Discrete-event control of stochastic networks

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Author: Eitan Altman | Bruno Gaujal | Arie Hordijk

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0.45 a b c d e f

0.4

0.35

0.3

0.25

0.2

0.15

0.1

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0.22

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0.26

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B 3 E62; C +Z B :G> + y3$(!4 q0m P+ +4" E6A!("9 ]"$A +# Z s( !4 AP!#" I3X2 +43G!#" s")( & 6 !4 q"9A!4 +6"83 +:+4";:S3G2B!4" ED"9( A 6 7 & ( Z . i"$F +4 + "hE3 ++#"9:S32 !#" ?F:SE3 +4Cu'( ?! 3!@!4 1! G!3 22 $3G5 7 <6(xg +'( E! GZ 2;2|!U +#L(4 2;!(FP +#L(#LA!#65 "9s!4 V * $F +4uE3 +4E0m 3G= !*= (IP !43!4" A (13GJ5s%5 q G!1( j> +Fg +'( 3$AC<% E +4"9=632 +++4(6Z L 7N")(1( E3G2;29 +'!4J3G 5 G!4!#L529"9 A!I2; {! E3$AlC +43G!#" $( E3 +4 AP!#" I32Z +E") (4!43GJ=0 i +I3292 H * K JZy"9(PJE3 +4!?")( G!1( ")s!4 * AF +4GZv I3")D"9A!#> +#L(#!F29";L( ";!")D!u+Q $ 5J( 7 +4H! AP!#" I3K2 +43G!#" *"9(B " <GZ $P!4" I3G`2 +3!4" & !3G6((4 :$ +32 +43G!#" $ 3G2v:S32)6( ,+3G$"9Ig +) ! I3$ 5O6J5"9 ";!l!4 *"9A!";!4";:$c +3!4" H!l(#> +#:<")=0-!4" EL( +Z L5 7 i!4 6 "9(E!4 A29CP $"9A! "9O!(4!43GQ ";29";!hoC +#G" $|Z x6:G> + < +2 B +@:S3G29L( i!4 *2 A3G5 J";! ")(H'( (E 3! (4F +FP +4"9(4")q!4J3!! $P!4" I3m2 +3!#" =k 3$GL( "9!4&!4w2 A3G53$ 556:A")3!46(*g +) !4V +43G!#" x!4I(#> +#:<")=0*!4" EL( +Z+E=k 3G$6( ! AP!#" I3s2 +43G!#" = =6F + 3G= #= +5")O! (4 :$ +3{2 +463G'( $J( Z ++1 m(3 IP2; S:G6(13 @3fClg +'( A29C 6 $ !,( $! G]!*(#!3GQ"929";!hrC +4 $" A Y!4")( & = =6F +(3.! 7 (4Z 1(3 E P 6% (E6 A = = A +4( 6 #% IP (N3 @3fC g +) ")=k = = A +4( 6 7 JZ N!4 G!4 + 3$ 5 & !*=k 3$G^g +'( J &E! J <=6= F +( "92; J &

K 7L

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optimal ratios bounds 3/14

0.25

0.24

0.23

0.22

0.21

0.2

\

0.8

RTSU VNV NV

0.85

0.9

0.95

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1

K K K #

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R GZX * AF +4F5 "9(P2)3fC<(B!1P $"9A!4( +4'!1:S32) =k 3GG6(6Z +40 +# AL3G=kN=062;2\#= ++#L(4+P $J5 ('! u= A P296( K b Fb "9!4N!4 *(4& (3 ? AP!#" " E3G2 +43G!#" JZ+ ?2)E3 + G> +H=062;2 & ( x#= ++#L(4+P $ 5J( ! N!4 +4 $" A +4?! +A 5 +'$Q")+P G29"9= C7"9( $P!4" I3G2YZ '(4"9!#" EN% ( $66";!hC \!4 AP!#" I3m2 +43G!#" $( 32 $!29"9 Q $295q2;")*")0 i" AF +4 +#:$6(x F +4!4 > +")$:$6(4!#"$3G!#" $|Z ]"$A +# Z &*"9(!4^(4(3 ?'3G( i"$F +4 Z _ +4 !4F= A(#!! Q+ AP!#" E" 665")(x% G! !4-3f:$ +3$ @3"9!#")l!#" E !4u=6 (4! (E> +4(1Q !*!4w3f:G +3$?'( "AF +4D!#" E !4-=6 (4! (E> +4( Gb K * K b 3b Z] $P !#" E3G2 g+43$=0!#" $ i= (4! )?> +4(B (46A!@! A 6 F= k 3G$6(6Z $A2;Cu (P+63A") +4( 6O!4 ( "$F +w!#" ?"9(' (#L5l3G(!#= $(4!! QJ E")" E"

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2

P % PO

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1

α opt = 1/2

1/S 2

0

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0

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RTSU&V NVXV

1

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αopt = 0

2

αopt = 1/4

α opt= 1/3

α opt= 1/2

1/S 2 = c 1/S 1

α opt=1

Instability domain

10

1

9 6 N

RTSU V NV NV

V

1/S

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