Birth-Death Processes. Outline. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Relationship Among Stochastic Processes.
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1 EEC 686/785 Modelig & Perforace Evaluatio of Couter Systes Lecture Webig Zhao Deartet of Electrical ad Couter Egieerig Clevelad State Uiversity based o Dr. Raj jai s lecture otes Relatioshi Aog Stochastic Processes Markov rocess is broader tha birth-death rocess ad birth-death rocess is broader tha Poisso rocess The Poisso rocess ca be odeled as a ure birth rocess with costat birth rate All birth-death rocesses are Markov rocesses with the restrictio that the trasitios are restricted to eighborig states 3 3 Noveber 5 EEC686/785 Webig Zhao Outlie irth-death Processes 4 Review of lecture Aalysis of a sigle queue art II State-trasitio diagra j j- 3 j I state : j j j j j j j j New arrivals take lace at a rate The service rate is oth the iterarrival ties ad service ties are assued exoetially distributed 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao
2 Theore: State Probability 5 Results for M/M/ Queues 7 The steady-state robability of a birth-death rocess beig i state i.e., there are jobs i the syste is give by:,,..., Here, is the robability of beig i the zero state o job i syste, i.e., syste is idle Utilizatio of the server: Mea uber of jobs i the syste: Variace of the uber of jobs: Mea resose tie: U E Var / Er Cuulative distributio fuctio of the resose r tie: F r e 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Results for M/M/ Queues j- j j irth-death rocesses with,,,...,,,..., Probability of jobs i the syste,,..., /: traffic itesity 6 Results for M/M/ Queues Cuulative distributio fuctio of the waitig tie F w e w This is a trucated exoetial distributio. Its q-ercetile is give by w q l q The above forula alies oly if q is greater tha -. All lower ercetiles are w w q ax, l q 8 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao
3 Results for M/M/ Queues 9 Exale 3. Mea uber of jobs i the queue q usy eriod: the tie iterval betwee two successive idle itervals Arrival rate 5 s Service rate /. 5 s Gateway utilizatio /.5 Probability of ackets i the gateway Mea uber of ackets i the gateway /.5/ Mea tie set i the gateway /- /5/ illisecods 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Exale 3. Exale 3. O a etwork gateway, easureets show that the ackets arrive at a ea rate of 5 ackets er secod s ad the gateway takes about two illisecods to forward the. Usig a M/M/ odel, aalyze the gateway What is the robability of buffer overflow if the gateway had oly 3 buffers? How ay buffers do we eed to kee acket loss below oe acket er illio? Probability of buffer overflow Pore tha 3 ackets i gateway ackets er billio ackets To liit the robability of loss to less tha -6 : -6 or: > log -6 /log We eed about buffers The last two results about buffer overflow are aroxiate 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 3
4 3 5 Exale 3. M/M/ Queue Strictly seakig, the gateway should actually be odeled as a fiite buffer M/M// queue Sice the utilizatio is low ad the uber of buffers is far above the ea queue legth, the results obtaied are a close aroxiatio Used to odel ultirocessor systes or devices that have several idetical servers ad all jobs waitig for these server are ket i oe queue idetical servers each with a service rate of jobs er uit tie The arrival rate is jobs er uit tie For a M/M/ queue to be stable, the traffic itesity ust be less tha If ay of the servers are idle, the arrivig job is serviced iediately If all servers are busy, the arrivig jobs wait i a queue 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 4 6 Exale 3. M/M/ Queue The state of the syste is rereseted by the uber of jobs i the syste Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 4
5 5 3 Noveber 5 EEC686/785 7 Webig Zhao Aalysis of M/M/ Queue Nuber of jobs i the syste is a birth-death rocess:,...,,,,...,,,,..., 3 Noveber 5 EEC686/785 8 Webig Zhao Aalysis of M/M/ Queue Probability of jobs i the syste: I ters of the traffic itesity /, we have:,...,,,,...,,...,,,,..., 3 Noveber 5 EEC686/785 9 Webig Zhao Aalysis of M/M/ Queue Probability of zero jobs i the syste: 3 Noveber 5 EEC686/785 Webig Zhao Aalysis of M/M/ Queue Probability that a arrivig job has to wait i the queue really should be, ca t iut it i PoitPoit : This is kow as Erlag s C forula. Notice that for, jobs P
6 6 3 Noveber 5 EEC686/785 Webig Zhao Aalysis of M/M/ Queue Mea uber of jobs i the queue q : E q Aalysis of M/M/ Queue Exected uber of jobs i service sice s E 3 Noveber 5 EEC686/785 3 Webig Zhao Aalysis of M/M/ Queue Exected uber of jobs i the syste: Variace of ad q ca siilarly be show as E E E s q q Var Var 3 Noveber 5 EEC686/785 4 Webig Zhao Aalysis of M/M/ Queue I T secods Total uber of jobs arrivig ad gettig service will be T Total busy tie of servers to service these jobs will be T/ Utilizatio of each server: T T U / / tie Total usy tie er server
7 Aalysis of M/M/ Queue 5 Aalysis of M/M/ Queue 7 Mea resose tie usig Little s law / r Mea waitig tie q w Cuulative distributio fuctio of the waitig tie: F w e w sice w has a trucated exoetial distributio fuctio, its q-ercetile: w q ax, l q If the robability of queueig is less tha -q/, the secod ter i the above equatio ca be egative. The correct aswer i those cases is 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Aalysis of M/M/ Queue Cuulative distributio fuctio of tie resose tie: r r r e e e / F r r r e re / r > Notice that the resose tie r is ot exoetially distributed uless I geeral, the coefficiet of variatio of r is less tha C.o.v stadard deviatio / exected value 6 Exale 3. Studets arrive at the uiversity couter ceter i a Poisso aer at a average rate of te er hour. Each studet seds a average of iutes at the terial ad the tie ca be assued to be exoetially distributed. The ceter curretly has five terials. Soe studets have bee colaiig that waitig ties are too log. Let us aalyze the ceter usage usig a queueig odel M/M/5 queueig syste with /6 er iute / er iute 8 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 7
8 Exale 3. 9 Exale 3. 3 Traffic itesity /.67/ Probability of all terials beig idle is: Average uber of studets waitig i the queue is: q Average uber of studets usig the terials is: s q 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Exale 3. 3 Exale 3. 3 Probability of all terials beig busy is: Average terial utilizatio.67 Average uber of studets i the ceter is: E The ea ad variace of the tie set i the ceter are:.33 r Var r Thus, each studet seds a average of 4 iutes i the ceter iutes workig 4 iutes waitig Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 8
9 Exale 3. Mea waitig tie:.33 w ercetile of the waitig tie is: w 4 ax, l ax, l % of the studets wait ore tha 4 iutes 33 Exale 3.3 With 6 terials: Traffic itesity:.67/ Probability of all terials beig idle.346 Probability of all terials beig busy.5 Average waitig tie w. iutes The 9-ercetile of waitig tie is:. ax, l > With just oe ore terial we will be able to satisfy the studets deads 35 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Exale Exale The studets would like to liit their waitig tie to a average of two iutes ad o ore tha five iutes i 9% of the cases. Is it feasible? If yes the how ay terials are required?.67 ad.5 Cosider what would have haeed if the five terials i exale 3. were located i five differet locatios o the caus, thereby eedig a searate queue for each Five searate M/M/ queues,.67/5.333, ad.5 Traffic itesity:.333/ Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 9
10 Exale 3.4 The ea tie set i the terial roo is: / /.5 r 6.67 The variace of the tie set i the terial roo is: / /.5 Var r Coare this to the ea of 4 iutes ad a variace of 479 i exale 3. > sigle queue alterative is better 37 M/M/ Queue Ifiite servers. Jobs ever wait Resose tie service tie Mea resose tie ea service tie regardless of the arrival rate > Called delay ceters Used to rereset dedicated resources, such as terials i tiesharig systes Proerties of such queues ca be easily derived fro those for M/M/ queues Also see results for M/G/ queues 39 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Exale M/M// Queue with Fiite uffers 4 Walkig tie was igored. Havig several terial roos distributed across the caus ay reduce the walkig tie cosiderably If all jobs are idetical, it is better to have just oe queue tha to have ultile queues If soe studets eed very short terial sessios ad others eed very log sessios, searate queues ay be better Siilar to the M/M/ queue but uber of buffers is fiite All arrivals, after buffers are full, are lost is greater tha or equal to ; otherwise soe servers will ever be able to oerate > M/M// queue 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao
11 3 Noveber 5 EEC686/785 4 Webig Zhao Aalysis of M/M// Queue irth-death rocess: -,...,,,,...,,,,..., 3 Noveber 5 EEC686/785 4 Webig Zhao Aalysis of M/M// Queue Probability of jobs i the syste: I ters of the traffic itesity /:,...,,,,...,,...,,,,..., 3 Noveber 5 EEC686/ Webig Zhao Aalysis of M/M// Queue Probability of zero jobs i the syste: 3 Noveber 5 EEC686/ Webig Zhao Aalysis of M/M// Queue Mea uber of jobs i the syste : Mea uber of jobs i the queue q : E q E
12 Aalysis of M/M// Queue 45 Aalysis of M/M// Queue 47 Variace ad other statistics o ad q ca be siilarly couted All arrivals occurrig whe the syste is i the state are lost. Rate of the jobs actually eterig the syste, called effective arrival rate, is: ' The differece reresets the loss rate Utilizatio of each server: usy tie er server ' T / / ' U Total tie T Probability of the full syste 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Aalysis of M/M// Queue 46 Aalysis of M/M// Queue 48 Mea resose tie usig Little s law: For a M/M// syste, loss robability is: r ' Siilarly, the ea waitig tie is: q q w ' / j j j This is Erlag s loss forula Used to coute the robability of lost hoe calls. Valid also for M/G// queues 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao
13 Exale Exale Cosider the gateway of exale 3. agai. Let us aalyze the gateway assuig it has oly two buffers. The arrival rate ad the service rate, as before, are 5 s ad 5 s, resectively. I this case: 5, 5,, ad Traffic itesity: /5/ 5.5 Mea uber of jobs i the syste: Mea uber of jobs i the queue: q Effective arrival rate i the syste: ' s 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao Exale Exale for,,, are:.5 Packet loss rate 596s Mea resose tie.5.65 is deteried by suig all robabilities: >.5.65 / Substitutig for i, we get: r.4 ' 9 Mea tie waitig i the queue q.476 w 4. ' 9 3 secods 4 secods 3 Noveber 5 EEC686/785 Webig Zhao 3 Noveber 5 EEC686/785 Webig Zhao 3
14 Exale Variace ad other statistics for the uber of jobs i the syste ca also be couted sice the colete robability ass fuctio is kow. For exale: Var Noveber 5 EEC686/785 Webig Zhao 4
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