Simulation of Discrete Event Systems

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Simulatio of Discrete Evet Systems Uit 13 Evet Schedulig Scheme ad Output Aalysis Fall Witer 2017/2018 Prof. Dr.-Ig. Dipl.-Wirt.-Ig. Sve Tackeberg Beedikt Adrew Latos M.

1 Simulatio of Discrete Evet Systems Uit 13 Evet Schedulig Scheme ad Output Aalysis Fall Witer 2017/2018 Prof. Dr.-Ig. Dipl.-Wirt.-Ig. Sve Tackeberg Beedikt Adrew Latos M.Sc.RWTH Chair ad Istitute of Idustrial Egieerig ad Ergoomics RWTH Aache Uiversity Bergdriesch Aache phoe:

2 Cotets 1. Evet schedulig scheme - Approach - Fuctioal model - Process-orieted simulatio scheme 2. Output aalysis - Termiatig simulatios - No-termiatig simulatios 13-2

3 Focus of lecture ad exercise model static dyamic time-varyig time-ivariat liear oliear cotiuous states time-drive discrete states Focus of lecture ad exercise evet-drive determiistic stochastic discrete-time cotiuous-time 13-3

4 1. Evet Schedulig Scheme 1. Evet Schedulig Scheme 13-4

5 Itroductio If real-world problems are trasferred to algebraic structures, these models are ofte too complicated to obtai aalytical results Desig, develop ad deploy computer-supported simulatio to compare alterative system desigs Computer-supported simulatio models ca be cosidered as electroic couterpart of empirical laboratory experimets 13-5

6 Approach (I) Remember Def.: A Stochastic Timed Automato is a six-tuple: (, ε, (x), p, p 0, V), where: ε coutable state space, coutable evet set, (x) set of feasible or eabled evets, defied for all x with (x) ε, p(x ; x, e ) state trasitio probability defied for all x, x ad e ε, ad such that p(x ; x, e ) = 0 for all e (x), p 0 (x) probability mass fuctio P(X 0 = x), x, w.r.t the iitial state X 0, V stochastic timig structure. The automato geerates a stochastic state sequece {X 0, X 1, } through a trasitio mechaism (based o observatios X = x, E = e ): X = x with probability p(x ; x, e ) ad it is drive by a stochastic evet sequece {E 1,E 2, } geerated through E Y with the stochastic clock values Y i, i ε arg mi. i( x) i 13-6

7 Approach (II) The formal cocept of a automato (, ε, (x), p, p 0, V) ca be trasferred ito a computer based simulatio model. 1 Evet schedulig scheme o a digital computer is as follows Time variable t = 0 is iitialized Iitial system state x 0 is defied by samplig the iitial probability mass fuctio p 0 The clock value of every feasible evet is set to a evet lifetime 1. Note: I order to sample this distributio the computer may geerate a radom umber with a uiform probability desity fuctio over the iterval [0; 1] ad the evaluate i which of the bis (discrete itervals) the computed value lies i. Radomly a value betwee 0.0 ad 1.0 is determied e.g.: ത ത The lifetime is sampled from a predefied cotiuous probability distributio (e.g. expoetial) usig the radom umber geerator of the computer. 13-7

8 Approach (III) 2 The set of feasible evets (x) is determied Valuatio of evet i: - If i (x) is a feasible evet, we associate with it a clock value, which represets the amout of time required util evet i occurs. Deotatio of clock values for evet i by y i ad their sampled lifetimes by v i. At t = 0 we set y i = v i for all i (x) All clock values y i where i (x) are cosidered The triggerig evet e is the evet with the smallest clock value: e arg mi y * i mi i( x) y i ( x) Update of the state of the system accordig to the state trasitio mechaism The ew state will be x with probability p(x ; x, e ) The value of x is determied by a simulated radom experimet o the basis of the coditioal distributio of the probability mass. Calculatig the iterevet time based o the amout of time spet at state x: y i 13-8

9 Approach (IV) 6 Updatig the system time by settig: t = t + y* 7 8 Updatig the clock values for all feasible evets i the ew state x (from 4 ) Two cases have to be cosidered: - If a evet i (x), i e remais feasible i the ew state, the remaiig time util its occurrece is simply give by: y i = y i - y* - The secod case applies to e itself if e (x ) ad to all other evets which were ot feasible i x, but become feasible i x. Lifetime values are defied by the radom umber geerator. If there are additioal evets to be processed the computer goes back to step 2 ad uses state x istead of x 0. Istead of time values y i ad the correspodig feasible evets i (x), the most computer-based simulatio systems use a simplified scheduled evet list (SEL): SEL = {(e k, t k )}; k = 1,..., m L ; m L <= m; m L : Number of feasible evets i curret state m: Number of evets i the set t k =t k-1 + v i : ext evet schedulig The SEL is always ordered o a smallest-scheduled-time-first basis. The first evet e 1 o the list is always the triggerig evet. 13-9

10 Fuctioal model of evet schedulig scheme i Computer simulatio (I) Iitialize Settig the state x to its iitial value x 0 Settig the simulatio time t to zero Defiitio of the seed values for the radom umber geerator ad specifies the probability distributios Program segmet: Idetificatio triggerig evet Update the stored system state Radom umber geerator System state variables x x Update state p(x ; x, e 1 ) Geerates pseudo radom umbers accordig to predefied distributio fuctios SEL e 1 t 1 e 2 t Data structure of simulatio time t Update time t = t 1 x t SEL: Ordered list of scheduled evets (smallest-scheduledtime-first) Delete ifeasible (e k, t k ) New evet lifetime v k Add ew feasible (e k, t + v k ) ad reorder t Program segmet: Idetificatio ext evet Icremetig the stored simulatio time 13-10

11 Fuctioal model of evet schedulig scheme i Computer simulatio (II) Simulatio procedure: Radom umber geerator System state variables x x Update state p(x ; x, e 1 ) x Delete ifeasible (e k, t k ) New evet lifetime v k Iitialize SEL e 1 t 1 e 2 t Add ew feasible (e k, t + v k ) ad reorder Data structure of simulatio time t t Update time t = t 1 t 1. Remove the first etry (e 1, t 1 ) from the list of scheduled evets SEL 2. Update the simulatio time t by advacig it to the ew evet time t 1 3. Update the state accordig to the state trasitio mechaism p(x ; x, e 1 ) 4. Delete from SEL ay etries correspodig to ifeasible evets i the ew state, that is, delete all (e k, t k ) SEL with such that e k (x ) 5. Add to SEL ay feasible evet which is ot already scheduled (possibly icludig the triggerig evet e 1 removed i Step 1). Scheduled evet time for i is give by t + v i, where t was set i step 2 ad v i is a lifetime from the radom umber geerator 6. Reorder the updated SEL based o a smallest-scheduled-time-first scheme

12 The process-orieted simulatio scheme The itroduced evet schedulig scheme is ofte hidde i moder simulatio tools ad istead graphical editors are used to desig, verify, ad validate the system processes Process-orieted simulatio scheme. Thik of etities such as workpieces as udergoig a process as they flow through the DES. Process: Sequece of evets i terms of activities separated by time itervals The behavior of the DES is described through processes, oe process for each type of system s etity of iterest

13 Example of a process-orieted simulatio scheme M / M / 3 Subject or object occur (Evet a) Subject or object is part of the set which wait for processig (Queue) Service (Server) Subject or object is fully processed (Evet d) Sigle type of etity i this system (e.g. a workpiece) Sigle type of resource (the server) Process: The workpiece arrives The workpiece eters the queue The workpiece requests service from the server; If the server is idle, the etity seizes that resource, If the server is busy, it remais i the queue util the server becomes idle If the server hadles the workpiece, it remais i service for some period of time correspodig to the service time Whe the service is complete, it releases the server It leaves the system 13-13

14 Compoets of the process-orieted simulatio scheme Etities Attributes Objects of the simulatio system to be processed Each type of etity is characterized by a particular process (e.g. workpieces i a maufacturig system) Iformatio to characterize a particular idividual etity of ay type A uique record to each etity that cosist of etity s attributes is attached (e.g. for a workpiece i a maufacturig system: attributes are arrival time, its type etc.) Process fuctios Resources Queues Descriptio of the processed trasformatios for a etity Logical fuctios ca be classified: - spotaeous without time cosumptio - certai time cosumptio of the fuctio Objects providig service to etities Time delays experieced by a etity: - waitig for a particular resource - receivig service (e.g. machies) Sets of etities with some commo characteristics, e.g. waitig for the use of a particular resource. If a etity is curretly ot i progress it must be i a queue or i a magazie

15 2. Output Aalysis 2. Output Aalysis 13-15

16 Approach Computer-based simulatio models geerate data output through repeated simulatio rus. Termiatig simulatio: The simulatio rus termiate after a fiite umber of time steps (fiite time horizo) The output ca directly be aalyzed with the help of: - descriptive statistical methods (mea, sample variace etc.) - iferetial statistics (tests of sigificace etc.) No-Termiatig simulatio: The model simulates a system i steady state ad is therefore by desig o-termiatig Compute statioary probability distributios of the covered state variables We cosider the performace variables geeratig the output data as radom variables. The radom performace variables X 1,..., X idetically distributed are idepedet from each other ad are Note: I the followig we assume that radom variables are idepedetly draw from idetical probability distributios (iid sequece) with mea ad variace

17 Poit estimatio (I) Oly a search of the total populatio leads to a complete descriptio of the distributio of a radom variable X i a give populatio. A poit estimate describes a method of determiig a sigle value to estimate a ukow parameter The ukow mea value μ of a metric radom variable X of a total populatio is estimated based o a radom sample size. The arithmetic average of the samplig elemets represets the estimated value μ for the ukow average value. x 1, x 2,, x Θ = 1 x i Estimated value of a poit estimatio of a variable is a radom variable which ca have differet values Θ Θ Θ = X 1,, X True value withi the populatio Estimated value of the parameter Estimatio fuctio 13-17

18 Poit estimatio (II) Ubiasedess of the estimates A estimator is ubiased if the expected value correspods to the true value of the parameter to be estimated: E( ˆ ) Variace describes the deviatio from the mea value. The empirically determied variace of the sample s 2 is used for the estimated value for the ukow variace of the basic populatio σ 2. biased estimator The empirically determied variace of the sample s 2 is calculated as follows: Bias is described by the factor: Deviatio of the sample s 2 = 1 x i xҧ 2 1 σ 2 = 1 1 i=1-1: umber of degrees of freedom Due to a correctio with its reciprocal: x i xҧ

19 Poit estimatio (III) The sample variace is a ubiased poit estimator of the variace of the sample mea, because it satisfies: Var( ˆ ) ˆ 2 S 1 ˆ X i ( 1) i1 2 Example of developmet of the probability desity fuctio of ˆ : f p ( ˆ ) is growig is growig further True value withi the populatio ˆ Estimated value of the parameter For all differet umbers of, the mea (ceter of gravity) of f p is. However, the variace is decreasig as more data are collected. For goig to ifiity the probability mass is cetered aroud like a Dirac impulse

20 Iterval estimatio (I) Iterval estimatio is a systematic way to defie a iterval cotaiig withi which we ca fid the true value with a predefied cofidece level (1-α). The variable is the probability that the true values lies outside the cofidece iterval. Based o the Cetral Limit Theorem, a stadardized radom variable is defied: Z ˆ ˆ E ˆ Var 2 ˆ / pdf Z ( x) ( x) as where ( x) represets the stadard ormal distributio with the pdf : 1 ( x) 2 e x 2 /2 Θ Θ σ 2 True value withi the populatio Estimated value of the parameter Deviatio of the sample Cetral Limit Theorem: For the probability desity fuctio (pdf) of the stadardized radom variable Z the followig holds: 13-20

21 Iterval estimatio (II) Due to the fact that we have a stadard ormal distributio we are able to defie symmetric iterval limits -z /2 ad z /2, so that area uder the Gaussia bell curve equals the cofidece level 1-. ( x) 1 For large we have: P z Z z / 2 / 2 1 x z /2 z /2 ˆ 2 2 P ˆ ˆ z / 2 z / 2 1 P z / 2 / z / 2 / 1 2 / Clearly, the sample mea, the variace, ad the sample size are the idepedet parameters to estimate the iterval limits i a iterval estimatio

22 Iterval estimatio (III) e.g. distributio of productio time The variace of the distributio of the most real-world problems is ukow! I that case, we have to cosider the sample variace: P ˆ z Sˆ / ˆ z Sˆ / /2 /2 e.g. simulated productio time e.g. simulated productio time e.g. simulatio rus If the output data X 1,..., X, are give ad is large eough The equatio above leads to a cosistet estimatio of the iterval of the expected value. How large should be so that it is sufficietly large for the Cetral Limit Theorem to hold Solutio: If the sequece X 1,..., X, is ormally distributed tha T ˆ P t1, /2 T t 1, /2 1 with T S / P ˆ t Sˆ / ˆ t Sˆ / , /2 1, /2 2 is based o t -1 distributio The first parameter of the t -1 -distributio is the corrected sample size (-1) idicatig the degrees of freedom (df) of X 1,..., X. The secod parameter is the error level which is divided ito upper ad lower limits /2 of the cofidece iterval

23 Iterval estimatio (IV) The probability desity fuctio of t-distributio is described here algebraic graphical Studet p f x df 1 / 2 1 / /2 x 1 1 e.g. experimetal df = 2 observatios df = 10 df = 100 I egieerig scieces, values of = 0.05 or = 0.01 are typical

24 Output aalysis of termiatig simulatios (1/2) Termiatig simulatios have well-defied termiatio criteria. Because the values are computed o the basis of a fuctioal system model with associatig evets: We caot expect that the sequece of radom variables X 1,..., X M is idepedet ad idetically distributed! By the use of simulatio models: we are i geeral ot iterested i sigle data poits X 1,..., X M we are iterested i aggregated performace measures L = f(x 1,..., X M ) Clearly, we ca repeat a simulatio with the same iitial coditios ad termiatig evets times, ad collect data L 1, L 2,, L. These data are a idepedet sequece as log as each simulatio ru is performed idepedetly from each other (e.g. differet seeds of the radom umber geerator) If the sample size is sufficietly large, the Cetral Limit Theorem ca be used to estimate the mea ad variace

25 Output aalysis of termiatig simulatios (2/2) A commo performace measure is the mea of the distributio characterizig the idepedet sequece {L 1, L 2,, L }. Let this mea be ad let the variace be 2. The estimates are as follows: ˆ L Sˆ j ˆ 2 Lj 1 j1 j1 P ˆ t Sˆ / ˆ t Sˆ / , /2 1, /

26 Example of output aalysis of termiatig simulatios (I) Task A compay wats to assess the assembly time of 200 egies Assembly steps of oe egie are 100 A stochastic-timed automato model is developed Procedure Step 1 Step 2 Simulatio of the executio time of 100 assembly steps (x i, i = 1,,100) As a aggregated performace measure the accumulated assembly time is used: L j 100 x i1 Based o the aggregated performace measure the mea assembly time of the lot is computed: ˆ 200 i j1 Cofidece level for a iterval estimatio is defied as: 1% L j Step 3 Iverse t-distributio with df = -1 = ad /2 = iterval limits: t 2.60 t , ,

27 Example of output aalysis of termiatig simulatios (II) The system simulatio computes the followig mea ad sample variace: ˆ [ s] S [ s ] Fially, we have the iterval estimate: P ˆ t Sˆ / 200 ˆ t Sˆ / , , [ s] 12026[ s] 2 2 PP ˆ 200 t ˆ ˆ ˆ 199,0.005 (2,60 S / Θ t199, (2,60 S / P 9864[ s] 12026[ s] 0.99 Therefore, the probability of fidig the mea assembly time of the lot i the iterval [9864 s; s] is 99%

28 Output aalysis of o-termiatig simulatios Ufortuately, i o-termiatig simulatios there are o model-related stoppig criteria ad therefore the data output as realizatios of the radom variables X 1,..., X grows over all limits. I this case we are usually iterested i estimatig parameters of a statioary distributio P(X = x), where X is a state variable of iterest i steady state. There is o guaratee that this distributio exists, but it is ofte the case that P(X k <= x) P(X <= x) as k. I the followig we assume that the statioary distributio exists. We try to obtai performace measures, which are represeted by parameters of the distributio such as the mea. A simple approach to deal with a o-termiatig simulatio is to select a appropriate widow over the data startig by step r ad edig by step m: (X r,..., X m ). For istace, the modeler ca elimiate the effect of the trasiet part of the system s behavior ad cocetrate o the steady state. This approach called a iitial data deletio or warmig up of the simulatio: 0 r m sample size Warmup iterval where data is igored Data collectio ad used for parameter estimatio 13-28

29 Regeerative simulatio (I) The warmup approach ofte is ot sufficiet for a reliable parameter estimatio. A alterative approach is referred to as a regeerative simulatio. The basic idea is that a stochastic process may be characterized by radom poits i time whe it regeerates itself ad becomes idepedet of its past history. Let R 1, R 2,... be process regeeratio poits i time. The, [R j ; R j+1 ), j = 1,2,... is called a regeerative iterval. If these itervals exist, the part of the stochastic process defied over such iterval is idepedet from the past defied over aother iterval. I the regeerative itervals the values of the cumulative performace measure L j are acquired, where N j deotes the umber of values i the j-th iterval. Due to the radom time poits of the regeerative itervals, N j ca be also cosidered as a radom variable. Because of the regeeratio properties the followig coditios hold: L1, L2,... is a iid sequece with expectatio E L N, N,... is a iid sequece with expectatio E N 1 2 However, L j i geeral is ot idepedet of N j! 13-29

30 Regeerative simulatio (II) A basic property of stochastic processes is that the mea defied as before for the statioary distributio of {X 1, X 2,...} is give by: lim E X k k E L E N If we set the total umber of data obtaied over regeerative itervals The we have M( ) X 1 j j1 j1 M( ) 1 j1 L N j j M ( ) N j1 j Now we ca come up with a poit estimate of usig data collected over regeerative itervals: 1 Lj Lj ˆ j1 j1 1 N j N j j1 j

31 Regeerative simulatio (III) For the poit estimate ca be show that it is a strogly cosistet estimator of : as ˆ However, the estimator is ot ubiased. Due to the legth limitatio of this itroductory course the correspodig iterval estimatio ca ot be derived. The iterested reader ca fid it i Law, A.M. ud Kelto, W.D. (1991): Simulatio Modelig ad Aalysis. New York: McGraw-Hill

32 Example of regeerative simulatio (I) A airlie has developed a G/G/1 queueig model for the customer service at their check-i desks. Iterarrival time: Give as empirical frequecy distributios time G Service time: Give as empirical frequecy distributios time G G / G / 1 Subject or object occur (Evet a) Subject or object is part of the set which wait for processig (Queue) Service (Server) Subject or object is fully processed (Evet d) Note: We assume the system is stable (arrival rate smaller tha service rate). Objective: Estimate the mea time the customer speds at the check-i desk at steady state, deoted by system time S. For this system the crucial observatio is that every arrivig customer fidig the system empty defies a regeeratio poit. The behavior of the system over the busy period that follows is idepedet of the behavior i ay other busy period

33 Example of regeerative simulatio (II) Every arrivig customer fidig the system empty defies a regeeratio poit. 1. Regeerative iterval: N 1 = 3 customers served 2. Regeerative iterval: N 2 = 2 customers served M 1 = 3 M 2 = 5 System time S i ( i = 1, 2,...): System time of the i-th customer i a simulatio ru j-th busy period: Starts with the arrival of the (M j-1 + 1) th customer ad cosists of N j = M j M j-1 customers (M 0 = 0) Specific busy period: Cumulative system time experieced by all N i customers: Estimator of the mea customer system time: busy periods are observed: Sˆ L j j1 j1 M j j1 1 im L N j j S i

34 Refereces CASSANDRAS, C.,G.; LAFORTUNE, S. (2008): Itroductio to Discrete Evet Systems. 2 d editio. Bosto (MA): Spriger Sciece+Busiess Media. Baks, J.B. (Ed.) (1998): Hadbook of Simulatio. New York (NY): Joh Wiley & Sos

35 Questios? Ope Questios??? 13-35

Simulation of Discrete Event Systems

Simulation of Discrete Event Systems Simulatio of Discrete Evet Systems Uit 9 Queueig Models Fall Witer 2014/2015 Uiv.-Prof. Dr.-Ig. Dipl.-Wirt.-Ig. Christopher M. Schlick Chair ad Istitute of Idustrial Egieerig ad Ergoomics RWTH Aache Uiversity