11. Simulation. Contents. Alternative to what? What is simulation?

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1 Cotets Itroductio Geeratio of traffic process realizatios Geeratio of radom variable realizatios Collectio of data Statistical aalsis Lect.ppt S Itroductio to Teletraffic Theor Sprig 005 What is simulatio? Alterative to what? Simulatio is at least from the teletraffic poit of view a statistical method to estimate the performace or some other importat characteristic of the sstem uder cosideratio. It tpicall cosists of the followig four phases: Modellig of the sstem real or imagiar as a damic stochastic process Geeratio of the realizatios of this stochastic process observatios Such realizatios are called simulatio rus Collectio of data measuremets Statistical aalsis of the gathered data, ad drawig coclusios I previous lectures, we have looked at a alterative wa to determie the performace: mathematical aalsis We cosidered the followig two phases: Modellig of the sstem as a stochastic process. I this course, we have restricted ourselves to birth-death processes. Solvig of the model b meas of mathematical aalsis The modellig phase is commo to both of them However, the accurac faithfuless of the model that these methods allow ca be ver differet ulike simulatio, mathematical aalsis tpicall requires heavil restrictive assumptios to be made 3 4

2 Performace aalsis of a teletraffic sstem Aalsis vs. simulatio Realimagiar sstem Mathematical model as a stochastic process Performace aalsis Mathematical aalsis modellig validatio of the model Simulatio Pros of aalsis Results produced rapidl after the aalsis is made Exact accurate results for the model Gives isight Optimizatio possible but tpicall hard Cos of aalsis Requires restrictive assumptios the resultig model is tpicall too simple e.g. ol statioar behavior performace aalsis of complicated sstems impossible Eve uder these assumptios, the aalsis itself ma be extremel hard 5 6 Aalsis vs. simulatio Steps i simulatig a stochastic process Pros of simulatio No restrictive assumptios eeded i priciple performace aalsis of complicated sstems possible Modellig straightforward Cos of simulatio Productio of results time-cosumig simulatio programs beig tpicall processor itesive Results iaccurate however, the ca be made as accurate as required b icreasig the umber of simulatio rus, but this takes eve more time Does ot ecessaril offer a geeral isight Optimizatio possible ol betwee ver few alteratives parameter combiatios or cotrols 7 Modellig of the sstem as a stochastic process This has alread bee discussed i this course. I the sequel, we will take the model that is: the stochastic process for grated. I additio, we will restrict ourselves to simple teletraffic models. Geeratio of the realizatios of this stochastic process Geeratio of radom umbers Costructio of the realizatio of the process from evet to evet discrete evet simulatio Ofte this step is uderstood as THE simulatio, however this is ot geerall the case Collectio of data Trasiet phase vs. stead state statioarit, equilibrium Statistical aalsis ad coclusios Poit estimators Cofidece itervals 8

3 Implemetatio Other simulatio tpes Simulatio is tpicall implemeted as a computer program Simulatio program geerall comprises the followig phases excludig modellig ad coclusios Geeratio of the realizatios of the stochastic process Collectio of data Statistical aalsis of the gathered data Simulatio program ca be implemeted b a geeral-purpose programmig laguage e.g. C or C++ most flexible, but tedious ad proe to programmig errors utilizig simulatio-specific program libraries e.g. CNCL utilizig simulatio-specific software e.g. OPNET, BONeS, NS i part based o p-libraries most rapid ad reliable depedig o the sw, but rigid What we have described above, is a discrete evet simulatio the simulatio is discrete evet-based, damic evolvig i time ad stochastic icludig radom compoets i.e. how to simulate the time evolvemet of the mathematical model of the sstem uder cosideratio, whe the aim is to gather iformatio o the sstem behavior We cosider ol this tpe of simulatio i this lecture Other tpes: cotiuous simulatio: state ad parameter spaces of the process are cotiuous; descriptio of the sstem tpicall b differetial equatios, e.g. simulatio of the trajector of a aircraft static simulatio: time plas o role as there is o process that produces the evets, e.g. umerical itegratio of a multi-dimesioal itegral b Mote Carlo method determiistic simulatio: o radom compoets, e.g. the first example above 9 0 Cotets Geeratio of traffic process realizatios Itroductio Geeratio of traffic process realizatios Geeratio of radom variable realizatios Collectio of data Statistical aalsis Assume that we have modelled as a stochastic process the evolutio of the sstem Next step is to geerate realizatios of this process. For this, we have to: Geerate a realizatio value for all the radom variables affectig the evolutio of the process takig properl ito accout all the statistical depedecies betwee these variables Costruct a realizatio of the process usig the geerated values These two parts are overlappig, the are ot dome separatel Realizatios for radom variables are geerated b utilizig pseudo radom umber geerators The realizatio of the process is costructed from evet to evet discrete evet simulatio

4 Discrete evet simulatio Discrete evet simulatio Idea: simulatio evolves from evet to evet If othig happes durig a iterval, we ca just skip it! basic evets modif somehow the state of the sstem e.g. arrivals ad departures of customers i a simple teletraffic model extra evets related to the data collectio icludig the evet for stoppig the simulatio ru or collectig data Evet idetificatio: occurrece time whe evet is hadled ad evet tpe what ad how evet is hadled Evets are orgaized as a evet list Evets i this list are ordered ascedigl b the occurrece time first: the evet occurrig ext Evets are hadled oe-b-oe i this order while at the same time geeratig ew evets to occur later Whe the evet has bee processed, it is removed from the list Simulatio clock tells the occurrece time of the ext evet progressig b jumps Sstem state tells the curret state of the sstem 3 4 Discrete evet simulatio 3 Example Geeral algorithm for a sigle simulatio ru: Iitializatio simulatio clock 0 sstem state give iitial value for each evet tpe, geerate ext evet wheever possible costruct the evet list from these evets Evet hadlig simulatio clock occurrece time of the ext evet hadle the evet icludig geeratio of ew evets ad their additio to the evet list updatig of the sstem state delete the evet from the evet list 3 Stoppig test if positive, the stop the simulatio ru; otherwise retur to Task: Simulate the MM queue more precisel: the evolutio of the queue legth process from time 0 to time T assumig that the queue is empt at time 0 ad omittig a data collectio Sstem state at time t queue legth t iitial value: 0 0 Basic evets: customer arrivals customer departures Extra evet: stoppig of the simulatio ru at time T Note. No collectio of data i this example 5 6

5 Example Example 3 Iitializatio: iitialize the sstem state: 0 0 geerate the time till the first arrival from the Expλ distributio Hadlig of a arrival evet occurrig at some time t: update the sstem state: t t + if t, the geerate the time t + S till the ext departure, where S is from the Expµ distributio geerate the time t + I till the ext arrival, where I is from the Expλ distributio Hadlig of a departure evet occurig at some time t: update the sstem state: t t if t > 0, the geerate the time t + S till the ext departure, where S is from the Expµ distributio Stoppig test: t > T geeratio of the evets arrival ad departure times queue legth time time 0 T 7 8 Cotets Geeratio of radom variable realizatios Itroductio Geeratio of traffic process realizatios Geeratio of radom variable realizatios Collectio of data Statistical aalsis Based o pseudo radom umber geerators First step: geeratio of idepedet uiforml distributed betwee 0 ad, i.e. U0, distributed, radom variables usig radom umber geerators Step from the U0, distributio to the desired distributio: rescalig Ua,b discretizatio Beroullip, Bi,p, Poissoa, Geomp iverse trasform Expλ other trasforms N0, Nµ, acceptace-rejectio method for a cotiuous radom variable defied i a fiite iterval whose desit fuctio is bouded two idepedet U0, distributed radom variables eeded 9 0

6 Radom umber geerator Radom umber geerator tpes Radom umber geerator is a algorithm geeratig pseudo radom itegers Z i i some iterval 0,,, m The sequece geerated is alwas periodic goal: this period should be as log as possible Strictl speakig, the umbers geerated are ot radom at all, but totall predictable thus: pseudo I practice, however, if the geerator is well desiged, the umbers appear to be IID with uiform distributio iside the set {0,,,m} Validitio of a radom umber geerator ca be based o empirical statistical ad theoretical tests: uiformit of the geerated empirical distributio idepedece of the geerated radom umbers o correlatio Liear cogruetial geerator most simple ext radom umber is based o just the curret oe: Z i+ fz i period at most m Multiplicative cogruetial geerator a special case of the first tpe Other: Additive cogruetial geerators Shufflig, etc. Liear cogruetial geerator LCG Multiplicative cogruetial geerator MCG Liear cogruetial geerator LCG uses the followig algorithm to geerate radom umbers belogig to {0,,, m}: Zi + azi + c mod m Here a, c ad m are fixed o-egative itegers a < m, c < m I additio, the startig value seed Z 0 < m should be specified Remarks: Parameters a, c ad m should be chose with care, otherwise the result ca be ver poor B a right choice of parameters, it is possible to achieve the full period m e.g. m b, c odd, a 4k + b ofte 48 3 Multiplicative cogruetial geerator MCG uses the followig algorithm to geerate radom umbers belogig to {0,,, m}: Zi + azi mod m Here a ad m are fixed o-egative itegers a < m I additio, the startig value seed Z 0 < m should be specified Remarks: MCG is clearl a special case of LCG: c 0 Parameters a ad m should still be chose with care I this case, it is ot possible to achieve the full period m e.g. if m b, the the maximum period is b However, for m prime, period m is possible b a proper choice of a PMMLCG prime modulus multiplicative LCG e.g. m 3 ad a 6,807 or 630,360,06 4

7 U0, distributio Ua,b distributio Let Z deote a pseudo radom umber belogig to {0,,, m} The approximatel Let U U0, The U m Z U0, a + b a U U a, b This is called the rescalig method 5 6 Discretizatio method Iverse trasform method Let U U0, Assume that Y is a discrete radom variable with value set S {0,,,} or S {0,,, } Deote: Fx P{Y x}, the Let U U0, Assume that Y is a cotiuous radom variable Assume further that Fx P{Y x} is strictl icreasig Let F deote the iverse of the fuctio Fx, the mi{ x S F x U} Y F U Y This is called the discretizatio method a special case of the iverse trasform method Example: Beroullip distributio 0,, if U p Beroulli p if U > p 7 This is called the iverse trasform method Proof: Sice P{U u} u for all u 0,, we have P { x} P{ F U x} P{ U F x} F x 8

8 Expλ distributio N0, distributio Let U U0, The also U U0, Let Y Expλ Fx P{Y x} e λx is strictl icreasig The iverse trasform is F λ log Thus, b the iverse trasform method, F U log U Exp λ λ Let U U0, ad U U0, be idepedet The, b so called Box-Müller method, the followig two trasformed radom variables are idepedet ad ideticall distributed obeig the N0, distributio: log U siπu N0, log U cosπu N0, 9 30 Nµ, distributio Cotets Let N0, The, b the rescalig method, Y µ + N µ, Itroductio Geeratio of traffic process realizatios Geeratio of radom variable realizatios Collectio of data Statistical aalsis 3 3

9 Collectio of data Trasiet phase characteristics Our startig poit was that simulatio is eeded to estimate the value, sa α, of some performace parameter This parameter ma be related to the trasiet or the stead-state behaviour of the sstem. Examples & trasiet phase characteristics average waitig time of the first k customers i a MM queue assumig that the sstem is empt i the begiig average queue legth i a MM queue durig the iterval [0,T] assumig that the sstem is empt i the begiig Example 3 stead-state characteristics the average waitig time i a MM queue i equilibrium Example : Cosider e.g. the average waitig time of the first k customers i a MM queue assumig that the sstem is empt i the begiig Each simulatio ru ca be stopped whe the kth customer eters the service The sample based o a sigle simulatio ru is i this case: k k i Here W i waitig time of the ith customer i this simulatio ru Each simulatio ru ields oe sample, sa, describig somehow the Multiple IID samples, parameter uder cosideratio,,, ca be geerated b the method of idepedet replicatios: For drawig statisticall reliable coclusios, multiple samples,,,, are eeded preferabl IID multiple idepedet simulatio rus usig idepedet radom umbers W i Trasiet phase characteristics Stead-state characteristics Example : Cosider e.g. the average queue legth i a MM queue durig the iterval [0,T] assumig that the sstem is empt i the begiig Each simulatio ru ca be stopped at time T that is: simulatio clock T The sample based o a sigle simulatio ru is i this case: T T Q t dt 0 Here Qt queue legth at time t i this simulatio ru Note that this itegral is eas to calculate, sice Qt is piecewise costat Multiple IID samples,,,, ca agai be geerated b the method of idepedet replicatios 35 Collectio of data i a sigle simulatio ru is i priciple similar to that of trasiet phase simulatios Collectio of data i a sigle simulatio ru ca tpicall but ot alwas be doe ol after a warm-up phase hidig the trasiet characteristics resultig i overhead extra simulatio bias i estimatio eed for determiatio of a sufficietl log warm-up phase Multiple samples,,,, ma be geerated b the followig three methods: idepedet replicatios batch meas 36

10 Stead-state characteristics Cotets Method of idepedet replicatios: multiple idepedet simulatio rus of the same sstem usig idepedet radom umbers each simulatio ru icludes the warm-up phase iefficiec samples IID accurac Method of batch meas: oe ver log simulatio ru divided artificiall ito oe warm-up phase ad equal legth periods each of which represets a sigle simulatio ru ol oe warm-up phase efficiec samples ol approximatel IID iaccurac, choice of, the larger the better Itroductio Geeratio of traffic process realizatios Geeratio of radom variable realizatios Collectio of data Statistical aalsis Parameter estimatio Example As metioed, our startig poit was that simulatio is eeded to estimate the value, sa α, of some performace parameter Each simulatio ru ields a radom sample, sa i, describig somehow the parameter uder cosideratio Sample i is called ubiased if E[ i ] α Assumig that the samples i are IID with mea α ad variace The the sample average : i i is ubiased ad cosistet estimator of α, sice E[ ] i E[ i ] α D [ ] [ ] i D i 0 as Cosider the average waitig time of the first 5 customers i a MM queue with load ρ 0.9 assumig that the sstem is empt i the begiig Theoretical value: α. o-trivial Samples i from te simulatio rus 0:.05, 6.44,.65, 0.80,.5, 0.55,.8,.8, 0.4,.3 Sample average poit estimate for α: i i K

11 4 Cofidece iterval Defiitio: Iterval, + is called the cofidece iterval for the sample average at cofidece level if Idea: with probabilit,the parameter α belogs to this iterval Assume the that samples i, i,,, are IID with ukow mea α but kow variace B the Cetral Limit Theorem see Lecture 5, Slide 48, for large, N0, : Z α α } { P 4 Cofidece iterval Let z p deote the p-fractile of the N0, distributio That is: P{Z z p } p, where Z N0, Example: for 5% 95% z z Propositio: The cofidece iterval for the sample average at cofidece level is Proof: B defiitio, we have to show that z ± α } { z P 43 α } { P z z x x x Z P x P P α α Φ Φ Φ Φ Φ Φ Φ Φ ] [ }] { : [ } { } { 44 Cofidece iterval 3 I geeral, however, the variace is ukow i additio to the mea α It ca be estimated b the sample variace: It is possible to prove that the sample variace is a ubiased ad cosistet estimator of : : i i i i S 0 ] [ ] [ S D S E

12 Cofidece iterval 4 Example cotiued Assume that samples i are IID obeig the Nα, distributio with ukow mea α ad ukow variace The it is possible to show that T : α S Studet Let t,p deote the p-fractile of the Studet distributio That is: P{T t,p } p, where T Studet Example : 0 ad 5%, t, t 9, Example : 00 ad 5%, t, t 99, Thus, the cof. iterval for the sample average at cof. level is S ± t, Cosider the average waitig time of the first 5 customers i a MM queue with load ρ 0.9 assumig that the sstem is empt i the begiig Theoretical value: α. Samples i from te simulatio rus 0:.05, 6.44,.65, 0.80,.5, 0.55,.8,.8, 0.4,.3 Sample average.98 ad the square root of the sample variace: S K So, the cofidece iterval that is: iterval estimate for α at cofidece level 95% is S ± t ±.98 ±.7, 0 0.7, Observatios Literature Simulatio results become more accurate that is: the iterval estimate for α becomes arrower whe the umber of simulatio rus is icreased, or the variace of each sample is reduced b ruig loger idividual simulataio rus variace reductio methods Give the desired accurac for the simulatio results, the umber of required simulatio rus ca be determied damicall I. Mitrai 98 Simulatio techiques for discrete evet sstems Cambridge Uiversit Press, Cambridge A.M. Law ad W. D. Kelto 98, 99 Simulatio modelig ad aalsis McGraw-Hill, New York 47 48

9. Simulation lect09.ppt S Introduction to Teletraffic Theory - Fall

9. Simulation lect09.ppt S Introduction to Teletraffic Theory - Fall lect09.ppt S-38.145 - Itroductio to Teletraffic Theory - Fall 2001 1 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical