Polling on a graph with general arrival and service time distribution
Texte intégral
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(138) Q ] ¡. Ð ¡ 5( ] ¡25 = 1 5 ] 02pXUZ[02Cb\ 1&Vd361$1&VHC:5HCIH136DSD02vt3hZ rO0}Z[Z5HT1$TcEcIGHD=025>1&VHC cIGHD&DSCE51:cIocSZ2C5HTD=0251&VHC5HCIH1 T5HCy MP5daCECba'@'0251&Vd361 cE3\C@'1&VHCA1&T143hZ¥ZYCE5HF(1&VT6{cIocSZ2Cb\ Ð 365da ] 0\ Ð ¡ 5( Ð 2¡ 5 ] ¡ 5 = 1 B Ð 2¡ 5( ] 2¡ 5 = 1 5 ] \m025dcI8C Ð ¡ sy )31. . . CE5dcIC Q Ð ¡365da Q ] ¡02pXUZ2o81&Vd361 k ¡ ^ y & k ¡0W\=1&VHCE58F602vCE5>io & k ¡ & Ð ¡25 )31 Ð ¡ 5 Ð ¡25( ] ¡ 5 = 1 7u\&\.GHpXC:5HT6r 1&Vd361 0W\\GdcSV>1SVd361 ¡ ^ @0Py Cy , (*) 1 Ð ¡ 5( ] ¡ 5 = 1 5 ] , 4( Ð ¡ ) , ¡ ( ] 5 ) 1 = 1 Ð ¡ ] ¡ ;$VHCE58{~TD365o Ð ¡ k ¡ I0 Ð ¡&¡ '(*)31 Ð 2 x k ¡ ¡ 5 ] ¡ 5 = 1 5 ] ,'( Ð ¡ ) , 0 Ð ¡S¡ & k ¡ (*)31 Ð 2¡ 5 ] ¡25 = 1 5 ] , ( Ð ¡ ) , 0 X Ð ¡S¡ & Ð 2¡ 5 )31 Ð 2¡ 5 Ð 2¡ 5( ] ¡25 = 1 (*)31 Ð 2 ¡ 5( ] 2¡ 5 = 1 5 ] , ( Ð ¡ ) , 0 x Ð ¡&¡ 025{ 5 )31 5 Ð ¡ 5( ] 2¡ 5 = 1 (*)31 Ð ¡25( ] ¡25 = 1 5 ] , ( Ð ¡ ) , 0 x Ð ¡&¡ ^ GH50}{~TD&pZYo{~TD3hZ[Z x Ð ¡ y ;$V0W\u{|T6Z[Z2T6r\={~D&Tp 1&VHCXaCId502102T5T6{$1&VHCpXCb3\GHD&C 365da ~{ D&Tp cIT5da(02102T5 702v¡4y . . . . .
(139) .
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(142) . 6 ? 6:#6 $ 7[6 % Ð / y'7=Z21&ps365@RBdyJT\&\E@/yN:yNR02CEVZ=3`5daBdyB10WaVd36p>@ 361&Vr$0W\C8B1I36i0[Z}021o365da CED{~TD&¨ ps365dcIC^RTGH5daH\R{~TD CE5HCEDI3hZ[0 JECba g 3cE3`102T5>365da T`Z}Z[025HFBo\1SCEps\ H@H<365Gd\&cID02UH1by ] /yR7=ZY1Sps365 365da y T6{~ps365@ hB36pXUZ2C 361&V ^RTGH5daH\x{~TD CED{|T(D&ps365dcIC<>Cb3(\GHD&Cb\xT6{ ;¥TwCE5>NR025HF\ @ \GHiHp0Y1S1&Cba81&T =ZYT(i cITp jk y . ]].
(143) k /y 7=Z21&ps365@ y T(5d\143651&TUTGZ2T\E@ y902G@ hB1436i0[Z[021o@X<T5HT(1&T50WcS021o¤365da Mm5v636D0W3651 Gd365102102Cb\ 025 CE5HCED43hZ T6Z[Z}025HFBo\1SCEps\ H@Y/:/($)/+ :2 1 (0#" ( >!/#)R#"$"%($)/+ :,# " E@H/aH\Ey Ayd;36wt3`F60CE1:Ly=^RThps3@UHUy kf ¨ f @ Ðbjj] y _ /yR7=ZY1Sps365 365d a y$9'CEvo@ GHCEGHCI025HF025 \Ud3cIC
(144) @:BGHiHp021&1&Cba 1&TP# )('" "-($>, .# .# (*"-(0 ty f /y 7Z21&ps3`5 365da yb902G@ IMmpXUHD&T6v025HF 1&VHC\.1436i0[Z[0Y1ocSVd36D43cI1SCED0W\10WcE\T`{3\.o5dcSVHD&T5HTGd\§1&D43hznc 025J M1&T(wCE5>D025HF\ H@ ;¥Tn3`UHUdCb36D0Y5 4>,#($)/+ C (*)6 4)( ($#)6#"3#) :)( #)
(145) (%+ D >, &( E@dQRVd36UdCI#Z $0[Z[ZP @ B7@'LcI1&TiCED Ðbj(jk y e /y'7=Z21&ps365365daJ$yBU02CEwH\ps3H@ CETpxCE1&D0Wc/D&FTHa(0WcS021o365da<TpXCE51B1436i0[Z[021o T`{B1436¨ 102T5>;R02pXCb\0Y5 T`Z}Z[025HF8BoH\1&CEps\ ;¥CbcSVH50cE3tZ N$CEUTD&1?=TdyH;$l ¨ j] ¨ ^j @9'CI0WaCE5 =502vCED4\m021o@ =T6Z[Z3`5da'y BdyH7:\pGd\&\CE5= @ "-($>, .# .# (*"-(0 #)=,
(146) Y !E@Hl 0}Z2CEo@ Ðbj y ;:yd9 0Y5dav63hZ[ZPy #) #* "-(*)+ 4 :,@Hl 0[Z2CEo@ Ðbj(j] y j 7yd7y ^RTDSTtvwT6v'@ E902p021:1&VHCETDSCEps\${~TD:£GHCEGHCI025HF5HCE1r$TDSw\ @ 7 # " @
(147) d@ UHUy _#Ðhk ¨ _] @ Ðbj e y Ðb^ 7y7 y ^T(D&T6vwT6v>365daN:yBcSVd3\&\iCED&FCEDb@ E/DSFTa(0WcS021oT6{O3 UdT6Z[Z[025HF\oH\1&CEp H@\.GHiHp021&1&Cba 1&T 4 2 (& > >!! ')=, ;"%(0 (*')PEy ÐqÐ 7y GH1b@ 7 , ')=,## #" ( E@'BUHD025HFCED&¨ g CEDZW36F@ Ðbj y Ðb] 9y D&CETD&F60W3a(0W\E@'lyB JEUd365Hw(Ttr \w0 hB143`i0}Z[021oT6{O;¥TwCE5 3\&\0Y5HF8NR025HF\ @Y (*)+ I 7 P :(&'" ( #)
(148) #"$"-(*)+ :,# " E@H/aH\Ey Ayd;36wt3`F60 365daL=y ^RThHps3H@ Ðbjj] y Ðbk QydJD0Wc&w(CED365da8<y¢N:y3h02i0P@ E<T5HT1ST50WcS0Y1os3`5da>\143`i0}Z[021osT`{KUCED02Ta(0WcAUdT6Z[Z[025HF8pXTHaCIZ\ H@ / CC6@ CEUd3`D&1&pXCE51T6{K/RcIT5HTp0WcE\E@';R0[Z2iHGHD&F 502vCED4\0Y1o@';R0[Z2iHGHD&Fd@ Ðbj(j] y ÐE_ y y DSTCb\C365da g yBcSVHp0Wa1b@ I7*cIT51.0Y5GHTGd\ T6Z[Z}025HFBo\1SCEp r$021&V CE5HCED43hZ$BCED&v(0WcIC ;R02pXCb\ =@ )=)6#" ;"%($, # .# (*"-(0 A'@d?Ty _ @UHUy j(^e ¨ j] @ Ðbjj(] y Ðbf y y DSTCb\C365da g y BcSVHp0Wa1b@ hB6025HFCIZ2¨BCED&vCED GHCEGHCb\8rO0Y1SV BUd3`10W3hZ[ZYo =0\.1&D02iHGH1&Cba 7 D&D02vt3hZW\ @BGHiHp021&1SCba'y Ðbe Ayt9'CEvo@<y6B60Wa(0P@Lyy ^RThp3H@ Tp025d365dcICN$CIZW36102T5d\¥025 T6Z[Z[0Y5HFxBo\.1&CEps\ H@Y ($)/+ :2 E@ ¥@HUHUy Ðbf(f ¨ Ð ] @ Ðbjj^ y . . . . ]k.
(149) Ð dB y y<CEo5 365da N:y9y;$r$CECba(02C@ hB1436i0[Z[021o T6{ <36DSwT6v0W365 UHD&TcICb\S\Cb\M cID021&CED.03{~TD a(0W\&cID&CE1&C:1.0YpxCc&Vd3h025d\ @= ,4#)6> ($)I "-(*, # .# (*"-(0 6@=A,>?Ty k @UHUy f6_] ¨ f _ @ Ðbjj] y Ð /y ?GHpXpxCIZ}025@ )= #" < >,#, (#!"- ( '(*)W #), ') + (' @(Q36pX¨ iHD0WaFC 502v'y D&Cb\&\E@ Ðhj _ y Ðbj Hy7 y§QydN$Cb\m025HFd @ T6Z[Z[025HFn\.o\1SCEps\365da p GZ2102¨~1oUdCiHD4365dcSV025HFsUHD&THcICb\&\Cb\H@ @¥Qy ly M4y@7 ps\1SCED4aH36p>y ]^ Ayd; 3`wt36F60P= @ )(#" :( #"$"-(*)+ I 2 @H;$VHC<M; D&Cb\S\E@ Ðbj e y ].Ð Ay;36wt36F`0 @ GHCEG025HF75d3hZ2oH\m0W\RT6{ T6Z[Z[025HFX<>THaCIZW3\ 365 =U aH361S
(150) C H@6025 4 # (0R)(#" :!( #@ ')=, #/)(& ($#)P I 2 E@ Ay; 36w636F60P@/Ra'y[¡4@/KZ\.CEv02CED BcS02CE5dcIC GHiy2@ UHUy ]e ¨ k.Ð @ Ðbj(j^ y ]] N:y9y;$r$CECba(02C @ hQD.0Y1SCED0W3{~TDcSZW3\&\m0[{~o025HFFCE5HCED43hZ<36D&wT6vQVd3t0Y5d\ H@ , " .# @UHUy k ¨ Ð @ Ðbj e y ]k N:y9y;$r$CECba(02C@ hQD.0Y1SCED0W3{~TDD43`1&Cb\:T6{cIT5vCED&F(CE5dcICsT6{<36D&w(Ttvc&Vd3t0Y5d\ @r$021&V36UHUZ[0cE3`¨ 102T581&T£GHCEG025HF8365da8\1&TDI36FC1&VHCET(D&o @ .# #!($"%(0
(151) ( # (0! #)=,)6#" !( @(Cba'y Hy¢J$y[Q y =025HFps365X365da y¢/y AybN$CE51&CEDb@9§T5daT5X<361&VXBTHc6y69'CbcI1&GHD&C?=T1&Cb\ BCED02Cb\ j @Q36piHD0WaFC =50Yv'y D&Cb\&\ @HUHUy ]e^ ¨ ] e @ Ðbj k y ]6_ N:y9y;$r$CECba(02C @ E;$VHC$CI0W\1&CE5dcIC T6{ pXT(pXCE51I\'{~TD\143`102T5d36D&o<36D&w(TtvcSVd3h025d\ H@ "-($>, .# .# (*"-(0 8A@HUHUy Ðbj.Ð ¨ Ðbj(e @ Ðbj k y ]f g y'Bdy dVdaH365HT6v'@H/yH7yB36w\.T5HT6v @ hQT(5da(0Y1.0YT(5d\T6{/0\.1&CE5dcICT6{$B1&Cb3ao¨B14361SC<>THaCb\R025 QoHcSZ[0c GHCEG025HF8Bo\.1&CEps\ HP@ , # ($#)#)=, #)6
(152) '"}@ ¥@?=Tdy ] @(UHUy Ð e ¨ Ð _ @ Ðbj j y . . . ]6_.
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