AN INTRODUCTION TO DISCRETE-EVENT SIMULATION

Transcription

1 AN INTRODUCTION TO DISCRETE-EVENT SIMULATION Peter W. Glynn 1 Peter J. Haas 2 1 Dept. of Management Science and Engineering Stanford University 2 IBM Almaden Research Center San Jose, CA

2 CAVEAT: WE ARE NOT BIOLOGISTS OR CHEMISTS!

3 Outline Overview of discrete-event simulation Basic models for discrete-event stochastic systems Generalized semi-markov processes (GSMPs) Continuous-time Markov chains (CTMCs) Stochastic Petri nets (SPNs) Some important techniques for simulation Efficiency improvement Sensitivity estimation Parallel simulation 3 Glynn and Haas

4 Part I Overview 4 Glynn and Haas

5 Discrete-Event Stochastic Systems Finite or countably infinite set of states System changes state when events occur Stochastic state changes At strictly increasing random times Underlying stochastic process { X (t): t 0 } X (t) = state of system at time t (a random variable) Ex: X i (t) = # molecules of ith species at time t 5 Glynn and Haas

6 Simulation: A Tool for Studying DESS Characteristics Characteristics often of the form α = E [Y ] Y = f ( X (t) ), e.g., Y = I (X (t) = x) Y = (1/t) t 0 f ( X (u) ) du Y = min{ t > 0: X (t) A } 6 Glynn and Haas

7 Other Characteristics Nonlinear functions of means: g(α 1, α 2,..., α k ) Apply Delta Method, jackknifing, bootstrapping, etc. Steady-state characteristics Time-average limits: 1 t α = lim f ( X (u) ) du t t 0 Steady-state means: α = E [f (X )], where X (t) X Apply regenerative method, batch means, STS methods, etc. Key issues of stability and validity For more details: buy us a beer 7 Glynn and Haas

8 Part II Models for Discrete-Event Systems 8 Glynn and Haas

9 Generalized Semi-Markov Processes Classical model for discrete-event stochastic systems Building blocks S = set of states (finite or countably infinite) E = set of events (finite) E(s) = active events in state s p(s ; s, e ) = state-transition probability One clock per event: records remaining time until occurrence 9 Glynn and Haas

10 Clocks (Event Scheduling) Active events compete to trigger state transition The clock that runs down to 0 first is the winner Can have simultaneous event occurrence: p(s ; s, E ) At a state transition s e s : three kinds of events New events: Clock for e set according to F (x; s, e, s, e ) Old events: Clocks continue to run down Cancelled events: Clock readings are discarded Clocks run down at state-dependent speeds r(s, e) 10 Glynn and Haas

11 The GSMP Process The continuous-time process: { X (t) : t 0 } X (t) = the state at time t A very complicated process Defined in terms of Markov chain { (S n, C n ) : n 0 } System observed after the nth state transition Sn = the state Cn = (C n,1,..., C n,m ) = the clock-reading vector Chain defined via GSMP building blocks 11 Glynn and Haas

12 Definition of the GSMP 12 Glynn and Haas

13 Generation of the GSSMC { (S n, C n ): n 0 } 1. [Initialization] Set n = 0. Generate state S 0 and clock readings C 0,i for e i E(S 0 ); set C 0,i = 1 for e i E(S 0 ). 2. Determine holding time t (S n, C n ) and trigger event e n. 3. Generate new state S n+1 according to p( ; S n, e n). 4. Set clock-reading C n+1,i for each new event e i according to F ( ; S n+1, e i, S n, e n). 5. Set clock-reading C n+1,i for each old event e i as C n+1,i = C n,i t (S n, C n )r(s n, e i ). 6. Set clock-reading C n+1,i equal to 1 for each cancelled event e i. 7. Set n n + 1 and go to Step 2. Can compute GSMP { X (t): t 0 } from GSSMC 13 Glynn and Haas

14 Implementation Considerations for Large-Scale GSMPs Use event lists (e.g., heaps) to determine e O(1) computation of e ( ) O log m update time, where m = # of active events 14 Glynn and Haas

15 Continuous-Time Markov Chains Time-homogeneous Markov property: P { X (t + h) = s X (t) = s, X (u) = x(u) for 0 u < t } = P { X (t + h) = s X (t) = s } = p h (s, s ) CTMC specified by intensity matrix Q Q(s, s p h (s, s ) ) = lim, s s h 0 h Q(s, s) = s s Q(s, s ) Define state intensity q(s) = Q(s, s) = s s Q(s, s ) 15 Glynn and Haas

16 Simulating a CTMC (Gillespie Algorithm) Thm (path structure of a CTMC): Sequence of visited states is a DTMC Given states, holding times are indep. and exponential 1. [Initialization] Set n = 0. Generate state S Generate holding time t (S n ) as Exponential ( q(s n ) ). 3. Generate next state S n+1 ( S n ) according to Q(S n, )/q(s n ). 4. Set n n + 1 and go to Step 2. A CTMC can be viewed as a GSMP with one exponential clock! 16 Glynn and Haas

17 CTMCs and GSMPs Theorem: Let { X (t) : t 0 } be a GSMP such that F (x; s, e, s, e ) 1 e λ(e )x for each e Then { X (t) : t 0 } is a CTMC with intensity matrix Q(s, s ) = e E(s) λ(e)r(s, e)p(s ; s, e), s s Under conditions of theorem, GSMP simulation algorithm can simplify to CTMC algorithm! OR can simulate CTMC using one exponential clock per event GSMPs with unit exponential clocks and time-varying speeds Glasserman (1991, Ch. 6): hazard-rate construction Anderson (2007): application to chemical reaction systems 17 Glynn and Haas

18 Stochastic Petri Nets 18 Glynn and Haas

19 Stochastic Petri Nets D = finite set of places 18 Glynn and Haas

20 Stochastic Petri Nets D = finite set of places E = finite set of transitions (timed and immediate) 18 Glynn and Haas

21 Stochastic Petri Nets D = finite set of places E = finite set of transitions (timed and immediate) 18 Glynn and Haas

22 Stochastic Petri Nets D = finite set of places E = finite set of transitions (timed and immediate) marking = assignment of token counts to places 18 Glynn and Haas

23 Stochastic Petri Nets d 1 d 2 s = (2, 1, 1) d 3 D = finite set of places E = finite set of transitions (timed and immediate) marking = assignment of token counts to places 18 Glynn and Haas

24 Transition Firing The marking changes when an enabled transition fires 19 Glynn and Haas

25 Transition Firing The marking changes when an enabled transition fires 19 Glynn and Haas

26 Transition Firing The marking changes when an enabled transition fires Removes 1 token per place from random subset of input places Deposits 1 token per place in random subset of output places 19 Glynn and Haas

27 Transition Firing The marking changes when an enabled transition fires Removes 1 token per place from random subset of input places Deposits 1 token per place in random subset of output places 19 Glynn and Haas

28 Transition Firing The marking changes when an enabled transition fires Removes 1 token per place from random subset of input places Deposits 1 token per place in random subset of output places 19 Glynn and Haas

29 Transition Firing The marking changes when an enabled transition fires Removes 1 token per place from random subset of input places Deposits 1 token per place in random subset of output places 19 Glynn and Haas

30 Transition Firing p(s ; s, e ) The marking changes when an enabled transition fires Removes 1 token per place from random subset of input places Deposits 1 token per place in random subset of output places 19 Glynn and Haas

31 SPNs and GSMPs SPN marking = GSMP state SPN transition firing = GSMP event occurrence Differences SPN has specific form of state SPN has restrictions on allowable state transitions SPN has immediate transitions SPN and GSMP have same modeling power 20 Glynn and Haas

32 Bottom-Up and Top-Down Modeling 21 Glynn and Haas

33 Bottom-Up and Top-Down Modeling 21 Glynn and Haas

34 Bottom-Up and Top-Down Modeling 21 Glynn and Haas

35 Bottom-Up and Top-Down Modeling 21 Glynn and Haas

36 Bottom-Up and Top-Down Modeling 21 Glynn and Haas

37 Bottom-Up and Top-Down Modeling 21 Glynn and Haas

38 Other Modeling Features 22 Glynn and Haas

39 Other Modeling Features Concurrency: 22 Glynn and Haas

40 Other Modeling Features Concurrency: Synchronization: 22 Glynn and Haas

41 Other Modeling Features Concurrency: Synchronization: 22 Glynn and Haas

42 Other Modeling Features Concurrency: Synchronization: Precedence: 22 Glynn and Haas

43 Other Modeling Features Concurrency: Synchronization: Precedence: 22 Glynn and Haas

44 Other Modeling Features Concurrency: Synchronization: Precedence: 22 Glynn and Haas

45 Other Modeling Features Concurrency: Synchronization: Precedence: Priority: 22 Glynn and Haas

46 Implementation Advantages for Large-Scale SPNs Updating the state is often simpler in an SPN than a GSMP Efficient techniques for event scheduling [Chiola91] Encode transitions potentially affected by firing of ei Parallel simulation of subnets E.g., Adaptive Time Warp [Ferscha & Richter PNPM97] Guardedly optimistic Slows down local firings based on history of rollbacks 23 Glynn and Haas

47 For Further Details: Asmussen and Glynn: Stochastic Simulation: Algorithms and Analysis, Springer, 2007 MS-level lecture notes: Haas: Stochastic Petri Nets: Modelling, Stability, Simulation, Springer, Glynn and Haas