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

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1 lect09.ppt S Itroductio to Teletraffic Theory - Fall

2 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical aalysis 2

3 What is it? Simulatio is (at least from the teletraffic poit of view) a statistical method to estimate the performace (or some other importat characteristic) of the system uder cosideratio. It typically cosists of the followig four phases: Modellig of the system (real or imagiary) as a dyamic stochastic process Geeratio of the realizatios of this stochastic process ( observatios ) Such realizatios are called simulatio rus Collectio of data ( measuremets ) Statistical aalysis of the gathered data, ad drawig of the coclusios 3

4 Alterative to what? I previous lectures, we have got familiar with a alterative way to determie the performace: mathematical aalysis It icludes oly the followig two phases: Modellig of the system as a stochastic process. (I this course, we have restricted ourselves to birth-death processes.) Solvig of the model by meas of mathematical aalysis The modellig phase is commo to both of them However, the accuracy (faithfuless) of the model that these methods allow ca be very differet ulike simulatio, mathematical aalysis typically requires (heavily) restrictive assumptios to be made 4

5 Performace aalysis of a teletraffic system Real/imagiary system modellig Mathematical model (as a stochastic process) Performace aalysis validatio of the model Mathematical aalysis Simulatio 5

6 Aalysis vs. simulatio (1) Pros of aalysis Results produced rapidly (after the aalysis is made) Exact (accurate) results (for the model) Gives isight Optimizatio possible (but typically hard) Cos of aalysis Requires restrictive assumptios the resultig model is typically too simple (ot capturig all essetial features of the system uder cosideratio) performace aalysis of complicated systems impossible Eve uder these assumptios, the aalysis itself may be (extremely) hard 6

7 Aalysis vs. simulatio (2) Pros of simulatio No restrictive assumptios eeded (i priciple) performace aalysis of complicated systems possible Modellig straightforward Cos of simulatio Productio of results time-cosumig (simulatio programs beig typically processor itesive) Results iaccurate (however, they ca be made as accurate as required by icreasig the umber of simulatio rus, but this takes eve more time) Does ot ecessarily offer a geeral isight (?) Optimizatio possible oly betwee very few alteratives (parameter combiatios or cotrols) 7

8 Simulatio of a stochastic process Modellig of the system as a stochastic process This has already 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) Collectio of data Trasiet phase vs. steady state (statioarity, equilibrium) Statistical aalysis ad coclusios Poit estimators Cofidece itervals 8

9 Implemetatio Simulatio is typically implemeted as a computer program Simulatio program geerally comprises the followig phases (excludig modellig ad coclusios) Geeratio of the realizatios of the stochastic process Collectio of data Statistical aalysis of the gathered data Simulatio program ca be implemeted by 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 most rapid ad reliable (depedig o s/w), but rigid 9

10 Simulatio types What we have described above, is discrete (evet-based), dyamic (evolvig i time) ad stochastic (icludig radom compoets) simulatio This is called discrete evet simulatio This is also what we will cosider later o i this lecture Other types: cotiuous simulatio: state ad parameter spaces of the process are cotiuous; descriptio of the system typically by differetial equatios, e.g. simulatio of the trajectory of a aircraft static simulatio: time plays o role, e.g. umerical itegratio of a multidimesioal itegral by Mote Carlo method determiistic simulatio: o radom compoets, e.g. the first example above 10

11 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical aalysis 11

12 Geeratio of the realizatios of the traffic process Assume that the modellig part of simulatio has bee doe So the stochastic process describig the evolutio of the system is kow 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 properly ito accout all the (statistical) depedecies betwee these variables) Costruct a realizatio of the process (usig the geerated values) These two parts are overlappig (thus, aythig else but sequetial) Realizatios for radom variables are geerated by utilizig (pseudo) radom umber geerators The realizatio of the process is costructed from evet to evet (discrete evet simulatio) 12

13 Discrete evet simulatio (1) Idea: simulatio evolves from evet to evet If othig happes durig a iterval, we ca just skip it! Evet classes: basic evets modifyig (somehow) the state of the system 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 Iside these evet classes, there are various evet types Evet idetificatio: occurrece time (whe) ad evet type (what) 13

14 Discrete evet simulatio (2) Evets are orgaized as a evet list Evets i this list are ordered (ascedigly) by the occurrece time first: the evet occurrig ext Evets are hadled oe-by-oe (i this order) Simulatio clock tells the occurrece time of the ext evet progressig by jumps System state tells the curret state of the system 14

15 Discrete evet simulatio (3) Geeral algorithm for a sigle simulatio ru: 1 Iitializatio simulatio clock = 0 system state = give iitial value for each evet type, geerate ext evet (wheever possible) costruct the evet list from these evets 2 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 system state delete the evet from the evet list 3 Stoppig test if positive, the stop the simulatio ru; otherwise retur to 2 15

16 Example (1) Task: Simulate the M/M/1 queue (more precisely: the evolutio of the queue legth process) from time 0 to time T assumig that the queue is empty at time 0 ad omittig ay data collectio System state (at time t) = queue legth X t iitial value: X 0 = 0 Basic evets: customer arrivals customer departures Extra evet: stoppig of the simulatio ru at time T 16

17 Example (2) Iitializatio: iitialize the system state: X 0 0 geerate the time till the first arrival from the Exp() distributio Hadlig of a arrival evet (occurrig at some time t): if X t 0, the geerate the time till the ext departure from the Exp() distributio geerate the time till the ext arrival from the Exp() distributio update the system state: X t X t 1 Hadlig of a departure evet (occurig at some time t): update the system state: X t X t 1 if X t 0, the geerate the time till the ext departure from the Exp() distributio Stoppig test: t T 17

18 Example (3) geeratio of the evets arrival ad departure times aika queue legth time T 18

19 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical aalysis 19

20 Geeratio of realizatios of radom variables Based o radom umber geerators First step: geeratio of idepedet U(0,1) distributed radom variables Step from the U(0,1) distributio to the desired distributio: rescalig ( U(a,b)) discretizatio ( Beroulli(p), Bi(,p), Poisso(a), Geom(p)) iverse trasform ( Exp()) other trasforms ( N(0,1) N(, 2 )) acceptace-rejectio method (for ay cotiuous radom variable defied i a fiite iterval whose desity fuctio is bouded) two idepedet U(0,1) distributed radom variables eeded 20

21 Radom umber geerator Radom umber geerator is a algorithm geeratig (pseudo) radom itegers Z i i some iterval 0,1,, m 1 The resultig umbers Z i are called (pseudo) radom umbers The sequece geerated is always periodic (goal: this period to be as log as possible) Strictly speakig, the umbers geerated are ot radom at all, i the sese of beig upredictable (thus: pseudo) I practice, however, the umbers appear to be IID with uiform distributio, provided that the radom umber geerator is desiged carefully Validitio of a radom umber geerator ca be based o empirical (statistical) ad theoretical tests: uiformity of the geerated empirical distributio idepedece of the geerated radom umbers 21

22 Radom umber geerator types Liear cogruetial geerator most simple ext radom umber is based o just the curret oe: Z i1 f(z i ) period at most m Multiplicative cogruetial geerator a special case of the first type Additive cogruetial geerators Shufflig, etc. 22

23 Liear cogruetial geerator (LCG) Liear cogruetial geerator (LCG) uses the followig algorithm to geerate radom umbers belogig to {0,1,, m1}: Z i 1 ( 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 very poor By a right choice of parameters, it is possible to achieve the full period m e.g. m 2 b, c odd, a 4k +1 23

24 Multiplicative cogruetial geerator (MCG) Multiplicative cogruetial geerator (MCG) uses the followig algorithm to geerate radom umbers belogig to {0,1,, m1}: Here a ad m are fixed o-egative itegers (a m) I additio, the startig value (seed) Z 0 m should be specified Remarks: Zi 1 ( az MCG is clearly 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 2 b, the the maximum period is 2 b2 However, for m prime, period m1 is possible (by a proper choice of a) PMMLCG = prime modulus multiplicative LCG e.g. m ad a 16,807 (or 630,360,016) i ) mod m 24

25 U(0,1) distributio Let Z deote a (pseudo) radom umber belogig to {0,1,, m1} The (approximately) U m Z U(0,1) 25

26 U(a,b) distributio Let U U(0,1) The X a ( b a) U U( a, b) This is called the rescalig method 26

27 Discretizatio method Let U U(0,1) Assume that Y is a discrete radom variable with value set S {0,1,,} or S {0,1,2, } Deote: F(x) P{Y x} The X mi{ x S F( x) U} Y This is called the discretizatio method (cf. iverse trasform method) Example: Beroulli(p) distributio 0, if U 1 p X 1, if U 1 p Beroulli( p) 27

28 Iverse trasform method Let U U(0,1) Assume that Y is a cotiuous radom variable Assume further that F(x) P{Y x} is strictly icreasig Let F 1 (y) deote the iverse of the fuctio F(x) The X F 1 ( U ) Y This is called the iverse trasform method Proof: Sice P{U u} u for all u, we have P{ X x} P{ F 1 ( U ) x} P{ U F( x)} F( x) 28

29 Exp() distributio Let U U(0,1) The also 1U U(0,1) Let Y Exp() F(x) P{Y x} 1e x is strictly icreasig The iverse trasform is F 1 (y) = (1/ log(1y) Thus, by the iverse trasform method, X 1 ( 1 F 1U ) log( U ) Exp( ) 29

30 N(0,1) distributio Let U 1 U(0,1) ad U 2 U(0,1) be idepedet The, by so called Box-Müller method, the followig two (trasformed) radom variables are idepedet ad idetically distributed obeyig the N(0,1) distributio: X1 2log( U1) si(2u 2) N(0,1) X 2 2log( U1) cos(2u 2) N(0,1) 30

31 N(, 2 ) distributio Let X N(0,1) The, by the rescalig method, Y X N(, 2 ) 31

32 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical aalysis 32

33 Collectio of data Our startig poit was that simulatio is eeded to estimate the value, say, of some performace parameter This parameter may be related to the trasiet or the steady-state behaviour of the system. Examples 1 & 2 (trasiet phase characteristics) average waitig time of the first k customers i a M/M/1 queue assumig that the system is empty i the begiig average queue legth i a M/M/1 queue durig the iterval [0,T] assumig that the system is empty i the begiig Example 3 (steady-state characteristics) the average waitig time i a M/M/1 queue i equilibrium Each simulatio ru yields oe sample, say X, describig somehow the parameter uder cosideratio For drawig statistically reliable coclusios, multiple samples, X 1,,X, are eeded (preferably IID) 33

34 Trasiet phase characteristics (1) Example 1: Cosider e.g. the average waitig time of the first k customers i a M/M/1 queue assumig that the system is empty i the begiig Each simulatio ru ca be stopped whe the kth customer eters the service The sample X based o a sigle simulatio ru is i this case: X k 1 k i1 Here W i = waitig time of the ith customer i this simulatio ru Multiple IID samples, X 1,,X, ca be geerated by the method of idepedet replicatios: multiple idepedet simulatio rus (usig idepedet radom umbers) W i 34

35 Trasiet phase characteristics (2) Example 2: Cosider e.g. the average queue legth i a M/M/1 queue durig the iterval [0,T] assumig that the system is empty i the begiig Each simulatio ru ca be stopped at time T (that is: simulatio clock = T) The sample X based o a sigle simulatio ru is i this case: X T 1 T Q( t) 0 Here Q(t) = queue legth at time t i this simulatio ru Note that this itegral is easy to calculate, sice Q(t) is piecewise costat Multiple IID samples, X 1,,X, ca agai be geerated by the method of idepedet replicatios dt 35

36 Steady-state characteristics (1) Collectio of data i a sigle simulatio ru ca typically (but ot always) be doe oly after a warm-up phase (hidig the trasiet characteristics) resultig i overhead bias i estimatio eed for determiatio of a sufficietly log warm-up phase Multiple samples, X 1,,X, may be geerated by the followig three methods: idepedet replicatios batch meas regeerative method The first two methods require a warm-up phase, but the last oe (that is: regeerative method) does ot 36

37 Steady-state characteristics (2) Method of idepedet replicatios: multiple idepedet simulatio rus (usig idepedet radom umbers) each simulatio ru icludes the warm-up phase iefficiecy samples IID accuracy Method of batch meas: oe (very) log simulatio ru divided (artificially) ito oe warm-up phase ad equal legth periods (each of which represets a sigle simulatio ru) oly oe warm-up phase efficiecy samples oly approximately IID iaccuracy, choice of 37

38 Steady-state characteristics (3) Regeerative method: possible oly if the traffic process is a regeerative stochastic process G/G/1 queue (ad, thus also M/M/1 ad M/G/1) is regeerative (a ew cycle starts wheever a ew customer arrives i a empty system) all Markov processes are regeerative (a ew cycle starts wheever the process eters some fixed state) oe (very) log simulatio ru divided ito IID cycles (each of which represets a sigle simulatio ru) o warm-up phase efficiecy samples IID accuracy the problem is that the cycle legths, which are radom variables, ca be (much too) log iefficiecy 38

39 Cotets Itroductio Geeratio of realizatios of the traffic process Geeratio of realizatios of radom variables Collectio of data Statistical aalysis 39

40 Parameter estimatio As metioed, our startig poit was that simulatio is eeded to estimate the value, say, of some performace parameter Each simulatio ru yields a (radom) sample, say X i, describig somehow the parameter uder cosideratio Sample X i is called ubiased if E[X i ] = Deote the sample average by 1 X : 1 X Assumig that the samples X i are IID with mea ad variace 2, the sample average is ubiased ad cosistet estimator of, sice E[ X D 2 [ X ] 1 ] 1 i1 2 E[ X i1 D i ] 2 [ X i i ] 1 i 2 0 (as ) 40

41 Example Cosider the average waitig time of the first 25 customers i a M/M/1 queue with load 0.9 assumig that the system is empty i the begiig Theoretical value: 2.12 Samples X i from te simulatio rus ( 10): 1.05, 6.44, 2.65, 0.80, 1.51, 0.55, 2.28, 2.82, 0.41, 1.31 Sample average (poit estimate for ): X 1 1 X ( ) 1.98 i 1 i 10 41

42 Cofidece iterval (1) Defiitio: Iterval (X y, X y) is called the cofidece iterval for the sample average at cofidece level 1 if P{ X y} 1 Idea: with probability 1 the parameter belogs to this iterval Assume the that samples X i, i 1,,, are IID with ukow mea but kow variace 2 By the Cetral Limit Theorem (see Lecture 5, Slide 48), for large, Z : X / N(0,1) 42

43 Cofidece iterval(2) Let z p deote the p-fractile of the N(0,1) distributio That is: P{Z z p } p, where Z N(0,1) Example: for 5% (1 95%) z 1(/2) z Propositio: The cofidece iterval for the sample average at cofidece level 1 is Proof: By defiitio, we have to show that X z 1 2 P{ X z 1 2 } 1 43

44 44 9. Simulatio 1 } { y X P y y y y y y y X y y X z y z x x x Z P x P P / 2 / / / / / / / / / / 1 ) ( )] ( 1 ) ( [ 1 )) ( (1 ) ( }] { ) : ( [ 1 ) ( ) ( 1 } { 1 } {

45 45 9. Simulatio Cofidece iterval (3) I geeral, however, the variace 2 is ukow (i additio to the mea ) It ca be estimated by the sample variace: It is possible to prove that the sample variace is a ubiased ad cosistet estimator of 2 : ) ( ) ( : i i i i X X X X S ) ( 0 ] [ ] [ S D S E

46 Cofidece iterval (4) Assume that samples X i are IID obeyig the N(, 2 ) distributio with ukow mea ad ukow variace 2 The it is possible to show that X T : S / Studet( 1) Let t 1,p deote the p-fractile of the Studet(1) distributio That is: P{T t 1,p } p, where T Studet(1) Example 1: 10 ad 5%, t 1,1(/2) t 9, Example 2: 100 ad 5%, t 1,1(/2) t 99, Thus, the cofidece iterval for the sample average at cofidece level 1 is ow as follows: X t 1,1 2 S 46

47 Example (cotiued) Cosider the average waitig time of the first 25 customers i a M/M/1 queue with load 0.9 assumig that the system is empty i the begiig X Theoretical value: 2.12 Samples X i from te simulatio rus ( 10): 1.05, 6.44, 2.65, 0.80, 1.51, 0.55, 2.28, 2.82, 0.41, 1.31 Sample average = 1.98 ad the square root of the sample variace: S 2 1 (( ) ( ) ) So, the cofidece iterval (that is: iterval estimate for ) at cofidece level 95% is t 1,1 2 S (0.71,3.25) 47

48 Observatios Simulatio results become more accurate (that is: the iterval estimate for becomes arrower) whe the umber of simulatio rus is icreased, or the variace 2 of each sample is reduced Give the desired accuracy for the simulatio results, the umber of required simulatio rus ca be determied dyamically 48

49 Literature I. Mitrai (1982) Simulatio techiques for discrete evet systems Cambridge Uiversity Press, Cambridge A.M. Law ad W. D. Kelto (1982, 1991) Simulatio modelig ad aalysis McGraw-Hill, New York 49

50 THE END 50

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