Availability and Cost Analysis of Exemplary System The considered exemplary system at the variable operating conditions is analyzed in Sect. 3.6.1 of Chap. 3. Its reliability, renewal and availability characteristics are found there. From these results, it follows that it is possible to improve these characteristics by changing the parameters of its operation process and after that its operation cost analysis can also be performed.
6.4.1.1 Optimization of Exemplary System Operation Process The objective function defined by (6.1), in this case, as the exemplary system critical state is r ¼ 2, takes the form lð2Þ ¼ p1 483:87 þ p2 694:44 þ p3 383:04 þ p4 253:88: ^
ð6:66Þ
_
Arbitrarily assumed, the lower pb and upper pb bounds of the unknown transient probabilities pb ; b ¼ 1; 2; 3; 4; respectively are: ^
^
_
_
^
^
p1 ¼ 0:201; p2 ¼ 0:03; p3 ¼ 0:245; p4 ¼ 0:309; _
_
p1 ¼ 0:351; p2 ¼ 0:105; p3 ¼ 0:395; p4 ¼ 0:459: Therefore, according to (6.2–6.3), we assume the following bound constraints 0:201 p1 0:351; 0:03 p2 0:105; 0:245 p3 0:395; 0:309 p4 0:459:
ð6:67Þ
284
6 Complex Technical Systems Operation, Reliability, Availability, Safety 4 X
pb ¼ 1;
ð6:68Þ
b¼1
Now, before we find optimal values p_ b of the transient probabilities pb ; b ¼ 1; 2; 3; 4; that maximize the objective function (6.66), w arranges the system conditional lifetime mean values lb ð2Þ; b ¼ 1; 2; 3; 4; in non-increasing order l2 ð2Þ l1 ð2Þ l3 ð2Þ l4 ð2Þ: Further, according to (6.6), we substitute x1 ¼ p2 ;
x2 ¼ p1 ;
x3 ¼ p3 ;
x 4 ¼ p4 ;
ð6:69Þ
and ^
^
^
x1 ¼ p2 ¼ 0:03;
^
^
x2 ¼ p1 ¼ 0:201;
^
x 3 ¼ p3 ¼ 0:245;
^
^
x4 ¼ p4 ¼ 0:309; ð6:70Þ
_
_
_
x1 ¼ p2 ¼ 0:105;
_
x2 ¼ p1 ¼ 0:351;
_
_
x 3 ¼ p3 ¼ 0:395;
_
_
x4 ¼ p4 ¼ 0:459; ð6:71Þ
and we maximize with respect to xi ; i ¼ 1; 2; 3; 4; the linear form (6.66) which according to (6.7–6.9) takes the form lð2Þ ¼ x1 694:44 þ x2 483:87 þ x3 383:04 þ x4 253:88;
ð6:72Þ
with the following bound constraints 0:03 x1 0:105; 0:201 x2 0:351; 0:245 x3 0:395; 0:309 x4 0:459: 4 X
ð6:73Þ
xi ¼ 1:
ð6:74Þ
4 X ^ ^ xi ¼ 0:785; ^y ¼ 1 x ¼ 1 0:785 ¼ 0:215
ð6:75Þ
i¼1
According to (6.11), we calculate ^
x¼
i¼1
and according to (6.12), we determine ^0
x ¼ 0;
_0
x ¼ 0;
^1
x ¼ 0:03;
^2
x ¼ 0:231;
_0
^0
x x ¼ 0;
_1
x ¼ 0:105; _2
x ¼ 0:456;
_1
^1
x x ¼ 0:075; _2
^2
x x ¼ 0:225;
6.4 Applications
285
^3
x ¼ 0:476:
^4
x ¼ 0:785;
_3
x ¼ 0:851; _4
_3 _4
x ¼ 1:31;
^3
x x ¼ 0:375; ^4
x x ¼ 0:525:
ð6:76Þ
From the above, as according to (6.75), the inequality (6.13) takes the form _I
^I
x x \0:215;
ð6:77Þ
it follows that the largest value I 2 f0; 1; 2; 3; 4g such that this inequality holds is I ¼ 1: Therefore, we fix the optimal solution that maximizes the linear function (6.72) according to rule (6.15). Namely, we get _
x_ 1 ¼ x 1 ¼ 0:105; _1
^1
^
x_ 2 ¼ ^y x þ x þ x2 ¼ 0:215 0:105 þ 0:03 þ 0:201 ¼ 0:341; ^
^
x_ 3 ¼ x 3 ¼ 0:245; x_ 4 ¼ x4 ¼ 0:309:
ð6:78Þ
Finally, after making the inverse substitution to (6.69), we get the optimal transient probabilities p_ 2 ¼ x_ 1 ¼ 0:105; p_ 1 ¼ x_ 2 ¼ 0:341; p_ 3 ¼ x_ 3 ¼ 0:245; p_ 4 ¼ x_ 4 ¼ 0:309; ð6:79Þ that maximize the exemplary system mean lifetime lð2Þ in the reliability state subset f2; 3g expressed by the linear form (6.66) giving, according to (6.18) and (6.79), its optimal value _ lð2Þ ¼ p_ 1 483:87 þ p_ 2 694:44 þ p_ 3 383:04 þ p_ 4 253:88 ¼ 0:341 483:87 þ 0:105 694:44 þ 0:245 383:04 þ 0:309 253:88 ffi 410:21
ð6:80Þ
6.4.1.2 Optimal Reliability Characteristics of Exemplary System Substituting the optimal solution (6.79) into the formula (6.19), we obtain the optimal solution for the mean values of the exemplary system unconditional lifetimes in the reliability state subsets f1; 2; 3g and f3g; that are as follows _ lð1Þ ¼ p_ 1 505 þ p_ 2 744:05 þ p_ 3 405:56 þ p_ 4 271:08 ¼ 0:341 505 þ 0:105 744:05 þ 0:245 405:56 þ 0:309 271:08 ffi 433:46;
ð6:81Þ
286
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ lð3Þ ¼ p_ 1 468:73 þ p_ 2 651:04 þ p_ 3 370:67 þ p_ 4 237:05 ¼ 0:341 468:73 þ 0:105 651:04 þ 0:245 370:67 þ 0:309 237:05 ffi 392:26;
ð6:82Þ
and according to (6.23), the optimal values of the mean values of the system unconditional lifetimes in the particular reliability states 1, 2 and 3, respectively are _ ð1Þ ¼ lð1Þ _ _ _ ð2Þ ¼ lð2Þ _ _ l lð2Þ ¼ 23:25; l lð3Þ ¼ 17:95; _ lð3Þ ¼ lð3Þ ¼ 392:26:
ð6:83Þ
Moreover, according to (6.20–6.21), the corresponding optimal unconditional multistate reliability function of the system is of the form _ Þ ¼ ½1; Rðt; _ 1Þ; Rðt; _ 2Þ Rðt; _ 3Þ; Rðt;
t 0;
ð6:84Þ
with the coordinates given by _ 1Þ ¼ 0:341 ½Rðt; 1Þð1Þ þ 0:105 ½Rðt; 1Þð2Þ þ 0:245 ½Rðt; 1Þð3Þ Rðt; þ 0:309 ½Rðt; 1Þð4Þ
for t 0
ð6:85Þ
_ 2Þ ¼ 0:341 ½Rðt; 2Þð1Þ þ 0:105 ½Rðt; 2Þð2Þ þ 0:245 ½Rðt; 2Þð3Þ Rðt; þ 0:309 ½Rðt; 2Þð4Þ
for t 0;
ð6:86Þ
_ 3Þ ¼ 0:341 ½Rðt; 3Þð1Þ þ 0:105 ½Rðt; 3Þð2Þ þ 0:245 ½Rðt; 3Þð3Þ Rðt; þ 0:309 ½Rðt; 3Þð4Þ
for t 0;
ð6:87Þ
where ½Rðt; 1ÞðbÞ ; ½Rðt; 2ÞðbÞ ; ½Rðt; 3ÞðbÞ ; b ¼ 1; 2; 3; 4; are fixed in Sect. 3.6.1. _ Þ given by The graph of the exemplary system optimal reliability function Rðt; (6.87) is presented in Fig. 6.1. Further, by (6.22), the corresponding optimal variances and standard deviations of the system unconditional lifetimes in the system reliability state subsets are 2
r_ ð1Þ ¼ 2
Z1
2 _ 1Þdt ½lð1Þ _ t Rðt; ffi 110365:45;
_ rð1Þ ffi 332:21;
ð6:88Þ
2 _ 2Þdt ½lð2Þ _ t Rðt; ffi 101665:04;
_ rð2Þ ffi 318:85;
ð6:89Þ
_ rð3Þ ffi 305:65;
ð6:90Þ
0
2
r_ ð2Þ ¼ 2
Z1 0
2
r_ ð3Þ ¼ 2
Z1 0
2 _ 3Þdt ½lð3Þ _ t Rðt; ffi 93424:64;
6.4 Applications
287
_ Þ coordinates Fig. 6.1 The graph of the exemplary system optimal reliability function Rðt;
Fig. 6.2 The graph of the exemplary system optimal risk function r_ ðtÞ
_ 1Þ; Rðt; _ 2Þ; Rðt; _ 3Þ are given by (6.85–6.87) and lð1Þ; _ _ _ where Rðt; lð2Þ; lð3Þ; are given by (6.80–6.82). As the critical reliability state is r = 2, the exemplary system optimal system risk function, according to (6.24), is given by (Fig. 6.2) _ 2Þ for t 0; r_ ðtÞ ¼ 1 Rðt;
ð6:91Þ
_ 2Þ is given by (6.86). where Rðt; Hence and considering (6.25), the moment when the optimal system risk function exceeds a permitted level, for instance d = 0.05, is s_ ¼ r_ 1 ðdÞ ffi 80:
ð6:92Þ
288
6 Complex Technical Systems Operation, Reliability, Availability, Safety
6.4.1.3 Optimal Renewal and Availability Characteristics of Exemplary System To make the estimation of the renewal and availability characteristics of the exemplary system after its operation process optimization, we distinguish two cases of the system renovation, one with ignored time of renovation and the second with non-ignored time of renovation. In the case when the exemplary system renovation time is ignored, applying Proposition 6.1, we determine the following its optimal characteristics: (a) the optimal time S_ N ð2Þ until the Nth exceeding by the system the reliability critical state 2, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð410:20N; 318:85 N Þ, i.e., _F ðNÞ ðt; 2Þ ¼ PðS_ N ð2Þ\tÞ ffi FNð0;1Þ t 410:20N pffiffiffiffi ; t 2 ð1; 1Þ; 318:85 N (b) the expected value and the variance of the optimal time S_ N ð2Þ until the Nth exceeding by the system the reliability critical state 2, for sufficiently large N; respectively are E½S_ N ð2Þ ¼ 410:20N; D½S_ N ð2Þ ¼ 101665:32N; _ 2Þ of exceedings by the system of the reliability (c) the optimal number Nðt; critical state 2 up to the moment t; t 0; for sufficiently large t, has distribution approximately of the form 410:20ðN þ 1Þ t 410:20N t _ pffi pffi FNð0;1Þ ; PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 15:74 t 15:74 t N ¼ 0; 1; . . .; (d) the expected value and the variance of the optimal number Nðt; 2Þ of exceedings by the system the reliability critical state 2 up to the moment t; t 0; for sufficiently large t, respectively are _ 2Þ ¼ 0:0024t; Hðt;
_ 2Þ ¼ 0:0015t: Dðt;
ð6:93Þ
To make the estimation of the renewal and availability characteristics of the exemplary system in the case when the time of renovation is non-ignored, assuming the mean value of the system renovation time l0 ð2Þ ¼ 10 and the standard deviation of the system renovation time r0 ð2Þ ¼ 5 and applying Proposition 6.2, we determine the following optimal characteristics: S_ N ð2Þ until the Nth exceeding by the system of the reliability (a) the optimal time critical state 2, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð410:20N þ 10ðN 1Þ; 101665:32N þ 25ðN 1Þ, i.e.,
6.4 Applications
289
t 420:20N þ 10 _ ðNÞ ðt; 2Þ ¼ Pð F S_ N ð2Þ\tÞ ¼ FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 101690:32N 25
t 2 ð1; 1Þ
(b) the expected value and the variance of the optimal time S_ N ð2Þ until the Nth exceeding by the system of the reliability critical state 2, for sufficiently large N, respectively are _ E½S_ N ð2Þ ffi 410:20N þ 10ðN 1Þ; D½Sð2Þ ffi 101665:32N þ 25ðN 1Þ; _ 2Þ of exceeding by the system of the reliability (c) the optimal number Nðt; critical state 2 up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 1Þ t 10 _ 2Þ ¼ NÞ ffi FNð0;1Þ 420:20ðN þ pffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 15:56 t þ 10 420:20N t 10 pffiffiffiffiffiffiffiffiffiffiffiffiffi ; N ¼ 0; 1; . . .; FNð0;1Þ 15:56 t þ 10 _ 2Þ of exceedings (d) the expected value and the variance of the optimal number Nðt; by the system of the reliability critical state 2 up to the moment t; t 0; for sufficiently large t, respectively are _ 2Þ ffi t þ 10 ; Dðt; _ 2Þ ffi 0:0014ðt þ 10Þ; Hðt; 420:20 : ¼ (e) the optimal time SN ð2Þ until the Nth system’s renovation, for sufficiently large pffiffiffiffi N, has approximately normal distribution N(420.20N,318.89 N ), i.e., : : ¼ ¼ t 420:20N ðNÞ pffiffiffiffi ; t 2 ð1; 1Þ; F ðt; 2Þ ¼ P SN ð2Þ\t ffi FNð0;1Þ 318:89 N N ¼ 0; 1; . . .; _ (f) the expected value and the variance of the optimal time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, respectively are _ E½ SN ð2Þ ffi 420:20N;
_ ðNÞ D½ S ð2Þ ffi 101690:32N;
_ 2Þ of system’s renovations up to the moment (g) the optimal number Nðt; t, t C 0, for sufficiently large t, has approximately distribution of the form 420:20ðN þ 1Þ t 420:20N t _ 2Þ ¼ NÞ ffi F p ffi p ffi F ; PðNðt; Nð0;1Þ Nð0;1Þ 15:56 t 15:56 t N ¼ 0; 1; . . .;
290
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ 2Þ of system’s (h) the expected value and the variance of the optimal number Nðt; renovations up to the moment t; t 0; for sufficiently large t, respectively are _ 1Þ ffi 0:0024t; Hðt; (i)
_ 1Þ ffi 0:0014t; Dðt;
the optimal steady availability coefficient of the system at the moment t; t 0; for sufficiently large t, is _ 2Þ ffi 0:98; Aðt;
(j)
ð6:94Þ
t 0;
the optimal steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; t 0; for sufficiently large t, is _ s; 2Þ ffi 0:0024 Aðt;
Z1
_ 2Þdt; t 0; Rðt;
s [ 0;
s
_ 2Þ is given by (6.86). where Rðt; 6.4.1.4 Optimal Sojourn Times at Operation States of Exemplary System and Operation Strategy Substituting the exemplary operation process optimal transient probabilities at operation states p_ 1 ¼ 0:341;
p_ 2 ¼ 0:105;
p_ 3 ¼ 0:245;
p_ 4 ¼ 0:309;
given by (6.79) and the steady probabilities p1 ffi 0:236; p2 ffi 0:169; p3 ffi 0:234; p4 ffi 0:361: determined by (2.31) in Sect. 2.3.1 into (6.41), we get the following system of _ b of the exemplary system equations with the unknown optimal mean values M operation process unconditional sojourn times at the operation states we are looking for _ 1 þ 0:057629M _ 2 þ 0:079794M _ 3 þ 0:123101M _4¼0 0:155524M _ 1 0:151255M _ 2 þ 0:02457M _ 3 þ 0:037905M _4¼0 0:02478M _ 2 0:17667M _ 3 þ 0:088445M _4¼0 _ 1 þ 0:041405M 0:05782M
ð6:95Þ
_ 2 þ 0:072306M _ 3 0:249451M _ 4 ¼ 0: _ 1 þ 0:052221M 0:072924M The determinant of the main matrix of the above homogeneous system of equations is equal to zero and therefore there are non-zero solutions of this system of equations that are ambiguous and dependent on one or more parameters. Thus, we may fix some of them and determine the remaining ones. To show the way of
6.4 Applications
291
solving this system of equations, we may suppose that we are arbitrarily interested _ 4 and we put in fixing the value of M _ 4 ¼ 400: M Substituting the above value into the system of equations (6.95), we get _ 2 þ 0:079794M _ 3 ¼ 49:2404 _ 1 þ 0:057629M 0:155524M _ 2 þ 0:02457M _ 3 ¼ 15:1620 _ 1 0:151255M 0:02478M _ 1 þ 0:041405M _ 2 0:17667M _ 3 ¼ 35:3780 0:05782M _ 1 þ 0:052221M _ 2 þ 0:072306M _ 3 ¼ 99:7804; 0:072924M _ 2 and M _ 3 , after omitting its last equation. _ 1, M and we solve it with respect to M In this way, we obtained solutions that satisfy (6.95), are _ 1 ffi 675; M _ 2 ffi 290; M _ 3 ffi 490; M _ 4 ¼ 400: M
ð6:96Þ
It can be seen that these solutions differ much from the values M1 ; M2 ; M3 and M4 ; estimated in Sect. 2.3.1 and given by (2.27–2.30). _ bl of Having these solutions, it is also possible to look for the optimal values M the mean values Mbl of the exemplary system operation process conditional _ b instead of sojourn times at operation states. Namely, substituting the values M Mb, the probabilities 2 3 0 0:22 0:32 0:46 6 0:20 0 0:30 0:50 7 7 ½pbl ¼ 6 4 0:12 0:16 0 0:72 5 0:48 0:22 0:30 0 of the exemplary system operation process transitions between the operation states _ bl in (6.42), we get the given by (2.25) in Sect. 2.3.1 and replacing Mbl by M following system of equations _ 12 þ 0:32M _ 13 þ 0:46M _ 14 ¼ 675 0:22M _ 23 þ 0:50M _ 24 ¼ 290 _ 21 þ 0:30M 0:20M _ 31 þ 0:16M _ 32 þ 0:72M _ 34 ¼ 490 0:12M
ð6:97Þ
_ 42 þ 0:30M _ 14 ¼ 400 _ 41 þ 0:22M 0:48M _ bl we want to find. with the unknown optimal values M As the solutions of the above system of equations are ambiguous, we fix some of them, because of practically important reasons, and we find the remaining ones. For instance: _ 13 ¼ 500 and we find M _ 14 ffi 1024; _ 12 ¼ 200; M • we fix in the first equation M _ _ 24 ffi 480; _ • we fix in the second equation M21 ¼ 100; M23 ¼ 100 and we find M
292
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ 31 ¼ 900; M _ 32 ¼ 500 and we find M _ 34 ffi 419; • we fix in the third equation M _ _ • we fix in the fourth equation M41 ¼ 300; M42 ¼ 500 and we find _ 43 ffi 487: M
ð6:98Þ
It can be seen that these solutions differ much from the mean values of the exemplary system conditional sojourn times at the particular operation states before its operation process optimization given by (2.26). Another very useful and much easier to be applied in practice tool that can help in planning the operation process of the exemplary system are the system operation process optimal mean values of the total sojourn times at the particular operation states during the system operation time which by the same assumption as in Sect. 2.3.1 is equal to h ¼ 1year = 365 days. Under this assumption, after applying (6.43), we get the optimal values of the exemplary system operation process total sojourn times at the particular operation states during 1 year ^2 ¼ p_ 2 h ¼ 0:105 365 ffi 38:3; _ h _ ^h1 ¼ p_ 1 h ¼ 0:341 365 ffi 124:5; E½ E½ _ ^h3 ¼ p_ 3 h ¼ 0:245 365 ffi 89:4; E½ _ ^ E½ h4 ¼ p_ 4 h ¼ 0:309 365 ffi 112:8; ð6:99Þ which differ much from the values of E½^ h1 ; E½^h2 ; E½^h3 ; E½^h4 determined by (2.33) in Sect. 2.3.1. _ b; M _ bl and E½ _ ^hb given In practice, knowledge of the optimal values of M respectively by (6.96), (6.98), (6.99), can be very important and helpful for the system operation process planning and improving in order to make the system operation more reliable and safer.
6.4.1.5 Exemplary System Operation Cost Analysis In Sects. 2.3.1 and 3.6.1, it is fixed that the exemplary system is composed of n ¼ 14 components and that the numbers of the system components operating at the various operation states zb ; b ¼ 1; 2; 3; 4; are different. Namely, there are operating six system components at the operation states z1, eight system components at the operation states z2 and 14 system components at the operation states z3 and z4. According to the arbitrary assumption, the approximate mean operation costs of the single basic components of the considered exemplary system that are used during the operation time h = 1 year, are independent of the operation states zb ; b ¼ 1; 2; 3; 4; and they amount to ci ð1; bÞ ¼ 100 PLN,
b ¼ 1; 2; 3; 4;
i ¼ 1; 2; . . .; 14;
whereas, the cost of each system’s single basic component that is not used is equal to 0. In the case when the exemplary system is repaired after exceeding the critical reliability state r ¼ 2 and its renovation time is ignored, we assume that the
6.4 Applications
293
approximate cost of the system’s singular renovation is cig ¼ 1000 PLN. Similarly, in the case when the exemplary system is repaired after exceeding the critical reliability state r ¼ 2 and its renewal time is not ignored, we assume that the approximate cost of the system singular renovation is cnig ¼ 1500 PLN. First, under the above assumptions, we analyze the system operation cost before its operation process optimization. Thus, according to (6.44), the total operation cost of the non-failed exemplary system during the operation time h = 1 year amounts to Cð1Þ ffi 0:214 100 6 þ 0:038 100 8 þ 0:293 100 14 þ 0:455 100 14 ¼ 1206 PLN:
ð6:100Þ
In the case when the exemplary system is repaired after exceeding the critical reliability state r ¼ 2 and its renovation time is ignored, according to (3.163) from Sect. 3.6.1, we know that the mean value of the number of exceedings by the exemplary system of the critical reliability state during the operation time h = 1 year is Hð1; 2Þ ffi 0:0028: Thus, by (6.45). the total operation cost of the repairable exemplary system ignoring its renewal time during the operation time h = 1 year, amounts to Cig ð1Þ ffi 1206 þ 1000 0:0028 ¼ 1206 þ 2:8 ¼ 1208:8 PLN:
ð6:101Þ
In the case when the exemplary system is repaired after exceeding the critical reliability state r ¼ 2 and its renewal time is not ignored, according to (3.165) from Sect. 3.6.1, we know that the mean value of the number of the system renovations after exceeding the critical reliability state during the operation time h = 1 year is 2Þ ffi 0:0027: Hð1; Thus, by (6.46), the total operation cost of the renewed exemplary system ignoring its renewal time during the operation time h = 1 year amounts to Cnig ð1Þ ffi 1206 þ 1500 0:0027 ¼ 1206 þ 4:05 ¼ 1210:05 PLN:
ð6:102Þ
Now, we analyze the system operation cost after its operation process optimization. Proceeding afterwards, according to (6.47), the total operation cost of the nonfailed exemplary system during the operation time h = 1 year is given by _ Cð1Þ ffi 0:341 100 6 þ 0:105 100 8 þ 0:245 100 14 þ 0:309 100 14 ¼ 1064:2 PLN:
ð6:103Þ
In the case when the exemplary system is repaired after exceeding the critical reliability state r ¼ 2 and its renewal time is ignored, from (6.93), we know that
294
6 Complex Technical Systems Operation, Reliability, Availability, Safety
the mean value of the number of exceedings by the exemplary system of the critical reliability state during the operation time h = 1 year is _ Hð1; 2Þ ffi 0:0024: Thus, according to (6.48), the total operation cost of the repairable exemplary system ignoring its renewal time during the operation time h = 1 year amounts to C_ ig ð1Þ ffi 1064:2 þ 1000 0:0024 ¼ 1064:2 þ 2:4 ¼ 1066:6 PLN:
ð6:104Þ
In the case when the exemplary system is repaired after exceeding the critical reliability state r ¼ 2 and its renewal time is not ignored, from (6.95), we know that the mean value of the number of system renovations after exceeding the critical reliability state during the operation time h = 1 year is respectively given by _ Hð1; 2Þ ffi 0:0024: Thus, according to (6.49), the total operation cost of the repairable exemplary system by non-ignoring its renewal time during the operation time h = 1 year amounts to C_ nig ð1Þ ffi 1064:2 þ 1500 0:0024 ¼ 1064:2 þ 3:6 ¼ 1067:8 PLN:
ð6:105Þ
The comparison of the results (6.100–6.102) with the results (6.103–6.105) justifies the sensibility of the exemplary system operation process optimization.
6.4.2 Optimization of Operation, Reliability and Availability and Cost Analysis of Port Oil Piping Transportation System The considered port oil piping transportation system at the variable operation conditions is analyzed in Sect. 3.6.2, where its reliability, renewal and availability characteristics are found. From these results, it follows that it is possible to improve these characteristics by changing the parameters of its operation process and after that its operation cost analysis can be performed as well.
6.4.2.1 Optimization of Port Oil Piping Transportation System Operation Process The objective function defined by (6.1), in this case, as the port oil piping transportation system critical state is r ¼ 1, takes the form
6.4 Applications
295
lð1Þ ¼ p1 0:364 þ p2 0:805 þ p3 0:307 þ p4 0:207 þ p5 0:307 þ p6 0:207 þ p7 0:364: ^
ð6:106Þ
_
The lower pb and upper pb bounds of the unknown transient probabilities pb ; b ¼ 1; 2; . . .; 7; coming from experts respectively are: ^
p1 ¼ 0:31;
p2 ¼ 0:04;
^
^
p6 ¼ 0:04;
^
p7 ¼ 0:25;
_
p1 ¼ 0:46;
p2 ¼ 0:08;
_
_
_
p6 ¼ 0:08;
p3 ¼ 0:002;
^
^
p4 ¼ 0:001;
^
_
_
_
p3 ¼ 0:006;
p5 ¼ 0:15;
p4 ¼ 0:004;
p5 ¼ 0:26;
p7 ¼ 0:40:
Therefore, according to (6.2–6.4), we assume the following bound constraints 0:31 p1 0:46; 0:04 p2 0:08; 0:002 p3 0:006; 0:001 p4 0:004; 0:15 p5 0:26; 0:04 p6 0:08; 0:25 p7 0:40; 7 X
pb ¼ 1:
ð6:107Þ
b¼1
Now, before we find optimal values p_ b of the transient probabilities pb ; b ¼ 1; 2; . . .; 7; that maximize the objective function (6.1), we arrange the system conditional lifetime mean values lb ð1Þ; b ¼ 1; 2; . . .; 7; in non-increasing order l2 ð1Þ l1 ð1Þ l7 ð1Þ l3 ð1Þ l5 ð1Þ l4 ð1Þ l6 ð1Þ: Next, according to (6.6), we substitute x 1 ¼ p2 ;
x 2 ¼ p1 ;
x3 ¼ p7 ;
x4 ¼ p3 ;
x 5 ¼ p5 ;
x 6 ¼ p4 ;
x7 ¼ p6 ; ð6:108Þ
and ^
x1 ¼ p2 ¼ 0:04;
^
^
^
^
x5 ¼ p5 ¼ 0:15;
^
^
^
_
x1 ¼ p2 ¼ 0:08;
_
_
_
_
_
_
_
x5 ¼ p5 ¼ 0:26;
x2 ¼ p1 ¼ 0:31;
^
^
x6 ¼ p4 ¼ 0:001; x2 ¼ p1 ¼ 0:46; x6 ¼ p4 ¼ 0:004;
^
x3 ¼ p7 ¼ 0:25;
^
^
x4 ¼ p3 ¼ 0:002;
^
x 7 ¼ p6 ¼ 0:04;
_
_
x3 ¼ p7 ¼ 0:40; _
_
x 7 ¼ p6 ¼ 0:08;
ð6:109Þ _
_
x4 ¼ p3 ¼ 0:006; ð6:110Þ
and we maximize with respect to xi ; i ¼ 1; 2; . . .; 7; the linear form (6.106) which according to (6.7–6.9) takes the form
296
6 Complex Technical Systems Operation, Reliability, Availability, Safety
lð1Þ ¼ x1 0:805 þ x2 0:364 þ x3 0:364 þ x4 0:307 þ x5 0:307 þ x6 0:207 þ x7 0:207; ð6:111Þ with the following bound constraints 0:04 x1 0:08;
0:31 x2 0:46;
0:25 x3 0:40;
0:002 x4 0:006;
0:15 x5 0:26;
0:001 x6 0:004;
0:04 x7 0:08; 7 X
xi ¼ 1:
ð6:112Þ
i¼1
According to (6.11), we calculate 7 X ^ ^ ^ xi ¼ 0:793; ^y ¼ 1 x ¼ 1 0:793 ¼ 0:207 x¼
ð6:113Þ
i¼1
and according to (6.12), we find ^0
_0
_0
^0
x ¼ 0; x ¼ 0; x x ¼ 0;
^1
_1
_1
^1
^2
_2
_2
^2
^3
_3
_3
^3
x ¼ 0:04; x ¼ 0:08; x x ¼ 0:04;
x ¼ 0:35; x ¼ 0:54; x x ¼ 0:19;
ð6:114Þ
x ¼ 0:60; x ¼ 0:94; x x ¼ 0:34;
^7
_7
_7
^7
x ¼ 0:793; x ¼ 1:29; x x ¼ 0:497:
From the above, as according to (6.113), the inequality (6.13) takes the form _I
^I
x x \0:207;
ð6:115Þ
then it follows that the largest value I 2 f0; 1; . . .; 7g such that this inequality holds is I ¼ 2: Therefore, we fix the optimal solution that maximizes the linear function (6.111) according to the rule (6.15). Namely, we get _
_
x_ 1 ¼ x1 ¼ 0:08; x_ 2 ¼ x 2 ¼ 0:46; _2
^2
^
x_ 3 ¼ ^y x þ x þ x3 ¼ 0:207 0:54 þ 0:35 þ 0:25 ¼ 0:267; ^
^
^
^
x_ 4 ¼ x4 ¼ 0:002; x_ 5 ¼ x5 ¼ 0:15; x_ 6 ¼ x 6 ¼ 0:001; x_ 7 ¼ x 7 ¼ 0:04:
ð6:116Þ
6.4 Applications
297
Finally, after making the substitution inverse to (6.108), we get the optimal transient probabilities p_ 2 ¼ x_ 1 ¼ 0:08; p_ 1 ¼ x_ 2 ¼ 0:46; p_ 7 ¼ x_ 3 ¼ 0:267; p_ 3 ¼ x_ 4 ¼ 0:002; p_ 5 ¼ x_ 5 ¼ 0:15; p_ 4 ¼ x_ 6 ¼ 0:001; p_ 6 ¼ x_ 7 ¼ 0:04;
ð6:117Þ
that maximize the pipeline system mean lifetime lð1Þ in the reliability state subset f1; 2g expressed by the linear form (6.106) giving, according to (6.18) and (6.117), its optimal value _ lð1Þ ¼ p_ 1 0:364 þ p_ 2 0:805 þ p_ 3 0:307 þ p_ 4 0:207 þ p_ 5 0:307 þ p_ 6 0:207 þ p_ 7 0:364 ¼ 0:46 0:364 þ 0:08 0:805 þ 0:002 0:307 þ 0:001 0:207 þ 0:15 0:307 þ 0:04 0:207 þ 0:267 0:364 ffi 0:384:
ð6:118Þ
6.4.2.2 Optimal Reliability Characteristics of Port Oil Piping Transportation System Further, substituting the optimal solution (6.117) into the formula (6.19), we obtain the optimal solution for the mean value of the port oil piping transportation system unconditional lifetime in the reliability state subset f2g _ lð2Þ ¼ p_ 1 0:302 þ p_ 2 0:665 þ p_ 3 0:217 þ p_ 4 0:156 þ p_ 5 0:217 þ p_ 6 0:156 þ p_ 7 0:302 ¼ 0:46 0:302 þ 0:08 0:665 þ 0:002 0:217 þ 0:001 0:156 þ 0:15 0:217 þ 0:04 0:156 þ 0:267 0:302 ffi 0:312;
ð6:119Þ
and according to (6.23), the optimal values of the mean values of the system unconditional lifetimes in the particular reliability states 1 and 2, respectively are _ ð1Þ ¼ lð1Þ _ _ l lð2Þ ¼ 0:072;
_ ð2Þ ¼ lð2Þ _ l ¼ 0:312:
ð6:120Þ
Moreover, according to (6.20–6.21) the corresponding optimal unconditional multistate reliability function of the pipeline system is of the form _ Þ ¼ ½1; Rðt; _ 1Þ; Rðt;
_ 2Þ; Rðt;
ð6:121Þ
with the coordinates given by _ 1Þ ¼ 0:46 ½Rðt; 1Þð1Þ þ 0:08 ½Rðt; 1Þð2Þ Rðt; þ 0:002 ½Rðt; 1Þð3Þ þ 0:001 ½Rðt; 1Þð4Þ 1Þð5Þ þ 0:04 ½Rðt; 1Þð6Þ þ 0:15 ½Rðt; þ 0:267 ½Rðt; 1Þð7Þ
for t 0;
ð6:122Þ
298
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Fig. 6.3 The graph of the port oil piping transportation system optimal reliability function _ Þ coordinates Rðt;
_ 2Þ ¼ 0:46 ½Rðt; 2Þð1Þ þ 0:08 ½Rðt; 2Þð2Þ þ 0:002 ½Rðt; 2Þð3Þ þ 0:001 Rðt; ½Rðt; 2Þð4Þ þ 0:15 ½Rðt; 2Þð5Þ þ 0:04 ½Rðt; 2Þð6Þ þ 0:267 ½Rðt; 2Þð7Þ for t 0;
ð6:123Þ
where ½Rðt; 1ÞðbÞ ; ½Rðt; 2ÞðbÞ ; b ¼ 1; 2; . . .; 7; are fixed in Sect. 3.6.2 (Fig. 6.3). Further, by (6.22) the corresponding optimal variances and standard deviations of the system unconditional lifetime in the system reliability state subsets are 2
r_ ð1Þ ¼ 2
Z1
2 _ 1Þdt ½lð1Þ _ t Rðt; ffi 0:102;
_ rð1Þ ffi 0:319;
ð6:124Þ
2 _ 2Þdt ½lð2Þ _ t Rðt; ffi 0:068;
_ rð2Þ ffi 0:261;
ð6:125Þ
0
2
r_ ð2Þ ¼ 2
Z1 0
_ 1Þ; Rðt; _ 2Þ are given by (6.122–6.123) and lð1Þ; _ _ where Rðt; lð2Þ, are given by (6.118–6.119). As the port oil piping transportation system critical safety reliability is r = 1, its optimal system risk function, according to (6.24), is given by (Fig. 6.4) _ 1Þ for t 0; r_ ðtÞ ¼ 1 Rðt;
ð6:126Þ
_ 1Þ is given by (6.122). where Rðt; Hence, and considering (6.25), the moment when the optimal system risk function exceeds a permitted level, for instance d = 0.05, is s_ ¼ r_ 1 ðdÞ ffi 0:059 year:
ð6:127Þ
6.4 Applications
299
Fig. 6.4 The graph of the port oil piping transportation system optimal risk function r_ ðtÞ
6.4.2.3 Optimal Renewal and Availability Characteristics of Port Oil Piping Transportation System To make the estimation of the renewal and availability characteristics of the port oil piping transportation system after its operation process optimization, we distinguish two cases of the system renovation, one with ignored time of renovation and the other with non-ignored time of renovation. In the case when the port oil piping transportation system renovation time is ignored, applying Proposition 6.1, we determine its following optimal characteristics: (a) the optimal time S_ N ð1Þ until the Nth exceeding by the system the reliability critical state 1, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð0:384N; 0:319 N Þ, i.e., _F ðNÞ ðt; 1Þ ¼ PðS_ N ð1Þ\tÞ ffi FNð0;1Þ t 0:384N pffiffiffiffi ; t 2 ð1; 1Þ; 0:319 N (b) the expected value and the variance of the optimal time S_ N ð1Þ until the Nth exceeding by the system the reliability critical state 1, for sufficiently large N, respectively are E½S_ N ð1Þ ¼ 0:384N;
D½S_ N ð1Þ ¼ 0:102N;
_ 1Þ of exceeding by the system the reliability critical (c) the optimal number Nðt; state 1 up to the moment t; t 0; for sufficiently large t, has distribution approximately of the form 0:384ðN þ 1Þ t 0:384N t _ pffi pffi ; FNð0;1Þ PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ 0:515 t 0:515 t N ¼ 0; 1; . . .;
300
6 Complex Technical Systems Operation, Reliability, Availability, Safety
(d) the expected value and the variance of the optimal number Nðt; 1Þ of exceeding by the system the reliability critical state 1 up to the moment t; t 0; for sufficiently large t, respectively are _ 1Þ ¼ 2:604t; Hðt;
_ 1Þ ¼ 1:801t: Dðt;
ð6:128Þ
To make the estimation of the renewal and availability of the port oil piping transportation system in the case when the time of renovation is non-ignored, assuming the mean value of the system renovation time l0 ð1Þ ¼ 0:005 year and the standard deviation of the system renovation time r0 ð1Þ ¼ 0:005 year and applying Proposition 6.2, we determine its following optimal characteristics: S_ N ð1Þ until the Nth exceeding by the system the reliability (a) the optimal time critical state 1, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð0:384N þ 0:005ðN 1Þ; 0:102N þ 0:000025ðN 1ÞÞ, i.e., t 0:389N þ 0:005 _F ðNÞ ðt; 1Þ ¼ PðS_ ð1Þ\tÞ ¼ F N Nð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0:102025N 0:000025
t 2 ð1; 1Þ;
(b) the expected value and the variance of the optimal time S_ N ð1Þ until the Nth exceeding by the system the reliability critical state 1, for sufficiently large N, respectively are E½S_ N ð1Þ ffi 0:384N þ 0:005ðN 1Þ;
_ D½Sð1Þ ffi 0:102N þ 0:000025ðN 1Þ;
_ 1Þ of exceeding by the system the reliability critical (c) the optimal number Nðt; state 1 up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form 0:389ðN þ 1Þ t 0005 _ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ 0:512 t þ 0:005 0:389N t 0:005 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; N ¼ 0; 1; . . .; FNð0;1Þ 0:512 t þ 0:005 _ 1Þ of (d) the expected value and the variance of the optimal number Nðt; exceeding by the system the reliability critical state 1 up to the moment t; t 0; for sufficiently large t, respectively are _ 1Þ ffi 2:571ðt þ 0:005Þ; Hðt;
_ 1Þ ffi 1:733ðt þ 0:005Þ; Dðt;
ð6:129Þ
_ (e) the optimal time SN ð1Þ until the Nth system’s renovation, for sufficiently large pffiffiffiffi N, has approximately normal distribution Nð0:389N; 0:319 N Þ, i.e.,
6.4 Applications
301
t 0:389N _ _ ðNÞ ðt; 1Þ ¼ Pð pffiffiffiffi ; SN ð1Þ\tÞ ffi FNð0;1Þ F 0:319 N N ¼ 0; 1; . . .;
t 2 ð1; 1Þ;
_ (f) the expected value and the variance of the optimal time SN ð1Þ until the Nth system’s renovation, for sufficiently large N, respectively are _ SN ð1Þ ffi 0:389N; E½
_ ðNÞ S ð1Þ ffi 0:102025N; D½
_ 1Þ of system’s renovations up to the moment t; t 0; (g) the optimal number Nðt; for sufficiently large t, has approximately distribution of the form 0:389ðN þ 1Þ t 0:389N t _ 1Þ ¼ NÞ ffi F p ffi p ffi F ; PðNðt; Nð0;1Þ Nð0;1Þ 0:512 t 0:512 t N ¼ 0; 1; . . .; _ 1Þ of system’s (h) the expected value and the variance of the optimal number Nðt; renovations up to the moment t; t 0; for sufficiently large t, respectively are _ 1Þ ffi 2:571t; Hðt; (i)
_ 1Þ ffi 1:733t; Dðt;
the optimal steady availability coefficient of the system at the moment t; t 0; for sufficiently large t, is _ 1Þ ffi 0:987; Aðt;
(j)
ð6:130Þ
t 0;
the optimal steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; t 0; for sufficiently large t, is _ s; 1Þ ffi 2:571 Aðt;
Z1
_ 1Þdt; Rðt;
t 0;
s [ 0;
s
_ 1Þis given by (6.122). where Rðt;
6.4.2.4 Optimal Sojourn Times at Operation States of Port Oil Transportation System and Operation Strategy To obtain the optimal mean sojourn times at the particular operation states maximizing the mean lifetime lð1Þ of the port oil piping transportation system in the reliability states subset f1; 2g, we substitute the optimal transient probabilities p_ b determined by (6.117) and the steady probabilities pb determined by (2.37) into the system of Eq. 6.41 and we get its following form
302
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ 1 þ 0:01288M _ 2 þ 0:00276M _ 3 þ 0:00322M _ 4 þ 0:13892M _5 0:15552M _ 6 þ 0:10258M _7¼0 þ 0:06716M _ 2 þ 0:00048M _ 3 þ 0:00056M _ 4 þ 0:02416M _5 _ 1 0:02576M 0:02304M _ 6 þ 0:01784M _7¼0 þ 0:01168M _ 1 þ 0:000056M _ 2 0:005988M _ 3 þ 0:000014M _ 4 þ 0:000604M _5 0:000576M _ 6 þ 0:000446M _7¼0 þ 0:000292M _ 1 þ 0:000028M _ 2 þ 0:000006M _ 3 0:006993M _ 4 þ 0:000302M _5 0:000288M _ 6 þ 0:000223M _7¼0 þ 0:000146M _ 1 þ 0:0042M _ 2 þ 0:0009M _ 3 þ 0:00105M4 0:2567M _ 5 þ 0:0219M _6 0:0432M _7¼0 þ 0:03345M _ 2 þ 0:00024M _ 3 þ 0:00028M _ 4 þ 0:01208M _5 _ 1 þ 0:00112M 0:01152M _ 6 þ 0:00892M _7¼0 0:14016M _ 1 þ 0:007476M _ 2 þ 0:001602M _ 3 þ 0:001869M _ 4 þ 0:080634M _5 0:076896M _ 6 0:163459M _7¼0 þ 0:038982M ð6:131Þ _ b of the system unconditional sojourn with the unknown optimal mean values M times at the operation states we are looking for. Since the determinant of the main matrix of the homogeneous system of equations (6.131) is equal to 0, its rank is less than 7 and there are non-zero solutions of this system of equations that are ambiguous and dependent on one or more parameters. Thus, we may fix some of them and determine the remaining ones. In our case, according to (2.36), after considering expert opinion, we conclude that it is sensible to assume _ 4 ffi 380: M
ð6:132Þ
After this the system of equations (6.131) takes the form _ 2 þ 0:00276M _ 3 þ 0:13892M _ 5 þ 0:06716M _6 _ 1 þ 0:01288M 0:15552M _ 7 ¼ 1:2236 þ 0:10258M _ 1 0:02576M _ 2 þ 0:00048M _ 3 þ 0:02416M _ 5 þ 0:01168M _6 0:02304M _ 7 ¼ 0:2128 þ 0:01784M _ 1 þ 0:000056M _ 2 0:005988M _ 3 þ 0:000604M _ 5 þ 0:000292M _6 0:000576M _ 7 ¼ 0:00532 þ 0:000446M _ 2 þ 0:000006M _ 3 þ 0:000302M _ 5 þ 0:000146M _6 _ 1 þ 0:000028M 0:000288M _ 7 ¼ 2:65734 þ 0:000223M
6.4 Applications
303
_ 1 þ 0:0042M _ 2 þ 0:0009M _ 3 0:2567M _ 5 þ 0:0219M _6 0:0432M _ 7 ¼ 0:399 þ 0:03345M _ 1 þ 0:00112M _ 2 þ 0:00024M _ 3 þ 0:01208M _ 5 0:14016M _6 0:01152M _ 7 ¼ 0:1064 þ 0:00892M _ 2 þ 0:001602M _ 3 þ 0:080634M _ 5 þ 0:038982M _6 _ 1 þ 0:007476M 0:076896M _ 7 ¼ 0:71022 0:163459M ð6:133Þ and its solutions are _ 1 ffi 4248:611; M _ 2 ffi 7600; M _ 3 ffi 886:666; M _ _ M6 ffi 728:767; M7 ffi 3184:843:
_ 5 ffi 1321:192; M
ð6:134Þ
Hence and considering (6.132), we get the following final solution of Eq. 6.131 _ 1 ffi 4248:611; M _ 5 ffi 1321:192; M
_ 2 ffi 7600; M _ 3 ffi 886:666; M _ 4 ffi 380; M _ 6 ffi 728:767; M _ 7 ffi 3184:843: M
ð6:135Þ
_ b of the system Now, substituting into (6.42) the above mean values M unconditional sojourn times at the particular operation states and the known probabilities pbl of the system operation process transitions between the operation _ bl of the states given in the matrix (2.34), we may look for the optimal values M mean values of the system conditional sojourn times in the particular operation states that maximizes the mean lifetime of the port oil piping transportation system _ bl ; b; l ¼ 1; 2; . . .; 7; in the reliability state subset f1; 2g. The optimal values M b 6¼ l; should satisfy the following system of equations obtained in this way _ 12 þ 0:022M _ 13 þ 0:534M _ 15 þ 0:111M _ 16 þ 0:311M _ 17 ¼ 4248:611 0:022M _ 21 þ 0:8M _ 27 ¼ 7600 0:2M _ 31 ¼ 886:666 1M _ 47 ¼ 380 1M _ 52 þ 0:023M _ 54 þ 0:233M _ 56 þ 0:233M _ 57 ¼ 1321:192 _ 51 þ 0:023M 0:488M _ 61 þ 0:667M _ 65 þ 0:23867 ¼ 728:767 0:095M _ 72 þ 0:219M _ 75 þ 0:188M _ 76 ¼ 3184:843: _ 71 þ 0:062M 0:531M ð6:136Þ As the solutions of the above system of equations are ambiguous, we may fix some of them, because of practically important reasons, and we find the remaining ones. For instance: _ 13 ¼ 500; M _ 15 ¼ 2000; M _ 16 ¼ 1000 _ 12 ¼ 2000; M • we fix in the first equation M _ 17 ffi 9693:283; and we find M _ 27 ffi 7000; _ 21 ¼ 10000 and we find M • we fix in the second equation M
304
• • • • • • •
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ 31 ¼ 886:666; we find from the third equation M _ 47 ¼ 380; we find from the fourth equation M _ 51 ¼ 900; M _ 52 ¼ 500; M _ 54 ¼ 300; M _ 56 ¼ 400; we fix in the fifth equation M _ 57 ffi 3306:4; and we find M _ 61 ¼ 350; M _ 65 ¼ 500; we fix in the sixth equation M _ 67 ffi 1521:08; and we find M _ 72 ¼ 500; M _ 75 ¼ 2600; and _ 71 ¼ 900; M we fix in the seventh equation M we find _ 76 ffi 11205:02: ð6:137Þ M
Another very useful and much easier to be applied in practice tool that can help in planning the operation process of the port oil transportation system are the system operation process optimal mean values of the total sojourn times at the particular operation states during the system operation time h. Assuming as in Sect. 2.3.1 the system operation time h ¼ 1year = 365 days, after applying (6.43), we get their values _ ^ _ ^h1 ¼ p_ 1 h ¼ 0:46 365 ¼ 167:9; E½ h2 ¼ p_ 2 h ¼ 0:08 365 ¼ 29:2; E½ ^4 ¼ p_ 4 h ¼ 0:001 365 ¼ 0:365; _ h _ h^3 ¼ p_ 3 h ¼ 0:002 365 ¼ 0:73; E½ E½ _ ^h5 ¼ p_ 5 h ¼ 0:15 365 ¼ 54:75; E½ _ ^ E½ h6 ¼ p_ 6 h ¼ 0:04 365 ¼ 14:6; _E½^h7 ¼ p_ 7 h ¼ 0:267 365 ¼ 97:46; ð6:138Þ
that differ much from the values of E½^ hi ; i ¼ 1; 2; . . .; 7; determined by (2.39) in Sect. 2.3.2. _ bl and E½ _ ^hb given _ b; M In practice, the knowledge of the optimal values of M respectively by (6.135, 6.137, 6.138), can be very important and helpful for the port oil piping transportation system operation process planning and improving in order to make the system operation more reliable and safer. 6.4.2.5 Port Oil Piping Transportation System Operation Cost Analysis In Sects. 2.3.1 and 3.6.2 it is fixed that the port oil transportation system is composed of n ¼ 2880 components and the numbers of the system components operating at the various operation states zb ; b ¼ 1; 2; . . .; 7; are different. Namely, there are operating 1086 system components at the operation states z1 ; z2 and z7 ; 1794 system components at the operation states z3 and z5 ; and 2880 system components at the operation states z4 and z6. According to the information from experts, the approximate mean operation cost of the single basic component of the considered pipeline transportation system used during the operation time h = 1 year, independently of the operation state zb ; b ¼ 1; 2; . . .; 7; amounts to
6.4 Applications
ci ð1; bÞ ¼ 9:6 PLN,
305
b ¼ 1; 2; . . .; 7;
i ¼ 1; 2; . . .; 2880:
whereas, the cost of each system singular basic component that is not used is equal to 0. In the case when the pipeline transportation system is repaired after exceeding the critical reliability state r ¼ 1 and its renovation time is ignored, according to expert opinion, we assume that the approximate cost of the system singular renovation is cig ¼ 88500 PLN. Similarly, in the case when the pipeline transportation system is repaired after exceeding the critical reliability state r ¼ 1 and its renewal time is not ignored, according to expert opinion, we assume that the approximate cost of the system singular renovation is cnig ¼ 90000 PLN. First, we will analyze the system operation cost before its operation process optimization. Thus, under the assumptions, the total operation cost of the non-failed pipeline transportation system during the operation time h = 1 year, according to (6.44), is given by Cð1Þ ffi 0:389 9:6 1086 þ 0:062 9:6 1086 þ 0:003 9:6 1794 þ 0:002 9:6 2880 þ 0:20 9:6 1794 þ 0:058 9:6 2880 þ 0:286 9:6 1086 ffi 12839 PLN: ð6:139Þ In the case when the pipeline transportation system is repaired after exceeding the critical reliability state r ¼ 1 and its renovation time is ignored, from (3.233) given in Sect. 3.6.2, we know that the mean value of the number of exceedings by the system the critical reliability state during the operation time h = 1 year is Hð1; 1Þ ffi 2:702: Thus, the total operation cost of the repairable pipeline transportation system ignoring its renewal time during the operation time h = 1 year, according to (6.45), amounts to Cig ð1Þ ffi 12839 þ 88500 2:702 ¼ 12839 þ 239127 ¼ 251966 PLN: ð6:140Þ In the case when the pipeline transportation system is repaired after exceeding the critical reliability state r ¼ 1 and its renovation time is not ignored, from (3.235) given in Sect. 3.6.2, the mean value of the number of system renovations after exceeding by the system the critical reliability state during the operation time h = 1 year is 1Þ ffi 2:667: Hð1; Thus, the total operation cost of the repairable pipeline transportation system non-ignoring its renovation time during the operation time h = 1 year, according to (6.46), amounts to
306
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Cnig ð1Þ ffi 12839 þ 90000 2:667 ¼ 12839 þ 240030 ¼ 252869 PLN: ð6:141Þ Now, we analyze the system operation cost after its operation process optimization. Proceeding similarly as before, the total operation cost of the non-failed pipeline transportation system during the operation time h = 1 year, according to (6.47), is given by _ Cð1Þ ffi 0:46 9:6 1086 þ 0:08 9:6 1086 þ 0:002 9:6 1794 þ 0:001 9:6 2880 þ 0:15 9:6 1794 þ 0:04 9:6 2880 þ 0:267 9:6 1086 ffi 12165 PLN:
ð6:142Þ
In the case when the pipeline transportation system is repaired after exceeding the critical reliability state r ¼ 1 and its renewal time is ignored, from (6.128), we know that the mean value of the number of exceedings of the critical reliability state during the operation time h = 1 year is _ Hð1; 1Þ ffi 2:604: Thus, the total operation cost of the repairable pipeline transportation system ignoring its renewal time during the operation time h = 1 year, according to (6.48), amounts to C_ ig ð1Þ ffi 12165 þ 88500 2:604 ¼ 12165 þ 230454 ¼ 242619 PLN: ð6:143Þ In the case when the pipeline transportation system is repaired after exceeding the critical reliability state r ¼ 1 and its renewal time is not ignored, from (6.130), we know that the mean value of the number of system renovations after exceeding the critical reliability state during the operation time h = 1 year is respectively given by _ Hð1; 1Þ ffi 2:571: Thus, the total operation cost of the repairable pipeline transportation system non-ignoring its renewal time during the operation time h = 1 year, according to (6.49), amounts to C_ nig ð1Þ ffi 12165 þ 90000 2:571 ¼ 12165 þ 231390 ¼ 243555 PLN: ð6:144Þ The comparison of the results (6.139–6.141) with the results (6.142–6.144) justifies the sensibility of the port oil piping transportation system operation process optimization.
6.4 Applications
307
6.4.3 Optimization of Operation, Safety and Availability of Maritime Ferry Technical System The considered ferry technical system at the variable operation conditions is analyzed in Sect. 3.6.3 and its safety characteristics are found. From these results, it follows that it is possible to improve these characteristics by changing the parameters of its operation process. 6.4.3.1 Optimization of Ferry Technical System Operation Process The objective function given by (6.1), in this case, as the ferry technical system critical safety state is r ¼ 2, takes the form lð2Þ ¼ p1 6:45 þ p2 2:43 þ p3 3:90 þ p4 3:80 þ p5 3:80 þ p6 3:24 þ p7 2:43 þ p8 7:69 þ p9 7:69 þ p10 2:43 þ p11 3:37 þ p12 3:80 þ p13 3:80 þ p14 3:80 þ p15 3:90 þ p16 3:37 þ p17 2:43 þ p18 6:45: ð6:145Þ ^
_
The lower pb and upper pb bounds of the unknown transient probabilities pb, b ¼ 1; 2; . . .; 18; from experts, respectively are: ^
^
^
^
^
^
p1 ¼ 0:0006; p2 ¼ 0:001; p3 ¼ 0:018; p4 ¼ 0:027; p5 ¼ 0:286; p6 ¼ 0:018; ^
p7 ¼ 0:002; ^ p13 ¼ 0:286; _ p1 ¼ 0:056; _ p7 ¼ 0:018; _
^
^
^
^
^
p8 ¼ 0:001; p9 ¼ 0:001; p10 ¼ 0:001; p11 ¼ 0:002; p12 ¼ 0:013; ^ ^ ^ ^ ^ p14 ¼ 0:025; p15 ¼ 0:018; p16 ¼ 0:002; p17 ¼ 0:002; p18 ¼ 0:001; _ _ _ _ _ p2 ¼ 0:002; p3 ¼ 0:027; p4 ¼ 0:056; p5 ¼ 0:780; p6 ¼ 0:024; _ _ _ _ _ p8 ¼ 0:018; p9 ¼ 0:056; p10 ¼ 0:003; p11 ¼ 0:004; p12 ¼ 0:024; _
_
_
_
_
p13 ¼ 0:780; p14 ¼ 0:043; p15 ¼ 0:024; p16 ¼ 0:004; p17 ¼ 0:007; p18 ¼ 0:018: Therefore, according to (6.2–6.4), we assume the following bound constraints 0:0006 p1 0:056; 0:027 p4 0:056; 0:002 p7 0:018; 0:001 p10 0:003; 0:286 p13 0:780; 0:002 p16 0:004; 18 X
pb ¼ 1:
0:001 p2 0:002; 0:286 p5 0:780; 0:001 p8 0:018; 0:002 p11 0:004; 0:025 p14 0:043; 0:002 p17 0:007;
0:018 p3 0:027; 0:018 p6 0:024; 0:001 p9 0:056; 0:013 p12 0:024; 0:018 p15 0:024; 0:001 p18 0:018; ð6:146Þ
b¼1
Now, before we find optimal values p_ b of the transient probabilities pb ; b ¼ 1; 2; . . .; 18; that maximize the objective function (6.145), we arrange the system conditional lifetime mean values lb ð2Þ; b ¼ 1; 2; . . .; 18; in non-increasing order
308
6 Complex Technical Systems Operation, Reliability, Availability, Safety
l8 ð2Þ l9 ð2Þ l1 ð2Þ l18 ð2Þ l3 ð2Þ l15 ð2Þ l4 ð2Þ l5 ð2Þ l12 ð2Þ l13 ð2Þ l14 ð2Þ l11 ð2Þ l16 ð2Þ l6 ð2Þ l2 ð2Þ l7 ð2Þ l10 ð2Þ l17 ð2Þ: Next, according to (6.6), we substitute x1 ¼ p8 ; x2 ¼ p9 ; x3 ¼ p1 ; x4 ¼ p18 ; x5 ¼ p3 ; x6 ¼ p15 ; x7 ¼ p4 ; x8 ¼ p5 ; x9 ¼ p12 ; x10 ¼ p13 ; x11 ¼ p14 ; x12 ¼ p11 ; x13 ¼ p16 ; x14 ¼ p6 ; x15 ¼ p2 ; x16 ¼ p7 ; x17 ¼ p10 ; x18 ¼ p17 ; ð6:147Þ and ^
x1 ¼ p8 ¼ 0:001;
^
^
^
^
^
^
^
^
^
x5 ¼ p3 ¼ 0:018;
x6 ¼ p15 0:018; ^
x9 ¼ p12 ¼ 0:013; ^
^
^
^
^
x7 ¼ p4 ¼ 0:027;
^
^
^
x14 ¼ p6 ¼ 0:018;
x 17 ¼ p10 ¼ 0:001; _
^
^
x 13 ¼ p16 ¼ 0:002;
^
x 3 ¼ p1 ¼ 0:0006;
x 10 ¼ p13 ¼ 0:286;
^
_
^
x2 ¼ p9 ¼ 0:001;
^
^
^
^
^
^
x 4 ¼ p18 0:001;
^
^
x 8 ¼ p5 ¼ 0:286;
x 11 ¼ p14 ¼ 0:025; x 15 ¼ p2 ¼ 0:001;
^
^
x12 ¼ p11 ¼ 0:02;
^
^
x 16 ¼ p7 ¼ 0:002;
x18 ¼ p17 ¼ 0:002;
_
_
_
ð6:148Þ _
_
_
x1 ¼ p8 ¼ 0:018; x2 ¼ p9 ¼ 0:056; x3 ¼ p1 ¼ 0:056; x4 ¼ p18 ¼ 0:018; _ _ _ _ _ _ _ _ x5 ¼ p3 ¼ 0:027; x6 ¼ p15 ¼ 0:024; x7 ¼ p4 ¼ 0:056; x8 ¼ p5 ¼ 0:780; _ _ _ _ _ _ _ _ x9 ¼ p12 ¼ 0:024; x 10 ¼ p13 ¼ 0:780; x11 ¼ p14 ¼ 0:043; x12 ¼ p11 ¼ 0:004; _ _ _ _ _ _ _ _ x13 ¼ p16 ¼ 0:004; x14 ¼ p6 ¼ 0:024; x15 ¼ p2 ¼ 0:002; x16 ¼ p7 ¼ 0:018; _
_
x17 ¼ p10 ¼ 0:003;
_
_
x18 ¼ p17 ¼ 0:007;
ð6:149Þ
and we maximize with respect to xi ; i ¼ 1; 2; . . .; 18; the linear form (6.145) which according to (6.7–6.9) takes the form lð2Þ ¼ x1 7:69 þ x2 7:69 þ x3 6:45 þ x4 6:45 þ x5 3:90 þ x6 3:90 þ x7 3:80 þ x8 3:80 þ x9 3:80 þ x10 3:80 þ x11 3:80 þ x12 3:37 þ x13 3:37 þ x14 3:24 þ x15 2:43 þ x16 2:43 þ x17 2:43 þ x18 2:43;
ð6:150Þ
with the following bound constraints 0:001 x1 0:018; 0:001 x4 0:018;
0:001 x2 0:056;
0:0006 x3 0:056;
0:018 x5 0:027; 0:018 x6 0:024; 0:027 x7 0:056; 0:286 x8 0:780; 0:013 x9 0:024; 0:286 x10 0:780; 0:025 x11 0:043; 0:002 x12 0:004;
0:002 x13 0:004;
0:018 x14 0:024;
0:001 x15 0:002;
0:002 x16 0:018;
0:001 x17 0:003;
0:002 x18 0:007:
18 X i¼1
xi ¼ 1;
ð6:151Þ
6.4 Applications
309
According to (6.11), we calculate ^
x¼
18 X ^ ^ x i ¼ 0:7046; ^y ¼ 1 x; ¼ 1 0:7046 ¼ 0:2954
ð6:152Þ
i¼1
and according to (6.12), we determine ^0
_0
x ¼ 0;
x ¼ 0;
_0
^0
x x ¼ 0;
^1
_1
_1
^1
^2
_2
_2
^2
_3
^3
x ¼ 0:001; x ¼ 0:002;
x ¼ 0:018;
x x ¼ 0:017;
x ¼ 0:074;
x x ¼ 0:072;
^3
_3
^4
_4
_4
^4
^5
_5
_5
^5
^6
_6
_6
^6
^7
_7
_7
^7
^8
_8
_8
^8
x ¼ 0:0026; x ¼ 0:0036; x ¼ 0:0216; x ¼ 0:0396; x ¼ 0:0666; x ¼ 0:3526;
x ¼ 0:13;
x x ¼ 0:1274;
x ¼ 0:148; x ¼ 0:175; x ¼ 0:199; x ¼ 0:255; x ¼ 1:035;
x x ¼ 0:1444; x x ¼ 0:1534; x x ¼ 0:1594; x x ¼ 0:1884; x x ¼ 0:6824:
ð6:153Þ
From the above, as according to (6.152), the inequality (6.13) takes the form _I
^I
x x \0:2954;
ð6:154Þ
it follows that the largest value I 2 f0; 1; . . .; 18g such that this inequality holds is I ¼ 7: Therefore, we fix the optimal solution that maximize linear function (6.150) according to the rule (6.15). Namely, we get _
_
_
_
_
_
_
x_ 1 ¼ x1 ¼ 0:018; x_ 2 ¼ x2 ¼ 0:056; x_ 3 ¼ x3 ¼ 0:056; x_ 4 ¼ x4 ¼ 0:018; x_ 5 ¼ x5 ¼ 0:027; x_ 6 ¼ x6 ¼ 0:024; x_ 7 ¼ x7 ¼ 0:056; _7
^7
^
x_ 8 ¼ ^y x þ x þ x8 ¼ 0:2954 0:255 þ 0:0666 þ 0:286 ¼ 0:393; ^
^
^
^
x_ 9 ¼ x9 ¼ 0:013; x_ 10 ¼ x 10 ¼ 0:286; x_ 11 ¼ x 11 ¼ 0:025; x_ 12 ¼ x 12 ¼ 0:002; ^
^
^
^
^
^
x_ 13 ¼ x 13 ¼ 0:002; x_ 14 ¼ x14 ¼ 0:018; x_ 15 ¼ x 15 ¼ 0:001; x_ 16 ¼ x16 ; ¼ 0:002; x_ 17 ¼ x 17 ¼ 0:001; x_ 18 ¼ x18 ¼ 0:002:
ð6:155Þ
Finally, after making the inverse to (6.147) substitution, we get the optimal transient probabilities
310
6 Complex Technical Systems Operation, Reliability, Availability, Safety
p_ 8 ¼ x_ 1 ¼ 0:018; p_ 9 ¼ x_ 2 ¼ 0:056; p_ 1 ¼ x_ 3 ¼ 0:056; p_ 18 ¼ x_ 4 ¼ 0:018; p_ 3 ¼ x_ 5 ¼ 0:027; p_ 15 ¼ x_ 6 ¼ 0:024; p_ 4 ¼ x_ 7 ¼ 0:056; p_ 5 ¼ x_ 8 ¼ 0:393; p_ 12 ¼ x_ 9 ¼ 0:013; p_ 13 ¼ x_ 10 ¼ 0:286; p_ 14 ¼ x_ 11 ¼ 0:025; p_ 11 ¼ x_ 12 ¼ 0:002; p_ 16 ¼ x_ 13 ¼ 0:002; p_ 6 ¼ x_ 14 ¼ 0:018; p_ 2 ¼ x_ 15 ¼ 0:001; p_ 7 ¼ x_ 16 ¼ 0:002; ð6:156Þ p_ 10 ¼ x_ 17 ¼ 0:001; p_ 17 ¼ x_ 18 ¼ 0:002; that maximize the system mean lifetime lð2Þ in the safety state subset f2; 3; 4g expressed by the linear form (6.145) giving, according to (6.18) and (6.156), its optimal value _ lð2Þ ffi 0:056 6:45 þ 0:001 2:43 þ 0:027 3:90 þ 0:056 3:80 þ 0:393 3:80 þ 0:018 3:24 þ 0:002 2:43 þ 0:018 7:69 þ 0:056 7:69 þ 0:001 2:43 þ 0:002 3:37 þ 0:013 3:80 þ 0:286 3:80 þ 0:025 3:80 þ 0:024 3:90 þ 0:002 3:37 þ 0:002 2:43 þ 0:018 6:45 ¼ 4:27:
ð6:157Þ
6.4.3.2 Optimal Safety Characteristics of Ferry Technical System Substituting the optimal solution (6.156) into the formulae (6.19), we obtain the optimal solution for the mean values of the system unconditional lifetime in the safety state subset f1; 2; 3; 4g; f3; 4g and f4g; that respectively amount to: _ lð1Þ ffi 0:056 8:13 þ 0:001 2:86 þ 0:027 4:94 þ 0:056 4:2 þ 0:393 4:2 þ 0:018 4:01 þ 0:002 2:86 þ 0:018 9:71 þ 0:056 9:71 þ 0:001 2:86 þ 0:002 3:91 þ 0:013 4:2 þ 0:286 4:2 þ 0:025 4:2 þ 0:024 4:94 þ 0:002 3:91 þ 0:002 2:86 þ 0:018 8:13 ¼ 4:92;
ð6:158Þ
_ lð3Þ ffi 0:056 5:71 þ 0:001 2:14 þ 0:027 3:44 þ 0:056 3:38 þ 0:393 3:38 þ 0:018 2:88 þ 0:002 2:14 þ 0:018 6:89 þ 0:056 6:89 þ 0:001 2:14 þ 0:002 3:07 þ 0:013 3:38 þ 0:286 3:38 þ 0:025 3:38 þ 0:024 3:44 þ 0:002 3:07 þ 0:002 2:14 þ 0:018 5:71 ¼ 3:79; ð6:159Þ _ lð4Þ ffi 0:056 5:00 þ 0:001 1:93 þ 0:027 3:1 þ 0:056 3:05 þ 0:552 3:05 þ 0:018 2:61 þ 0:002 1:93 þ 0:018 6:25 þ 0:056 6:25 þ 0:001 1:93 þ 0:002 2:76 þ 0:013 3:05 þ 0:286 3:05 þ 0:025 3:05 þ 0:024 3:10 þ 0:002 2:76 þ 0:002 1:93 þ 0:018 5:00 ¼ 3:42;
ð6:160Þ
and according to (6.23), the optimal solutions for the mean values of the system unconditional lifetimes in the particular safety states 1, 2, 3 and 4, respectively are _ ð1Þ ¼ lð1Þ _ _ _ ð2Þ ¼ lð2Þ _ _ l lð2Þ ¼ 0:65; l lð3Þ ¼ 0:48; _ ð3Þ ¼ lð3Þ _ _ _ ð4Þ ¼ lð4Þ _ l lð4Þ ¼ 0:37; l ¼ 3:42;
ð6:161Þ
6.4 Applications
311
Moreover, according to (6.20–6.21), the corresponding optimal unconditional multistate safety function of the system is of the form s_ ðt; Þ ¼ ½1; s_ ðt; 1Þ; s_ ðt; 2Þ; s_ ðt; 3Þ; s_ ðt; 4Þ;
t 0;
ð6:162Þ
with the coordinates given by s_ ðt; 1Þ ¼ 0:056 ½sðt; 1Þð1Þ þ 0:001 ½sðt; 1Þð2Þ þ 0:027 ½sðt; 1Þð3Þ þ 0:056 ½sðt; 1Þð4Þ þ 0:393 ½sðt; 1Þð5Þ þ 0:018 ½sðt; 1Þð6Þ þ 0:002 ½sðt; 1Þð7Þ þ 0:018 ½sðt; 1Þð8Þ þ 0:056 ½sðt; 1Þð9Þ þ 0:001 ½ sðt; 1Þð10Þ þ 0:002 ½sðt; 1Þð11Þ þ 0:013 ½sðt; 1Þð12Þ þ 0:286 ½sðt; 1Þð13Þ þ 0:025 ½sðt; 1Þð14Þ þ 0:024 ½sðt; 1Þð15Þ þ 0:002 ½sðt; 1Þð16Þ þ 0:002 ½sðt; 1Þð17Þ þ 0:018 ½sðt; 1Þð18Þ ; ð6:163Þ s_ ðt; 2Þ ¼ 0:056 ½sðt; 2Þð1Þ þ 0:001 ½sðt; 2Þð2Þ þ 0:027 ½sðt; 2Þð3Þ þ 0:056 ½sðt; 2Þð4Þ þ 0:393 ½sðt; 2Þð5Þ þ 0:018 ½sðt; 2Þð6Þ þ 0:002 ½sðt; 2Þð7Þ þ 0:018 ½sðt; 2Þð8Þ þ 0:056 ½sðt; 2Þð9Þ þ 0:001 ½sðt; 2Þð10Þ þ 0:002 ½sðt; 2Þð11Þ þ 0:013 ½sðt; 2Þð12Þ þ 0:286 ½sðt; 2Þð13Þ þ 0:025 ½sðt; 2Þð14Þ þ 0:024 ½sðt; 2Þð15Þ þ 0:002 ½sðt; 2Þð16Þ þ 0:002 ½sðt; 2Þð17Þ þ 0:018 ½sðt; 2Þð18Þ ; ð6:164Þ s_ ðt; 3Þ ¼ 0:056 ½sðt; 3Þð1Þ þ 0:001 ½sðt; 3Þð2Þ þ 0:027 ½sðt; 3Þð3Þ þ 0:056 ½sðt; 3Þð4Þ þ 0:393 ½sðt; 3Þð5Þ þ 0:018 ½sðt; 3Þð6Þ þ 0:002 ½sðt; 3Þð7Þ þ 0:018 ½sðt; 3Þð8Þ þ 0:056 ½sðt; 3Þð9Þ þ 0:001 ½sðt; 3Þð10Þ þ 0:002 ½sðt; 3Þð11Þ þ 0:013 ½sðt; 3Þð12Þ þ 0:286 ½sðt; 3Þð13Þ þ 0:025 ½sðt; 3Þð14Þ þ 0:024 ½sðt; 3Þð15Þ þ 0:002 ½sðt; 3Þð16Þ þ 0:002 ½sðt; 3Þð17Þ þ 0:018 ½sðt; 3Þð18Þ ;
ð6:165Þ
s_ ðt; 4Þ ¼ 0:056 ½sðt; 4Þð1Þ þ 0:001 ½sðt; 4Þð2Þ þ 0:027 ½sðt; 4Þð3Þ þ 0:056 ½sðt; 4Þð4Þ þ 0:393 ½sðt; 4Þð5Þ þ 0:018 ½sðt; 4Þð6Þ þ 0:002 ½sðt; 4Þð7Þ þ 0:018 ½sðt; 4Þð8Þ þ 0:056 ½sðt; 4Þð9Þ þ 0:001 ½ sðt; 4Þð10Þ þ 0:002 ½sðt; 4Þð11Þ þ 0:013 ½sðt; 4Þð12Þ þ 0:286 ½sðt; 4Þð13Þ þ 0:025 ½sðt; 4Þð14Þ þ 0:024 ½sðt; 4Þð15Þ þ 0:002 ½sðt; 4Þð16Þ þ 0:002 ½sðt; 4Þð17Þ þ 0:018 ½sðt; 4Þð18Þ
for t 0;
ð6:166Þ
312
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Fig. 6.5 The graph of the ferry technical system optimal safety function s_ ðt; Þ coordinates
where sðt; uÞðbÞ ; u ¼ 1; 2; 3; 4; b ¼ 1; 2; . . .; 18; are fixed in Sect. 3.6.3 (Fig. 6.5). Further, according to (6.22), the corresponding optimal variances and standard deviations of the system unconditional lifetimes in the system safety state subsets are r_ 2 ð1Þ ¼ 2
Z1
2 _ t s_ ðt; 1Þdt ½lð1Þ ffi 21:16;
_ rð1Þ ffi 4:60;
ð6:167Þ
2 _ t s_ ðt; 2Þdt ½lð2Þ ffi 13:54;
_ rð2Þ ¼ 3:68;
ð6:168Þ
2 _ t s_ ðt; 3Þdt ½lð3Þ ffi 11:69;
_ rð3Þ ffi 3:42;
ð6:169Þ
_ rð4Þ ¼ 3:01;
ð6:170Þ
0
2
r_ ð2Þ ¼ 2
Z1 0
2
r_ ð3Þ ¼ 2
Z1 0
2
r_ ð4Þ ¼ 2
Z1
2 _ t s_ ðt; 4Þdt ½lð4Þ ffi 9:06;
0
_ _ where s_ ðt; 1Þ; s_ ðt; 2Þ; s_ ðt; 3Þ; s_ ðt; 4Þ; are given by (6.163–6.166) and lð1Þ; lð2Þ; _ _ lð3Þ; lð4Þ;are given by (6.157–6.160). As the critical safety state is r = 2, the system risk function, according to (6.24) is given by (Fig. 6.6) r_ ðtÞ ¼ 1 s_ ðt; 2Þ
for t 0;
ð6:171Þ
where s_ ðt; 2Þ is given by (6.164). Hence, considering (6.25) the moment when the optimal system risk function exceeds a permitted level, for instance d = 0.05, is
6.4 Applications
313
Fig. 6.6 The graph of the ferry technical system optimal risk function r_ ðtÞ
s_ ¼ r_ 1 ðdÞ ffi 0:22 year:
ð6:172Þ
6.4.3.3 Optimal Renewal and Availability Characteristics of Ferry Technical System To determine the optimal renewal and availability characteristics of the ferry technical system after its operation process optimization, we use the results of the system safety characteristics optimization performed in Sects. 6.4.3.1–6.4.3.2 and the results of Sect. 6.2.3. In the case when the ferry technical system renovation time is ignored, con_ _ sidering the optimal values lð2Þ determined by (6.157) and rð2Þ determined by (6.168) and applying Proposition 6.1 from Sect. 6.2.3, we determine its following optimal characteristics: (a) the optimal time S_ N ð2Þ until the Nth exceeding by the system the safety critical state 2, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð4:27N; 3:68 N Þ, i.e., _F ðNÞ ðt; 2Þ ¼ PðS_ N ð2Þ\tÞ ffi FNð0;1Þ t 4:27N pffiffiffiffi ; t 2 ð1; 1Þ; 3:68 N (b) the expected value and the variance of the optimal time S_ N ð2Þ until the Nth exceeding by the system the safety critical state 2, for sufficiently large N, respectively are E½S_ N ð2Þ ¼ 4:27N;
D½S_ N ð2Þ ¼ 13:542N;
_ 2Þ of exceeding by the system the safety critical state (c) the optimal number Nðt; 2 up to the moment t; t 0; for sufficiently large t, has distribution approximately of the form
314
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ 2Þ ¼ NÞ ffi FNð0;1Þ PðNðt;
4:27ðN þ 1Þ t 4:27N t pffi pffi ; FNð0;1Þ 1:781 t 1:781 t
N ¼ 0; 1; . . .; (d) the expected value and the variance of the optimal number Nðt; 2Þ of exceeding by the system the safety critical state 2 up to the moment t; t 0; for sufficiently large t, respectively are _ 2Þ ¼ 0:234t; Dðt; _ 2Þ ¼ 0:174t: Hðt; To make the estimation of the renewal and availability of the ferry technical system in the case when the time of renovation is non-ignored, considering the _ _ optimal values lð2Þ determined by (6.157) and rð2Þ determined by (6.168), assuming the mean value of the system renovation time l0 ð2Þ ¼ 0:019 year and the standard deviation of the system renovation time r0 ð2Þ ¼ 0:0095 year and applying Proposition 6.2 from Sect. 6.2.3, we determine its following optimal characteristics: (a) the optimal time S_ N ð2Þ until the Nth exceeding by the system the safety critical state 2, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð4:27N þ 0:019ðN 1Þ; 13:542N þ 0:000093ðN 1ÞÞ, i.e., t 4:289N þ 0:019 _S ð2Þ\tÞ ¼ F _F ðNÞ ðt; 2Þ ¼ Pð N Nð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 13:542093N 0:000093 t 2 ð1; 1Þ; (b) the expected value and the variance of the optimal time S_ N ð2Þ until the Nth exceeding by the system the safety critical state 2, for sufficiently large N, respectively are _ Sð2Þ ffi 13:542N þ 0:000093ðN 1Þ; E½S_ ð2Þ ffi 4:27N þ 0:019ðN 1Þ; D½ N
_ 2Þ of exceeding by the system the safety critical state (c) the optimal number Nðt; 2 up to the moment t; t 0; for sufficiently large t, has approximately distribution of the form t 0:019 _ 2Þ ¼ NÞ ffi FNð0;1Þ 4:289ðN þp1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 1:778 t þ 0:019 4:289N t 0:019 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; N ¼ 0; 1; . . .; FNð0;1Þ 1:778 t þ 0:019 _ 2Þ of (d) the expected value and the variance of the optimal number Nðt; exceeding by the system the safety critical state 2 up to the moment t; t 0; for sufficiently large t, respectively are _ 2Þ ffi 0:1716ðt þ 0:019Þ; _ 2Þ ffi 0:233ðt þ 0:019Þ; Dðt; Hðt; _ (e) the optimal time SN ð2Þ until the Nth system’s renovation, for sufficiently pffiffiffiffi large N, has approximately normal distribution Nð4:289N; 3:680 N Þ, i.e.,
6.4 Applications
315
t 4:89N _ ðNÞ ðt; 2Þ ¼ PðS_ ð2Þ\tÞ ffi F p ffiffiffiffi ; F N Nð0;1Þ 3:680 N
t 2 ð1; 1Þ;
N ¼ 0; 1; . . .;
_ (f) the expected value and the variance of the optimal time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, respectively are h i h ðNÞ i _ _ S ð2Þ ffi 13:542093N; E SN ð2Þ ffi 4:289N; D _ 2Þ of system’s renovations up to the moment t; t 0; (g) the optimal number Nðt; for sufficiently large t, has approximately distribution of the form 4:289ðN þ 1Þ t 4:289N t _ 2Þ ¼ NÞ ffi F p ffi p ffi PðNðt; F ; Nð0;1Þ Nð0;1Þ 1:778 t 1:778 t N ¼ 0; 1; . . .; _ 2Þ of system’s (h) the expected value and the variance of the optimal number Nðt; renovations up to the moment t; t 0; for sufficiently large t, respectively are _ 2Þ ffi 0:233t; Hðt; (i)
_ 2Þ ffi 0:1716t; Dðt;
the optimal steady availability coefficient of the system at the moment t; t 0; for sufficiently large t, is _ 2Þ ffi 0:996; Aðt;
(j)
t 0;
the optimal steady availability coefficient of the system in the time interval ht; t þ sÞ; s [ 0; t 0; for sufficiently large t, is _ s; 2Þ ffi 0:233 Aðt;
Z1
s_ ðt; 2Þdt;
t 0;
s [ 0;
s
where s_ ðt; 2Þ is given by (6.164).
6.4.3.4 Optimal Sojourn Times at Operation States of Ferry Technical System and Operation Strategy To obtain the optimal mean sojourn times at the particular operation states maximizing the mean lifetime lð2Þ of the ferry technical system in the safety state subset f2; 3; 4g, we substitute the optimal transient probabilities p_ b determined by (6.156) and the steady probabilities pb determined by (2.47) into the system of Eq. 6.41 and get its following form
316
6 Complex Technical Systems Operation, Reliability, Availability, Safety
_ 1 þ 0:056ðM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9þM _ 10 0:944M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _1þM _3þM _4þM _5þM _6þM _7þM _8þM _9þM _ 10 _ 2 þ 0:001ðM 0:999M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _ 3 þ 0:027ðM _1þM _2þM _4þM _5þM _6þM _7þM _8þM _9þM _ 10 0:973M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _ 4 þ 0:056ðM _1þM _2þM _3þM _5þM _6þM _7þM _8þM _9þM _ 10 0:944M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _ 5 þ 0:393ðM _1þM _2þM _3þM _4þM _6þM _7þM _8þM _9þM _ 10 0:607M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _1þM _2þM _3þM _4þM _5þM _7þM _8þM _9þM _ 10 _ 6 þ 0:018ðM 0:982M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _ 7 þ 0:002ðM _1þM _2þM _3þM _4þM _5þM _6þM _8þM _9þM _ 10 0:998M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _ 8 þ 0:018ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _9þM _ 10 0:982M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _ 10 _ 9 þ 0:056ðM 0:944M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _ 10 þ 0:001ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:999M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 _ 11 þ 0:002ðM 0:998þM _ 10 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _ 12 þ 0:013ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:987M _ 10 þ þM _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 _ 13 þ 0:286ðM 0:714M _ 10 þ þM _ 12 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 _ 11 þ M þM _ 14 þ 0:025ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:975M _ 10 þ M _ 11 þ M _ 12 þ M _ 13 þ M _ 15 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _ 15 þ 0:024ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:976M _ 10 þ M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 16 þ M _ 17 þ M _ 18 Þ ¼ 0 þM _ 16 þ 0:002ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:998M _ 10 þ M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 17 þ M _ 18 Þ ¼ 0 þM
6.4 Applications
317
_ 17 þ 0:002ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:998M _ 10 þ M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 18 Þ ¼ 0 þM _ 18 þ 0:018ðM _1þM _2þM _3þM _4þM _5þM _6þM _7þM _8þM _9 0:982M _ 10 þ M _ 11 þ M _ 12 þ M _ 13 þ M _ 14 þ M _ 15 þ M _ 16 þ M _ 17 Þ ¼ 0 þM ð6:173Þ From the above we get _ _ 1 ¼ 0:056M; M _ 5 ¼ 0:393M; _ M _ _ 9 ¼ 0:056M; M _ 13 ¼ 0:286M; _ M _ 17 ¼ 0:002M; _ M _ ¼ where M
18 P
_ 2 ¼ 0:001M; _ M _ 6 ¼ 0:018M; _ M _ 10 ¼ 0:001M _ M _ 14 M _ 18 M
_ 3 ¼ 0:027M; _ _ 4 ¼ 0:056M; _ M M _ 7 ¼ 0:002M; _ _ 8 ¼ 0:018M; _ M M _ 11 ¼ 0:002M; _ _ 12 ¼ 0:013M; _ M M _ _ 15 ¼ 0:024M _ M _ 16 ¼ 0:002M; _ ¼ 0:025M; M _ ¼ 0:018M;
ð6:174Þ
_ b. M
b¼1
_ is equal to the total time of one voyage given in the ferry Assuming that M _ timetable, i.e. M ¼ 1350 min, we get the following solution of the system of equations _ 1 ¼ 75:60; M _ 6 ¼ 24:30; M _ 11 ¼ 2:70; M _ 15 M
_ 2 ¼ 1:35; M _ 3 ¼ 36:45; M _ 4 ¼ 75:60; M _ 5 ¼ 530:55; M _ 7 ¼ 2:70; M _ 8 ¼ 24:30; M _ 9 ¼ 75:60; M _ 10 ¼ 1:35; M _ 12 ¼ 17:55; M _ 13 ¼ 386:10; M _ 14 ¼ 33:75; M _ 16 ¼ 2:70; M _ 17 ¼ 2:70; M _ 18 ¼ 24:30: ¼ 32:40; M ð6:175Þ
_ b of the system unconditional Substituting into (6.42) the above mean values M sojourn times at the particular operation states and the known probabilities pbl of the system operation process transitions between the operation states given in the _ bl of the mean values of the matrix (2.40), we may look for the optimal values M system conditional sojourn times at the particular operation states that maximize the mean lifetime lð2Þ of the ferry technical system in the safety states subset _ bl ; b; l ¼ 1; 2; . . .; 18; b 6¼ l; should satisfy the f2; 3; 4g. The optimal values M following system of equations obtained in this way _ 12 ¼ 75:60; M _ 23 ¼ 1:35; M _ 34 ¼ 36:45; M _ 45 ¼ 75:60; M _ 56 ¼ 530:55; M _ 67 ¼ 24:30; M _ 78 ¼ 2:70; M _ 89 ¼ 24:30; M _ 910 ¼ 75:60; M _ 1011 ¼ 1:35; M _ 1213 ¼ 17:55; M _ 1314 ¼ 386:10; M _ 1415 ¼ 33:75; _ 1112 ¼ 2:70; M M _ 1516 ¼ 32:40; M _ 1617 ¼ 2:70; M _ 1718 ¼ 2:70; M _ 181 ¼ 24:30: M ð6:176Þ Another very useful and much easier to be applied in practice tool that can help in planning the operation process of the ferry technical systems is the system
318
6 Complex Technical Systems Operation, Reliability, Availability, Safety
operation process optimal mean values of the total sojourn times at the particular operation states during the system operation time h. Assuming as in Sect. 2.3.1 the system operation time h ¼ 1 year = 365 days, after applying (6.43), we get their values _ ^h1 ¼ p_ 1 h ¼ 0:056 365 ¼ 20:44; E½ _ ^h3 ¼ p_ 3 h ¼ 0:027 365 ¼ 9:855; E½
_ ^ E½ h2 ¼ p_ 2 h ¼ 0:001 365 ¼ 0:365; _E½^ h4 ¼ p_ 4 h ¼ 0:056 ¼ 20:44;
_ ^ _ ^h5 ¼ p_ 5 h ¼ 0:393 365 ¼ 143:445; E½ h6 ¼ p_ 6 h ¼ 0:018 365 ¼ 6:57; E½ _ ^h7 ¼ p_ 7 h ¼ 0:002 365 ¼ 0:73; E½ _ ^ E½ h8 ¼ p_ 8 h ¼ 0:018 365 ¼ 6:57; _ ^ _ ^h9 ¼ p_ 9 h ¼ 0:056 365 ¼ 20:44; E½ h10 ¼ p_ 10 h ¼ 0:001 365 ¼ 0:365; E½ _ ^h11 ¼ p_ 11 h ¼ 0:002 365 ¼ 0:73; E½ _ ^ E½ h12 ¼ p_ 12 h ¼ 0:013 365 ¼ 4:745; ^ _ ^ _ h13 ¼ p_ 13 h ¼ 0:286 365 ¼ 104:39; E½ h14 ¼ p_ 14 h ¼ 0:025 365 ¼ 9:125; E½ _ ^h15 ¼ p_ 15 h ¼ 0:024 365 ¼ 8:76; E½ _ ^h17 ¼ p_ 17 h ¼ 0:002 365 ¼ 0:73; E½
_ ^ E½ h16 ¼ p_ 16 h ¼ 0:002 365 ¼ 0:73; _ ^ E½ h18 ¼ p_ 18 h ¼ 0:018 365 ¼ 6:57; ð6:177Þ
that differ from the values of E½^ hi ; i ¼ 1; 2; . . .; 18; determined by (2.49) in Sect. 2.3.3. _ bl ; and E½ _ ^hb given _ b; M In practice, the knowledge of the optimal values of M respectively by (6.175–6.177), can be very important and helpful for the system operation process planning and improving in order to make the system operation more reliable and safer.
6.4.4 Corrective and Preventive Maintenance Policy of Exemplary Technical System 6.4.4.1 Corrective and Preventive Maintenance Policy Maximizing Exemplary Technical System Availability To optimize the exemplary system corrective and preventive maintenance policy maximizing its availability, we use its following reliability and renewal parameters: • the system and components critical reliability state r = 2, • the 2nd coordinate of the system unconditional reliability function Rðt; Þ Rðt; 2Þ ¼ 0:214 ½Rðt; 2Þð1Þ þ 0:038 ½Rðt; 2Þð2Þ þ 0:293 ½Rðt; 2Þð3Þ þ 0:455 ½Rðt; 2Þð4Þ
for t 0;
ð6:178Þ
6.4 Applications
319
where ½Rðt; 2ÞðbÞ ; b ¼ 1; 2; 3; 4; are respectively given by (3.108), (3.115), (3.130), (3.145). • the derivative of the 2nd coordinate of the system unconditional reliability function Rðt; Þ R0 ðt; 2Þ ¼ 0:214 ½R0 ðt; 2Þð1Þ þ 0:038 ½R0 ðt; 2Þð2Þ þ 0:293 ½R0 ðt; 2Þð3Þ þ 0:455 ½R0 ðt; 2Þð4Þ
for t 0;
ð6:179Þ
• the mean value of the system corrective renovation time l0 ð2Þ ¼ 10; • the mean value of the system preventive renovation time l1 ð2Þ ¼ 5: Moreover, to apply the algorithm proposed in Sect. 6.3.1, we fix: • the measure of the method of secants accuracy e ¼ 0:001; • the number of values of the system preventive maintenance period g for which we find the values of the availability coefficient of the exemplary system in cases when there is no optimal value j ¼ 20; • the values of the system preventive maintenance period g for which we find the values of the availability coefficient of the system in cases when there is no optimal value gi ¼ ði 1Þ 0:2lð2Þ ¼ ði 1Þ0:2 358;
i ¼ 1; 2; . . .; 20:
ð6:180Þ
where lð2Þ ffi 358, is determined by (3.157). Since l0 ð2Þ ¼ 10 [ l1 ð2Þ ¼ 5; we are looking for the optimal value g_ of the preventive maintenance period g that maximizes the availability coefficient of the system Aðg; rÞ given by (6.52) in Sect. 6.3.1 by determining, if it exists, its approximate value from Eq. 6.58 by applying the method of secants in the interval ha; bi as follows: • we define, obtained after the transformation of Eq. 6.58, the function f ðgÞ ¼ kðg; 2Þ
Zg
Rðt; 2Þdt þ Rðg; 2Þ
0
¼ kðg; 2Þ
Zg
l0 ð2Þ l0 ð2Þ l1 ð2Þ
Rðt; 2Þdt þ Rðg; 2Þ 2 for
0
where Rðt; 2Þ is given by (6.178) and kðg; 2Þ ¼
R0 ðg; 2Þ ; Rðg; 2Þ
g 0;
320
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Table 6.1 The values of the availability coefficient of the exemplary system
g
Aðg; 2Þ
g
Aðg; 2Þ
0.0 71.6 143.2 214.8 286.4 358.0 429.6 501.2 572.8 644.4
0.0 0.9304 0.9574 0.9650 0.9681 0.9696 0.97054 0.97110 0.97152 0.97180
716.0 787.6 859.2 930.8 1002.4 1074.0 1145.6 1217.2 1288.8 1360.4
0.97203 0.97220 0.97233 0.97243 0.97250 0.97256 0.97262 0.97265 0.97268 0.97271
• we define the interval ha; bi assuming a ¼ 0 and finding b such that f ðbÞ [ 0, • we use the recurrent formula g0 ¼ a; gkþ1 ¼ gk
f ðgk Þ ðb gk Þ for k ¼ 0; 1; . . .; K; f ðbÞ f ðgk Þ
where K is such that f ðgKþ1 Þ\e and e ¼ 0:001 is the measure of the method of secants accuracy, • we fix the optimal value g_ of the preventive maintenance period g assuming g_ ¼ gKþ1 :
As a result of computer calculations, we recognize that there is no optimal value g_ of the exemplary system preventive maintenance period g that maximizes the value of its availability coefficient. The values of the system preventive maintenance period g defined by (6.180) and the corresponding values of the availability coefficient of the exemplary system are given in Table 6.1.
6.4.4.2 Corrective and Preventive Maintenance Policy Minimizing Exemplary Technical System Renovation Cost To optimize the exemplary system corrective and preventive maintenance policy minimizing its cost of renovation, we use its following reliability and operation cost parameters: • the system and components critical reliability state r = 2, • the 2nd coordinate Rðt; 2Þ of the system unconditional reliability function Rðt; Þ given by (3.178),
6.4 Applications
321
• the derivative R0 ðt; 2Þ of the 2nd coordinate of the system unconditional reliability function Rðt; Þ given by (3.179), • the mean value of the cost of the exemplary system corrective renovation c0 ð1Þ ¼ 1000 PLN, • the mean value of the cost of the exemplary system preventive renovation c1 ð1Þ ¼ 800 PLN, • the measure of the method of secants accuracy e ¼ 0:001; • the number of values of the system age 1 for which we find the values of the system renovation cost in cases when there is no optimal value j ¼ 20; • the values of the system age 1 for which we find the values of the system renovation cost in cases when there is no optimal value 1i ¼ i 0:2lð2Þ ¼ i 0:2 358;
i ¼ 1; 2; . . .; 20
ð6:181Þ
where lð2Þ ffi 358, is determined by (3.157). After fixing the above system reliability and operation cost input parameters, we use the procedure described in Sect. 6.3.2. Since c0 ð2Þ ¼ 1000 [ c1 ð2Þ ¼ 800, we are looking for the optimal value 1_ of the system age 1 at which the system preventive renovation is performed that minimizes the system renovation cost per unit time Cð1; 2Þ given by (6.62) in Sect. 6.3.2 by determining, if it exists, its approximate value from Eq. 6.65 by applying the method of secants in the interval ha; bi as follows: • we define, obtained after the transformation of Eq. 6.65, the function 0
0
wð1Þ ¼ R2 ð1; 2Þ 1 R ð1; 2ÞRð1; 2Þ
c0 ð2ÞRð1; 2Þ c1 ð2ÞR ð1; 2ÞLð1; 2Þ c0 ð2Þ c1 ð2Þ
for 1 [ 0; where Rðt; 2Þ is given by (6.178) and R1 1Rð1; 2Þ þ 0 Rð1; 2Þdt ; Lð1; 2Þ ¼ 1 Rð1; 2Þ • we define the interval ha; bi assuming a ¼ 0 and finding b such that wðbÞ [ 0; • we use the recurrent formula 10 ¼ a; 1kþ1 ¼ 1k
wðgk Þ ðb 1k Þ for k ¼ 0; 1; . . .; K; wðbÞ wðgk Þ
where K is such that
322
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Table 6.2 The values of the renovation cost of the port oil pipeline transportation system
1
Cð1; 1Þ
1
Cð1; 1Þ
71.6 143.2 214.8 286.4 358.0 429.6 501.2 572.8 644.4 716.0
11.5295 6.27350 4.68207 3.96252 3.57631 3.34259 3.19124 3.08762 3.01558 2.96299
787.6 859.2 930.8 1002.4 1074.0 1145.6 1217.2 1288.8 1360.4 1432.0
2.92419 2.89513 2.87310 2.85626 2.84328 2.83309 2.82536 2.81921 2.81438 2.81056
wð1Kþ1 Þ\e and e is the measure of the method of secants accuracy, • we fix the optimal value 1_ of the system age 1 assuming 1_ ¼ 1Kþ1 ; As a result of computer calculations, we recognize that there is no optimal value 1_ of the exemplary system age 1 at which the system preventive renovation is performed that minimizes the system renovation cost. The exemplary values of the system age 1 at which the system preventive renovation is performed defined by (6.181) and the corresponding values of the renovation cost of the exemplary system are given in Table 6.2.
6.4.5 Corrective and Preventive Maintenance Policy of Port Oil Transportation System 6.4.5.1 Corrective and Preventive Maintenance Policy Maximizing Port Oil Transportation System Availability To optimize the port oil pipeline transportation system corrective and preventive maintenance policy maximizing its availability, we use its following reliability and renewal parameters: • the oil pipeline system and its components’ critical reliability state r = 1, • the 1st coordinate of the oil pipeline system unconditional reliability function Rðt; Þ Rðt; 1Þ ¼ 0:395 ½Rðt; 1Þð1Þ þ 0:060 ½Rðt; 1Þð2Þ þ 0:003 ½Rðt; 1Þð3Þ þ 0:002 ½Rðt; 1Þð4Þ þ 0:2 ½Rðt; 1Þð5Þ þ 0:058 ½Rðt; 1Þð6Þ þ 0:282 ½Rðt; 1Þð7Þ
for t 0;
ð6:182Þ
6.4 Applications
323
where ½Rðt; 1ÞðbÞ ; b ¼ 1; 2; . . .; 7; are respectively given by (3.167), (3.173), (3.185), (3.200), (3.206), (3.212), (3.218). • the derivative of the 1st coordinate of the oil pipeline system unconditional reliability function Rðt; Þ R0 ðt; 1Þ ¼ 0:395 ½R0 ðt; 1Þð1Þ þ 0:060 ½R0 ðt; 1Þð2Þ þ 0:003 ½R0 ðt; 1Þð3Þ þ 0:002 ½R0 ðt; 1Þð4Þ þ 0:2 ½R0 ðt; 1Þð5Þ þ 0:058 ½R0 ðt; 1Þð6Þ þ 0:282 ½R0 ðt; 1Þð7Þ
for t 0;
ð6:183Þ
• the mean value of the oil pipeline system corrective renovation time l0 ð1Þ ¼ 0:005; • the mean value of the oil pipeline system preventive renovation time l1 ð1Þ ¼ 0:003: Moreover, to apply the algorithm proposed in Sect. 6.3.1, we fix: • the measure of the method of secants accuracy e ¼ 0:001; • the number of values of the system preventive maintenance period g for which we find the values of the availability coefficient of the oil pipeline system in cases when there is no optimal value j ¼ 20; • the values of the system preventive maintenance period g for which we find the values of the availability coefficient of the oil pipeline system in cases when there is no optimal value gi ¼ ði 1Þ 0:2lð1Þ ¼ ði 1Þ 0:2 0:370;
i ¼ 1; 2; . . .; 20;
ð6:184Þ
where lð1Þ ffi 0:370, is determined by (3.226). After fixing the above reliability and renewal input parameters, we use the procedure described in Sect. 6.3.1. Since l0 ð1Þ ¼ 0:005 [ l1 ð1Þ ¼ 0:003; we are looking for the optimal value g_ of the preventive maintenance period g that maximizes the availability coefficient of the system Aðg; 1Þ given by (6.52) in Sect. 6.3.1 by determining, if it exists, its approximate value from Eq. 6.58 by applying the method of secants in the interval ha; bi as follows: • we define, obtained after the transformation of Eq. 6.58, the function f ðgÞ ¼ kðg; 1Þ
Zg 0
¼ kðg; 1Þ
Zg 0
Rðt; 1Þdt þ Rðg; 1Þ
l0 ð1Þ l0 ð1Þ l1 ð1Þ
Rðt; 1Þdt þ Rðg; 1Þ 2:5
for g 0;
324
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Table 6.3 The values of the availability coefficient of the port oil transportation system
g
Aðg; 1Þ
g
Aðg; 1Þ
0.0 0.074 0.148 0.222 0.296 0.370 0.444 0.518 0.592 0.666
0.0 0.95865 0.97566 0.98081 0.98306 0.98428 0.98503 0.98550 0.98581 0.98603
0.74 0.814 0.888 0.962 1.036 1.110 1.184 1.258 1.332 1.406
0.98619 0.98632 0.98641 0.98647 0.98653 0.98657 0.98660 0.98663 0.98665 0.98667
where Rðt; 1Þis given by (6.182) and kðg; 1Þ ¼
R0 ðg; 1Þ ; Rðg; 1Þ
• we define the interval ha; bi assuming a ¼ 0 and finding b such that f ðbÞ [ 0; • we use the recurrent formula g0 ¼ a; gkþ1 ¼ gk
f ðgk Þ ðb gk Þ f ðbÞ f ðgk Þ
for k ¼ 0; 1; . . .; K;
where K is such that f ðgKþ1 Þ\e and e ¼ 0:001 is the measure of the method of secants accuracy, • we fix the optimal value g_ of the preventive maintenance period g assuming g_ ¼ gKþ1 :
As a result of the computer calculations, we recognize that there is no optimal value g_ of the oil pipeline transportation system preventive maintenance period g that maximizes the value of its availability coefficient. The exemplary values of the system preventive maintenance period g defined by (6.184) and the values of the availability coefficient of the exemplary system are given in Table 6.3.
6.4 Applications
325
6.4.5.2 Corrective and Preventive Maintenance Policy Minimizing Port Oil Transportation System Renovation Cost To optimize the port oil pipeline transportation system corrective and preventive maintenance policy minimizing its cost of renovation, we use its following reliability and operation cost parameters: • the oil pipeline system and components critical reliability state is r = 1, • the 1st coordinate Rðt; 1Þ of the system unconditional reliability function Rðt; Þ given by (6.182), • the derivative R0 ðt; 1Þ of the 1st coordinate of the system unconditional reliability function Rðt; Þ given by (6.183), • the mean value of the cost of the oil pipeline system corrective renovation c0 ð1Þ ¼ 90000 PLN, • the mean value of the cost of the oil pipeline system preventive renovation c1 ð1Þ ¼ 80000 PLN, • the measure of the method of secants accuracy e ¼ 0:001; • the number of values of the system age 1 for which we find the values of the system renovation cost in the cases when there is no its optimal value j ¼ 20; • the values of the system age 1 for which we find the values of the system renovation cost in cases when there is no optimal value ð6:185Þ 1i ¼ i 0:2lð1Þ ¼ i 0:2 0:370; i ¼ 1; 2; . . .; 20 where lð1Þ ffi 0:370 is determined by (3.226). After fixing the above system reliability and operation cost input parameters, we use the procedure described in Sect. 6.3.2. Since c0 ð2Þ ¼ 90000 [ c1 ð2Þ ¼ 80000; we are looking for the optimal value 1_ of the system age 1 at which the system preventive renovation is performed that minimizes the system renovation cost per unit time Cð1; 1Þ given by (6.62) in Sect. 6.3.2 by determining, if it exists, its approximate value from Eq. 6.65 by applying the method of secants in the interval ha; bi as follows: • we define, obtained after the transformation of Eq. 6.65, the function 0
0
wð1Þ ¼ R2 ð1; 1Þ 1 R ð1; 1ÞRð1; 1Þ
c0 ð1ÞRð1; 1Þ c1 ð1ÞR ð1; 1ÞLð1; 1Þ c0 ð1Þ c1 ð1Þ
for 1 [ 0; where Rðt; 1Þ is given by (6.182) and R1 1Rð1; 1Þ þ Rð1; 1Þdt Lð1; 1Þ ¼
0
1 Rð1; 1Þ
;
326
6 Complex Technical Systems Operation, Reliability, Availability, Safety
Table 6.4 The values of the 1 renovation cost of the port oil piping transportation system 0.074 0.148 0.222 0.296 0.37 0.444 0.518 0.592 0.666 0.74
Cð1; 1Þ
1
Cð1; 1Þ
12181207 2721949 1250203 785079 574767 462143 395952 353887 325210 305135
0.814 0.888 0.962 1.036 1.110 1.184 1.258 1.332 1.406 1.48
290550 279928 272036 265898 261223 257615 254704 252445 250608 249121
• we define the interval ha; bi assuming a ¼ 0 and finding b such that wðbÞ [ 0; • we use the recurrent formula 10 ¼ a; 1kþ1 ¼ 1k
wðgk Þ ðb 1k Þ for k ¼ 0; 1; . . .; K; wðbÞ wðgk Þ
where K is such that wð1Kþ1 Þ\e and e is the measure of the method of secants accuracy, • we fix the optimal value 1_ of the system age 1 assuming 1_ ¼ 1Kþ1 . As a result of the computer calculations, we recognize that there is no optimal value 1_ of the port oil pipeline system age 1 at which the system preventive renovation is performed that minimizes the system renovation cost. The exemplary values of the system age 1 at which the system preventive renovation is performed defined by (6.185) and the values of the renovation cost of the oil pipeline system are given in Table 6.4.
6.4.6 Corrective and Preventive Maintenance Policy of Ferry Technical System 6.4.6.1 Corrective and Preventive Maintenance Policy Maximizing Ferry Technical System Availability To optimize the maritime ferry technical system corrective and preventive maintenance policy maximizing its availability, we use its following safety and renewal parameters:
6.4 Applications
327
• the ferry technical system and components critical safety state r = 2, • the 2nd coordinate of the ferry technical system unconditional safety function sðt; Þ sðt; 2Þ ¼ 0:038 ½sðt; 2Þð1Þ þ 0:002 ½sðt; 2Þð2Þ þ 0:026 ½sðt; 2Þð3Þ þ 0:036 ½sðt; 2Þð4Þ þ 0:363 ½sðt; 2Þð5Þ þ 0:026 ½sðt; 2Þð6Þ þ 0:005 ½sðt; 2Þð7Þ þ 0:016 ½sðt; 2Þð8Þ þ 0:037 ½sðt; 2Þð9Þ þ 0:002 ½sðt; 2Þð10Þ þ 0:003 ½sðt; 2Þð11Þ þ 0:016 ½sðt; 2Þð12Þ þ 0:351 ½sðt; 2Þð13Þ þ 0:034 ½sðt; 2Þð14Þ þ 0:024 ½sðt; 2Þð15Þ þ 0:003 ½sðt; 2Þð16Þ þ 0:005 ½sðt; 2Þð17Þ þ 0:013 ½sðt; 2Þð18Þ for t 0; ð6:186Þ where ½sðt; ÞðbÞ ; b ¼ 1; 2; . . .; 18; are respectively given by (3.248), (3.271), (3.289), (3.312), (3.320), (3.343), (3.351), (3.369), (3.377), (3.385), (3.403), (3.411), (3.420), (3.427), (3.435), (3.443), (3.451), (3.459). • the derivative of the 2nd coordinate of the ferry technical system unconditional safety function sðt; Þ s0 ðt; 2Þ ¼ 0:038 ½s0 ðt; 2Þð1Þ þ 0:002 ½s0 ðt; 2Þð2Þ þ 0:026 ½s0 ðt; 2Þð3Þ þ 0:036 ½s0 ðt; 2Þð4Þ þ 0:363 ½s0 ðt; 2Þð5Þ þ 0:026 ½s0 ðt; 2Þð6Þ þ 0:005 ½s0 ðt; 2Þð7Þ þ 0:016 ½s0 ðt; 2Þð8Þ þ 0:037 ½s0 ðt; 2Þð9Þ þ 0:002 ½ s0 ðt; 2Þð10Þ þ 0:003 ½s0 ðt; 2Þð11Þ þ 0:016 ½s0 ðt; 2Þð12Þ þ 0:351 ½s0 ðt; 2Þð13Þ þ 0:034 ½s0 ðt; 2Þð14Þ þ 0:024 ½s0 ðt; 2Þð15Þ þ 0:003 ½s0 ðt; 2Þð16Þ þ 0:005 ½s0 ðt; 2Þð17Þ þ 0:013 ½s0 ðt; 2Þð18Þ for t 0; • the mean value of the ferry technical system corrective renovation time l0 ð2Þ ¼ 0:019; • the mean value of the ferry technical system preventive renovation time l1 ð2Þ ¼ 0:016: Moreover, to apply the algorithm proposed in Sect. 6.3.1, we fix: • the measure of the method of secants accuracy e ¼ 0:001; • the number of values of the system preventive maintenance period g for which we find the values of the availability coefficient of the ferry technical system in the cases when there is no optimal value j ¼ 20; • the values of the system preventive maintenance period g for which we find the values of the availability coefficient of the ferry technical system in cases when there is no optimal value gi ¼ ði 1Þ 0:2lð2Þ ¼ ði 1Þ 0:2 4:11;
i ¼ 1; 2; . . .; 20:
ð6:187Þ
328
6 Complex Technical Systems Operation, Reliability, Availability, Safety
where lð2Þ ffi 4:11is determined by (3.472). After fixing the above safety and renewal input parameters, we use the procedure described in Sect. 6.3.1. Since l0 ð2Þ ¼ 0:019 [ l1 ð2Þ ¼ 0:016; we are looking for the optimal value g_ of the preventive maintenance period g that maximizes the availability coefficient of the system Aðg; 2Þ given by (6.52) in Sect. 6.3.1 by determining, if it exists, its approximate value from Eq. 6.58 by applying the method of secants in the interval ha; bi as follows: • we define, obtained after the transformation of the Eq. 6.58, the function f ðgÞ ¼ kðg; 2Þ
Zg
sðt; 2Þdt sðg; 2Þ
0
¼ kðg; 2Þ
Zg
l0 ð2Þ l0 ð2Þ l1 ð2Þ
sðt; 2Þdt þ sðg; 2Þ 6:33 for g 0;
0
where sðt; 2Þ is given by (6.186) and kðg; 2Þ ¼
s0 ðg; 2Þ ; sðg; 2Þ
• we define the interval ha; bi assuming a ¼ 0 and finding b such that f ðbÞ [ 0; • we use the recurrent formula g0 ¼ a; gkþ1 ¼ gk
f ðgk Þ ðb gk Þ f ðbÞ f ðgk Þ
for k ¼ 0; 1; . . .; K;
where K is such that f ðgKþ1 Þ\e and e ¼ 0:001 is the measure of the method of secants accuracy, • we fix the optimal value g_ of the preventive maintenance period g assuming g_ ¼ gKþ1 : As a result of computer calculations, we recognize that there is no optimal value g_ of the ferry technical system preventive maintenance period g that maximizes the
6.4 Applications Table 6.5 The values of the availability coefficient of the ferry technical system
329 g
Aðg; 2Þ
g
Aðg; 2Þ
0 0.822 1.644 2.466 3.288 4.110 4.932 5.754 6.576 7.398
0.0 0.97829 0.98762 0.99071 0.99225 0.99313 0.99371 0.99411 0.99440 0.99461
8.220 9.042 9.864 10.686 1.508 12.330 13.152 13.974 14.796 15.618
0.99476 0.99489 0.994981 0.99506 0.99511 0.99516 0.99520 0.99523 0.99526 0.99528
value of its availability coefficient. The exemplary values of the system preventive maintenance period g defined by (6.187) and the values of the availability coefficient of the ferry technical system are given in Table 6.5.
6.5 Summary In this chapter tools useful in reliability, availability and safety optimization and operation cost analysis of a very wide class of real technical systems operating at varying conditions that have an influence on changing their reliability and safety structures and their components’ reliability and safety characteristics are presented. The results achieved are interesting for reliability practitioners from the maritime transport industry and from other industrial sectors as well. The testing of the tools performed in the maritime industry for reliability, risk, availability, safety and cost optimization of the port oil piping transportation system and the maritime ferry technical system justifies their practical utility.
References 1. Helvacioglu S, Insel M (2008) Expert system applications in marine technologies. Ocean Eng 35:1067–1074 2. Klabjan D, Adelman D (2006) Existence of optimal policies for semi-Markov decision processes using duality for infinite linear programming. SIAM J Contr Optim 44(6):2104–2122 3. Kolowrocki K, Soszynska J (2009) Reliability, risk and availability based optimization of complex technical systems operation processes. Part 1. Theoretical backgrounds. Electron J Reliab Risk Anal Theory Appl 2(44):141–152 4. Kolowrocki K, Soszynska J (2009) Reliability, risk and availability based optimization of complex technical systems operation processes. Part 2. Application in port transportation. Electron J Reliab Risk Anal Theory Appl 2(4):153–167 5. Kolowrocki K, Soszynska J (2010) Reliability, availability and safety of complex technical systems: modelling–identification–prediction–optimization. Summer safety & reliability seminars. J Pol Saf Reliab Assoc 4(1):133–158
330
6 Complex Technical Systems Operation, Reliability, Availability, Safety
6. Kolowrocki K, Soszynska J (2010) Reliability modeling of a port oil transportation system’s operation processes. Int J Perform Eng 6(1):77–87 7. Kolowrocki K, Soszynska J (2010) Safety and risk optimization of a ferry technical system Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(1):159–172 8. Kuo W, Prasad VR (2000) An annotated overview of system-reliability optimization. IEEE Trans Reliab 49(2):176–187 9. Kuo W, Zuo M J (2003) Optimal reliability modeling: principles and applications. Wiley, Hoboken 10. Levitin G, Lisnianski A (2000) Optimisation of imperfect preventive maintenance for multistate systems. Reliab Eng Sys Saf 67:193–203 11. Levitin G, Lisnianski A (2003) Optimal replacement scheduling in multi-state series-parallel systems. Qual Reliab Eng Int 16:157–162 12. Lisnianski A, Levitin G (2003) Multi-State System Reliability Assessment Optimisation and Applications. World Scientific Publishing Co. Pte. Ltd, Singapore 13. Suich RC, Patterson RL (1991) k-out-of-n:G system: some cost considerations. IEEE Trans Reliab 40(3):259–264 14. Tang H, Yin BQ, Xi HS (2007) Error bounds of optimization algorithms for semi-Markov decision processes. Int J Sys Sci 38(9):725–736 15. Merrick JRW, van Dorp R (2006) Speaking the truth in maritime risk assessment. Risk Anal 26(1):223–237 16. Vercellis S (2009) Data mining and optimization for decision making. John Wiley & Sons Ltd, Indianapolis 17. Zio E (2006) An introduction to the basics of reliability and risk analysis. World Scientific Publishing Co Pte. Ltd, Singapore
Chapter 7
Additional Applications
7.1 Introduction The applications of some theoretical results of the book are not illustrated in the previous chapters. The direct applications of the results included in the propositions of Chaps. 1 and 4 to much simpler systems than that considered in the appliqué sections of these chapters like series, parallel [3–7, 10] and consecutive ‘‘m out of n: F’’ [1, 9] systems operating at the variable conditions are not illustrated. The application of the asymptotic approach [2, 5, 10] to the large system reliability analysis [11–13] also is not illustrated in the previous chapters. The complementary applications of the results presented in the book to these systems, reliability and safety analysis are included in this chapter. Moreover, the application of the results of the book to the reliability, renewal and availability identification, prediction and optimization of the container gantry crane [8] is illustrated to give the reader the method of using these tools for an overall reliability or safety analysis of any real complex technical system.
7.2 Operation, Reliability, Availability, Safety and Cost Analysis of Complex Technical Systems 7.2.1 Reliability and Availability of Series System We consider a series system composed of components Ei ; i ¼ 1; 2; 3; 4; operating at two operation states z1 and z2, i.e. m = 2, with the same reliability structure presented in Fig. 7.1. We arbitrarily assume that the transient probabilities of the system at the particular operation states z1 and z2 respectively are
K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8_7, Springer-Verlag London Limited 2011
331
332
7 Additional Applications
E2
E1
E4
E3
Fig. 7.1 The scheme of the series system reliability structure at the operation states z1 and z2
p1 ¼ 0:25;
p2 ¼ 0:75:
Moreover, we distinguish four reliability states of the system components 0, 1, 2, 3, i.e. z = 3. We fix that the critical reliability state is r = 2 and we assume that the components’ reliability characteristics are different at the operation states z1 and z2. Consequently, we define the four-state conditional reliability functions of the system components Ei, i = 1, 2, 3, 4, at the operation state z1 in the form of the vector ½Ri ðt; Þð1Þ ¼ ½1; ½Ri ðt; 1Þð1Þ ; ½Ri ðt; 2Þð1Þ ; ½Ri ðt; 3Þð1Þ ;
i ¼ 1; 2; 3; 4;
with the exponential coordinates ½Ri ðt; uÞð1Þ ¼ exp½2ut; u ¼ 1; 2; 3;
i ¼ 1; 2; 3; 4:
and the four-state conditional reliability functions of the system components Ei, i = 1, 2, 3, 4, at the operation state z2 in the form of the vector ½Ri ðt; Þð2Þ ¼ ½1; ½Ri ðt; 1Þð2Þ ; ½Ri ðt; 2Þð2Þ ; ½Ri ðt; 3Þð2Þ ; i ¼ 1; 2; 3; 4; with the exponential coordinates ½Ri ðt; uÞð2Þ ¼ exp½ut; u ¼ 1; 2; 3; i ¼ 1; 2; 3; 4: After direct application of the formula (3.18) stated in Proposition 3.1, we get the unconditional system reliability function Rðt; Þ ¼ ½1; Rðt; 1Þ; Rðt; 2Þ; Rðt; 3Þ; t 0;
ð7:1Þ
where Rðt; uÞ ¼ R4 ðt; uÞ ¼
"
2 X
4 X pb exp ½ki ðuÞðbÞ t
b¼1
i¼1
#
¼ p1 exp½4 2ut þ p2 exp½4 ut ¼ 0:25 exp½8ut þ 0:75 exp½4ut; u ¼ 1; 2; 3; and particularly (Fig. 7.2) Rðt; 1Þ ¼ 0:25 exp½8t þ 0:75 exp½4t;
ð7:2Þ
Rðt; 2Þ ¼ 0:25 exp½16t þ 0:75 exp½8t;
ð7:3Þ
Rðt; 3Þ ¼ 0:25 exp½24t þ 0:75 exp½12t:
ð7:4Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
333
Fig. 7.2 The graph of the series system reliability function R(t, •) coordinates
The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3}, calculated from the results given by (7.2–7.4), according to (3.7–3.9), respectively are: 1 1 lð1Þ ¼ 0:25 þ 0:75 ffi 0:219; rð1Þ ffi 0:232; 8 4
ð7:5Þ
1 1 þ 0:75 ffi 0:109; rð2Þ ffi 0:116; 16 8
ð7:6Þ
1 1 þ 0:75 ffi 0:073; rð3Þ ffi 0:077: 24 12
ð7:7Þ
lð2Þ ¼ 0:25 lð3Þ ¼ 0:25
Consequently, considering (3.10) and (7.5–7.7), the mean values of the system unconditional lifetimes in the particular reliability states 1, 2, 3, respectively are: ð1Þ ¼ lð1Þ lð2Þ ¼ 0:110; l ð2Þ ¼ lð2Þ lð3Þ ¼ 0:036; l ð3Þ ¼ lð3Þ ¼ 0:073: l
ð7:8Þ
Since the critical reliability state is r = 2, the system risk function, according to (3.11), is given by rðtÞ ¼ 1 Rðt; 2Þ ¼ 1 0:25 exp½16t 0:75 exp½8t for t 0:
ð7:9Þ
Hence, by (3.12), the moment when the system risk function exceeds a permitted level, for instance d = 0.05, is (Fig. 7.3) ð7:10Þ s ¼ r1 ðdÞ ffi 0:00515: Further, assuming that the system is repaired after the exceeding its critical reliability state r = 2 and that the time of the system renovation is ignored and applying Proposition 3.4, we obtain the following results:
334
7 Additional Applications
Fig. 7.3 The graph of the series system risk function r(t)
(a) the time SN(2) until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð0:109 N; 0:116 N Þ, i.e., t 0:109 N pffiffiffiffi ; t 2 ð1; 1Þ; F ðNÞ ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ 0:116 N (b) the expected value and the variance of the time SN(2) until the Nth exceeding by the system the reliability critical state r = 2 are respectively given by E½SN ð2Þ ffi 0:109 N; D½SN ð2Þ ffi 0:0135 N; (c) the number N(t, 2) of exceedings by the system the reliability critical state r = 2 up to the moment t, t C 0, for sufficiently large t, approximately has the distribution of the form 0:109ðN þ 1Þ t 0:109 N t pffi pffi ; FNð0;1Þ PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 0:351 t 0:351 t N ¼ 0; 1; . . .; (d) the expected value and the variance of the number N(t, 2) of exceedings by the system the reliability critical state r = 2 up to the moment t, t C 0, for sufficiently large t, approximately are respectively given by Hðt; 2Þ ffi 9:174t; Dðt; 2Þ ffi 10:424t: Assuming that the system is repaired after exceeding its reliability critical state r = 2 and that the time of the system renovation is non-ignored and it has the mean value l0 ð2Þ ¼ 0:05 and the standard deviation r0 ð2Þ ¼ 0:02 and applying Proposition 3.5, we obtain the following results:
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
335
(a) the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð0:109 N þ 0:05ðN 1Þ; 0:0135 N þ 0:0004ðN 1Þ, i.e., t 0:159 N þ 0:05 ðNÞ F ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0:0139 N 0:0004
t 2 ð1; 1Þ;
(b) the expected value and the variance of the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, are respectively given by E½SN ð2Þ ffi 0:109 N þ 0:05ðN 1Þ; D½ SN ð2Þ ffi 0:0135 N þ 0:0004ðN 1Þ; 2Þ of exceedings by the system the reliability critical state r = 2 (c) the number Nðt; up to the moment t, t C 0, for sufficiently large t, has approximately distribution of the form 2Þ ¼ NÞ PðNðt;
0:159ðN þ 1Þ t 0:05 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:296 t þ 0:05 0:159 N t 0:05 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; FNð0;1Þ 0:296 t þ 0:05 N ¼ 0; 1; . . .;
ffi FNð0;1Þ
2Þ of exceedings by the (d) the expected value and the variance of the number Nðt; system the reliability critical state r = 2 up to the moment t, t C 0, for sufficiently large t, are respectively given by 2Þ ffi 3:458ðt þ 0:05Þ; 2Þ ffi t þ 0:05 ; Dðt; Hðt; 0:159 (e) the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffi approximately normal distribution Nð0:159 N; 0:118 N Þ, i.e., t 0:159 NÞ ðNÞ ðt; 2Þ ¼ Pð pffiffiffiffi ; t 2 ð1; 1Þ; F SN ð2Þ\tÞ ffi FNð0;1Þ 0:118 N (f) the expected value and the variance of the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, are respectively given by SN ð2Þ ffi 0:159 N; D½ E½ SN ð2Þ ffi 0:0139 N; 2Þ of the system’s renovations up to the moment t, t C 0, for (g) the number Nðt; sufficiently large t, has approximately distribution of the form
336
7 Additional Applications
0:159ðN þ 1Þ t 0:159 N t pffi pffi ; FNð0;1Þ PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 0:296 t 0:296 t N ¼ 0; 1; . . .; 2Þ of system’s reno(h) the expected value and the variance of the number Nðt; vations up to the moment t, t C 0, for sufficiently large t, are respectively given by 2Þ ffi 6:289t; Hðt;
2Þ ffi 3:458t; Dðt;
(i) the steady availability coefficient of the system at the moment t, t C 0, for sufficiently large t, is given by Aðt; 2Þ ffi 0:686; t 0; (j) the steady availability coefficient of the system in the time interval ht; t þ sÞ; si0; for sufficiently large t, is given by Aðt; s; 2Þ ffi 6:289
Z1
Rðt; 2Þdt; t 0; s [ 0;
s
where R(t, 2) is given by (7.3).
7.2.2 Reliability and Availability of Parallel System We consider a parallel system composed of components Ei, i = 1, 2, 3, 4, operating at two operation states z1 and z2, i.e. m = 2. We assume that the system reliability structure and the system components’ reliability characteristics are changing at the various operation states. At the operation state z1 the system is composed of two components Ei, i = 1, 2, with the reliability structure presented in Fig. 7.4. At the operation state z2 the system is composed of four components Ei, i = 1, 2, 3, 4, with the reliability structure illustrated in Fig. 7.5. We arbitrarily assume that the transient probabilities of the system at the particular operation states z1 and z2 respectively are p1 ¼ 0:25;
p2 ¼ 0:75:
Moreover, we distinguish three reliability states of the system components 0, 1, 2, i.e. z = 2, and we fix that the critical reliability state is r = 1. Consequently, we define the three-state reliability functions of the system components Ei, i = 1, 2, at the operation state z1 in the form of the vector
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
337
Fig. 7.4 The scheme of the parallel system reliability structure at the operation state z1
E1
E2
Fig. 7.5 The scheme of the parallel system reliability structure at the operation state z2
E1
E2
E3
E4
½Ri ðt; Þð1Þ ¼ ½1; ½Ri ðt; 1Þð1Þ ; ½Ri ðt; 2Þð1Þ ;
i ¼ 1; 2;
with the exponential coordinates ½Ri ðt; uÞð1Þ ¼ exp½2ut;
u ¼ 1; 2;
i ¼ 1; 2;
and the three-state reliability functions of the system components Ei, i = 1, 2, at the operation state z2 in the form of the vector ½Ri ðt; Þð2Þ ¼ ½1; ½Ri ðt; 1Þð2Þ ; ½Ri ðt; 2Þð2Þ ;
i ¼ 1; 2; 3; 4;
with the exponential coordinates ½Ri ðt; uÞð2Þ ¼ exp½ut;
u ¼ 1; 2;
i ¼ 1; 2; 3; 4:
After considering Remark 3.1 given at the end of Sect. 3.2 and direct application of the formula (3.20) stated in Proposition 3.1, we get the system reliability function Rðt; Þ ¼ ½1; Rðt; 1Þ; Rðt; 2Þ; t 0; where
ð7:11Þ
338
7 Additional Applications
Fig. 7.6 The graph of the parallel system reliability function R(t,) coordinates
Rðt; uÞ ¼ 1
2 X b¼1
pb
nðbÞ Y
½1 exp½½ki ðuÞðbÞ t
i¼1
¼ 1 p1 ½1 exp½2ut2 p2 ½1 exp½ut4 ¼ 1 0:25½1 2 exp½2ut þ exp½4ut 0:75½1 4 exp½ut þ 6 exp½2ut 4 exp½3ut þ exp½4ut ¼ 0:25½2 exp½2ut exp½4ut þ 0:75½4 exp½ut 6 exp½2ut þ 4 exp½3ut exp½4ut ¼ 3 exp½ut 4 exp½2ut þ 3 exp½3ut exp½4ut;
u ¼ 1; 2;
and particularly (Fig. 7.6) Rðt; 1Þ ¼ 3 exp½t 4 exp½2t þ 3 exp½3t exp½4t;
ð7:12Þ
Rðt; 2Þ ¼ 3 exp½2t 4 exp½4t þ 3 exp½6t exp½8t:
ð7:13Þ
The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets {1,2}, {2}, calculated from the results given by (7.12–7.13), according to (3.7–3.9), respectively are: 1 1 1 1 lð1Þ ¼ 3 4 þ 3 1 ffi 1:750; rð1Þ ffi 1:070; 1 2 3 4
ð7:14Þ
1 1 1 1 lð2Þ ¼ 3 4 þ 3 1 ffi 0:875; rð2Þ ffi 0:608; 2 4 6 8
ð7:15Þ
and further, considering (3.10) and (7.14–7.15), the mean values of the unconditional lifetimes in the particular reliability states 1, 2, respectively are: ð1Þ ¼ lð1Þ lð2Þ ¼ 0:875; l
ð2Þ ¼ lð2Þ ¼ 0:875: l
ð7:16Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
339
Fig. 7.7 The graph of the parallel system risk function r(t)
Since the critical reliability state is r = 1, the system risk function, according to (3.11), is given by (Fig. 7.7) rðtÞ ¼ 1 Rðt; 1Þ ¼ 1 ½3 exp½t 4 exp½2t þ 3 exp½3t exp½4t
for t 0:
ð7:17Þ
Hence, by (3.12), the moment when the system risk function exceeds a permitted level, for instance d = 0.05, is s ¼ r1 ðdÞ ffi 0:285:
ð7:18Þ
Further, assuming that the system is repaired after the exceeding its reliability critical state r = 1 and that the time of the system renovation is ignored and applying Proposition 3.4, we obtain the following results: (a) the time SN(1) until the Nth exceeding by the system the reliability critical state r = 1, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð1:750 N; 1:070 N Þ, i.e., t 1:750 N pffiffiffiffi ; t 2 ð1; 1Þ; F ðNÞ ðt; 1Þ ¼ PðSN ð1Þ\tÞ ffi FNð0;1Þ 1:070 N (b) the expected value and the variance of the time SN (1) until the Nth exceeding by the system the reliability critical state r = 1 are respectively given by E½SN ð1Þ ffi 1:750 N;
D½SN ð1Þ ffi 1:1449 N;
(c) the number N(t, 1) of exceedings by the system the reliability critical state r = 1 up to the moment t, t C 0, for sufficiently large t, approximately has the distribution of the form 1:750ðN þ 1Þ t 1:750 N t pffi pffi ; FNð0;1Þ PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ 0:809 t 0:809 t N ¼ 0; 1; . . .;
340
7 Additional Applications
(d) the expected value and the variance of the number N(t, 1) of exceedings by the system the reliability critical state r = 1 up to the moment t, t C 0, for sufficiently large t, approximately are respectively given by Hðt; 1Þ ffi 0:571t; Dðt; 1Þ ffi 0:214t: Assuming that the system is repaired after the exceeding of its reliability critical state r = 1 and that the time of the system renovation is not ignored and it has the mean value l0 ð1Þ ¼ 0:10 and the standard deviation r0 ð1Þ ¼ 0:05 and applying Proposition 3.5, we obtain the following results: (a) the time SN ð1Þ until the Nth exceeding by the system the reliability critical state r = 1, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð1:750 N þ 0:1ðN 1Þ; 1:1449 N þ 0:0025ðN 1Þ, i.e., 1:850 N þ 0:1 ðNÞ ðt; 1Þ ¼ PðSN ð1Þ\tÞ ffi FNð0;1Þ ptffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ; t 2 ð1; 1Þ; 1:1474 N 0:0025 (b) the expected value and the variance of the time SN ð1Þ until the Nth exceeding by the system the reliability critical state r = 1, for sufficiently large N, are respectively given by E½SN ð1Þ ffi 1:750 N þ 0:10ðN 1Þ;
D½ SN ð1Þ ffi 1:1449 N þ 0:0025ðN 1Þ;
1Þ of exceeding by the system the reliability critical state (c) the number Nðt; r = 1 up to the moment t, t C 0, for sufficiently large t, has approximately distribution of the form 1Þ ¼ NÞ PðNðt;
1:850ðN þ 1Þ t 0:10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi FNð0;1Þ 0:786 t þ 0:10 1:850 N t 0:10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; FNð0;1Þ 0:786 t þ 0:10 N ¼ 0; 1; . . .;
1Þ of exceeding by the (d) the expected value and the variance of the number Nðt; system the reliability critical state r = 1 up to the moment t, t C 0, for sufficiently large t, are respectively given by 1Þ ffi t þ 0:10 ; Hðt; 1:85
1Þ ffi 0:181ðt þ 0:10Þ; Dðt;
(e) the time SN ð1Þ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffi approximately normal distribution Nð1:850 N; 1:071 N Þ, i.e.,
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
t 1:850 NÞ ðNÞ ðt; 1Þ ¼ Pð pffiffiffiffi ; F SN ð1Þ\tÞ ffi FNð0;1Þ 1:071 N
341
t 2 ð1; 1Þ;
(f) the expected value and the variance of the time SN ð1Þ until the Nth system’s renovation, for sufficiently large N, are respectively given by SN ð1Þ ffi 1:850 N; D½ E½ SN ð1Þ ffi 1:1474 N; 1Þ of the system’s renovations up to the moment t, t C 0, for (g) the number Nðt; sufficiently large t, has approximately distribution of the form 1:850ðN þ 1Þ t 1:850 N t p ffi p ffi FNð0;1Þ ; PðNðt; 1Þ ¼ NÞ ffi FNð0;1Þ 0:786 t 0:786 t N ¼ 0; 1; . . .; 1Þ of system’s reno(h) the expected value and the variance of the number Nðt; vations up to the moment t, t C 0, for sufficiently large t, are respectively given by 1Þ ffi 0:181t; 1Þ ffi 0:541t; Dðt; Hðt; (i) the steady availability coefficient of the system at the moment t, t C 0, for sufficiently large t, is given by Aðt; 1Þ ffi 0:946; t 0; (j) the steady availability coefficient of the system in the time interval ht; t þ sÞ; si0; for sufficiently large t, is given by Aðt; s; 1Þ ffi 0:541
Z1
Rðt; 1Þdt; t 0; s [ 0;
s
where R(t, 1) is given by (7.12).
7.2.3 Reliability of Large Parallel-Series System We consider a parallel-series system composed of components Eij, i = 1, 2, 3, j ¼ 1; 2; . . .; 36; operating at three operation states z1, z2 and z3, i.e. m = 3. We assume that the system reliability structure and the system components’ reliability characteristics are changing at the various operation states. At the operation state z1 the system is composed of one parallel subsystem composed of components Eij, i = 1, j ¼ 1; 2; . . .; 36; with the reliability structure presented in Fig. 7.8.
342
7 Additional Applications
Fig. 7.8 The scheme of the parallel-series system reliability structure at the operation state z1
E 11
E 12
E 13
E 136
Fig. 7.9 The scheme of the parallel-series system reliability structure at the operation state z2
E11
E 21
E12
E 22
E13
E 23
E136
E 236
At the operation state z2 the system is composed of two parallel subsystems linked in series and composed of components Eij, i = 1, 2, j ¼ 1; 2; . . .; 36; with the reliability structure presented in Fig. 7.9. At the operation state z3, the system is composed of three parallel subsystems linked in series and composed of components Eij, i = 1, 2, 3, j ¼ 1; 2; . . .; 36; with the reliability structure presented in Fig. 7.10. We arbitrarily assume that the transient probabilities of the system at particular operation states z1, z2 and z3 respectively are
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
343
E11
E21
E31
E12
E22
E32
E13
E23
E33
E 136
E236
E336
Fig. 7.10 The scheme of the parallel-series system structure at the operation state z3
p1 ¼ 0:2;
p2 ¼ 0:4;
p3 ¼ 0:4:
Moreover, we distinguish three reliability states of the system components 0, 1, 2, i.e. z = 2, and we fix that the critical reliability state is r = 1. Consequently, we define the three-state reliability functions of the system components Eij, i = 1, j ¼ 1; 2; . . .; 36; at the operation state z1 in the form of the vector ½Rij ðt; Þð1Þ ¼ ½1; ½Rij ðt; 1Þð1Þ ; ½Rij ðt; 2Þð1Þ ;
i ¼ 1;
j ¼ 1; 2; . . .; 36;
with the exponential coordinates ½Rij ðt; uÞð1Þ ¼ exp½ut; u ¼ 1; 2; i ¼ 1; j ¼ 1; 2; . . .; 36; and the three-state reliability functions of the system components Eij, i = 1, 2, j ¼ 1; 2; . . .; 36; at the operation state z2 in the form of the vector ½Rij ðt; Þð2Þ ¼ ½1; ½Rij ðt; 1Þð2Þ ; ½Rij ðt; 2Þð2Þ ; i ¼ 1; 2; j ¼ 1; 2; . . .; 36; with the exponential coordinates ½Rij ðt; uÞð2Þ ¼ exp½2ut;
u ¼ 1; 2;
i ¼ 1; 2;
j ¼ 1; 2; . . .; 36;
and the three-state reliability functions of the system components Eij, i = 1, 2, 3, j ¼ 1; 2; . . .; 36; at the operation state z3 in the form of the vector ½Rij ðt; Þð3Þ ¼ ½1; ½Rij ðt; 1Þð3Þ ; ½Rij ðt; 2Þð3Þ ; with the exponential co-ordinates
i ¼ 1; 2; 3;
j ¼ 1; 2; . . .; 36;
344
7 Additional Applications
½Rij ðt; uÞð3Þ ¼ exp½3ut; u ¼ 1; 2; 3 i ¼ 1; 2; 3; j ¼ 1; 2; . . .; 36: Since the shape parameters of the considered parallel-series system are: ð1Þ
• kn ! kð1Þ ¼ 1; lð1Þ ¼ 36; at the operation state z1, ð2Þ • kn ! kð2Þ ¼ 2; lð2Þ ¼ 36; at the operation state z2, ð3Þ • kn ! kð3Þ ¼ 3; lð3Þ ¼ 36; at the operation state z3, the application of the second part of Proposition 3.3 is suitable. Thus, applying directly the formulae (3.53–3.54), we get the system reliability function Rðt; Þ ¼ ½1; Rðt; 1Þ; Rðt; 2Þ; t 0;
ð7:19Þ
where
Rðt; uÞ ffi
3 X
ðbÞ k pb ½1 exp½lðbÞ n exp½½kðuÞ t
ðbÞ
b¼1 ð1Þ
ð1Þ k ð2Þ k ¼ p1 ½1 exp½lð1Þ þ p2 ½1 exp½lð2Þ n exp½½kðuÞ t n exp½½kðuÞ t ð3Þ k þ p3 ½1 exp½lð3Þ n exp½½kðuÞ t
ð2Þ
ð3Þ
¼ 0:2½1 exp½36 exp½ut1 þ 0:4½1 exp½36 exp½2ut2 þ 0:4½1 exp½36 exp½3ut3 ¼ 1 0:2 exp½36 exp½ut 0:8 exp½36 exp½2ut þ 0:4 exp½72 exp½2ut 1:2 exp½36 exp½3ut þ 1:2 exp½72 exp½3ut 0:4 exp½108 exp½3ut for t 0; u ¼ 1; 2;
and particularly Rðt; 1Þ ffi 1 0:2exp½36exp½t 0:8 exp½36 exp½2t þ 0:4 exp½72 exp½2t 1:2exp½36exp½3t þ 1:2 exp½72 exp½3t 0:4exp½108exp½3t; t 0;
ð7:20Þ
Rðt; 2Þ ffi 10:2exp½36exp½2t0:8exp½36exp½4tþ0:4exp½72exp½4t 1:2exp½36exp½6tþ1:2exp½72exp½6t 0:4exp½108exp½6t; for t0:
ð7:21Þ
The graphs of the approximate coordinates of the parallel-series system reliability function are presented in Fig. 7.11.
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
345
Fig. 7.11 The graph of the parallel-series system reliability function R(t, ) coordinates
The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets {1, 2}, {2}, calculated from the results given by (7.20–7.21), according to (3.7–3.9), respectively are: lð1Þ ffi 1:939; rð1Þ ffi 0:924;
ð7:22Þ
lð2Þ ffi 0:969; rð2Þ ffi 0:462;
ð7:23Þ
and further, considering (3.10) and (7.20–7.21), the mean values of the unconditional lifetimes in the particular reliability states 1, 2, respectively are: ð1Þ ¼ lð1Þ lð2Þ ¼ 0:970; l
ð2Þ ¼ lð2Þ ¼ 0:969: l
ð7:24Þ
Since the critical reliability state is r = 1, the system risk function, according to (3.11), is given by (Fig. 7.12) rðtÞ ¼ 1 Rðt; 1Þ ffi 0:2 exp½36 exp½t þ 0:8 exp½36 exp½2t 0:4 exp½72 exp½2t þ 1:2 exp½36 exp½3t 1:2 exp½72 exp½3t þ 0:4 exp½108 exp½3t; t 0; for t 0: ð7:25Þ Hence, by (3.12), the moment when the system risk function exceeds a permitted level, for instance d = 0.05, is s ¼ r1 ðdÞ ffi 0:812:
ð7:26Þ
346
7 Additional Applications
Fig. 7.12 The graph of the parallel-series system risk function r(t)
7.2.4 Safety of Steel Cover We consider the safety of the steel cover composed of n = 24 arranged identical sheets E1 ; E2 ; . . .; E24 . We assume that z = 4, i.e. the steel cover and the sheets it is composed of may be in one of the safety states from the safety state set {0, 1, 2, 3, 4}. The cover is out of the safety state subset fu; u þ 1; . . .; 4g; u = 1, 2, 3, 4, if at least m = 2 of its neighboring sheets is out of this safety state subset. Suppose that the considered steel cover critical safety state is r = 2, we conclude that the steel cover is failed, i.e. it is out of the safety state subset {2, 3, 4}, if at least 2 of its neighboring sheets from 24 sheets are out of the safety state subset {2, 3, 4}. Thus, the considered steel cover is the five-state consecutive ‘‘2 out of 24: F’’ system, and according to formulae (1.32–1.33) from Sect. 1.2, after replacing in them the reliability notations R(t, ), CR224 ðt; Þby the safety notations sðt; Þ; Cs224 ðt; Þ, we get the following expression for the steel cover safety function, Cs224 ðt; Þ ¼ ½1; Cs224 ðt; 1Þ; Cs224 ðt; 2Þ; Cs224 ðt; 3Þ; Cs224 ðt; 4Þ;
ð7:27Þ
where Cs224 ðt; uÞ ¼ sðt; uÞCs223 ðt; uÞ þ sðt; uÞFðt; uÞCs222 ðt; uÞ for t 2 \0; 1Þ; u ¼ 1; 2; 3; 4; and
ð7:28Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
Cs2n ðt; uÞ ¼
8 1 > > > > n Y > > > > Fi ðt; uÞ 1 > > > i¼1 > < > sn ðt; uÞCs2n1 ðt; uÞ þ > > > > > > > n Y > > > > Fj ðt; uÞ :
347
for n\2; for n ¼ 2; 1 X
sni ðt; uÞCs2ni1 ðt; uÞ
ð7:29Þ
i¼1
for n [ 2
j¼niþ1
for t C 0, u ¼ 1; 2; . . .; z; n ¼ 2; 3; . . .; 24: In the particular case, when the lifetimes Ti(u), u ¼ 1; 2; 3; 4; of the sheets Ei, i ¼ 1; 2; . . .; 24; in the safety state subsets have the exponential safety functions given by the vector sðt; Þ ¼ ½1; sðt; 1Þ; sðt; 2Þ; sðt; 3Þ; sðt; 4Þ; t 2 \0; 1Þ; with the coordinates sðt; 1Þ ¼ exp½0:01t; sðt; 2Þ ¼ exp½0:02t; sðt; 3Þ ¼ exp½0:05t; sðt; 4Þ ¼ exp½0:10t for t 0; considering (7.28–7.29), we get the following recurrent formulae for the cover safety function coordinates: • Cs224 ðt; 1Þ is determined by the formulae Cs22 ðt; 1Þ ¼ 1 ½1 exp½0:01t2 for t 2 h0; 1Þ; Cs2n ðt; 1Þ ¼ exp½0:01tCs2n1 ðt; 1Þ þ exp½0:01t½1 exp½0:01tCs2n2 ðt; 1Þ for t 2 h0; 1Þ; n ¼ 3; 4; . . .; 24; ð7:30Þ • Cs224 ðt; 2Þ is determined by the formulae Cs22 ðt; 2Þ ¼ 1 ½1 exp½0:02t2 for t 2 h0; 1Þ; Cs2n ðt; 2Þ ¼ exp½0:02tCs2n1 ðt; 2Þ þ exp½0:02t½1 exp½0:02tCs2n2 ðt; 2Þ for t 2 h0; 1Þ; n ¼ 3; 4; . . .; 24; ð7:31Þ • Cs224 ðt; 3Þ is determined by the formulae Cs22 ðt; 3Þ ¼ 1 ½1 exp½0:05t2 for t 2 \0; 1Þ; Cs2n ðt; 3Þ ¼ exp½0:05tCs2n1 ðt; 3Þ þ exp½0:05t½1 exp½0:05tCs2n2 ðt; 3Þ for t 2 \0; 1Þ; n ¼ 3; 4; . . .; 24; ð7:32Þ
348
7 Additional Applications
Fig. 7.13 The graphs of the steel cover five-state safety function coordinates
• Cs224 ðt; 4Þ is determined by the formulae (Fig. 7.13) Cs22 ðt; 4Þ ¼ 1 ½1 exp½0:10t2 for t 2 \0; 1Þ; Cs2n ðt; 4Þ ¼ exp½0:10tCs2n1 ðt; 4Þ þ exp½0:10t½1 exp½0:10tCs2n2 ðt; 4Þ for t 2 \0; 1Þ; n ¼ 3; 4; . . .; 24:
ð7:33Þ
The expected values and standard deviations of the system unconditional lifetimes in the reliability state subsets {1, 2, 3, 4}, {2, 3, 4}, {3, 4}and {4}, calculated from the results given by (7.30–7.32), according to (3.7–3.9), and using the computer programme respectively are: lð1Þ ¼ 22:969;
rð1Þ ffi 14:086;
ð7:34Þ
lð2Þ ¼ 11:485;
rð2Þ ffi 7:043;
ð7:35Þ
lð3Þ ¼ 4:594;
rð3Þ ffi 2:817;
ð7:36Þ
lð4Þ ¼ 2:297;
rð4Þ ffi 1:409;
ð7:37Þ
and further, considering (3.10) and (7.34–7.37), the mean values of the unconditional lifetimes in the particular reliability states 1, 2, 3, 4 respectively are: ð1Þ ¼ lð1Þ lð2Þ ffi 11:484; l ð2Þ ¼ lð2Þ lð3Þ ffi 6:891; l ð3Þ ¼ lð3Þ lð4Þ ffi 2:297; l ð4Þ ¼ lð4Þ ffi 2:297: l
ð7:38Þ
Since the critical reliability state is r = 2, the system risk function, according to (3.11) and (7.31), is given by (Fig. 7.14) rðtÞ ¼ 1 Cs224 ðt; 2Þ ¼ 1 exp½0:02tCs223 ðt; 2Þ exp½0:02t½1 exp½0:02tCs223 ðt; 2Þ for t 2 \0; 1Þ:
ð7:39Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
349
Fig. 7.14 The graph of the steel cover risk function
Hence, by (3.12), the moment when the system risk function exceeds a permitted level, for instance d = 0.05, is s ¼ r1 ðdÞ ffi 2:5:
ð7:40Þ
7.2.5 Identification, Prediction and Optimization of Container Gantry Crane Operation, Reliability and Availability 7.2.5.1 Container Gantry Crane System Analysis We analyze the reliability of the container gantry crane operating at the container terminal placed at the seashore. The considered container terminal is engaged in trans-shipment of containers. The loading of containers is carried out by using the gantry cranes called Ship-To-Shore (STS). We consider the STS container gantry crane that is composed of 5 basic subsystems S1, S2, S3, S4 and S5 having an essential influence on its reliability. The subsystems are as follows: S1 S2 S3 S4 S5
— — — — —
the the the the the
power supply subsystem, control and monitoring subsystem, arm getting up and getting down subsystem, transferring subsystem, loading and unloading subsystem.
The gantry crane power supply subsystem S1 consists of: • a high voltage cable delivering the energy from the substation to the gantry ð1Þ crane E1 ;
350
7 Additional Applications ð1Þ
• a drum allowing the cable unreeling during the crane transferring E2 ; ð1Þ E3 ;
• an inner crane power supply cable • a device transmitting the energy from the high voltage cable to the inner crane ð1Þ cable E4 ; ð1Þ
• main and supporting voltage transformers E5 ; ð1Þ
• a low voltage power supply cable E6 ; ð1Þ
• relaying and protective electrical components E7 : The gantry crane control and monitoring subsystem S2 consists of: • a crane software controller precisely analyzing the situation and takes suitable ð2Þ actions in order to assure correct work of the crane E1 ; • a measuring and diagnostic device sending signals about the crane state to the ð2Þ software controller E2 ; • a transmitter of signals from the controller to elements executing the set of ð2Þ commands E3 ; • devices carrying out the controller’s orders (a permission to work, a blockade of ð2Þ work, etc.) E4 ; ð2Þ
• control panels (an engine room, an operator’s cabin, a crane arm cabin) E5 ; • control and steering cables’ connections
ð2Þ E6 :
The gantry crane arm getting up and getting down subsystem S3 consists of: • a propulsion unit (an engine, a rope drum, a transmission gear, a clutch, breaks, ð3Þ a rope) E1 ; ð3Þ
• a set of rollers and multi-wheels E2 ; ð3Þ
• a crane arm (joints, hooks fastening the arm) E3 : The gantry crane transferring subsystem S4 consists of: • a driving unit (an engine, a clutch, breaks, a transmission gear, gantry crane ð4Þ wheels) E1 : The gantry crane loading and unloading subsystem S5 consists of the winch unit ð5Þ E1 composed of: • • • •
a a a a
propulsion unit (an engine, a clutch, breaks, a transmission gear, ropes), winch head (which a container grab is connected to), container’s grab, container’s grab stabilizing unit ð5Þ
and the cart unit E2 composed of: • a propulsion unit (an engine, a clutch, breaks, a transmission gear, cart wheels, ropes),
7.2 Operation, Reliability, Availability, Safety and Cost Analysis Fig. 7.15 General scheme of gantry crane reliability structure
S1
S2
351
. . .
S5
• rails which cart is moving on during the operation, • a crane cart. The subsystems S1, S2, S3, S4, S5 form a general series gantry crane reliability structure presented in Fig. 7.15.
7.2.5.2 Container Gantry Crane Operation Process Modelling The container gantry crane reliability structure and the subsystems and components reliability depend on the changing in time operation states. Taking into account expert opinion on the varying in time operation process of the considered container gantry crane we fix the number of system operation process states v = 6 and we distinguish the following as its six operation states: • operation state z1 — the crane standby with the power supply on and the control system off, • operation state z2 — the crane preparing either to start or finish the work with the crane arm angle position of 90o, • operation state z3 — the crane preparing either to start or finish the work with the crane arm angle position of 0o, • operation state z4 — the crane transferring either to or from the loading and unloading area with the crane arm angle position of 90o, • operation state z5 — the crane transferring either to or from the loading and unloading area with the crane arm angle position of 0o, • operation state z6 — loading and unloading of the containers with the crane arm angle position of 0o. Moreover, we fix that there are possible transitions between all system operation states. Thus, according to Sect. 2.2, the unknown parameters of the container gantry crane operation process semi-Markov model are: • the vector of probabilities ½pb ð0Þ16 of the container gantry crane operation process stay at the particular operation states at the initial moment t = 0, • the matrix of probabilities ½pbl 66 of the container gantry crane operation process transitions between the operation states • the matrix of conditional distribution functions ½Hbl ðtÞ66 of the container gantry crane operation process conditional sojourn times hbl at the operation states • the mean values Mbl of the conditional sojourn times hbl ;b; l ¼ 1; 2; . . .; 6; b 6¼ l.
352
7 Additional Applications
To identify all these parameters of the container gantry crane operation process the statistical data coming from this process realizations are needed. The statistical data collected by the system operators are described in the following section. 7.2.5.3 Container Gantry Crane Operation Process Identification Data Collection for Estimating Unknown Parameters of Container Gantry Crane Operation Process The statistical data necessary collected by experts to evaluate the probabilities of the container gantry crane operation process straying at the particular operation states at the initial moment t = 0, defined by (2.1), are: • the container gantry crane operation process observation/experiment time H ¼ 29 days; • the number of container gantry crane operation process realizations nð0Þ ¼ 25; • the vector of realizations n1 ð0Þ ¼ 1; n2 ð0Þ ¼ 18; n3 ð0Þ ¼ 1; n4 ð0Þ ¼ 0; n5 ð0Þ ¼ 2; n6 ð0Þ ¼ 3; of the number of strayings of the system operation process respectively at the operation states z1, z2, …, z6, at the initial moments t = 0 of all n(0) = 25 observed realizations of the system operation process ½nb ð0Þ ¼ ½1; 18; 1; 0; 2; 3; where n1 ð0Þ þ n2 ð0Þ þ n3 ð0Þ þ n4 ð0Þ þ n5 ð0Þ þ n6 ð0Þ ¼ 25: The collected statistical data necessary to evaluate the probabilities pbl of the container gantry crane transitions between the system operation states, defined by (2.2), are: • the matrix of realizations n11 ¼ 0; n12 ¼ 77; n13 ¼ 40; n14 ¼ 1; n15 ¼ 0; n16 ¼ 1; n21 ¼ 62; n22 ¼ 0; n23 ¼ 44; n24 ¼ 11; n25 ¼ 0; n26 ¼ 1; n31 ¼ 40; n32 ¼ 42; n33 ¼ 0; n34 ¼ 0; n35 ¼ 45; n36 ¼ 253; n41 ¼ 5; n42 ¼ 7; n43 ¼ 0; n44 ¼ 0; n45 ¼ 0; n46 ¼ 0; n51 ¼ 1; n52 ¼ 0; n53 ¼ 42; n54 ¼ 0 n55 ¼ 0; n56 ¼ 148; n61 ¼ 5; n62 ¼ 0; n63 ¼ 225; n64 ¼ 0; n65 ¼ 146; n66 ¼ 0;
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
353
of the number of container gantry crane operation process transitions from state zb into state zl during the experiment time H ¼ 29 days 2
0 77 6 62 0 6 6 40 42 ½nbl ¼ 6 6 5 7 6 4 1 0 5 0
40 44 0 0 42 255
1 11 0 0 0 0
0 0 45 0 0 146
3 1 1 7 7 253 7 7; 0 7 7 148 5 0
• the vector of realizations n1 ¼ n11 þ n12 þ n13 þ n14 þ n15 þ n16 ¼ 119; n2 ¼ n21 þ n22 þ n23 þ n24 þ n25 þ n26 ¼ 118; n3 ¼ n31 þ n32 þ n33 þ n34 þ n35 þ n36 ¼ 380; n4 ¼ n41 þ n42 þ n43 þ n44 þ n45 þ n46 ¼ 12; n5 ¼ n51 þ n52 þ n53 þ n54 þ n55 þ n56 ¼ 191; n6 ¼ n61 þ n62 þ n63 þ n64 þ n65 þ n66 ¼ 406; (the sums of the number of matrix ½nbl rows) of the number of container gantry crane operation process departures from the operation state zb during the experiment time H ¼ 29 days ½nb ¼ ½119; 118; 380; 12; 191; 406: The collected statistical data necessary to evaluate the unknown parameters of the distributions of the conditional sojourn times of the container gantry crane operation process in the particular operation states, defined by (2.3), are as follows: • the numbers nbl, b; l ¼ 1; 2; . . .; 6; b 6¼ l; of the realizations hkbl , k = 1,2, …, nbl, of the conditional sojourn times hbl of the container gantry crane operation process at the operation state zb when the next transition is to the operation state zl during the observation time; • the realizations hkbl , k = 1,2, …, nbl, of the conditional sojourn times hbl of the container gantry crane operation process at the operation state zb when the next transition is to the operation state zl during the observation time H= 29 days. For instance, the collected statistical data for the sojourn time h12 are as follows: • the number of realizations n12 = 77, • the realizations hk12 ; k ¼ 1; 2; . . .; 77 : 1480, 159, 356, 422, 369, 235, 47, 30, 7, 1304, 997, 1375, 2, 2, 12, 2301, 411, 8, 1, 1, 1, 1, 1, 1, 1, 1, 435, 631, 1, 304, 37, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 435, 631, 1, 304, 37, 1625, 286, 25, 1157, 11, 225, 22,
354
7 Additional Applications
5, 862, 127, 64, 978, 93, 1126, 82, 337, 144, 777, 401, 270, 1669, 60, 329, 16, 75, 663, 2203, 409, 219, 469, 2898.
Estimating Parameters of Container Gantry Crane Operation Process On the basis of the statistical data presented in ‘‘Data Collection for Estimating Unknown Parameters of Container Gantry Crane Operation Process’’, using respectively the formulae (4.1–4.3) and (4.4–6.6), we evaluate • the vector ½pð0Þ ¼ ½0:04; 0:72; 0:04; 0; 0:08; 0:08 of the probabilities pb ð0Þ, b ¼ 1; 2; . . .; 6; of the container gantry crane operation process stay at the particular states zb at the initial moment t = 0, • the matrix 2 3 0 0:648 0:336 0:008 0 0:008 6 0:525 0 0:373 0:093 0 0:009 7 6 7 6 0:105 0:111 0 0 0:118 0:666 7 7; ½pbl ¼ 6 ð7:41Þ 6 0:417 0:583 0 0 0 0 7 6 7 4 0:005 0 0:220 0 0 0:775 5 0:012 0 0:628 0 0:360 0 of the probabilities pbl, b; l ¼ 1; 2; . . .; 6; of the container gantry crane operation process transitions from the operation state zb to the operation state zl.
Estimating Parameters of Distributions of Container Gantry Crane Conditional Sojourn Times at Operation States On the basis of the statistical data partly presented in ‘‘Data Collection for Estimating Unknown Parameters of Container Gantry Crane Operation Process’’, using the procedure and the formulae given in Sect. 4.2.3, it is possible to determine the empirical characteristics of the realizations of the conditional sojourn time of the container gantry crane operation process at the particular operation states. To illustrate this procedure and the formulae application, we perform it for the conditional sojourn time h12, using their realizations presented in ‘‘Data Collection for Estimating Unknown Parameters of Container Gantry Crane Operation Process’’. The results for the conditional sojourn time h12 are: • defined by (4.7) the realization of the mean value h12 of the conditional sojourn time h12 of the container gantry crane operation process at the operation state z1 when the next transition is to the operation state z2
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
355
77 1 X hk ¼ 389:4; h12 ¼ 77 k¼1 12
• the number r12 of the disjoint intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; r12 that include the realizations hk12 , k ¼ 1; 2; . . .; 77; of the conditional sojourn time h12 at the operation state z1 when the next transition is to the operation state z2 pffiffiffiffiffi r12 ffi 77 ffi 9; • the length d12 of the intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; 9, which after considering 12 ¼ max hk12 min hk12 ¼ 2898 1 ¼ 2897 R 1 k 77
1 k 77
is d12 ¼
12 2897 R ¼ ffi 362; r12 1 8
• the ends ablj ; bblj , of the intervals Ij ¼ hablj ; bblj Þ; j ¼ 1; 2; . . .; 9, which after considering min hk12
1 k 77
d12 362 ¼1 ¼ 180; 2 2
are a112 ¼ maxf180; 0g ¼ 0; b112 ¼ a112 þ 362 ¼ 0 þ 362 ¼ 362; a212 ¼ b112 ¼ 362; b212 ¼ a112 þ 2 362 ¼ 0 þ 724 ¼ 724; a312 ¼ b212 ¼ 724; b312 ¼ a112 þ 3 362 ¼ 0 þ 1086 ¼ 1086; a412 ¼ b312 ¼ 1086; b412 ¼ a112 þ 4 362 ¼ 0 þ 1448 ¼ 1448; a512 ¼ b412 ¼ 1448; b512 ¼ a112 þ 5 362 ¼ 0 þ 1810 ¼ 1810; a612 ¼ b512 ¼ 1810; b612 ¼ a112 þ 6 362 ¼ 0 þ 2172 ¼ 2172; a712 ¼ b612 ¼ 2172; b712 ¼ a112 þ 7 362 ¼ 0 þ 2534 ¼ 2534; a812 ¼ b712 ¼ 2534; b812 ¼ a112 þ 8 362 ¼ 0 þ 2896 ¼ 2896; a912 ¼ b812 ¼ 2896; b912 ¼ a112 þ 9 362 ¼ 0 þ 3258 ¼ 3258; j of the realizations hk12 in particular intervals Ij ¼ hablj ; bblj Þ; • the numbers n12 j ¼ 1; 2; . . .; 9;
n112 ¼ 52; n212 ¼ 11; n312 ¼ 4; n412 ¼ 4; n512 ¼ 3; n612 ¼ 0; n712 ¼ 2; n812 ¼ 0; n912 ¼ 1:
356
7 Additional Applications
Table 7.1 The realization of the histogram of the conditional sojourn time h12 Histogram of the conditional sojourn time h12 j j Ij ¼ ha12 ; b12 Þ
0–362
362–724
724–1086
1086–1448
1448–1810
1810–2172
2172–2534
2534–2896
j n12 12 ðtÞ ¼ n j =n12 h 12
52
11
4
4
3
0
2
0
1
52/77
11/77
4/77
4/77
3/77
0/77
2/77
0/77
1/77
2896–3258
Fig. 7.16 The graph of the histogram of the conditional sojourn time h12
Identifying Distribution Functions of Container Gantry Crane Conditional Sojourn Times at Operation States Using the procedure given in Sect. 4.2.4 and the results partly presented in ‘‘Estimating Parameters of Distributions of Container Gantry Crane Conditional Sojourn Times at Operation States’’, we verify the hypotheses on the distributions of the system conditional sojourn times hbl , b; l ¼ 1; 2; . . .; 6; b 6¼ l; at the particular operation states. To make the procedure familiar to the reader, we perform it for the conditional sojourn time for h12, preliminary analyzed in ‘‘Estimating Parameters of Distributions of Container Gantry Crane Conditional Sojourn Times at Operation States’’. The realization h12 ðtÞ of the histogram of the conditional sojourn time h12 defined by (4.45) is presented in Table 7.1 and illustrated in Fig. 7.16. After analyzing and comparing the realization h12 ðtÞ of the histogram with the graphs of the density functions hbl ðtÞ of the previously distinguished in Sect. 2.2 distributions, we formulate the null hypothesis H0 in the following form: H0: The system conditional sojourn time h12 at the operation state z1 when the next transition is to the operation state z2, has the Weibull’s distribution with the density function defined by (2.10) in the following form
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
( h12 ðtÞ ¼
357
t [ x12
0; a12 b12 ðt x12 Þ
b12 1
b12
exp½a12 ðt x12 Þ ;
t x12 :
ð7:42Þ
We estimate the unknown parameters of the density function of the hypothetical Weibull’s distribution using the formulae (4.14) that take the following forms x12 ¼ a112 ¼ 0; a12 ¼ P77
77
k¼1
ðhkbl Þb12
; a12 ¼
P77 k 77 k¼1 lnðh12 x12 Þ b12 þ : P77 k bbl lnðhk12 x12 Þ k¼1 ðhbl Þ
ð7:43Þ
The approximate solution of the above system of equations (7.43) are a12 ¼ 0:097 and b12 ¼ 0:448: Substituting the above results into (7.42), we completely define the hypothetical density function in the form ( 0; t[0 h12 ðtÞ ¼ ð7:44Þ 0:4481 0:448 0:097 0:448t exp½0:097t ; t\0: Hence, the hypothetical distribution function H12(t) of the conditional sojourn time h12, after taking the integral of the hypothetical density function h12(t) given by (7.44), takes the following form H12 ðtÞ ¼
Zt 0
( h12 ðtÞdt ¼
0; 1 exp½0:097t
t\0 0:448
;
t 0:
ð7:45Þ
Next, we join the intervals defined in the realization of the histogram h12(t) that j have the numbers n12 ; of realizations less than 4 into new intervals performing the following steps: • we fix the new number of intervals r 12 ¼ 5, • we determine the new intervals I1 ¼ h0; 362Þ; I2 ¼ h362; 724Þ; I3 ¼ h724; 1086Þ; I4 ¼ h1086; 1448Þ; I5 ¼ h1448; 3258Þ; • we fix the number of realizations in the new intervals 212 ¼ 11; n 312 ¼ 4; n 412 ¼ 4; n512 ¼ 6; n112 ¼ 52; n • we calculate, using (4.46) and (7.45), the hypothetical probabilities that the variable h12 takes values from the new intervals p1 ¼ Pðh12 2 I1 Þ ¼ Pð0 h12 \362Þ ¼ H12 ð362Þ H12 ð0Þ ¼ ð1 exp½0:097 3620:448 Þ 0 ¼ 1 0:257 ¼ 0:734;
358
7 Additional Applications
p2 ¼ Pðh12 2 I2 Þ ¼ Pð362 h12 \724Þ ¼ H12 ð724Þ H12 ð362Þ ¼ ð1 exp½0:097 7240:448 Þ ð1 exp½0:097 3620:448 Þ ¼ 0:157 þ 0:257 ¼ 0:1; p3 ¼ Pðh12 2 I3 Þ ¼ Pð724 h13 \1086Þ ¼ H12 ð1086Þ H12 ð724Þ ¼ ð1 exp½0:097 10860:448 Þ ð1 exp½0:097 7240:448 Þ ¼ 0:108 þ 0:157 ¼ 0:049; p4 ¼ Pðh12 2 I4 Þ ¼ Pð1086 h13 \1448Þ ¼ H12 ð1448Þ H12 ð1086Þ ¼ ð1 exp½0:097 14480:448 Þ ð1 exp½0:097 10860:448 Þ ¼ 0:08 þ 0:108 ¼ 0:028; p5 ¼ Pðh12 2 I5 Þ ¼ Pð1448 h13 \; 1Þ ¼ 1 ½p1 þ p2 þ p3 þ p4 ¼ 1 ½0:743 þ 0:1 þ 0:049 þ 0:028 ¼ 1 0:92 ¼ 0:08 • we calculate, using (4.47), the realization of the v2 (chi-square)-Pearson’s statistics u77 ¼
j 5 X ð n12 n12 pj Þ2 ð52 77 0:743Þ2 ð11 77 0:1Þ2 ð4 77 0:049Þ2 þ þ ¼ n12 pj 77 0:743 77 0:1 77 0:049 j¼1
ð4 77 0:028Þ2 ð6 77 0:08Þ2 þ 77 0:028 77 0:08 ffi 0:47 þ 1:41 þ 0:06 þ 0:86 þ 0:004 ¼ 2:804; þ
ð7:46Þ • we assume the significance level a ¼ 0:05, • we fix the number of degrees of freedom r12 l 1 ¼ 5 2 1 ¼ 2; • we read from the Tables of the v2 -Pearson’s distribution the value ua for the fixed values of the significance level a ¼ 0:05 and the number of degrees of freedom r 12 l 1 ¼ 2; such that, according to (4.48), the following equality holds PðU77 [ ua Þ ¼ a ¼ 0:05 which amounts to ua ¼ 5:99 and we determine the critical domain in the form of the interval ð5:99; þ1Þ and the acceptance domain in the form of the interval \0; 5:99 [ (Fig. 7.17). • we compare the obtained in (7.46) value u77 ¼ 2:804 of the realization of the statistics U77 with that read from the Tables, critical value ua ¼ 5:99 of the chisquare random variable and since the value u24 ¼ 2:804 does not belong to the critical domain, i.e.
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
359
Fig. 7.17 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test
u77 ¼ 2:804 ua ¼ 5:99; we do not reject the hypothesis H0. For the remaining cases, when the realizations of conditional sojourn times hbl are more than 30, proceeding afterwards in an analogous way as in the case of the conditional sojourn time h12, we can get the following results: • the conditional sojourn time h13 has a chimney distribution with the density function 8 0; t\0 > > < 0:0218; 0 t\39 h13 ðtÞ ¼ 0:00076; 39 t\234 > > : 0; t 234: • the conditional sojourn time h21 has a chimney distribution with the density function 8 0; t\0 > > < 0:066923; 0 t\13 h21 ðtÞ ¼ 0:003769; 13 t\104 > > : 0; t 104: • the conditional sojourn time h23 has a function 8 0; > > < 0:09253; h23 ðtÞ ¼ 0:002436; > > : 0;
chimney distribution with the density t\0 0 t\9:33 9:33 t\65:33 t 65:33;
• the conditional sojourn time h31 has an exponential distribution with the density function 0; t\0 h31 ðtÞ ¼ 0:182 exp½0:182t; t 0;
360
7 Additional Applications
• the conditional sojourn time h32 has a chimney distribution with the density function 8 0; t\0 > > < 0:20677; 0 t\3:8 h32 ðtÞ ¼ 0:011278; 3:8 t\22:8 > > : 0; t 22:8; • the conditional sojourn time h35 has an exponential distribution with the density function 0; t\0 h35 ðtÞ ¼ 0:147 exp½0:147t; t 0; • the conditional sojourn time h36 has a chimney distribution with the density function 8 0; t\0 > > < 0:099; 0 t\7 h36 ðtÞ ¼ 0:0148081; 7 t\112 > > : 0; t 112; • the conditional sojourn time h53 has a chimney distribution with the density function 8 0; t\0:3 > > < 0:25510; 0:3 t\3:1 h53 ðtÞ ¼ 0:05102; 3:1 t\8:7 > > : 0; t 8:7; • the conditional sojourn time h56 has a function 8 0; > > < 0:02610; h56 ðtÞ ¼ 0:00000659; > > : 0;
chimney distribution with the density t\0 0 t\37:27 37:27 t\447:27 t 447:27;
• the conditional sojourn time h63 has Weibull’s distribution with the density function 0; t\0 h63 ðtÞ ¼ 0:061 0:904t0:9041 exp½0:061t0:904 ; t 0; • the conditional sojourn time h65 has Weibull’s distribution with the density function
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
h65 ðtÞ ¼
0; 0:081 0:854t0:8541 exp½0:081t0:854 ;
361
t\0 t 0:
For the distributions identified in this section, by application either the general formulae for the mean value given by (2.12) or the particular formulae (2.13–2.19), the mean values Mbl ¼ E½hbl ; b; l ¼ 1; 2; . . .; 6; b 6¼ l; of the container gantry crane operation process conditional sojourn times at the particular operation states can be determined and they amount to: M12 ¼ 456:978; M13 ¼ 36:860; M21 ¼ 7:887; M23 ¼ 9:121; M31 ¼ 5:5; M32 ¼ 4:343; M35 ¼ 6:822; M36 ¼ 7:857; M53 ¼ 2:899; M56 ¼ 24:681; ð7:47Þ M63 ¼ 23:117; M65 ¼ 20:512: In the remaining cases, when the number of realizations of the sojourn times are less than 30 and the distributions cannot be identified, using formula (4.7), it is possible to find the approximate empirical values of the mean values Mbl ¼ E½hbl of the conditional sojourn times at the particular operation states that are as follow: M14 ¼ 50; M16 ¼ 3; M24 ¼ 1:545; M26 ¼ 16; M41 ¼ 2; M42 ¼ 2:143; M51 ¼ 10; M61 ¼ 22:6:
ð7:48Þ
These approximate mean values given by (7.48) and previously fixed mean values given by (7.47) are used in Sect. 7.2.5.4 for the prediction of the characteristics of the container gantry crane operation process. As there are no realizations of conditional sojourn times h15 ; h25 ; h34 ; h43 ; h45 ; h46 ; h52 ; h54 ; h62 ; h64 ; it is impossible to estimate their empirical conditional mean values M15 ; M25 ; M34 ; M43 ; M45 ; M46 ; M52 ; M54 ; M62 ; M64 :
7.2.5.4 Container Gantry Crane Operation Process Prediction After considering the results (7.47–7.48) and applying the formulae (2.21), we conclude that the unconditional mean sojourn times of the container gantry crane operation process at the particular operation states are given by: M1 ¼ E½h1 ¼ p12 M12 þ p13 M13 þ p14 M14 þ p16 M16 ¼ 0:648 456:978 þ 0:336 36:860 þ 0:008 50 þ 0:008 3 ffi 308:93; ð7:49Þ
362
7 Additional Applications
M2 ¼ E½h2 ¼ p21 M21 þ p23 M23 þ p24 M24 þ p26 M26 ¼ 0:525 7:887 þ 0:373 9:121 þ 0:093 1:545 þ 0:009 16 ffi 7:83; ð7:50Þ M3 ¼ E½h3 ¼ p31 M31 þ p32 M32 þ p35 M35 þ p36 M36 ¼ 0:105 5:5 þ 0:111 4:343 þ 0:118 6:822 þ 0:666 7:857 ffi 7:09; ð7:51Þ M4 ¼ E½h4 ¼ p41 M41 þ p42 M42 ¼ 0:417 2 þ 0:583 2:143 ffi 2:08;
ð7:52Þ
M5 ¼ E½h5 ¼ p51 M51 þ p53 M53 þ p56 M56 ¼ 0:005 10 þ 0:220 2:899 þ 0:775 24:681 ffi 19:82;
ð7:53Þ
M6 ¼ E½h6 ¼ p61 M61 þ p63 M63 þ p65 M65 ¼ 0:012 22:6 þ 0:628 23:117 þ 0:360 20:512 ffi 22:17:
ð7:54Þ
Since, according to (2.23), from the system of equations ( ½p1 ; p2 ; p3 ; p4 ; p5 ; p6 ¼ ½p1 ; p2 ; p3 ; p4 ; p5 ; p6 ½pbl 66 p1 þ p2 þ p3 þ p4 þ p5 þ p6 ¼ 1; we get p1 ¼ 0:0951; p2 ¼ 0:1020; p3 ¼ 0:3100; p4 ¼ 0:0102; p5 ¼ 0:1547; p6 ¼ 0:3280:
ð7:55Þ
Then, the limit values of the transient probabilities pb(t) of the gantry crane operation process at the operation states zb, according to (2.22), are given by p1 ¼ 0:6874; p2 ¼ 0:0187; p3 ¼ 0:0515; p4 ¼ 0:0005; p5 ¼ 0:0717; p6 ¼ 0:1702:
ð7:56Þ
Hence, the expected values of the total sojourn times ^hb , b ¼ 1; 2; . . .; 6; of the container gantry crane operation process at the particular operation states zb, b ¼ 1; 2; . . .; 6; during the fixed operation time h ¼ 1 year ¼ 365 days; after applying (2.24), amount to: E½h^1 ¼ 0:6874 year ffi 251 days; E½^ h2 ¼ 0:0187 year ffi 7 days; h4 ¼ 0:0005 year ffi 0:2 day; E½^h3 ¼ 0:0515 year ffi 19 day; E½^ ^ E½h5 ¼ 0:0717 year ffi 26 days; E½^ h6 ¼ 0:1702 year ffi 62 days:
ð7:57Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis Fig. 7.18 The scheme of the container gantry crane at operation state z1
363
S1 E1(1)
E 2(1)
E3(1)
E4(1)
E 5(1)
E 6(1)
E 7(1)
7.2.5.5 Container Gantry Crane Components Reliability Modeling After discussion with experts, taking into account the effectiveness of the operation of the container gantry crane, we fix that the system and its components have four reliability states 0, 1, 2, 3, i.e. z = 3. And consequently, at all operation states zb, b ¼ 1; 2; . . .; 6, we distinguish the following reliability states of the system and its components: • a reliability state 3 — the gantry operation is fully effective, • a reliability state 2 — the gantry operation is less effective because of ageing, • a reliability state 1 — the gantry operation is less effective because of ageing and more dangerous, • a reliability state 0 — the gantry is destroyed. We assume that there are possible transitions between the components, reliability states only from better to worse and we fix that the system and components, critical reliability state is r = 2. Moreover, we assume that the changes of the operation states of the container gantry crane operation process have an influence on its reliability structure and its multistate components reliability as well. The container gantry crane operation process influence on the system reliability structure is expressed as follows. At the system operation state z1, the container gantry crane is composed of the ð1Þ subsystem S1 which is a series system composed of n = 7 components Ei ; i ¼ 1; 2; ::; 7 (subsystems) with the structure shown in Fig. 7.18. At the system operation states z2 and z3, the container gantry crane is composed of subsystems S1, S2 and S3 forming a series structure shown in Fig. 7.19. Subð1Þ system S1 is a series system composed of n = 7 components Ei ; i ¼ 1; 2; ::; 7, ð2Þ
subsystem S2 is a series system composed of n = 6 components Ei ; i ¼ 1; 2; ::; 6, and subsystem S3 is a series system composed of n = 3 components ð3Þ Ei ; i ¼ 1; 2; 3: At the system operation states z4 and z5, the container gantry crane is composed of subsystems S1, S2, S3 and S4 forming a series structure shown in Fig. 7.20. ð1Þ Subsystem S1 is a series system composed of n = 7 components Ei ; i ¼ 1; 2; ::; 7, ð2Þ
subsystem S2 is a series system composed of n = 6 components Ei ; i ¼ 1; 2; ::; 6, ð3Þ
subsystem S3 is a series system composed of n = 3 components Ei ; i ¼ 1; 2; 3, ð4Þ
and subsystem S4 consists of a component E1 : At the system operation state z6, the container gantry crane is composed of subsystems S1, S2, S3 and S5 forming a series structure shown in Fig. 7.21.
364
7 Additional Applications S2
S1 E1(1)
E7(1)
E2(1)
E1( 2)
E2( 2)
S3 E1(3)
E6( 2)
E2(3)
E3(3)
Fig. 7.19 The scheme of the container gantry crane at operation states z2 and z3 S2
S1 E1(1)
E2(1)
E1( 2)
E7(1)
S3 E1(3)
E6( 2)
E2( 2)
S4
E2(3)
E3(3)
E1( 4)
Fig. 7.20 The scheme of the container gantry crane at operation states z4 and z5 S1 E1(1)
E2(1)
S2 E1( 2)
E7(1)
E2( 2)
S3 ( 3) 1
( 3) 2
E
E6( 2)
E
S5 ( 3) 3
E1(5)
E
E2(5)
Fig. 7.21 The scheme of the container gantry crane at operation state z6 ð1Þ
Subsystem S1 is a series system composed of n = 7 components Ei ;i ¼ 1; 2; ::; 7, ð2Þ
subsystem S2 is a series system composed of n = 6 components Ei ;i ¼ 1; 2; ::; 6, ð3Þ
subsystem S3 is a series system composed of n = 3 components Ei ;i = 1, 2, 3 ð5Þ
and subsystem S5 is a series system composed of n = 2 components Ei ;i = 1, 2. The container gantry crane operation process, influence on its components reliability is expressed by the assumption that, its subsystems St ; t ¼ 1; 2; . . .5; are ðtÞ composed of four-state, i.e. z = 3, components Ei ; t ¼ 1; 2; . . .; 5 having the conditional four-state reliability functions ðtÞ
ðtÞ
ðtÞ
ðtÞ
½Ri ðt; ÞðbÞ ¼ ½1; ½Ri ðt; 1ÞðbÞ ; ½Ri ðt; 2ÞðbÞ ; ½Ri ðt; 3ÞðbÞ ; b ¼ 1; 2; . . .; 6; ð7:58Þ with the exponential co-ordinates ðtÞ
ðtÞ
ðtÞ
ðtÞ
½Ri ðt; 1ÞðbÞ ¼ exp½½ki ð 1ÞðbÞ t; ½Ri ðt; 2ÞðbÞ ¼ exp½½ki ð 2ÞðbÞ t; ðtÞ
ðtÞ
½Ri ðt; 3ÞðbÞ ¼ exp½½ki ð 3ÞðbÞ t; t 0; b ¼ 1; 2; . . .; 6; t ¼ 1; 2; . . .; 5; ð7:59Þ different at the various operation states zb, b ¼ 1; 2; . . .; 6; where ðtÞ
ðtÞ
ðtÞ
½ki ð 1ÞðbÞ ; ½ki ð 2ÞðbÞ ; ½ki ð 3ÞðbÞ ; b ¼ 1; 2; ::; 6; t ¼ 1; 2; . . .; 5; are the subsystems components, unknown intensities of departures respectively from the reliability state subsets {1, 2, 3}, {2, 3}, {3}.
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
365
7.2.5.6 Container Gantry Crane Components Reliability Identification Data Collections Coming from Experts To estimate existing in the formulae (7.58–7.59) the subsystem components, ðtÞ ðtÞ ðtÞ unknown intensities ½kij ð 1ÞðbÞ ; ½kij ð 2ÞðbÞ ; ½ki ð 3ÞðbÞ ;b ¼ 1; 2; ::; 6; t ¼ 1; 2; . . .; 5; of departure respectively from the reliability state subsets {1, 2, 3}, {2, 3}, {3} we have data from experts. The approximate realizations ðvÞ
½^ li ðuÞðbÞ ; t ¼ 1; 2; . . .; 5; u ¼ 1; 2; 3; b ¼ 1; 2; ::; 6; of the mean values ðvÞ
ðtÞ
½^ li ðuÞðbÞ ¼ E½½Ti ðuÞðbÞ ; t ¼ 1; 2; . . .; 5; u ¼ 1; 2; 3; b ¼ 1; 2; ::; 6; ðtÞ
of the conditional lifetimes ½Ti ðuÞðbÞ ; t ¼ 1; 2; . . .; 5; u = 1, 2, 3, b ¼ 1; 2; ::; 6; ðtÞ
in reliability state subsets fu; u þ 1; . . .; 3g, u = 1, 2, 3, of the component Ei of the container gantry crane subsystems St ; t ¼ 1; 2; . . .; 5; at the particular operation states zb, b ¼ 1; 2; . . .; 6; estimated on the basis of expert opinions are collected. For instance, in Table 7.2, there are data from experts concerning the compoð1Þ nents Ei ; i ¼ 1; 2; . . .; 7; of subsystem S1.
Evaluating Container Gantry Crane Components Intensities of Departures from Reliability State Subsets ðtÞ
ðtÞ
ðtÞ
To evaluate the approximate values ½^ ki ð1ÞðbÞ ; ½^ki ð2ÞðbÞ ; ½^ki ð3ÞðbÞ ; of subSt ; ðtÞ ðbÞ ½ki ð2Þ and systems
t ¼ 1; 2; ::; 5
components,
unknown
intensities
ðtÞ
½ki ð1ÞðbÞ ,
ðtÞ ½ki ð3ÞðbÞ
of departure respectively from the reliability states subsets {1, 2, 3}, {2, 3}, {3} while the container gantry crane is operating at the operation state zb, b ¼ 1; 2; . . .; 6; existing in (7.59), we can use statistical data from experts partly presented in ‘‘Data Collections Coming from Experts’’. The statistical data collected by experts operating the gantry crane and resulting from (5.22) formula ðtÞ
½ki ðuÞðbÞ ¼
1 ðtÞ ½^ li ðuÞðbÞ
;
t ¼ 1; 2; ::; 5;
u ¼ 1; 2; 3;
b ¼ 1; 2; . . .; 6; ð7:60Þ
application yield the approximate values of the system components, unknown intensities of departures.
366
7 Additional Applications ð1Þ
Table 7.2 The approximate mean values ½^ li ðuÞðbÞ of subsystem S1 components, conditional ð1Þ lifetimes ½Ti ðuÞðbÞ at operation states zb ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ Component of subsystem S1 E1 E2 E3 E4 E5 E6 E7 ð1Þ Operation state Reliability state subset Mean value ½^ li ðuÞðbÞ (in years) zb fu; u þ 1; . . .; 3g z1
{1,2,3} {2,3} {3} {1,2,3} {2,3} {3} {1,2,3} {2,3} {3} {1,2,3} {2,3} {3} {1,2,3} {2,3} {3} {1,2,3} {2,3} {3}
z2
z3
z4
z5
z6
50 30 20 50 30 20 50 30 20 35 25 15 35 25 15 50 30 20
25 20 15 25 20 15 25 20 15 20 15 10 20 15 10 25 20 15
25 20 15 25 20 15 25 20 15 25 20 15 25 20 15 25 20 15
50 30 20 50 30 20 50 30 20 50 30 20 50 30 20 50 30 20
47 39 23 45 37 21 45 37 21 45 37 21 45 37 21 42 34 18
55 35 25 55 35 25 55 35 25 53 33 23 53 33 23 50 30 20
30 25 20 30 25 20 28 23 18 28 23 18 28 23 18 27 22 17
For instance, substituting into (7.60) the values of the mean lifetimes ð1Þ
½^ l1 ð1Þð1Þ ¼ 50;
ð1Þ
½^ l1 ð2Þð1Þ ¼ 30;
ð1Þ
½^ l1 ð3Þð1Þ ¼ 20;
taken from the third, fourth and fifth rows of the third column of Table 7.2, we obtain the approximate evaluations of the unknown intensities of departure of the ð1Þ component E1 of subsystem S1 from the reliability states subset {1, 2, 3}, {2, 3}, {3} while the container gantry crane is operating at the operation state z1 that respectively amount to: ð1Þ
½k1 ð1Þð1Þ ¼ ð1Þ
½k1 ð3Þð1Þ ¼
1 ð1Þ ½^ l1 ð1Þð1Þ
1 ð1Þ ½^ l1 ð3Þð1Þ
1 1 1 ð1Þ ¼ 0:020; ½k1 ð2Þð1Þ ¼ ð1Þ ffi 0:033; ¼ ð1Þ 50 30 ½^ l1 ð2Þ 1 ¼ ¼ 0:050: 20 ¼
ð7:61Þ The evaluations of all unknown intensities of departure from the reliability states subset {1, 2, 3}, {2, 3}, {3} of components of the container gantry crane operating at various operation states, obtained in this way, allows to determine the system components’ reliability functions.
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
367
Substituting the evaluation given by (7.61) and the remaining evaluations of the intensities of the subsystems components, departures from the reliability state subsets respectively into the formulae (7.59), we get the exponential coordinates of the container gantry crane components, four-state reliability functions defined by (7.58) which after arbitrary acceptance in ‘‘Identifying Container Gantry Crane Components Exponential Reliability Functions’’ are used in Sect. 7.2.5.7 for the prediction of the system reliability characteristics. Identifying Container Gantry Crane Components, Exponential Reliability Functions As there are no data collected from the container gantry crane components reliability states changing processes, it is not possible to verify the hypotheses on the exponential forms of the container gantry crane components, conditional reliability functions. Consequently, we arbitrarily assume that these reliability functions are exponential and using the results partly presented in the previous section and the relationships (7.58–7.59) given in Sect. 7.2.5.5 we fix their forms. For instance, using evaluations (7.61) and expressions (7.58–7.59), we conclude ð1Þ that at the system operation state z1 subsystem S1 component E1 has the conditional reliability function ð1Þ
ð1Þ
ð1Þ
ð1Þ
½R1 ðt; Þð1Þ ¼ ½1; ½R1 ðt; 1Þð1Þ ; ½R1 ðt; 2Þð1Þ ; ½R1 ðt; 3Þð1Þ ; t 0; with the exponential coordinates ð1Þ
ð1Þ
½R1 ðt; 1Þð1Þ ¼ exp½0:020t; ½R1 ðt; 2Þð1Þ ¼ exp½0:033t; ð1Þ
½R1 ðt; 3Þð1Þ ¼ exp½0:050t: In this way, arbitrarily fixed the exponential four-state reliability functions of the container gantry crane components are used in Sect. 7.2.5.7 for the prediction of the system reliability characteristics. 7.2.5.7 Container Gantry Crane Reliability Prediction Considering the results of the system components reliability modeling from Sect. 7.2.5.5 concerned with the fixed system reliability structures and their shape parameters and with the assumed exponential models of the reliability functions of the system components at various operation states and the results of the evaluations of the system components, intensities of departures from the reliability state subsets partly presented in Sect. 7.2.5.6, we perform the prediction of the container gantry crane reliability characteristics. Thus, as we fixed in Sect. 7.2.5.5 that at the system operation state z1, the container gantry crane is identical with subsystem S1, which is a four-state series system with its structure shape parameter n = 7 and according to (1.22–1.23), its four-state reliability function is given by the vector
368
7 Additional Applications
½Rðt; Þð1Þ ¼ ½1; ½Rðt; 1Þð1Þ ; ½Rðt; 2Þð1Þ ; ½Rðt; 3Þð1Þ ; t 0; with the coordinates 7 Y ð1Þ 7 ðt; 1Þ ¼ ½Ri ðt; 1Þð1Þ ; ½Rðt; 1Þð1Þ ¼ R i¼i
7 ðt; 1Þ ¼ ½Rðt; 2Þð1Þ ¼ R
7 Y
ð1Þ
½Ri ðt; 2Þð1Þ ;
i¼i
7 ðt; 1Þ ¼ ½Rðt; 3Þð1Þ ¼ R
7 Y
ð1Þ
½Ri ðt; 3Þð1Þ :
i¼i
After substituting in the above expressions for the coordinates, the suitable evaluations of the container gantry crane components intensities of departures from the reliability state subsets partly found in Sect. 7.2.5.6, we get: ½Rðt; 1Þð1Þ ¼ exp½0:020texp½0:040texp½0:040texp½0:020t ð7:62Þ exp½0:020texp½0:018texp½0:033t ¼ exp½0:191t; ½Rðt; 2Þð1Þ ¼ exp½0:033texp½0:050texp½0:050texp½0:033t ð7:63Þ exp½0:030texp½0:028texp½0:040t ¼ exp½0:264t; ½Rðt; 3Þð1Þ ¼ exp½0:050texp½0:066texp½0:066texp½0:050t ð7:64Þ exp½0:040texp½0:040texp½0:050t ¼ exp½0:362t: The expected values of the container gantry crane conditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} at the operation state z1, calculated from the results given by (7.62–7.64), according to (3.8), respectively are: l1 ð1Þ ffi 5:24; l1 ð2Þ ffi 3:79; l1 ð3Þ ffi 2:76 years:
ð7:65Þ
At the system operation state z2, the container gantry crane is composed of subsystems S1, S2 and S3 forming a series structure. At this operation state, subsystem S1 is a four-state series system with its structure shape parameter n = 7 and according to (1.22–1.23), its four-state reliability function is given by the vector ½Rð1Þ ðt; Þð2Þ ¼ ½1; ½Rð1Þ ðt; 1Þð2Þ ; ½Rð1Þ ðt; 2Þð2Þ ; ½Rð1Þ ðt; 3Þð2Þ ; t 0; with the coordinates 7 Y ð1Þ 7 ðt; 1Þ ¼ ½Ri ðt; 1Þð2Þ ; ½Rð1Þ ðt; 1Þð2Þ ¼ R i¼i
7 ðt; 1Þ ¼ ½Rð1Þ ðt; 2Þð2Þ ¼ R
7 Y
ð1Þ
½Ri ðt; 2Þð2Þ ;
i¼i
7 ðt; 1Þ ¼ ½Rð1Þ ðt; 3Þð2Þ ¼ R
7 Y i¼i
ð1Þ
½Ri ðt; 3Þð2Þ :
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
369
After substituting in the above expressions for the coordinates, the suitable evaluations of the container gantry crane components intensities of departures from the reliability state subsets partly found in Sect. 7.2.5.6, we get: ½Rð1Þ ðt;1Þð2Þ ¼ exp½0:020texp½0:040texp½0:040texp½0:020t exp½0:022texp½0:018texp½0:033t ¼ exp½0:193t; ½Rð1Þ ðt;2Þð2Þ ¼ exp½0:033texp½0:050texp½0:050texp½0:033t exp½0:027texp½0:028texp½0:040t ¼ exp½0:261t;
ð7:66Þ
ð7:67Þ
½Rð1Þ ðt;3Þð2Þ ¼ exp½0:050texp½0:066texp½0:066texp½0:050t ð7:68Þ exp½0:048texp½0:040texp½0:050t ¼ exp½0:370t: Subsystem S2 at the operation state z2, is a four-state series system with its structure shape parameter n = 6 and according to (1.22–1.23), its four-state reliability function is given by the vector ½Rð2Þ ðt; Þð2Þ ¼ ½1; ½Rð2Þ ðt; 1Þð2Þ ; ½Rð2Þ ðt; 2Þð2Þ ; ½Rð2Þ ðt; 3Þð2Þ ; t 0; with the coordinates 6 ðt; 1Þ ¼ ½Rð2Þ ðt; 1Þð2Þ ¼ R
6 Y
ð2Þ
½Ri ðt; 1Þð2Þ ;
i¼i
6 ðt; 1Þ ¼ ½Rð2Þ ðt; 2Þð2Þ ¼ R
6 Y
ð2Þ
½Ri ðt; 2Þð2Þ ;
i¼i
6 ðt; 1Þ ¼ ½Rð2Þ ðt; 3Þð2Þ ¼ R
6 Y
ð2Þ
½Ri ðt; 3Þð2Þ :
i¼i
After substituting in the above expressions for the coordinates, the suitable evaluations of the container gantry crane components intensities of departures from the reliability state subsets, we get: ½Rð2Þ ðt; 1Þð2Þ ¼ exp½0:053texp½0:048texp½0:048texp½0:048texp½0:020t exp½0:018t ¼ exp½0:235t; ð7:69Þ ½Rð2Þ ðt; 2Þð2Þ ¼ exp½0:059texp½0:053texp½0:053texp½0:053texp½0:025t exp½0:029t ¼ exp½0:272t; ð7:70Þ ½Rð2Þ ðt; 3Þð2Þ ¼ exp½0:066texp½0:059texp½0:059texp½0:059texp½0:033t exp½0:040t ¼ exp½0:316t: ð7:71Þ
370
7 Additional Applications
Subsystem S3 at the operation state z2, is a four-state series system with its structure shape parameter n = 3 and according to (1.22–1.23), its four-state reliability function is given by the vector ½Rð3Þ ðt; Þð2Þ ¼ ½1; ½Rð3Þ ðt; 1Þð2Þ ; ½Rð3Þ ðt; 2Þð2Þ ; ½Rð3Þ ðt; 3Þð2Þ ;
t 0;
with the coordinates 3 ðt; 1Þ ¼ ½Rð3Þ ðt; 1Þð2Þ ¼ R
3 Y
ð3Þ
½Ri ðt; 1Þð2Þ ; ½Rð3Þ ðt; 2Þð2Þ
i¼i
3 ðt; 1Þ ¼ ¼R
3 Y
ð3Þ
½Ri ðt; 2Þð2Þ ;
i¼i
3 ðt; 1Þ ¼ ½Rð3Þ ðt; 3Þð2Þ ¼ R
3 Y
ð3Þ
½Ri ðt; 3Þð2Þ :
i¼i
After substituting in the above expressions for the coordinates, the suitable evaluations of the container gantry crane components intensities of departures from the reliability state subsets, we get: ½Rð3Þ ðt; 1Þð2Þ ¼ exp½0:025texp½0:033texp½0:033t ¼ exp½0:091t; ð7:72Þ ½Rð3Þ ðt; 2Þð2Þ ¼ exp½0:040texp½0:040texp½0:040t ¼ exp½0:120t; ð7:73Þ ½Rð3Þ ðt; 3Þð2Þ ¼ exp½0:066texp½0:066texp½0:066t ¼ exp½0:198t: ð7:74Þ Considering that the container gantry crane at the operation state z2 is a fourstate series system composed of subsystems S1, S2 and S3, after applying (1.22– 1.23), its conditional four-state reliability function is given by the vector ½Rðt; Þð2Þ ¼ ½1; ½Rðt; 1Þð2Þ ; ½Rðt; 2Þð2Þ ; ½Rðt; 3Þð2Þ ; t 0; with the coordinates 3 ðt; 1Þ ¼ ½Rð1Þ ðt; 1Þð2Þ ½Rð2Þ ðt; 1Þð2Þ ½Rð3Þ ðt; 1Þð2Þ ; ½Rðt; 1Þð2Þ ¼ R 3 ðt; 2Þ ¼ ½Rð1Þ ðt; 2Þð2Þ ½Rð2Þ ðt; 2Þð2Þ ½Rð3Þ ðt; 2Þð2Þ ; ½Rðt; 2Þð2Þ ¼ R 3 ðt; 2Þ ¼ ½Rð1Þ ðt; 3Þð2Þ ½Rð2Þ ðt; 3Þð2Þ ½Rð3Þ ðt; 3Þð2Þ : ½Rðt; 3Þð2Þ ¼ R After substituting in the above expressions for the coordinates the results (7.66– 7.68), (7.69–7.71), and (7.72–7.74), we get: ½Rðt; 1Þð2Þ ¼ exp½0:193texp½0:235texp½0:091t ¼ exp½0:519t;
ð7:75Þ
½Rðt; 2Þð2Þ ¼ exp½0:261texp½0:272texp½0:120t ¼ exp½0:653t;
ð7:76Þ
½Rðt; 3Þð2Þ ¼ exp½0:370texp½0:316texp½0:198t ¼ exp½0:884t:
ð7:77Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
371
The expected values of the container gantry crane conditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} at the operation state z2, calculated from the results given by (7.75–7.77), according to (3.8), respectively are: l2 ð1Þ ffi 1:93; l2 ð2Þ ffi 1:53; l2 ð3Þ ffi 1:13 year:
ð7:78Þ
After proceeding in the analogous way in the system reliability analysis and evaluation at the remaining operation states z3, z4, z5 and z6, we may determine the system conditional reliability function presented below. At the operation state z3, the container gantry crane conditional reliability function of the system is given by the vector ½Rðt; Þð3Þ ¼ ½1; ½Rðt; 1Þð3Þ ; ½Rðt; 2Þð3Þ ; ½Rðt; 3Þð3Þ ; t 0; with the coordinates ½Rðt; 1Þð3Þ ¼ exp½0:196texp½0:235texp½0:091t ¼ exp½0:522t;
ð7:79Þ
½Rðt; 2Þð3Þ ¼ exp½0:264texp½0:272texp½0:120t ¼ exp½0:656t;
ð7:80Þ
½Rðt; 3Þð3Þ ¼ exp½0:375texp½0:316texp½0:198t ¼ exp½0:889t:
ð7:81Þ
The expected values of the container gantry crane conditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} at the operation state z3, calculated from the results given by (7.79–7.81), according to (3.8), respectively are: l3 ð1Þ ffi 1:91; l3 ð2Þ ffi 1:52; l3 ð3Þ ffi 1:12 year:
ð7:82Þ
At the operation state z4, the container gantry crane conditional reliability function of the system is given by the vector ½Rðt; Þð4Þ ¼ ½1; ½Rðt; 1Þð4Þ ; ½Rðt; 2Þð4Þ ; ½Rðt; 3Þð4Þ ; t 0; with the coordinates ½Rðt; 1Þð4Þ ¼ exp½0:216texp½0:241texp½0:061texp½0:029t ¼ exp½0:547t; ½Rðt; 2Þð4Þ ¼ exp½0:289texp½0:278texp½0:091texp½0:04t ¼ exp½0:698t; ½Rðt; 3Þð4Þ ¼ exp½0:428texp½0:328texp½0:133texp½0:066t ¼ exp½0:955t:
ð7:83Þ
ð7:84Þ
ð7:85Þ
372
7 Additional Applications
The expected values and of the container gantry crane conditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} at the operation state z4, calculated from the results given by (7.83–7.85), according to (3.8), respectively are: l4 ð1Þ ffi 1:83; l4 ð2Þ ffi 1:43; l4 ð3Þ ffi 1:05 year:
ð7:86Þ
At the operation state z5, the container gantry crane conditional reliability function of the system is given by the vector ½Rðt; Þð5Þ ¼ ½1; ½Rðt; 1Þð5Þ ; ½Rðt; 2Þð5Þ ; ½Rðt; 3Þð5Þ ; t 0; with the coordinates ½Rðt; 1Þð5Þ ¼ exp½0:216texp½0:241texp½0:061texp½0:025t ¼ exp½0:543t;
ð7:87Þ
½Rðt; 2Þð5Þ ¼ exp½0:289texp½0:278texp½0:091texp½0:029t ¼ exp½0:687t;
ð7:88Þ
½Rðt; 3Þð5Þ ¼ exp½0:428texp½0:328texp½0:133texp½0:050t ¼ exp½0:939t:
ð7:89Þ
The expected values of the container gantry crane conditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} at the operation state z5, calculated from the results given by (7.87–7.89), according to (3.8), respectively are: l5 ð1Þ ffi 1:84; l5 ð2Þ ffi 1:46; l5 ð3Þ ffi 1:06 year:
ð7:90Þ
At the operation state z6, the container gantry crane conditional reliability function of the system is given by the vector ½Rðt; Þð6Þ ¼ ½1; ½Rðt; 1Þð6Þ ; ½Rðt; 2Þð6Þ ; ½Rðt; 3Þð6Þ ;
t 0;
with the coordinates ½Rðt; 1Þð6Þ ¼ exp½0:201texp½0:250texp½0:087texp½0:080t ¼ exp½0:618t;
ð7:91Þ
½Rðt; 2Þð6Þ ¼ exp½0:273texp½0:29texp½0:12texp½0:1t ¼ exp½0:783t;
ð7:92Þ
½Rðt; 3Þð6Þ ¼ exp½0:396texp½0:337texp½0:16texp½0:132t ¼ exp½1:025t:
ð7:93Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
373
Fig. 7.22 The graph of the container gantry crane reliability function R(t, ) coordinates
The expected values of the container gantry crane conditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} at the operation state z6, calculated from the results (7.91–7.93), according to (3.8), respectively are: l6 ð1Þ ffi 1:62;
l6 ð2Þ ffi 1:28;
l6 ð3Þ ffi 0:98 year:
ð7:94Þ
In the case when the operation time is large enough the unconditional four-state reliability function of the container gantry crane is given by the vector Rðt; Þ ¼ ½1; Rðt; 1Þ; Rðt; 2Þ; Rðt; 3Þ;
t 0;
ð7:95Þ
where according to (3.5–3.6) and considering (7.56), the vector coordinates are given respectively by Rðt; 1Þ ¼ 0:6874 ½Rðt; 1Þð1Þ þ 0:0187 ½Rðt; 1Þð2Þ þ 0:0515 ½Rðt; 1Þð3Þ þ 0:0005 ½Rðt; 1Þð4Þ þ 0:0717 ½Rðt; 1Þð5Þ þ 0:1702 ½Rðt; 1Þð6Þ for t 0;
ð7:96Þ Rðt; 2Þ ¼ 0:6874 ½Rðt; 2Þð1Þ þ 0:0187 ½Rðt; 2Þð2Þ þ 0:0515 ½Rðt; 2Þð3Þ þ 0:0005 ½Rðt; 2Þð4Þ þ 0:0717 ½Rðt; 2Þð5Þ þ 0:1702 ½Rðt; 2Þð6Þ for t 0;
ð7:97Þ Rðt; 3Þ ¼ 0:6874 ½Rðt; 3Þð1Þ þ 0:0187 ½Rðt; 3Þð2Þ þ 0:0515 ½Rðt; 3Þð3Þ þ 0:0005 ½Rðt; 3Þð4Þ þ 0:0717 ½Rðt; 3Þð5Þ þ 0:1702 ½Rðt; 3Þð6Þ for t 0;
ð7:98Þ
374
7 Additional Applications
Fig. 7.23 The graph of the container gantry crane risk function r(t)
and the coordinates ½Rðt; 1Þð1Þ ; ½Rðt; 1Þð2Þ ; ½Rðt; 1Þð3Þ ; ½Rðt; 1Þð4Þ ; ½Rðt; 1Þð5Þ ; ½Rðt; 1Þð6Þ ; are given by (7.62), (7.75), (7.79), (7.83), (7.87), (7.91), and ½Rðt; 2Þð1Þ ; ½Rðt; 2Þð2Þ ; ½Rðt; 2Þð3Þ ; ½Rðt; 2Þð4Þ ; ½Rðt; 2Þð5Þ ; ½Rðt; 2Þð6Þ ; are given by (7.63), (7.76), (7.80), (7.84), (7.88), (7.92), and ½Rðt; 3Þð1Þ ; ½Rðt; 3Þð2Þ ; ½Rðt; 3Þð3Þ ; ½Rðt; 3Þð4Þ ; ½Rðt; 3Þð5Þ ; ½Rðt; 3Þð6Þ ;are given by (7.64), (7.77), (7.81), (7.85), (7.89), (7.93). The graphs of the coordinates of the container gantry crane reliability function are presented in Fig. 7.22. The expected values and standard deviations of the container gantry crane unconditional lifetimes in the reliability state subsets {1, 2, 3}, {2, 3}, {3} calculated from the results given by (7.96–7.98), according to (3.7–3.9) and considering (7.56), (7.65), (7.78), (7.82), (7.86), (7.90), (7.94), respectively are: lð1Þ¼0:68745:24þ0:01871:93þ0:05151:91þ0:00051:83þ0:07171:84 þ0:17021:62ffi4:14years; ð7:99Þ
rð1Þffi4:71years;
lð2Þ¼0:68743:79þ0:01871:53þ0:05151:52þ0:00051:43þ0:07171:46 þ0:17021:28ffi3:04years; ð7:100Þ rð2Þ ffi 3:43years;
ð7:101Þ
lð3Þ ¼ 0:6874 2:76 þ 0:0187 1:13 þ 0:0515 1:12 þ 0:0005 1:05 þ 0:0717 1:06 þ 0:1702 0:98 ffi 2:22 years; rð3Þ ffi 2:50 years;
ð7:102Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
375
Further, considering (3.10) and (7.99), (7.100) and (7.102), the mean values of the unconditional lifetimes in the particular reliability states 1, 2, 3 respectively are: ð1Þ ¼ lð1Þ lð2Þ ¼ 1:10; l ð2Þ ¼ lð2Þ lð3Þ ¼ 0:82; l ð3Þ ¼ lð3Þ ¼ 2:22 years: l
ð7:103Þ
Since the critical reliability state is r = 2, the system risk function, according to (3.11), is given by rðtÞ ¼ 1 Rðt; 2Þ;
ð7:104Þ
where R(t, 2) is given by (7.97). Hence, the moment when the system risk function exceeds a permitted level, for instance d = 0.05, from (3.12), is s ¼ r1 ðdÞ ffi 0:126 year:
ð7:105Þ
The graph of the risk function r(t) of the container gantry crane operating at the variable conditions is given in Fig. 7.23.
7.2.5.8 Container Gantry Crane Renewal and Availability Prediction Using the results of the container gantry crane reliability prediction given by (7.100–7.101) and the results of the classical renew theory presented in Sect. 3.4, we may predict the renewal and availability characteristics of this system in the case when it is repairable and its time of renovation is either ignored or nonignored. First, assuming that the container gantry crane is repaired after the exceeding of its reliability critical state r = 2 and that the time of the system renovation is ignored and applying Proposition 3.4, we obtain the following results: (a) the time SN(2) until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð3:04 N; 3:43 N Þ, i.e., t 3:04 N pffiffiffiffi ; t 2 ð1; 1Þ; F ðNÞ ðt; 2Þ ¼ PðSN ð2Þ\tÞ ffi FNð0;1Þ 3:43 N (b) the expected value and the variance of the time SN(2) until the Nth exceeding by the system the reliability critical state r = 2 are respectively given by E½SN ð2Þ ffi 3:04 N; D½SN ð2Þ ffi 11:76 N; (c) the number N(t, 2) of exceedings by the system the reliability critical state r = 2 up to the moment t, t C 0, for sufficiently large t, approximately has the distribution of the form
376
7 Additional Applications
PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ
3:04ðN þ 1Þ t 3:04 N t pffi pffi ; FNð0;1Þ 1:967 t 1:967 t
N ¼ 0; 1; . . .; (d) the expected value and the variance of the number N(t, 2) of exceedings by the system the reliability critical state r = 2 up to the moment t, t C 0, for sufficiently large t, approximately are respectively given by Hðt; 2Þ ¼ 0:3289t; Dðt; 2Þ ¼ 0:419t: Further, assuming that the container gantry crane is repaired after the exceeding of its reliability critical state r = 2, the time of the system renovation is not ignored and it has the mean value l0 ð2Þ ¼ 0:0027, the standard deviation r0 ð2Þ ¼ 0:0014 and applying Proposition 3.5, we obtain the following results: (a) the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð3:04 N þ 0:0027ðN 1Þ; 11:76 N 0:0000019ðN 1ÞÞ, i.e., t 3:0427 N þ 0:0027 ðNÞ ðt; 2Þ ¼ Pð F SN ð2Þ\tÞ ffi FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 11:760002 N 0:0000019 t 2 ð1; 1Þ; (b) the expected value and the variance of the time SN ð2Þ until the Nth exceeding by the system the reliability critical state r = 2, for sufficiently large N, are respectively given by E½SN ð2Þ ffi 3:04 N þ 0:0027ðN 1Þ;
D½ SN ð2Þ ffi 11:76 N þ 0:0000019ðN 1Þ;
2Þ of exceeding by the system the reliability critical state (c) the number Nðt; r = 2 up to the moment t, t C 0, for sufficiently large t, has approximately distribution of the form 3:0427ðN þ 1Þ t 0:0027 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 1:966 t þ 0:0027 3:0427 N t 0:0027 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; N ¼ 0; 1; . . .; FNð0;1Þ 1:966 t þ 0:005 2Þ of exceedings by the (d) the expected value and the variance of the number Nðt; system the reliability critical state r = 2 up to the moment t, t C 0, for sufficiently large t, are respectively given by 2Þ ffi 0:329ðt þ 0:0027Þ; Hðt;
2Þ ffi 0:417ðt þ 0:0027Þ; Dðt;
(e) the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, has pffiffiffiffi approximately normal distribution Nð3:0427 N; 3:429 N Þ, i.e.,
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
377
t 3:0427 NÞ ðNÞ ðt; 2Þ ¼ Pð pffiffiffiffi F SN ð2Þ\tÞ ffi FNð0;1Þ ; t 2 ð1; 1Þ; 3:429 N (f) the expected value and the variance of the time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, are respectively given by E½ SN ð2Þ ffi 3:0427 N; D½ SN ð2Þ ffi 11:76002 N; 2Þ of the system’s renovations up to the moment t, t C 0, for (g) the number Nðt; sufficiently large t, has approximately distribution of the form 3:0427ðN þ 1Þ t 3:0427 N t 2Þ ¼ NÞ ffi F p ffi p ffi F ;N PðNðt; Nð0;1Þ Nð0;1Þ 1:966 t 1:966 t ¼ 0; 1; . . .; 2Þ of system’s reno(h) the expected value and the variance of the number Nðt; vations up to the moment t, t C 0, for sufficiently large t, are respectively given by 2Þ ffi 0:3286t; Hðt;
2Þ ffi 0:417t; Dðt;
(i) the steady availability coefficient of the system at the moment t, t C 0, for sufficiently large t, is given by Aðt; 2Þ ffi 0:9989; t 0; (j) the steady availability coefficient of the system in the time interval \t; t þ sÞ; s [ 0; for sufficiently large t, is given by Aðt; s; 2Þ ffi 0:329
Z1
Rðt; 2Þdt; t 0; s [ 0;
s
where R(t, 2) is given by (7.97).
7.2.5.9 Container Gantry Crane Operation Process Optimization Optimal Transient Probabilities of Container Gantry Crane Operation Process at Operation States Considering Eqs. 7.95–7.98, it is natural to assume that the container gantry crane operation process has a significant influence on the system reliability. This influence is also clearly expressed in the formulae (7.99), (7.100) and (7.102) for the mean values of the container gantry crane unconditional lifetimes in the reliability state subsets that can be used for this system operation process optimization performed in accordance with the procedure proposed in Sect. 6.2.1.
378
7 Additional Applications
The objective function defined by (6.1), in this case as the container gantry crane critical reliability state is r = 2, takes the form lð2Þ ¼ p1 3:79 þ p2 1:53 þ p3 1:52 þ p4 1:43 þ p5 1:46 þ p6 1:28: ð7:106Þ ^
_
The lower pb and upper pb bounds of the unknown transient probabilities pb, b ¼ 1; 2; . . .; 6; defined by (6.5), coming from experts respectively are: ^
^
^
^
^
^
_
_
_
_
_
_
p1 ¼ 0:50; p2 ¼ 0:01; p3 ¼ 0:03; p4 ¼ 0:0004; p5 ¼ 0:05; p6 ¼ 0:09; p1 ¼ 0:90; p2 ¼ 0:03; p3 ¼ 0:07; p4 ¼ 0:0007; p5 ¼ 0:09; p6 ¼ 0:30: Therefore, according to (6.2–6.3), we assume the following bound constraints 0:50 p1 0:90; 0:01 p2 0:03; 0:03 p3 0:07; 0:0004 p4 0:0007; 0:05 p5 0:09; 0:09 p6 0:30; 6 X
ð7:107Þ
pb ¼ 1:
b¼1
Now, in order to find the optimal values p_ b of the transient probabilities pb, b ¼ 1; 2; . . .; 6; that maximize the objective function (7.106), we arrange the system conditional lifetimes mean values lb ð2Þ; b ¼ 1; 2; . . .; 6; in non-increasing order l1 ð2Þ l2 ð2Þ l3 ð2Þ l5 ð2Þ l4 ð2Þ l6 ð2Þ: Next, according to (6.6), we substitute x1 ¼ p1 ; x2 ¼ p2 ; x3 ¼ p3 ; x4 ¼ p5 ; x5 ¼ p4 ; x6 ¼ p6 ;
ð7:108Þ
and ^
^
^
^
_
_
_
_
^
^
^
^
^
^
_
_
_
x1 ¼ p1 ¼ 0:50; x2 ¼ p2 ¼ 0:01; x 3 ¼ p3 ¼ 0:03; x4 ¼ p5 ¼ 0:05; ^
^
x 5 ¼ p4 ¼ 0:0004; x6 ¼ p6 ¼ 0:09; _
_
_
x1 ¼ p1 ¼ 0:90; x2 ¼ p2 ¼ 0:03; x 3 ¼ p3 ¼ 0:07; x4 ¼ p5 ¼ 0:09; _
ð7:109Þ
_
x 5 ¼ p4 ¼ 0:0007; x6 ¼ p6 ¼ 0:30;
and we maximize with respect to xi, i ¼ 1; 2; . . .; 6; the linear form (7.106) which according to (6.7–6.10) takes the form lð2Þ ¼ x1 3:79 þ x2 1:53 þ x3 1:52 þ x4 1:46 þ x5 1:43 þ x6 1:28; ð7:110Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
379
with the following bound constraints 0:50 x1 0:90; 0:01 x2 0:03; 0:03 x3 0:07; 0:05 x4 0:09; 0:0004 x5 0:0007; 0:09 x6 0:30; 6 X
xi ¼ 1:
ð7:111Þ
i¼1
According to (6.11), we calculate ^
x¼
6 X ^ ^ x i ¼ 0:6804; ^y ¼ 1 x ¼ 1 0:6804 ¼ 0:3196
ð7:112Þ
i¼1
and according to (6.12), we find ^0
_0
_0
^0
x ¼ 0; x ¼ 0; x x ¼ 0;
^1
_1
_1
^1
^2
_2
_2
^2
x ¼ 0:50; x ¼ 0:90; x x ¼ 0:40;
ð7:113Þ
x ¼ 0:51; x ¼ 0:93; x x ¼ 0:42;
^6
_6
_6
^6
x ¼ 0:6804; x ¼ 1:3907; x x ¼ 0:7103:
By (7.113), the inequality (6.13) takes the form _I
^I
x x \0:3196:
ð7:114Þ
Thus, from the above and from (7.113), it follows that the largest value I 2 f0; 1; . . .; 6g such that the inequality (7.114) is satisfied, is I = 0. Therefore, we fix the optimal solution that maximizes linear function (7.110) according to the rule (6.14). Namely, we get ^
x_ 1 ¼ ^y þ x1 ¼ 0:3196 þ 0:5 ¼ 0:8196; ^
^
^
x_ 2 ¼ x2 ¼ 0:01; x_ 3 ¼ x3 ¼ 0:03; x_ 4 ¼ x4 ¼ 0:05; ^
ð7:115Þ
^
x_ 5 ¼ x5 ¼ 0:0004; x_ 6 ¼ x 6 ¼ 0:09: Finally, after making the substitution inverse to (7.108), we get the optimal transient probabilities p_ 1 ¼ x_ 1 ¼ 0:8196; p_ 2 ¼ x_ 1 ¼ 0:01; p_ 3 ¼ x_ 3 ¼ 0:03; p_ 4 ¼ x_ 5 ¼ 0:0004; ð7:116Þ p_ 5 ¼ x_ 4 ¼ 0:05; p_ 6 ¼ x_ 6 ¼ 0:09; that maximize the system mean lifetime in the reliability state subset {2, 3} expressed by the linear form (7.106) giving, according (6.18) and (7.116), its optimal value
380
7 Additional Applications
_ lð2Þ ¼ p_ 1 3:79 þ p_ 2 1:53 þ p_ 3 1:52 þ p_ 4 1:43 þ p_ 5 1:46 þ p_ 6 1:28 ¼ 0:8196 3:79 þ 0:01 1:53 þ 0:03 1:52 þ 0:0004 1:43 þ 0:05 1:46 þ 0:09 1:28 ffi 3:36:
ð7:117Þ
Optimal Sojourn Times of Container Gantry Crane Operation Process at Operation States Having the values of the optimal transient probabilities determined by (7.116), it is possible to find the optimal conditional and unconditional mean values of the sojourn times of the container gantry crane operation process at the operation states and the optimal mean values of the total unconditional sojourn times of the container gantry crane operation process at the operation states during the fixed operation time as well. Substituting the optimal transient probabilities at operation states p_ 1 ¼ 0:8196; p_ 2 ¼ 0:01; p_ 3 ¼ 0:03; p_ 4 ¼ 0:0004; p_ 5 ¼ 0:05; p_ 6 ¼ 0:09; determined in (7.116) and the steady probabilities p1 ffi 0:0951; p2 ffi 0:1020; p3 ffi 0:3100; p4 ffi 0:0102; p5 ffi 0:1547; p6 ffi 0:3280: determined by (7.55) into (6.41), we get the following system of equations _ 1 þ 0:0835992M _ 2 þ 0:254076M _ 3 þ 0:0083599M _ 4 þ 0:1267921M _5 0:017156M _6¼0 þ 0:2688288M _ 1 0:10098M _ 2 þ 0:0031M _ 3 þ 0:000102M _ 4 þ 0:001547M _5 0:000951M _6¼0 þ 0:00328M _ 1 þ 0:00306M _ 2 0:3007M _ 3 þ 0:000306M _ 4 þ 0:004641M _5 0:002853M _6¼0 þ 0:00984M _ 2 þ 0:000124M _ 3 0:0101959M _ 4 þ 0:0000618M _5 _ 1 þ 0:0000408M 0:000038M _6¼0 þ 0:0001312M _ 1 þ 0:0051M _ 2 þ 0:0155M _ 3 þ 0:00051M4 0:146965M _5 0:004755M _6¼0 þ 0:0164M _ 1 þ 0:00918M _ 2 þ 0:0279M _ 3 þ 0:000918M _ 4 þ 0:013923M _5 0:008559M _6¼0 0:29848M ð7:118Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
381
_ b of the system unconditional sojourn with the unknown optimal mean values M times in the operation states we are looking for. Since the determinant of the main matrix of the homogeneous system of Eqs. 7.118 is equal to 0, its rank is less than 6 and there are non-zero solutions of this system of equations that are ambiguous and dependent on one or more parameters. Thus, we may fix some of them and determine the remaining ones. In our case, according to (7.52), after considering expert opinion, we conclude that it is sensible to assume _ 4 ffi 2: M
ð7:119Þ
Consequently, from (7.118), we get the system of equations _ 2 þ 0:254076M _ 3 þ 0:1267921M _5 _ 1 þ 0:0835992M 0:017156M _ 6 ¼ 0:0167198 þ 0:2688288M _ 2 þ 0:0031M _ 3 þ 0:001547M _5 _ 1 0:10098M 0:000951M _ 6 ¼ 0:000204 þ 0:00328M _ 2 0:3007M _ 3 þ 0:004641M _5 _ 1 þ 0:00306M 0:002853M _ 6 ¼ 0:000612 þ 0:00984M _ 2 þ 0:000124M _ 3 þ 0:0000618M _5 _ 1 þ 0:0000408M 0:000038M _ 6 ¼ 0:0203918 þ 0:0001312M _ 2 þ 0:0155M _ 3 0:146965M _5 _ 1 þ 0:0051M 0:004755M _ 6 ¼ 0:00102 þ 0:0164M _ 2 þ 0:0279M _3 _ 1 þ 0:00918M 0:008559M _ 5 0:29848M _ 6 ¼ 0:001836 þ 0:013923M _ 2; M _ 3; M _ 5 and M _ 6 . The solutions of the _ 1, M and we solve it with respect to M system of Eqs. 7.118 obtained in this way, are _ 1 ffi 439:9400; M _ 2 ffi 5:0046; M _ 3 ffi 4:9401; M _ 4 ffi 2; M _ 5 ffi 16:4988; M ð7:120Þ _ 6 ffi 14:0069: M It can be seen that these solutions differ much from the values M1, M2, M3, M4, M5 and M6 estimated in Sect. 7.2.5.4 and given by (7.49–7.54). _ bl of Having these solutions, it is also possible to look for the optimal values M the mean values Mbl of the conditional sojourn times at the operation states. Namely, substituting the probabilities of the system operation process transitions between the operation states, determined by (7.41) and the optimal mean values _ b given by (7.120) into (6.42), we get the following system of equations M
382
7 Additional Applications
_ 12 þ 0:336M _ 13 þ 0:008M _ 14 þ 0:008M _ 16 ¼ 439:9400 0:648M _ 23 þ 0:093M _ 24 þ 0:009M _ 26 ¼ 5:0046 _ 21 þ 0:373M 0:525M _ 31 þ 0:111M _ 32 þ 0:118M _ 35 þ 0:666M _ 36 ¼ 4:9401 0:105M _ 42 ¼ 2 _ 41 þ 0:583M 0:417M _ 51 þ 0:220M _ 53 þ 0:775M _ 56 þ ¼ 16:4988 0:005M _ 61 þ 0:628M _ 63 þ 0:360M _ 65 ¼ 14:0069: 0:012M _ bl we want to find. with the unknown optimal values M As the solutions of the above system of equations are ambiguous, we arbitrarily fix some of them, for practically important reasons, and we find the remaining ones. In this case we proceed as follows: _ 14 ¼ 50; M _ 16 ¼ 4 and we find _ 13 ¼ 40; M • we fix in the first equation M _ 12 ffi 657:5; M _ 24 ¼ 2;M _ 26 ¼ 16and we find _ 23 ¼ 9; M • we fix in the second equation M _ M21 ffi 2:51; _ 31 ¼ 6;M _ 32 ¼ 4;M _ 35 ¼ 7 and we find M _ 36 ffi 4:56; • we fix in the third equation M _ 41 ¼ 2 and we find M _ 42 ffi 2; • we fix in the fourth equation M _ 53 ¼ 3 and we find M _ 56 ffi 20:37; _ 51 ¼ 10;M • we fix in the fifth equation M _ 61 ¼ 23;M _ 65 ¼ 21 and we find • we fix in the sixth equation M _ 63 ffi 9:83: M
ð7:121Þ
Other very useful and much easier to be applied in practice tools that can help in planning the operation process of the ferry technical system are the system operation process optimal mean values of the total sojourn times at the particular operation states during the fixed system operation time h: Assuming as in Sect. 7.2.5.4, the system operation time h ¼ 1year = 365 days, after applying (6.43), we get their values _ ^h1 ¼ p_ 1 h ¼ 0:8196 365 ffi 299:15; E½ _ ^ E½ h2 ¼ p_ 2 h ¼ 0:01 365 ¼ 3:65; ^ ^ _ h4 ¼ p_ 4 h ¼ 0:0004 365 ffi 0:15; _ h3 ¼ p_ 3 h ¼ 0:03 365 ¼ 10:95; E½ E½ _ ^h5 ¼ p_ 5 h ¼ 0:05 365 ¼ 18:25; E½ _ ^ E½ h6 ¼ p_ 6 h ¼ 0:09 365 ¼ 32:85: ð7:122Þ that differ from the values of E½^ hi ; i ¼ 1; 2; . . .; 6; determined by (7.57). _ bl and E½ _ ^hb given _ b, M In practice, the knowledge of the optimal values of M respectively by (7.120–7.122), can be very important and helpful for the container gantry crane operation process planning and improving in order to make the system operation more reliable and safer.
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
383
7.2.5.10 Container Gantry Crane Reliability Optimization To make the optimization of the reliability of the container gantry crane we need the optimal values p_ b , b ¼ 1; 2; . . .; 6; of the transient probabilities pb, b ¼ 1; 2; . . .; 6; in particular operation states determined by (7.116). Substituting these optimal solutions into the formula (6.19), we obtain the optimal mean values of the container gantry crane unconditional lifetimes in the reliability state subset {1, 2, 3} and {3} that respectively are _ lð1Þ ¼ p_ 1 5:24 þ p_ 2 1:93 þ p_ 3 1:91 þ p_ 4 1:83 þ p_ 5 1:84 þ p_ 6 1:62 ¼ 0:8196 5:24 þ 0:01 1:93 þ 0:03 1:91 þ 0:0004 1:83 þ 0:05 1:84 þ 0:09 1:62 ffi 4:61:
ð7:123Þ _ lð3Þ ¼ p_ 1 2:76 þ p_ 2 1:13 þ p_ 3 1:12 þ p_ 4 1:05 þ p_ 5 1:06 þ p_ 6 0:98 ¼ 0:8196 2:76 þ 0:01 1:13 þ 0:03 1:12 þ 0:0004 1:05 þ 0:05 1:06 þ 0:09 0:98 ffi 2:45: ð7:124Þ According to (6.23), the optimal solutions for the mean values of the container gantry crane unconditional lifetimes in the particular reliability states 1, 2 and 3, respectively are _ ð1Þ ¼ lð1Þ _ _ _ ð2Þ ¼ lð2Þ _ _ l lð2Þ ¼ 1:25; l lð3Þ ¼ 0:91; _l ð3Þ ¼ lð3Þ _ ¼ 2:45:
ð7:125Þ
Moreover, according to (6.20–6.21), the corresponding optimal unconditional multistate reliability function of the container gantry crane is given by the vector _ Þ ¼ ½1; Rðt; _ 1Þ; Rðt; _ 2Þ; Rðt; _ 3Þ; Rðt;
ð7:126Þ
with the coordinates _ 1Þ ¼ 0:8196 ½Rðt; 1Þð1Þ þ 0:01 ½Rðt; 1Þð2Þ þ 0:03 ½Rðt; 1Þð3Þ Rðt; þ 0:0004 ½Rðt; 1Þð4Þ þ 0:05 ½Rðt; 1Þð5Þ þ 0:09 ½Rðt; 1Þð6Þ for t 0;
ð7:127Þ _ 2Þ ¼ 0:8196 ½Rðt; 2Þð1Þ þ 0:01 ½Rðt; 2Þð2Þ þ 0:03 ½Rðt; 2Þð3Þ Rðt; þ 0:0004 ½Rðt; 2Þð4Þ þ 0:05 ½Rðt; 2Þð5Þ þ 0:09 ½Rðt; 2Þð6Þ for t 0;
ð7:128Þ _ 3Þ ¼ 0:8196 ½Rðt; 3Þð1Þ þ 0:01 ½Rðt; 3Þð2Þ þ 0:03 ½Rðt; 3Þð3Þ Rðt; þ 0:0004 ½Rðt; 3Þð4Þ þ 0:05 ½Rðt; 3Þð5Þ þ 0:09 ½Rðt; 3Þð6Þ for t 0;
ð7:129Þ
384
7 Additional Applications
Fig. 7.24 The graph of the container gantry crane optimal reliability function R(t, ) coordinates
where ½Rðt; 1ÞðbÞ ; ½Rðt; 2ÞðbÞ ; ½Rðt; 3ÞðbÞ ; b ¼ 1; 2; . . .; 6; are fixed in Sect. 7.2.5.7, respectively by (7.62–7.64), (7.75–7.77), (7.79–7.81), (7.83–7.85), (7.87–7.89), (7.91–7.93). The graphs of the coordinates of the container gantry crane optimal reliability function are given in Fig. 7.24. Further, by (6.22), the corresponding optimal variances and standard deviations of the container gantry crane unconditional lifetime in the reliability state subsets are 2
r_ ð1Þ ¼ 2
Z1
2 _ 1Þdt ½lð1Þ _ _ t Rðt; ffi 24:758; rð1Þ ffi 4:9;
0
r_ 2 ð2Þ ¼ 2
Z1
2 _ 2Þdt ½lð2Þ _ _ t Rðt; ffi 12:923; rð2Þ ffi 3:59;
ð7:130Þ
0 2
r_ ð3Þ ¼ 2
Z1
2 _ 2Þdt ½lð3Þ _ _ t Rðt; ffi 6:893; rð3Þ ffi 2:63;
0
_ 1Þ; Rðt; _ 2Þ, Rðt; _ 3Þare given by (7.127–7.129) and lð1Þ, _ _ _ where Rðt; lð2Þ, lð3Þ are given by (7.117) and (7.123), (7.124). Since the critical reliability state is r = 2, the optimal system risk function, according to (6.24), is given by (Fig. 7.25) _ 2Þ for t 0; r_ ðtÞ ¼ 1 Rðt; _ 2Þ is given by (7.128). where Rðt;
ð7:131Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
385
_ Fig. 7.25 The graph of the container gantry crane optimal risk function rðtÞ
Hence, considering (6.25), the moment when the optimal system risk function exceeds a permitted level, for instance d = 0.05, is s_ ¼ r_ 1 ðdÞ ffi 0:149 year:
ð7:132Þ
7.2.5.11 Container Gantry Crane Renewal and Availability Optimization To determine the optimal renewal and availability characteristics of the container gantry crane after its operation process optimization, we use the results of the system reliability characteristics optimization performed in Sects. 7.2.5.9 and 7.2.5.10 and the results of Sect. 6.2.3. In the case when the container gantry crane renovation time is ignored, con_ _ sidering the optimal values lð2Þ ffi 3:36 determined by (7.117) and rð2Þ ffi 3:59 determined by (7.130) and applying Proposition 6.1, we determine its following optimal characteristics: (a) the optimal time S_ N ð2Þ until the Nth exceeding by the system the reliability critical state 2, for sufficiently large N, has approximately normal distribution pffiffiffiffi Nð3:36 N; 3:59 N Þ, i.e., N _F ðNÞ ðt; 2Þ ¼ PðS_ N ð2Þ\tÞ ffi FNð0;1Þ t 3:36 pffiffiffiffi ; t 2 ð1; 1Þ; 3:59 N (b) the expected value and the variance of the optimal time S_ N ð2Þ until the Nth exceeding by the system the reliability critical state 2, for sufficiently large N, respectively are E½S_ N ð2Þ ¼ 3:36 N;
D½S_ N ð2Þ ¼ 12:888 N;
386
7 Additional Applications
_ 2Þ of exceedings by the system the reliability critical (c) the optimal number Nðt; state 2 up to the moment t, t C 0, for sufficiently large t, has distribution approximately of the form 3:36ðN þ 1Þ t 3:36 N t _ pffi pffi ; N FNð0;1Þ PðNðt; 2Þ ¼ NÞ ffi FNð0;1Þ 1:959 t 1:959 t ¼ 0; 1; . . .; (d) the expected value and the variance of the optimal number N(t, 2) of exceedings by the system the reliability critical state 2 up to the moment t, t C 0, for sufficiently large t, respectively are _ 2Þ ¼ 0:298t; Dðt; _ 2Þ ¼ 0:339t: Hðt;
To make the estimation of the renewal and availability of the container gantry crane in the case when the time of renovation is non-ignored, considering the _ _ optimal values lð2Þ ffi 3:36 determined by (7.117) and rð2Þ ffi 3:59 determined by (7.130), assuming the mean value of the system renovation time l0 ð2Þ ¼ 0:0027 year and the standard deviation of the system renovation time r0 ð2Þ ¼ 0:0014 year and applying Proposition 6.2, we determine its following optimal characteristics: _ N ð2Þ until the Nth exceeding by the system the reliability (a) the optimal time S critical state 2, for sufficiently large N, has approximately normal distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nð3:36 N þ 0:0027ðN 1Þ; 12:888 N þ 0:0000019ðN 1ÞÞ, i.e., t 3:3627 N þ 0:0027 _ ðNÞ ðt; 2Þ ¼ Pð F S_ N ð2Þ\tÞ ¼ FNð0;1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 12:888002 N 0:0000019 t 2 ð1; 1Þ; (b) the expected value and the variance of the optimal time S_ N ð2Þ until the Nth exceeding by the system the reliability critical state 2, for sufficiently large N, respectively are _ Sð2Þ ffi 12:888N þ 0:0000019ðN 1Þ; E½S_ N ð2Þ ffi 3:36 N þ 0:0027ðN 1Þ; D½ _ 2Þ of exceedings by the system the reliability critical (c) the optimal number Nðt; state 2 up to the moment t, t C 0, for sufficiently large t, has approximately distribution of the form þ 1Þ t 0:0027 _ 2Þ ¼ NÞ ffi FNð0;1Þ 3:3627ðN p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðNðt; 1:98 t þ 0:0027 3:3627 N t 0:0027 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FNð0;1Þ ; N ¼ 0; 1; . . .; 1:98 t þ 0:0027
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
387
_ 2Þ of ex(d) the expected value and the variance of the optimal number Nðt; ceedings by the system the reliability critical state 2 up to the moment t, t C 0, for sufficiently large t, respectively are _ 2Þ ffi 0:297ðt þ 0:0027Þ; Dðt; _ 2Þ ffi 0:3389ðt þ 0:0027Þ; Hðt; _ SN ð2Þ until the Nth system’s renovation, for sufficiently large (e) the optimal time pffiffiffiffi N, has approximately normal distribution Nð3:3627 N; 3:5899 N Þ, i.e., t 3:3627 NÞ _ _ ðNÞ ðt; 2Þ ¼ Pð pffiffiffiffi ; t 2 ð1; 1Þ; F SN ð2Þ\tÞ ffi FNð0;1Þ 3:5899 N _ (f) the expected value and the variance of the optimal time SN ð2Þ until the Nth system’s renovation, for sufficiently large N, respectively are _ SN ð2Þ ffi 3:3627 N; E½
_ ðNÞ S ð2Þ ffi 12:888002 N; D½
_ 2Þ of system’s renovations up to the moment t, t C 0, (g) the optimal number Nðt; for sufficiently large t, has approximately distribution of the form. 3:3627ðN þ 1Þ t 3:3627 N t _ 2Þ ¼ NÞ ffi F p ffi p ffi F ; PðNðt; Nð0;1Þ Nð0;1Þ 1:98 t 1:98 t N ¼ 0; 1; . . .; _ 2Þ of system’s (h) the expected value and the variance of the optimal number Nðt; renovations up to the moment t, t C 0, for sufficiently large t, respectively are _ 2Þ ffi 0:297t; Hðt;
_ 2Þ ffi 0:3389t; Dðt;
(i) the optimal steady availability coefficient of the system at the moment t, t C 0, for sufficiently large t, is _ 2Þ ffi 0:9992; t 0; Aðt; (j) the optimal steady availability coefficient of the system in the time interval \t; t þ sÞ; s [ 0;t C 0, for sufficiently large t, is _ s; 2Þ ffi 0:297 Aðt;
Z1 s
_ 2Þ is given by (7.128). where Rðt;
_ 2Þdt; t 0; s [ 0; Rðt;
388
7 Additional Applications
7.2.5.12 Container Gantry Crane Corrective and Preventive Maintenance Policy Optimization Maintenance Policy Maximizing System Availability To optimize the container gantry crane corrective and preventive maintenance policy maximizing its availability, we use its following reliability and renewal parameters: • the container gantry crane and its components critical reliability state r = 2, • the 2nd coordinate of the container gantry crane unconditional reliability function Rðt; Þ Rðt; 2Þ ¼ 0:6874 ½Rðt; 2Þð1Þ þ 0:0187 ½Rðt; 2Þð2Þ þ 0:0515 ½Rðt; 2Þð3Þ þ 0:0005 ½Rðt; 2Þð4Þ þ 0:0717 ½Rðt; 2Þð5Þ þ 0:1702 ½Rðt; 2Þð6Þ
for
t 0;
ð7:133Þ where ½Rðt; 2ÞðbÞ ; b ¼ 1; 2; . . .; 6; are respectively given by (7.63), (7.76), (7.80), (7.84), (7.88), (7.92), • the derivative of the 2-nd coordinate of the container gantry crane unconditional reliability function Rðt; Þ R0 ðt; 2Þ ¼ 0:6874 ½R0 ðt; 2Þð1Þ þ 0:0187 ½R0 ðt; 2Þð2Þ þ 0:0515 ½R0 ðt; 2Þð3Þ þ 0:0005 ½R0 ðt; 2Þð4Þ þ 0:0717 ½R0 ðt; 2Þð5Þ þ 0:1702 ½R0 ðt; 2Þð6Þ
for t 0
• the mean value of the container gantry crane unconditional corrective renovation time l0 ð1Þ ¼ 0:0027; • the mean value of the container gantry crane unconditional preventive renovation time l1 ð1Þ ¼ 0:0025: Moreover, to apply the algorithm proposed in Sect. 6.3.1, we fix: • the measure of the method of secants accuracy e ¼ 0:001; • the number of values of the system preventive maintenance period g for which we find the values of the availability coefficient of the container gantry crane in cases when there is no optimal value j ¼ 20; • the values of the system preventive maintenance period g for which we find the values of the availability coefficient of the container gantry crane in cases when there is no optimal value gi ¼ ði 1Þ0:2lð2Þ ¼ ði 1Þ0:2 3:04; where lð2Þ ffi 3:04, is given by (7.100).
i ¼ 1; 2; . . .; 20:
ð7:134Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
389
After fixing the above reliability and renewal input parameters, we use the procedure described in Sect. 6.3.1.Since l0 ð2Þ ¼ 0:0027 [ l1 ð2Þ ¼ 0:0025 we are looking for the optimal value g_ of the preventive maintenance period g that maximizes the availability coefficient of the system Aðg; 1Þ given by (6.52) by determining, if it exists, its approximate value from Eq. 6.58 by applying the method of secants in the interval ha; bi as follows: • we define, obtained after the transformation of the Eq. 6.58, the function f ðgÞ ¼ kðg; 2Þ
Zg
Rðt; 2Þdt þ Rðg; 2Þ
0
¼ kðg; 2Þ
Zg
l0 ð2Þ l0 ð2Þ l1 ð2Þ
Rðt; 2Þdt þ Rðg; 2Þ 13:5
for g 0;
0
where Rðt; 2Þ is given by (7.133) and kðg; 2Þ ¼
R0 ðg; 2Þ ; Rðg; 2Þ
• we define the interval h a, b i assuming a = 0 and finding b such that f ðbÞ [ 0; • we use the recurrent formula g0 ¼ a; gkþ1 ¼ gk
f ðgk Þ ðb gk Þ f ðbÞ f ðgk Þ
for k ¼ 0; 1; . . .; K;
where K is such that f ðgKþ1 Þ\e and e ¼ 0:001 is the measure of the method of secants accuracy, • we fix the optimal value g_ of the preventive maintenance period g assuming g_ ¼ gKþ1 : As a result of computer calculations, we recognize that there is no optimal value g_ of the container gantry crane preventive maintenance period g that maximizes the value of its availability coefficient. The exemplary values of the system preventive maintenance period g defined by (7.134) and the values of the availability coefficient of the container gantry crane are given in Table 7.3.
390
7 Additional Applications
Table 7.3 The values of the availability coefficient of the container gantry crane
g
A(g, 2)
g
A(g, 2)
0.0 0.608 1.216 1.824 2.432 3.04 3.648 4.256 4.864 5.472
0.0 0.99532 0.99735 0.99802 0.99836 0.99856 0.99869 0.99878 0.99885 0.99890
6.080 6.688 7.296 7.904 8.512 9.120 9.728 10.336 0.944 11.552
0.99894 0.99897 0.99899 0.99901 0.99903 0.99904 0.99905 0.99906 0.99907 0.99908
7.2.5.13 Container Gantry Crane Operation and Reliability New Strategy Parameters and Characteristics of Container Gantry Crane Operation Process Before and After Its Optimization From Sect. 7.2.5.3, we have the values of the following container gantry crane operation process parameters before its optimization: • the conditional mean sojourn times of the container gantry crane at the particular operation states M12 ¼ 456:978; M13 ¼ 36:860; M14 ¼ 50; M16 ¼ 3; M21 ¼ 7:887; M23 ¼ 9:121; M24 ¼ 1:545; M26 ¼ 16; M31 ¼ 5:5; M32 ¼ 4:343; M35 ¼ 6:822; M36 ¼ 7:857;
ð7:135Þ
M41 ¼ 2; M42 ¼ 2:143; M51 ¼ 10; M53 ¼ 2:899; M56 ¼ 24:681; M61 ¼ 22:6; M63 ¼ 23:117; M65 ¼ 20:512:
From Sect. 7.2.5.9, we have the values of the following container gantry crane operation process parameters after its optimization: • the optimal conditional mean sojourn times of the container gantry crane at the particular operation states _ 12 ffi 657:5; M _ 13 ¼ 40; M _ 14 ¼ 50; M _ 16 ¼ 4; M _ 23 ¼ 9; M _ 24 ¼ 2; M _ 26 ¼ 16; _ 21 ffi 2:51; M M _ 31 ¼ 6; M _ 32 ¼ 4; M _ 35 ¼ 7; M _ 36 ffi 4:56; M _ 42 ffi 2; M _ 51 ¼ 10; M _ 53 ¼ 3; M _ 56 ffi 20; _ 41 ¼ 2; M M _ 61 ¼ 23; M _ 65 ¼ 21; M _ 63 ffi 9:83: M
ð7:136Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
391
From Sect. 7.2.5.4, we have the values of the following container gantry crane operation process characteristics before its optimization: • the unconditional mean sojourn times of the container gantry crane at the particular operation states M1 ¼ 308:93; M2 ¼ 7:83; M3 ¼ 7:09; M4 ¼ 2:08; M5 ffi 19:82; M6 ffi 22:17; ð7:137Þ • the transient probabilities of the container gantry crane operation process at the operational states p1 ¼ 0:6874; p2 ¼ 0:0187; p3 ¼ 0:0515; p4 ¼ 0:0005; p5 ¼ 0:0717; p6 ¼ 0:1702;
ð7:138Þ
• the total sojourn times of the container gantry crane operation process in particular operation states during the operation time h ¼ 1 year = 365 days E½^ h1 ¼ 251 days; E½^ h2 ¼ 7 days; E½^h3 ¼ 19 day; h5 ¼ 26 days; E½^h6 ¼ 62 days: E½^ h4 ¼ 0:2 day; E½^
ð7:139Þ
From Sect. 7.2.5.9, we have the values of the following container gantry crane operation process characteristics after its optimization: • the optimal unconditional mean sojourn times of the container gantry crane in the particular operation states _ 2 ffi 5:00; M _ 3 ffi 439:94; M _ 4 ffi 2; M _ 5 ffi 16:50; _ 1 ffi 439:94; M M _ 6 ffi 14:01; M
ð7:140Þ
• the optimal transient probabilities of the container gantry crane operation process at the operational states p_ 1 ¼ 0:8196; p_ 2 ¼ 0:01; p_ 3 0:03; p_ 4 ¼ 0:0004; p_ 5 ¼ 0:05; p_ 6 ¼ 0:09;
ð7:141Þ
• the optimal total sojourn times of the container gantry crane operation process in particular operation states during the operation time h ¼ 1 year = 365 days ^2 ¼ 3:65 days; E½ _ ^h1 ¼ 299:15 days; E½ _ h _ ^h3 ¼ 10:95days; E½ _ ^ _ ^ _ ^h4 ¼ 0:15 days; E½ h5 ¼ 18:25 days; E½ h6 ¼ 32:85days: E½
ð7:142Þ
392
7 Additional Applications
Characteristics of Container Gantry Crane Reliability Before and After Operation Process Optimization From Sect. 7.2.5.7, we have the values of the following container gantry crane reliability characteristics before its operation process optimization: • the expected values of the container gantry crane unconditional lifetimes respectively in the reliability state subsets {1, 2, 3},{2, 3}, {3} lð1Þ ¼ 4:14 years; lð2Þ ¼ 3:04 years; lð3Þ ¼ 2:22 years;
ð7:143Þ
• the mean values of the unconditional lifetimes respectively in the particular reliability states 1, 2, 3 ð1Þ ¼ 1:10; l
ð2Þ ¼ 0:82; l
ð3Þ ¼ 2:22 years; l
ð7:144Þ
• the moment when the system risk function exceeds a permitted level s ffi 0:126 year:
ð7:145Þ
From Sects. 7.2.5.9 and 7.2.5.10, we have the values of the following container gantry crane reliability parameters and characteristics after its operation process optimization: • the optimal expected values of the container gantry crane unconditional lifetimes respectively in the reliability state subsets {1, 2, 3}, {2, 3}, {3} _ _ _ lð1Þ ¼ 4:61 years; lð2Þ ¼ 3:36 years; lð3Þ ¼ 2:45 years;
ð7:146Þ
• the optimal mean values of the unconditional lifetimes respectively in the particular reliability states 1, 2, 3 _ ð1Þ ¼ 1:25; l _ ð2Þ ¼ 0:91; l _ ð3Þ ¼ 2:45 years; l
ð7:147Þ
• the optimal moment when the system risk function exceeds a permitted level s_ ffi 0:149 year:
ð7:148Þ
Characteristics of Container Gantry Crane Renewal and Availability Before and After Operation Process Optimization From Sect. 7.2.5.8, we have the values of the following container gantry crane renewal and availability characteristics before its operation process optimization: • the expected value of the number of exceedings by the system with ignored time of renovation the reliability critical state during 1 year = 365 days Hð365; 2Þ ¼ 120:05;
ð7:149Þ
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
393
• the expected value of the number of renovations of the system with non-ignored time of renovation during 1 year = 365 days Hð365; 2Þ ffi 119:94;
ð7:150Þ
• the steady availability coefficient of the system Aðt; 2Þ ffi 0:9989:
ð7:151Þ
From Sect. 7.2.5.11, we have the values of the following container gantry crane renewal and availability characteristics after its operation process optimization: • the optimal expected value of the number of exceeding by the system with ignored time of renovation the reliability critical state during 1 year = 365 days _ Hð365; 2Þ ¼ 108:77;
ð7:152Þ
• the optimal expected value of the number of renovations of the system with nonignored time of renovation during 1 year = 365 days _ Hð365; 2Þ ffi 108:41;
ð7:153Þ
• the optimal steady availability coefficient of the system _ 2Þ ffi 0:9992: Aðt;
ð7:154Þ
Results of Container Gantry Crane Corrective and Preventive Maintenance Policy Optimization In Sect. 7.2.5.12, we fixed that there is no optimal value g_ of the container gantry crane preventive maintenance period g that maximizes the value of its availability coefficient. The exemplary values of the system preventive maintenance period g and the values of the availability coefficient of the container gantry crane that are given in Table 7.3 justify that the container gantry crane coefficient is an increasing function of the preventive period. Analysis of Container Gantry Crane Operation and Reliability Characteristics and Maintenance Optimization Results The comparison of the values of the selected container gantry crane characteristics before the system operation process optimization given by (7.143–7.145) for the system reliability and by (7.149–7.151) for the system renewal and availability with their values after the system operation process optimization respectively given by (7.146–7.148) and by (7.152–7.154) justifies the sensibility of the performed system operation process optimization.
394
7 Additional Applications
The results of the system corrective and preventive optimization does not give the value of the preventive maintenance period that maximizes the system availability coefficient. The value of the system availability coefficient increases while the preventive maintenance period increases.
Suggestions on New Strategy of Container Gantry Crane Operation Process Organizing From the analysis performed of the results of the container gantry crane operation process optimization it can be suggested to organize the system operation process in a way that causes the replacing (or the approaching/convergence to) the conditional mean sojourn times Mbl of the system at the particular operation states _ bl after the before the optimization given by (7.135) by their optimal values M optimization given by (7.136). The possibility of fulfilling this suggestion of the operation process parameters changing is not easy and has to be checked in practice. It seems to be practically easier to, change the operation process characteristics which results in replacing (or the approaching/convergence to) the unconditional mean sojourn times Mb of the container gantry crane at the particular operation _ b after the states before the optimization given by (7.137) by their optimal values M optimization given by (7.140). The easiest way of the system operation process reorganizing is that leading to the replacing (or the approaching/convergence to) the total sojourn times E½^hb of the container gantry crane operation process at the particular operation states during the operation time h ¼ 1 year before the optimization given by (7.139) by _ ^ their optimal values E½ hb after the optimization given by (7.142).
Suggestions on New Strategy of Container Gantry Crane Maintenance Policy From the analysis of the results of the system corrective and preventive optimization it can only be suggested that in order to keep high availability of the system, the value of the preventive maintenance period of time that maximizes its availability coefficient should not be too small.
Other Suggestions The container gantry crane operating at the varying in time operation conditions reliability and availability evaluation and prediction are approximate because of non sufficiently exact input data. Although the container gantry crane operation process parameters are fixed with a high accuracy, their changing reliability structures are fixed on a high level of generality and the reliability parameters of its
7.2 Operation, Reliability, Availability, Safety and Cost Analysis
395
components are not sufficiently exact because of the lack of statistical data necessary for their estimation. Therefore, the achieved results may only be considered as an illustration of the possibilities of applications of the proposed methods and procedures to this container gantry crane operation, reliability and availability analysis, prediction and optimization. However, the obtained evaluation may be a very useful example in the container gantry cranes and other complex technical systems, reliability and availability characteristics, identification and prediction, especially during the design and when planning and improving their operation processes safety and effectiveness.
7.3 Summary The examples of applications, complementary to the appliqué parts of the previous chapters, are presented. They illustrate the possibilities of the theoretical results, applications in the book that are different from that presented in the previous chapters. Studying them makes much wider the range of real complex technical systems, whose reliability and safety identification, prediction and optimization can be performed with the methods and procedures presented in the book.
References 1. Guze S, Kołowrocki K (2008) Reliability analysis of multistate ageing consecutive ‘‘k out of n: F’’ systems. Int J Mat Struct Reliab 6(1):47–60 2. Kołowrocki K (2004) Reliability of large systems. Elsevier, Amsterdam 3. Kołowrocki K (2007) Reliability modelling of complex systems–Part 1. Int J Gnedenko e-Forum Reliab: Theory Appl 2(3–4):116–127 4. Kołowrocki K (2007) Reliability modelling of complex systems–Part 2. Int J Gnedenko e-Forum Reliab: Theory Appl 2(3–4):128–139 5. Kołowrocki K (2008) Reliability of large systems. In: Encyclopedia of quantitative risk analysis and assessment, vol 4. Wiley, New York, pp 1466–1471 6. Kołowrocki K, Soszyn´ska J (2006) Reliability and availability of complex systems. Qual Reliab Eng Int 22(1):79–99 7. Kolowrocki K, Soszynska J (2010) Reliability, availability and safety of complex technical systems: modelling–identification–prediction–optimization. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 4(1):133–158 8. Kolowrocki K, Soszynska J (2010) Preliminary statistical identification and orediction of the container gantry crane operation process. Summer Safety & Reliability Seminars. J Pol Saf and Reliab Assoc 4(1):173–181 9. Kossow A, Preuss W (1995) Reliability of linear consecutively-connected systems with multistate components. IEEE Trans Reliab 44:518–522 10. Soszyn´ska J (2004) Reliability of large parallel systems in variable operation conditions. Fac of Navig Res Works, vol 16, Gdynia, pp 168–180 11. Soszyn´ska J (2006) Reliability of large series-parallel system in variable operation conditions. Int J Autom Comput 3(2):199–206
396
7 Additional Applications
12. Soszyn´ska J (2007) Systems reliability analysis in variable operation conditions. PhD thesis, Gdynia Maritime University-System Research Institute Warsaw (in Polish) 13. Soszyn´ska J (2008) Asymptotic approach to reliability evaluation of large ‘‘m out of l’’–series system in variable operation conditions. Summer Safety & Reliability Seminars. J Pol Saf Reliab Assoc 2(2):323–346
Chapter 8
Summary
In this book the comprehensive approach to the analysis, identification, evaluation, prediction and optimization of complex technical systems, operation, reliability, availability and safety is presented. The results obtained may play the role of an easy-to-use guide necessary in reliability and safety evaluations of real complex technical systems, as well as during their operation and when they are designed. To make the results and the proposed methods and algorithms easy and useful tools for reliability practitioners their usage is illustrated by various examples of their application to the identification of the unknown parameters and to evaluation and optimization of the real system reliability and safety characteristics. The general analytical reliability, availability and safety models are the basis for the procedures and algorithms that allow to find the main and practically important reliability and safety and renewal and availability characteristics of the complex technical multistate systems. These models together with the linear pogramming are very useful in the complex technical systems operation, reliability, availability and safety improvement, optimization and their operation cost analysis. The methods and the procedures of identification of operation processes and the reliability and safety models of the complex technical systems presented in the book may be successfully applied to the real complex technical systems operation, reliability and safety unknown parameters estimation. All the tools are useful in the reliability, availability and safety optimization and operation cost analysis of a very wide class of real technical systems operating at the varying conditions that have an influence on changing their reliability and safety structures and their components, reliability and safety characteristics. In addition to the general solutions, theoretical results are illustrated by practical examples of their application in reliability and safety evaluation and optimization of complex technical systems. The identification of the real technical port and maritime transportation systems, unknown parameters are obtained on the basis of enough precise systems and components operation data and on nonprecise component reliability and safety data and therefore, first of all, they should
K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8_8, Springer-Verlag London Limited 2011
397
398
8 Summary
be treated as an illustration of a wide possibility of applications of the proposed approach in system reliability analysis. Reliability and safety data come from experts and are concerned with component’s mean lifetimes and their arbitrary assumed exponential multistate reliability or safety functions. These evaluations, despite not being precise may be a very useful, simple and quick tool in approximate reliability evaluation, especially during the design of large systems, and when planning and improving their safety and effectiveness operation processes. Nonetheless, the approximate optimisation of the operation processes and reliability and safety characteristics of the considered real complex technical systems also with respect to their operation costs may be performed with the application of the methods and procedures presented and used in the reorganization of the system operation and its new strategy planning. Thus, the results that can be achieved by the proposed tools application may be very interesting for reliability and safety practitioners from the port and maritime transport industry and from other industrial sectors as well. The results presented in the book suggest that it seems reasonable to continue the investigations focusing on: • methods and procedures of increasing the reliability and safety of complex multistate systems, • methods and procedures of reliability and safety optimization of complex technical multistate systems related to their operation costs, • availability and maintenance optimization of complex technical systems, • elaboration of universal practical tools in the form of the reliability and safety decision support system including the methods and procedures and the computer program packages addressed to the operators of the complex technical systems, allowing them to evaluate and optimize automatically these systems, operation, reliability, availability and safety, • elaboration of the offer of the addressed to industry training courses in the scope of operation, safety, reliability and availability of complex technical systems based on the methods and procedures and corresponding computer program packages included in the reliability and safety decision support system.
Appendix A: Table of Standard Normal Distribution—Laplace’s Function
The values of Laplace’s function t 0 1 2 3
4
5
6
7
8
9
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
0.0160 0.0557 0.0948 0.1331 0.1700 0.2054 0.2389 0.2704 0.2995 0.3264 0.3508 0.3729 0.3925 0.4099 0.4251 0.4382 0.4495 0.4591 0.4671 0.4738
0.0199 0.0596 0.0987 0.1368 0.1736 0.2088 0.2422 0.2734 0.3023 0.3489 0.3531 0.3749 0.3945 0.4115 0.4265 0.4394 0.4505 0.4599 0.4678 0.4744
0.0239 0.0636 0.1026 0.1406 0.1772 0.2123 0.2454 0.2764 0.3051 0.3315 0.3554 0.3770 0.3962 0.4131 0.4279 0.4406 0.4515 0.4608 0.4686 0.4750
0.0279 0.0675 0.1064 0.1443 0.1808 0.2157 0.2486 0.2794 0.3078 0.3340 0.3577 0.3790 0.3980 0.4147 0.4292 0.4418 0.4525 0.4616 0.4693 0.4756
0.0319 0.0714 0.1103 0.1480 0.1844 0.2190 0.2517 0.2823 0.3106 0.3365 0.3599 0.3810 0.3997 0.4162 0.4306 0.4429 0.4535 0.4625 0.4699 0.4761
0.0359 0.0753 0.1141 0.1517 0.1879 0.2224 0.2549 0.2752 0.3133 0.3389 0.3621 0.3830 0.4015 0.4177 0.4319 0.4441 0.4545 0.4633 0.4706 0.4767
0.0000 0.0398 0.0793 0.1179 0.1554 0.1915 0.2257 0.2580 0.2781 0.3159 0.3413 0.3643 0.3849 0.4032 0.4192 0.4332 0.4452 0.4554 0.4641 0.4713
0.0040 0.0438 0.0832 0.1217 0.1591 0.1950 0.2291 0.2611 0.2910 0.3186 0.3438 0.3665 0.3869 0.4049 0.4270 0.4345 0.4463 0.4564 0.4649 0.4719
0.0080 0.0478 0.0871 0.1255 0.1628 0.1985 0.2324 0.2642 0.2939 0.3212 0.3461 0.3686 0.3888 0.4066 0.4222 0.4357 0.4474 0.4573 0.4656 0.4726
0.0120 0.0517 0.0910 0.1293 0.1664 0.2019 0.2357 0.2673 0.2967 0.3238 0.3485 0.3708 0.3907 0.4082 0.4236 0.4370 0.4484 0.4582 0.4664 0.4732
(continued) K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8, Ó Springer-Verlag London Limited 2011
399
400
Appendix A: Table of Standard Normal Distribution
(continued) t 0
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
0.4772 0.4821 0.4861 0.4892 0.4918 0.4938 0.4953 0.4965 0.4974 0.4981 0.4987 0.4990 0.4993 0.4995 0.4997 0.4998
1
2
3
4
5
6
7
8
9
0.4778 0.4826 0.4864 0.4896 0.4920 0.4940 0.4955 0.4966 0.4975 0.4982 0.4987 0.4991 0.4993 0.4995 0.4997
0.4783 0.4830 0.4868 0.4898 0.4922 0.4941 0.4956 0.4967 0.4976 0.4982 0.4987 0.4991 0.4994 0.4995 0.4997
0.4788 0.4834 0.4871 0.4901 0.4925 0.4943 0.4957 0.4968 0.4977 0.4983 0.4988 0.4991 0.4994 0.4996 0.4997
0.4793 0.4838 0.4875 0.4904 0.4927 0.4945 0.4959 0.4969 0.4977 0.4984 0.4988 0.4992 0.4994 0.4996 0.4997
0.4798 0.4842 0.4878 0.4906 0.4929 0.4946 0.4960 0.4970 0.4978 0.4984 0.4989 0.4992 0.4994 0.4996 0.4997
0.4803 0.4846 0.4881 0.4909 0.4931 0.4948 0.4961 0.4971 0.4979 0.4985 0.4989 0.4992 0.4994 0.4996 0.4997
0.4808 0.4850 0.4884 0.4911 0.4932 0.4949 0.4962 0.4972 0.4979 0.4985 0.4989 0.4992 0.4995 0.4996 0.4997
0.4812 0.4854 0.4887 0.4913 0.4934 0.4951 0.4963 0.4973 0.4980 0.4986 0.4990 0.4993 0.4995 0.4996 0.4997
0.4817 0.4857 0.4890 0.4916 0.4936 0.4952 0.4964 0.4974 0.4981 0.4986 0.4990 0.4993 0.4995 0.4997 0.4998
The relationship between the standard normal distribution FN(0,1)(t) and the Laplace’s function UðtÞ 1 FNð0;1Þ ðtÞ ¼ PðT tÞ ¼ pffiffiffiffiffiffi 2p
Zt e 1
x2
1 dx ¼ 0:5 þ signðjtjÞ pffiffiffiffiffiffi 2p
Zt
2
ex dx
0
¼ 0:5 þ signðjtjÞPð0 T tÞ ¼ 0:5 þ signðtÞUðtÞ for t 2 ð1; 1Þ
Appendix B: Table of Chi-Square Distribution
The v2 with n degrees of freedom distribution of a variable U a 0.01 0.02 0.05 0.10 0.20 0.30 0.70 0.80 n
0.90
0.95
0.98
0.99
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
0.02 0.21 0.58 1.06 1.61 2.20 2.83 3.49 4.17 4.86 5.58 6.30 7.04 7.79 8.55 9.31 10.08 10.86 11.65 12.44 13.24 14.04
0.00 0.10 0.35 0.71 1.14 1.63 2.17 2.73 3.32 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.12 10.85 11.59 12.34
0.00 0.04 0.18 0.43 0.75 1.13 1.56 2.03 2.53 3.06 3.61 4.18 4.76 5.37 5.98 6.61 7.25 7.91 8.57 9.24 9.91 10.60
0.00 0.02 0.11 0.30 0.55 0.87 1.24 1.65 2.09 2.56 3.08 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54
6.63 9.21 11.34 13.28 15.09 16.81 18.47 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.80 36.19 37.57 38.93 40.29
5.41 7.82 9.35 11.67 13.39 15.03 16.62 18.17 19.68 21.16 22.62 24.05 25.47 26.87 28.26 29.63 30.99 32.35 33.69 35.02 36.34 37.66
3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.67 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.15 31.41 32.67 33.92
2.71 4.60 6.25 7.78 9.24 10.68 12.02 13.36 14.68 15.99 17.27 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41 29.61 30.81
1.64 3.22 4.64 5.99 7.29 8.56 9.80 11.03 12.24 13.44 14.63 15.81 16.98 18.15 19.31 20.46 21.61 22.76 23.90 25.04 26.17 27.30
1.07 2.41 3.66 4.88 6.06 7.23 8.38 9.52 10.66 11.78 12.90 14.01 15.12 16.22 17.32 18.42 19.51 20.60 21.69 22.77 23.86 24.94
0.15 0.71 1.42 2.19 3.00 3.83 4.67 5.53 6.39 7.27 8.15 9.03 9.93 10.82 11.72 12.62 13.53 14.44 15.35 16.27 17.18 18.10
0.06 0.45 1.00 1.65 2.34 3.07 3.82 4.59 5.38 6.18 6.99 7.81 8.63 9.47 10.31 11.15 12.00 12.86 13.72 14.59 15.45 16.31
(continued) K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8, Ó Springer-Verlag London Limited 2011
401
402
Appendix B: Table of Chi-Square Distribution
(continued) a 0.01 0.02 n
0.05
0.10
0.20
0.30
0.70
0.80
0.90
0.95
0.98
0.99
23 24 25 26 27 28 29 30
35.17 36.41 37.66 38.88 40.11 41.34 42.56 43.77
32.01 33.20 34.38 35.56 36.74 37.92 39.09 40.26
28.43 29.55 30.67 31.79 32.91 34.03 35.14 36.25
26.02 27.10 28.17 29.25 30.32 31.39 32.46 33.53
19.02 19.94 20.87 21.79 22.72 23.65 24.58 25.51
17.19 18.06 18.94 19.82 20.70 21.59 22.47 23.36
14.85 15.66 16.47 17.29 18.11 18.84 19.77 20.60
13.09 13.85 14.61 15.38 15.15 16.93 17.71 18.49
11.29 11.99 12.70 13.41 14.12 14.85 14.57 16.31
10.20 10.86 11.52 12.20 12.88 13.56 14.26 14.95
41.64 42.98 44.31 45.64 46.96 48.28 49.60 50.89
38.97 40.27 41.57 42.86 44.14 45.42 46.69 47.96
Appendix C: Table of k-Kolmogorov Distribution
The values of Q(k) function k Q(k) k
Q(k) k 1 P k ð1Þ exp½2k2 k2 The values of a function QðkÞ ¼
Q(k)
k
Q(k)
0.9387 0.9418 0.9449 0.9478 0.9505 0.9531 0.9556 0.9580 0.9603 0.9625 0.9646 0.9665 0.9684 0.9702 0.9718 0.9734 0.9750 0.9764 0.9778 0.9791 0.9803 0.9815 0.9826 0.9836 0.9846 0.9855 0.9864
1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08
0.9973 0.9975 0.9977 0.9979 0.9980 0.9981 0.9983 0.9984 0.9985 0.9986 0.9987 0.9988 0.9989 0.9990 0.9991 0.9991 0.9992 0.9993 0.9993 0.9994 0.9994 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996
k¼1
0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58
0.0000 0.0001 0.0002 0.0003 0.0005 0.0008 0.0013 0.0019 0.0028 0.0040 0.0055 0.0074 0.0097 0.0126 0.0160 0.0200 0.0247 0.0300 0.0361 0.0428 0.0503 0.0585 0.0675 0.0772 0.0876 0.0987 0.1104
0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08
0.4880 0.5038 0.5194 0.5347 0.5497 0.5645 0.5791 0.5933 0.6073 0.6209 0.6343 0.6473 0.6601 0.6725 0.6846 0.6964 0.7079 0.7191 0.7300 0.7406 0.7508 0.7608 0.7704 0.7798 0.7889 0.7976 0.8061
1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58
(continued) K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8, Ó Springer-Verlag London Limited 2011
403
Appendix C: Table of k-Kolmogorov Distribution
404 (continued) k Q(k)
k
0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81
1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31
Q(k) k 1 P k ð1Þ exp½2k2 k2 The values of a function QðkÞ ¼
Q(k)
k
Q(k)
0.9873 0.9880 0.9888 0.9895 0.9902 0.9908 0.9914 0.9919 0.9924 0.9929 0.9934 0.9938 0.9942 0.9946 0.9950 0.9953 0.9956 0.9959 0.9962 0.9965 0.9967 0.9969 0.9971
2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31
0.9997 0.9997 0.9997 0.9997 0.9998 0.9998 0.9998 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 1.0000
k¼1
0.1228 0.1357 0.1492 0.1632 0.1778 0.1927 0.2080 0.2236 0.2396 0.2558 0.2722 0.2888 0.3055 0.3223 0.3391 0.3560 0.3728 0.3896 0.4064 0.4230 0.4395 0.4559 0.4720
0.8143 0.8223 0.8299 0.8374 0.8445 0.8514 0.8580 0.8644 0.8706 0.8765 0.8823 0.8877 0.8930 0.8981 0.9030 0.9076 0.9121 0.9164 0.9206 0.9245 0.9283 0.9319 0.9354
1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81
Appendix D: Table of Gamma Function
The values of a gamma function p
CðpÞ
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
1.00000 0.97350 0.95135 0.93304 0.91817 0.90640 0.89747 0.89115 0.88726 0.88566 0.88623 0.88887 0.89352 0.90012 0.90864 0.91906 0.93138 0.94561 0.96177 0.97988 1.00000
The gamma function properties Cð1Þ ¼ 1; Cðp þ 1Þ ¼ pCðpÞ; p [ 0; Cðn þ 1Þ ¼ n!; n 2 N:
K. Kołowrocki and J. Soszyn´ska-Budny, Reliability and Safety of Complex Technical Systems and Processes, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-694-8, Ó Springer-Verlag London Limited 2011
405
![Reliability, maintainability and risk practical methods for engineers; [including reliability centred maintenance and safety-related systems]](https://epdf.tips/img/300x300/reliability-maintainability-and-risk-practical-met_5a4abcfcb7d7bc23216578e9.jpg)
