Multi-class Queues and Stochastic Networks

Transcription

1 Mucass Queues and Socasc Newors LNMB 06 Rcard J. Boucere & Werner Scenard dearmen of Aed Maemacs Unversy of Twene

2 Mucass Queues and Socasc Newors Deaed conen:. reversby saonary basc ueues ouu eorem feedforward newors. ara baance Jacson newor KeyWe newer arrva eorem 3. uasreversby cusomer yes BCM newors bandwd sarng newors 4. bocng aggregaon decomoson 5. oss newors nsensvy va suemenary varabes 6. soourn me dsrbuon n newors 7. MVA AMVA QNA 8. fud ueues basc modes 9. feedbac fud ueues newors of fud ueues

3 Mucass Queues and Socasc Newors 3 Deaed conen:. reversby saonary basc ueues ouu eorem feedforward newors [Key caer ]. ara baance Jacson newor KeyWe newer arrva eorem [Key caer ] 3. uasreversby cusomer yes BCM newors bandwd sarng newors [Key caer 3] 4. bocng aggregaon decomoson 5. oss newors nsensvy va suemenary varabes 6. soourn me dsrbuon n newors 7. MVA AMVA QNA 8. fud ueues basc modes 9. feedbac fud ueues newors of fud ueues

4 Mucass Queues and Socasc Newors 4 Leraure: R. Neson robaby Socasc rocesses and Queueng Teory 995 : Caer 0 F.. Key Reversby and Socasc Newors Wey 979 avaabe onne R.W. Woff Socasc Modeng and e Teory of Queues rence Ha 989. R.J. Boucere N.M. van D edors Queueng Newors A Fundamena Aroac Inernaona Seres n Oeraons Researc and Managemen Scence Vo 54 Srnger 0 Handous sdes and references o reevan addona eraure w be made avaabe a e ecures.

5 Mucass Queues and Socasc Newors 5 Excersses: Afer 3 rd ecure and afer 5 ecure. Deadne December 06 Mared by: January 6 07

6 Mucass Queues and Socasc Newors Today ecure : ueue eng based on Bure Neson sec Key sec.. Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues 6

7 7 Connuous me Marov can socasc rocess 0 evouon random varabe counabe or fne sae sace S{0 } Marov roery Marov can: Socasc rocess sasfyng Marov roery Transon robaby me omogeneous CamanKomogorov euaons... s s n n 0 s s å

8 8 Connuous me Marov can CamanKomogorov euaons ranson raes or um raes Komogorov forward euaons: REGULAR s s å ¹ m 0 ] [ ' ] [ ] [ å å å å ¹ ¹ ¹

9 Assume ergodc and reguar Connuous me Marov can 9 goba baance euaons eubrum euaons 0 å [ ] ¹ π s nvaran measure saonary measure souon a can be normased s saonary dsrbuon f saonary dsrbuon exss en s unue and s mng dsrbuon m 0

10 Mucass Queues and Socasc Newors Today ecure : 0 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

11 Brdea rocess Sae sace Marov can ranson raes Komogorov forward euaons Goba baance euaons ï ï î ï ï í ì oerwse rae dea rae br 0 S Í Z ] [ 0 ] [ d d

12 Mucass Queues and Socasc Newors Today ecure : Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

13 3 M/M/ ueue osson arrva rocess rae snge server exonena servce mes mean / Sae sace S{0 } Marov can? Assume nay emy: 00 Transon raes : ï ï î ï ï í ì > > 0 0 ] [ 0 ] [ o o o λ

14 4 M/M/ ueue Komogorov forward euaons >0 Goba baance euaons >0 ï ï î ï ï í ì > > 0 0 ] [ ] [ ] [ d d d d

15 5 M/M/ ueue λ λ Eubrum dsrbuon: < 0 / / / Saonary measure; summabe à eubrum dsrbuon roof: Inser no goba baance Deaed baance! Teorem: A dsrbuon a sasfes deaed baance s a saonary dsrbuon

16 Mucass Queues and Socasc Newors Today ecure : 6 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

17 Brdea rocess Sae sace S Í N {0...} Marov can ranson raes " $ $ # $ $ % λ br rae > 0 dea rae λ > 0 λ0 0 Defnon: Deaed baance euaons 7 Teorem: Assume a en 0 Õ r é 0 ê ë r r r r r r ù ú r r û < s e eubrum dsrbuon of e brdea rocess. å ÎS Õ r ÎS

18 Advanced Queueng Teory Today ecure : 8 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

19 Reversby; saonary Saonary rocess: A socasc rocess s saonary f for a n τ... n ~... n 9 Teorem: If e na dsrbuon of a Marov can s a saonary dsrbuon en a Marov can s saonary Reversbe rocess: A socasc rocess s reversbe f for a n τ... n ~... n

20 Reversby; saonary 0 Lemma: A reversbe rocess s saonary. Teorem: A saonary Marov can s reversbe f and ony f ere exss a coecon of osve numbers π S summng o uny a sasfy e deaed baance euaons Î S Wen ere exss suc a coecon π eubrum dsrbuon. roof S s e

21 Marov um can For Marov can: odng me n sae s exonenay dsrbued; s mnmum of exonena mes for eac ranson ou of sae say o sae w rae Mnmum as exonena dsrbuon w rae robaby a ranson o sae occurs uon dearure from sae s / Marov um can : Marov can w ranson raes secfed by and and s euvaen o orgna can For Marov um can we may generase e odng me n e saes o nonexonena mes: s does no affec e eubrum robaby for beng n sae

22 roof suose rocess reversbe Le rev.: en Conversey: suose ere exss π sasfyng deaed baance summng gves goba baance.

23 3 roof cd. Consder beavour of for T T sars a T n says random me before umng o says random me before umng o 3 and so on un arrves a m were says un T robaby densy for s a s densy w resec o m. Inegrae over m suc a m T. Deaed baance mes so a * s rob densy for a sarng T n m says random me m before umng o m and so on un arrves a says un T. Tus aso nvong saonary m m m m m m m m m e e e e e e m m m m m m m *... ~... n n

24 4 Komogorov s crera Teorem: A saonary Marov can s reversbe ff for eac fne seuence of saes. Furermore for eac seuence for wc e denomnaor s osve n n n n n n n n n n n n S n Î...

25 Mucass Queues and Socasc Newors Today ecure : 5 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

26 6 Tme reversed rocess reversbe Marov rocess à aso bu me omogeney no nered for nonsaonary rocess Lemma: If s a meomogeneous Marov rocess wc s non saonary en e reversed rocess s a Marov rocess wc s no even meomogeneous. roof. s a Marov rocessà easy nonmeomogeneous: observe does no deend on meom do deend on a eas for some and so aso e ranson robabes of e reversed rocess τ τ

27 Tme reversed rocess 7 Teorem: If s a saonary Marov rocess w ranson raes and eubrum dsrbuon π S en e reversed rocess τ s a saonary Marov rocess w ranson raes ' and e same eubrum dsrbuon. Î S Teorem: Key s emma Le be a saonary Marov rocess w ranson raes. If we can fnd a coecon of numbers suc a S and a coecon of osve numbers π S summng o uny suc a ' Î S en are e ranson raes of e mereversed rocess and π S s e eubrum dsrbuon of bo rocesses.

28 Mucass Queues and Socasc Newors Today ecure : 8 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

29 Truncaon of reversbe rocesses 9 Teorem: If e ranson raes of a reversbe Marov rocess w sae sace S and eubrum dsrbuon Î S are aered by cangng o c for Î A Î S \ A were c>0 en e resung Marov rocess s reversbe n eubrum and as eubrum dsrbuon B s e normazng consan. ì B í îbc Î Î S A A If c0 en e reversbe Marov rocess s runcaed o A and e resung Marov rocess s reversbe w eubrum dsrbuon å ÎA \ Î A A S\A

30 Advanced Queueng Teory Today ecure : 30 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

31 Exame: wo M/M/ ueues 3 Consder wo M/M/ ueues ueue w osson arrva rocess rae λ servce rae Indeendence: π Now nroduce a common caacy resrcon Queues no onger ndeenden bu ρ ρ 0 ρ λ / < J π B ρ ρ 0 J

32 Mucass Queues and Socasc NeworsToday ecure : 3 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

33 Ouu M/M/ ueue: Bure s eorem 33 M/M/ ueue osson arrvas exonena servce number of cusomers n M/M/ ueue: n eubrum reversbe Marov rocess. Forward rocess: uward ums osson λ Reversed rocess : uward ums osson downward um of forward rocess Downward um rocess of osson rocess λ λ λ

34 Ouu M/M/ ueue 34 Le 0 fxed. Arrva rocess osson us arrva rocess afer 0 ndeenden of number n ueue a 0. For reversed rocess : arrva rocess afer 0 ndeenden of number n ueue a 0 Reversby: on dsrbuon dearure rocess u o 0 and number n ueue a 0 for ave same dsrbuon as arrva rocess o u o 0 and number n ueue a 0. Bures eorem: In eubrum e dearure rocess from an M/M/ ueue s a osson rocess and e number n e ueue a me 0 s ndeenden of e dearure rocess ror o 0 Hods for eac reversbe Marov rocess w osson arrvas as ong as an arrva causes e rocess o cange sae

35 Mucass Queues and Socasc Newors Today ecure : 35 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

36 Tandem newor of M/M/ ueues M/M/ ueue osson arrvas exonena servce Eubrum dsrbuon r r ÎS {0...} r / < Tandem of J M/M/ ueues ex servce ueue number n ueue a me Queue n soaon: M/M/ ueue. λ Dearure rocess ueue osson us ueue n soaon: M/M/ ueue Sae 0 ndeenden dearure rocess ror o 0 bu s deermnes 0 J 0 ence 0 ndeenden 0 J 0. Smar 0 ndeenden 0 J 0. Tus 0 0 J 0 muuay ndeenden and 36 π... J J ρ ρ ρ λ / <

37 Exame: feed forward newor of M/M/ ueues arrva rae servce rae... 5 roung robabes π ρ ρ ρ λ / <

38 Mucass Queues and Socasc Newors Today ecure : 38 Connuous me Marov can Brdea rocess Exame: M/M/ ueue Brdea rocess: eubrum dsrbuon Reversby saonary Tme reversed rocess Truncaon of reversbe rocesses Exame: wo M/M/ ueues Ouu M/M/ ueue Tandem newor of M/M/ ueues Soourn me n a andem newor of M/M/ ueues

39 39 Wang me M/M/ ueue Consder sme ueue FCFS dscne W : wang me yca cusomer n M/M/ excudes servce me N cusomers resen uon arrva S r resdua servce me of cusomers resen ASTA Voor 0 r r e S N W N W W å å å > > > >! 0 0

40 40 r r r r r r r r r r r r r r r r 0 0 0!! > > > > > å å å å EW W W E e W W W e e e e e W Wang me M/M/ ueue Tus s exonena λ

41 Soourn me andem M/M/ ueues 4 Reca: In eubrum e dearure rocess from an M/M/ ueue s a osson rocess and e number n e ueue a me 0 s ndeenden of e dearure rocess ror o 0 Teorem: If servce dscne a eac ueue n andem of J M/M/ ueues s FCFS en n eubrum e wang mes of a cusomer a eac of e J ueues are ndeenden roof: Key. 38 Tandem M/M/s ueues: overang

42 Summary / nex: 4 Deaed baance or reversby and er conseuences Brdea rocess M/M/ ueue Truncaon Komogorov s crera Tandem newor Soourn me n andem newor Nex on MQST.. Quasreversby and ara baance and er conseuences Newor of ueues J Bocng roocos π Cusomer yes... J ρ ρ ρ λ / < Queue dscnes