8. Queueing systems lect08.ppt S Introduction to Teletraffic Theory - Fall

Transcription

1 8. Queueg sysems lec8. S Iroduco o Teleraffc Theory - Fall

2 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces

3 8. Queueg sysems Smle eleraffc model Cusomers arrve a rae cusomers er me u / = average er-arrval me Cusomers are served by arallel servers he busy a server serves a rae cusomers er me u / = average servce me of a cusomer There are m wag laces m 3

4 8. Queueg sysems ure wag sysem Ife umber of wag laces m If all servers are occued whe a cusomer arrves she occues oe of he wag laces No cusomers are los bu some of hem have o wa before geg served From he cusomer s o of vew s eresg o kow e.g. wha s he robably ha she has o wa oo log? 4

5 8. Queueg sysems Queueg dscle Cosder a sgle server queueg sysem Queueg dscle deermes he way he server serves he cusomers I ells wheher he cusomers are served oe-by-oe or smulaeously Furhermore f he cusomers are served oe-by-oe ells whch order hey are ake o he servce Ad f he cusomers are served smulaeously ells how he servce caacy s shared amog hem A queueg dscle s called work-coservg f cusomers are served wh full servce rae wheever he sysem s o-emy 5

6 8. Queueg sysems Varous work-coservg queueg dscles Frs I Frs Ou FIFO = Frs Come Frs Served FCFS he mos ordary queueg dscle queue cusomers served oe-by-oe wh full servce rae always serve he cusomer ha has bee wag for he loges me Las I Frs Ou LIFO = Las Come Frs Served LCFS sack cusomers served oe-by-oe wh full servce rae always serve he cusomer ha has bee wag for he shores me rocessor Sharg S far queueg cusomers served smulaeously whe cusomers he sysem each of hem served wh equal rae / 6

7 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces 7

8 8. Queueg sysems M/M/ queue Cosder he followg smle eleraffc model: Ife umber of deede cusomers k Ierarrval mes are IID ad exoeally dsrbued wh mea / so cusomers arrve accordg o a osso rocess wh esy Oe server Servce mes are IID ad exoeally dsrbued wh mea / Ife umber of wag laces m Defaul queueg dscle: FIFO Usg Kedall s oao hs s a M/M/ queue more recsely: M/M/-FIFO queue Noao: raffc load 8

9 8. Queueg sysems Ieresg radom varables X = umber of cusomers he sysem a a arbrary me = queue legh equlbrum X* = umber of cusomers he sysem a a ycal arrval me = queue legh see by a arrvg cusomer = wag me of a ycal cusomer S = servce me of a ycal cusomer D = + S = oal me he sysem of a ycal cusomer = delay 9

10 8. Queueg sysems Sae raso dagram Le X deoe he umber of cusomers he sysem a me Assume ha X a some me ad cosder wha haes durg a shor me erval h: wh rob. h oh a ew cusomer arrves sae raso f he wh rob. h oh a cusomer leaves he sysem sae raso rocess X s clearly a Markov rocess wh sae raso dagram Noe ha rocess X s a rreducble brh-deah rocess wh a fe sae sace S...}

11 8. Queueg sysems qulbrum dsrbuo Local balace equaos LB: Normalzg codo N: LB N f

12 8. Queueg sysems qulbrum dsrbuo Thus for a sable sysem he equlbrum dsrbuo exss ad s a geomerc dsrbuo: X Geom X X } D X Remarks: Ths resul s vald for ay work-coservg queueg dscle FIFO LIFO S... Ths resul s o sesve o he servce me dsrbuo as far as he FIFO queueg dscle s cocered However for ay symmerc queueg dscle such as LIFO or S he resul s deed sesve o he servce me dsrbuo

13 8. Queueg sysems Mea queue legh X vs. raffc load X Traffc load 3

14 8. Queueg sysems Recall Lle s formula Cosder a sysem where ew cusomers arrve a rae Assume sably: very ow ad he he sysem s emy Lle s formula: N T average r of cusomers he sysem average me a cusomer seds he sysem N T Very useful formula: does o requre ASTA roery works for all STABL sysems 4

15 8. Queueg sysems Mea delay Le D deoe he oal me delay he sysem of a ycal cusomer cludg boh he wag me ad he servce me S: D S Lle s formula: X D. Thus Remarks: D X The mea delay s he same for all work-coservg queueg dscles FIFO LIFO S... Bu he varace ad oher momes are dffere! 5

16 8. Queueg sysems Mea delay D vs. raffc load D Traffc load 6

17 8. Queueg sysems Mea wag me Le deoe he wag me of a ycal cusomer Sce D S we have D S Remarks: The mea wag me s he same for all work-coservg queueg dscles FIFO LIFO S... Bu he varace ad oher momes are dffere! 7

18 8. Queueg sysems ag me dsrbuo Le deoe he wag me of a ycal cusomer Le X* deoe he umber of cusomers he sysem a he arrval me ASTA: X* } X }. Assume ow for a whle ha X* Servce mes S S of he wag cusomers are IID ad x Due o he memoryless roery of he exoeal dsrbuo he remag servce me S * of he cusomer servce also follows x-dsrbuo ad s deede of everyhg else Due o he FIFO queueg dscle S * S S Cosruc a osso o rocess by defg S * ad S * S S. Now sce X* : S * S S 3 3 S S 8

19 9 8. Queueg sysems ag me dsrbuo Sce X* we have Deoe by A he osso couer rocess corresodg o I follows ha: A - O he oher had we kow ha A osso. Thus } } } * } * } } * } X X X j! } } j j e A 8

20 8. Queueg sysems ag me dsrbuo 3 By combg he revous formulas we ge j j j j j j j j e e e e e e j j j!!! } }

21 8. Queueg sysems ag me dsrbuo 4 ag me ca hus be reseed as a roduc JD of wo deede radom varables J Beroull ad D x: } } } } } D D J D J e D J J

22 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces

23 8. Queueg sysems M/M/ queue Cosder he followg smle eleraffc model: Ife umber of deede cusomers k Ierarrval mes are IID ad exoeally dsrbued wh mea / so cusomers arrve accordg o a osso rocess wh esy Fe umber of servers Servce mes are IID ad exoeally dsrbued wh mea / Ife umber of wag laces m Defaul queueg dscle: FCFS Usg Kedall s oao hs s a M/M/ queue more recsely: M/M/-FCFS queue Noao: raffc load 3

24 8. Queueg sysems Sae raso dagram Le X deoe he umber of cusomers he sysem a me Assume ha X a some me ad cosder wha haes durg a shor me erval h: wh rob. h oh a ew cusomer arrves sae raso f he wh rob. m}h oh a cusomer leaves he sysem sae raso rocess X s clearly a Markov rocess wh sae raso dagram Noe ha rocess X s a rreducble brh-deah rocess wh a fe sae sace S...} 4

25 5 8. Queueg sysems qulbrum dsrbuo 4 Local balace equaos LB for : Local balace equaos LB for : LB LB!!!

26 6 8. Queueg sysems qulbrum dsrbuo Normalzg codo N:!!!!!!!! Noao : f N 5

27 7 8. Queueg sysems qulbrum dsrbuo 3 Thus for a sable sysem ha s: he equlbrum dsrbuo exss ad s as follows: : : }!! X

28 8 8. Queueg sysems robably of wag Le deoe he robably ha a arrvg cusomer has o wa Le X* deoe he umber of cusomers he sysem a a arrval me A arrvg cusomer has o wa wheever all he servers are occued a her arrval me. Thus ASTA: X* } X }. Thus } * X!! } * X : : 7

29 9 8. Queueg sysems Mea umber of wag cusomers Le X deoe he umber of wag cusomers equlbrum The! X 3 : : X X

30 3 8. Queueg sysems Mea wag me Le deoe he wag me of a ycal cusomer Lle s formula: X. Thus X : :

31 3 8. Queueg sysems Mea delay Le D deoe he oal me delay he sysem of a ycal cusomer cludg boh he wag me ad he servce me S: D S The S D : : D D

32 3 8. Queueg sysems Mea queue legh Le X deoe he umber of cusomers he sysem queue legh equlbrum Lle s formula: X D. Thus D X : : X X

33 33 8. Queueg sysems ag me dsrbuo Le deoe he wag me of a ycal cusomer Le X* deoe he umber of cusomers he sysem a he arrval me The cusomer has o wa oly f X*. Ths haes wh rob.. Uder he assumo ha X* he sysem however looks lke a ordary M/M/ queue wh arrval rae ad servce rae. Le deoe he wag me of a ycal cusomer hs M/M/ queue Le X* deoe he umber of cusomers he sysem a he arrval me I follows ha } *' ' } * } * } } e X X X

34 34 8. Queueg sysems ag me dsrbuo ag me ca hus be reseed as a roduc JD of wo de. radom varables J Beroull ad D x: ' } ' } ' } } } D D J D J e D J J

35 8. Queueg sysems xamle rer roblem Cosder he followg wo dffere cofguraos: Oe rad rer IID rg mes x Two slower arallel rers IID rg mes x Crero: mmze mea delay D Oe rad rer M/M/ model wh /: D Two slower rers M/M/ model wh /: D D D 35

36 8. Queueg sysems xamle D /D Traffc load 36

37 8. Queueg sysems TH ND 37