EFFICIENT RARE-EVENT SIMULATION FOR MANY-SERVER LOSS SYSTEMS

Transcription

1 EFFICIENT RARE-EVENT SIMULATION FOR MANY-SERVER LOSS SYSTEMS Jose Blanchet IEOR Department Columbia University

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3 MANY-SERVER LOSS SYSTEM Loss : GI/GI/s/0 no waiting room customers are lost if all servers are busy Assume s = # of servers large Focus of this talk: Computing loss probabilities / overflow events Conditional distribution at the time of a loss Also discuss s with time varying / Markov modulated arrivals Rare-event simulation for stochastic 3

4 MANY-SERVER LOSS SYSTEM: THE MODEL Customer 4 Server 1 Server 2 Server 3 Server 4

5 MANY-SERVER LOSS SYSTEM Customer 5 Server 1 Server 2 Server 3 Server 4

6 APPLICATION 1: LONG DISTANCE LINES A local company sets up long-distance call lines Customers are the employees (can be over 5000 in big companies) Service times are the call holding times Rare-event simulation for stochastic How many call lines should be set up to guarantee a loss probability of less than, say 0.1%? 6

7 APPLICATION 2: TELECOMMUNICATION SWITCHES 7

8 APPLICATION 3: INSURANCE PORTFOLIO Rare-event simulation for stochastic 8

9 IMPORTANT FEATURES OF MANY-SERVER LOSS SYSTEM 9

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11 THE BASIC MODEL Server 1 11 Server 2 Server 3 Server 4

12 THE BASIC MODEL 12

13 MAIN GOALOFTHETALK 13

14 CRUDE MONTE CARLO SCHEME Arrival time 14 Required service time at arrival

15 CRUDE MONTE CARLO SCHEME Arrival time 15 Required service time at arrival

16 CRUDE MONTE CARLO SCHEME Arrival time 16 Required service time at arrival

17 CRUDE MONTE CARLO SCHEME Arrival time 17 Required service time at arrival

18 CRUDE MONTE CARLO SCHEME 18

19 A NUMERICAL EXAMPLE Parameters/Assumptio ns: Loss probability calculated from Erlang s loss formula Running time to simulate 1000 time units using crude Monte Carlo Approximate number of customers that can be simulated in this time Approximate time to simulate one loss event Approximate time to simulate 100 loss events Rare-event simulation for stochastic 19

20 OUR ALGORITHM 20

21 ORGANIZATION OF THE TALK 1. Introduce notions in rare-event simulation 2. Explain in detail our importance sampling scheme 3. Algorithmic efficiency 4. Generalizations e.g. renewal arrivals, Markovmodulation, time-varying Rare-event simulation for stochastic 21

22 LITERATURE REVIEW Central Limit Theorems / diffusion approximation: Iglehart (1965), Halfin and Whitt (1981), Reed (2007) Rare event analysis / large deviations: most papers are on queues with single server / several servers, e.g. Asmussen (1982), Anantharam (1988), Sadowsky (1991, 1993), Frater et al. (1989, 1990, 1991, 1994), Glasserman and Kou (1995), Dupuis et al. (2007), Lehtonen and Nyrhinen (1992), Chang et al. (1993, 1994) Many-server queues under heavy traffic: Glynn (1995), Szechtman and Glynn (2002), Ridder (2009) Rare-event simulation for stochastic 22

23 FUNDAMENTAL CHALLENGE OF RARE- EVENT SIMULATION 23

24 IMPORTANCE SAMPLING 24

25 IMPORTANCE SAMPLING 25

26 A SIMPLE (SIMPLEST) EXAMPLE 26

27 NOTES FROM THE EXAMPLE 27

28 LARGE DEVIATIONS AND IMPORTANCE SAMPLING 28

29 LARGE DEVIATIONS AND IMPORTANCE SAMPLING To find an optimal importance sampler for large deviations event Formulate Gartner-Ellis limit of the random object Decode from the limit the contributions of more elementary objects that lead to the rare event In many cases (but not all), the naturally suggested sampler is asymptotically optimal Rare-event simulation for stochastic 29

30 BACK TO OUR PROBLEM Do we know the Gartner-Ellis limit of the random object i.e. the steady-state loss distribution? Problem reformulation Rare-event simulation for stochastic 30

31 CRUDE MONTE CARLO REVISITED 31 Arrival time Require d service time at arrival

32 A REGENERATIVE VIEW 32

33 A REGENERATIVE VIEW Cycle starts 33

34 SPLITTING ALGORITHM 34

35 SPLITTING ALGORITHM Regeneration is hit. Split into two chains 35

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37 KEY QUESTIONS 37

38 A SIMPLER PROBLEM 38

39 SOLUTION TO THE SIMPLER PROBLEM 39

40 IMPORTANCE SAMPLING FOR THE SIMPLER PROBLEM Rare-event simulation for stochastic Required service time at arrival 40 Arrival time

41 IMPORTANCE SAMPLING FOR THE SIMPLER PROBLEM Rare-event simulation for stochastic Required service time at arrival 41 Arrival time

42 IMPORTANCE SAMPLING FOR THE SIMPLER PROBLEM Rare-event simulation for stochastic Required service time at arrival 42 Arrival time

43 INTUITION FOR OUR PROBLEM Idea 1: Before the first loss, the acts the same as if there are infinite number of servers Idea 2: Since time horizon for loss is not fixed, we shall randomize the time horizon (also preventing blowing up variance due to nonoptimal sample paths) Rare-event simulation for stochastic 43

44 Arrival time 44 Required service time at arrival

45 Arrival time 45 Required service time at arrival

46 Arrival time 46 Required service time at arrival

47 Arrival time 47 Required service time at arrival

48 Arrival time 48 Required service time at arrival

49 Arrival time 49 Required service time at arrival

50 Arrival time 50 Required service time at arrival

51 ASYMPTOTIC OPTIMALITY 51

52 ASYMPTOTIC OPTIMALITY 52 Arrival time Required service time at arrival

53 ASYMPTOTIC OPTIMALITY 53

54 ASYMPTOTIC OPTIMALITY 54

55 SIMPLIFICATION AND EXTENSIONS 55

56 SIMPLIFICATION AND EXTENSIONS 56