B. Maddah INDE 504 Simulation 09/02/17

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1 B. Maddah INDE 54 Simulaio 9/2/7 Queueig Primer Wha is a queueig sysem? A queueig sysem cosiss of servers (resources) ha provide service o cusomers (eiies). A Cusomer requesig service will sar service if he required server is o busy. Oherwise, he cusomer wais i queue uil he server is available. Compoes of a queueig sysem A queuig sysem ca be composed of oe or may service ceers or odes. Cusomers are roued from oe ode o he oher accordig o cerai rules. Each ode is characerized by hree compoes. (i) The arrival process ; (ii) The service process ; (iii) The queue disciplie.

2 The arrival process is specified hrough he radom variables A, A 2,, where A i is he ier-arrival ime bewee he (i ) s ad he i h cusomer. A ypical modelig assumpio is o assume ha A i s are idepede ad ideically disribued (iid). The, he arrival process is characerized by F A (x) he cdf of A. Impora parameers of he arrival process (i addiio o F A (.)) are he mea ier-arrival ime E[A], ad he arrival rae λ = /E[A]. The mos commoly assumed arrival process is he Poisso process, where he ier-arrival imes, A, have a expoeial disribuio wih mea λ. This assumpio is realisic (i may cases). I addiio, i grealy simplifies he aalysis. The service process is specified hrough he radom variables S, S 2,, where S i is he service ime he i h cusomer. The S i are also ypically assumed iid wih cdf F S (x). Impora parameers of he arrival process are he mea service ime E[S] ad he service rae μ = / E[S]. Service imes are also commoly assumed o be expoeial. 2

3 Aalyical mehods (uilizig queueig heory) are quie complex wihou he expoeial assumpio. Whe usig simulaio, however, i is o difficul o relax he expoeial assumpios. The queueig or service disciplie refers o he rule uilized o selec he ex cusomer from he queue whe a cusomer fiishes service. Typical queueig disciplies iclude firs-i, firs-ou (FIFO), las-i, firs ou (LIFO), processor sharig (PS), service i radom order (SIRO), ad prioriy (PR). Uder he iid assumpios, a sigle-ode queueig sysem is geerally deoed by GI/GI/c, where he firs he GI refer o iid ier-arrival ad service imes ad c is he umber of servers. If he ier-arrival ad service imes are iid expoeial he he queue is deoed by M/M/c, where he M refers o he Markovia or memoryless propery of he expoeial rv. Performace measures ad geeral relaios Cosider a GI/GI/c queue (o simplify higs). I he followig we defie seady sae measures, which are saisical measures gahered afer ha he sysem has bee operaioal for a ime which is large eough. A impora measure is he raffic iesiy, ρ = λ/(cμ). 3

4 If ρ, he i ca be show ha he queue legh will icrease idefiiely as ime passes. I sable sysems, ρ <. For a sigle-server sysem, ρ is he mea server uilizaio. The saioary (seady-sae) sysem size disribuio is P = lim P{ L( ) = }, where L() is he umber of cusomers i he sysem a ime. The mea umber i he sysem is L = P =. I ca be show ha L could esimaed differely as L = lim Lsds ( ). The mea waiig ime i he sysem is Wi i= W = lim, where W i is he waiig ime of cusomer i i queue plus he ime he cusomer sped i service. Amog he mos impora queueig heory resuls is Lile s law L= λw. () 4

5 Oher measures cocer waiig i queue. The mea umber i he queue is L = lim L ( s) ds, q q where L q (s) is he umber of cusomers i queue a ime s. The mea waiig ime i he queue (or simply he mea delay) is W q = i= lim, W q i where i Wq is he waiig ime of cusomer i i queue. Lile s law implies ha Furhermore, W ad W q are relaed by L q = λw q. (2) W = W q + /μ. (3) Muliplyig by λ ad applyig Lile s law we ge L = L q + λ/μ. (4) Here, λ/μ ca be see as he mea umber of busy servers. (This ca i fac be also prove by Lile s law.) Noe ha kowig oe of he four performace measures, L, W, L q, ad W q, allows deermiig he oher hree. This is useful i verifyig ad validaig he oupu of a queueig sysem simulaio. 5

6 The M/M/ queue Cosider a sigle-server queue wih iid expoeial ierarrival ad service imes (hece called M/M/). Le λ ad μ deoe he arrival ad service raes ad ρ = λ/μ. Assume ρ <. Birh-deah aalysis shows ha ρ λ L = =. ρ μ λ Oher performace measures are deermied by Lile s law. For example, suppose cusomers arrive a a bak accordig o a Poisso process wih rae / cusomers per miue ad reques service of a sigle eller. The eller has a expoeial service ime wih rae /9 cusomers per miue. This ca be modeled as a M/M/ wih λ = / ad μ = /9. The raffic iesiy is ρ = 9/ =.9. The mea umber i he sysem is L = ρ / ( ρ) =.9/. = 9 cusomers. The mea umber i he queue is L q = L ρ = 9.9 = 8. cusomers. The mea waiig ime i he sysem is W = L / λ = 9 / (/) = 9 miues. The mea delay is W q = L q / λ = 8. / (/) = 8 miues. 6

7 Fially, beware of he oliear behavior of queues! 5 4 L( ρ) ρ For example, if ρ is icreased from.9 o.945 (by 5%) i he M/M/ example, L icreases from 9 o 7.8 (by abou %). 7