Rare-Event Simulation for Many-Server Queues

1. Rare-Event Simulation for Many-Server Queues Henry Lam Department of Mathematics and Statistics, Boston University Joint work with J. Blanchet, X. Chen and P. W. Glynn

2. Many-Server Loss System Customer Server 1 Server 2 Server 3 Server 4 Server Efficient simulation for many-server queues

3. Many-Server Loss System Customer Server 1 Server 2 Server 3 Server 4 • Steady state distribution of loss? Server Efficient simulation for many-server queues

4. Many-Server Loss System: Quality-Driven Regime Server 1 Customers arrive according to a renewal process with rate i.e. interarrival times are i.i.d. with mean Server 2 Server 3 Service times are i.i.d. Server 4 • Traffic intensity • possess exponential moments • has moments up to infinite order Server Efficient simulation for many-server queues

5. Logarithmic Asymptotic • Applications in communications, call centers… • Many-server loss system (Ridder 2009, Blanchet, Glynn & L. 2009, Blanchet & L. 2012): where is the first time of loss • Similar for delay of many-server queue in the same regime • Can be further extended to non-homogeneous functional of system status -> application in insurance modeling etc. (Blanchet & L. 2011) • Steady-state phenomena (Blanchet & L. 2012): • Rate function depends on the initial configuration of the queue • Goal: construct asymptotically optimal importance sampler Efficient simulation for many-server queues

6. Model Dynamics Required service time at arrival Arrival time Efficient simulation for many-server queues

7. Model Dynamics Required service time at arrival Arrival time Efficient simulation for many-server queues

8. Model Dynamics Required service time at arrival Arrival time Efficient simulation for many-server queues

9. Model Dynamics Required service time at arrival Arrival time Efficient simulation for many-server queues

10. Model Dynamics Required service time at arrival Arrival time Efficient simulation for many-server queues

11. Markov Representation Required service time at arrival Arrival time Efficient simulation for many-server queues

12. Markov Representation Required service time at arrival Markov state: customers at time with residual service time > Arrival time Efficient simulation for many-server queues

13. Markov Representation Required service time at arrival Markov state: customers at time with residual service time > Arrival time Efficient simulation for many-server queues

14. Markov Representation Required service time at arrival Markov state: customers at time with residual service time > Arrival time Efficient simulation for many-server queues

15. Markov Representation Required service time at arrival Markov state: customers at time with residual service time > Arrival time Efficient simulation for many-server queues

16. A Numerical Example Parameters/Assumptions: , , Poisson arrival with rate Service time Erlang’sloss formula= Time to simulate 1000 time units Number of arrivals in this time Time to obtain 1 loss The next algorithm takes 10 seconds to simulate one loss event. Efficient simulation for many-server queues

17. Theoretical Performance Theorem (Blanchet and L. ‘12 & Blanchet, Glynn and L. ‘09): 1. Under current assumptions, the loss probability satisfies where , and is the large deviations rate of , i.e. where is the number of customers in an infinite-server system. 2. The algorithm we propose is asymptotically optimal for where and and is the rate function starting from any initial configuration . Efficient simulation for many-server queues

18. Steady-State Loss Probability • Suppose is a recurrent set of the system • Kac'sformula: Notations: • = expectation with initial state in steady-state conditional on being in • = number of loss before reaching again • = time units to reach again Efficient simulation for many-server queues

19. What is a good choice of set ? • is a - ball around the fluid limit of , given by where i.e. decays slower than the standard deviation exhibited by the diffusion limit of Efficient simulation for many-server queues

20. Brief Comments on Importance Sampling and Rare-Event Simulation • Want to estimate where is a rare event • Importance sampling (IS) identity: Given a suitable probability measure , • So IS estimator is Efficient simulation for many-server queues

21. Brief Comments on Importance Sampling and Rare-Event Simulation • If then IS gives zero variance: • Moral: Good IS mimics the conditional distribution given the rare event! • Use large deviations, but carefully (counter-examples in Glasserman and Kou ‘97) • Asymptotic optimality: Relative error does not grow exponentially Efficient simulation for many-server queues

22. What is a good choice of set ? • Visited infinitely often • Large deviation behavior is unique starting from every point in : where is any point in and • Possess good property of return time: for any Efficient simulation for many-server queues

23. Construction of Importance Sampler • For simplicity let us first concentrate on Poisson arrivals • Intuition: where is the first time to experience a loss Efficient simulation for many-server queues

24. Construction of Importance Sampler • Observation 1: is the same for -server and infinite-server system • Observation 2: Remarkably handy Efficient simulation for many-server queues

25. Importance Sampling Procedure STEP 1: Sample a random time over INDEPENDENT of the system Required service time at arrival Arrival time Efficient simulation for many-server queues

26. Importance Sampling Procedure STEP 2: Sample the path given Required service time at arrival Arrival time Efficient simulation for many-server queues

27. Importance Sampling Procedure STEP 2: Sample the path given Use Poisson point process representation Required service time at arrival Arrival time Efficient simulation for many-server queues

28. Importance Sampling Procedure STEP 2: Sample the path given First sample given Required service time at arrival Arrival time Efficient simulation for many-server queues

29. Importance Sampling Procedure STEP 2: Sample the path given First sample given Given, the points in triangle are distributed independently according to intensity Required service time at arrival Arrival time Efficient simulation for many-server queues

30. Importance Sampling Procedure STEP 2: Sample the path given First sample given Given, the points in triangle are distributed independently according to intensity Required service time at arrival Arrival time Efficient simulation for many-server queues

31. Importance Sampling Procedure STEP 2: Sample the path given The rest of points outside the triangle follow non-homogeneous spatial Poisson process Required service time at arrival Arrival time Efficient simulation for many-server queues

32. Importance Sampling Procedure STEP 3: Identify and continue the process with the original measure Required service time at arrival Arrival time Efficient simulation for many-server queues

33. Importance Sampling Procedure STEP 3: Identify and continue the process with the original measure Required service time at arrival Arrival time Efficient simulation for many-server queues

34. Importance Sampling Procedure STEP 3: Identify and continue the process with the original measure Required service time at arrival Arrival time Efficient simulation for many-server queues

35. Importance Sampling Procedure STEP 3: Identify and continue the process with the original measure Required service time at arrival Until time Arrival time Efficient simulation for many-server queues

36. Importance Sampling Procedure STEP 4: Output Required service time at arrival Until time Arrival time Efficient simulation for many-server queues

37. Change-of-Measure The measure in this IS scheme is given by where is an independent r. v. Efficient simulation for many-server queues

38. General Renewal Arrivals • Use exponential tilting to induce • Represent • Gartner-Ellis limit (Glynn ‘95) • Suggest state-dependent exponential tilting of each interarrival and service times according to an optimal (Szechtman and Glynn ‘02): Efficient simulation for many-server queues

39. Specifications for Importance Sampling Procedure • Distribution of random horizon: where • Likelihood ratio: where Efficient simulation for many-server queues

40. Proof of Efficiency • Goal: The second moment of the estimator satisfies • Main arguments: • The second moment is bounded approximately as Efficient simulation for many-server queues

41. Proof of Efficiency • If , the area of is • Poisson arrivals: use thinning property • General arrivals: Conditioned on arrivals times, probability of each customer lying on the area is independent and Efficient simulation for many-server queues

42. Lower Bound • Construct the optimal sample path • Use the more general Gartner-Ellis limit where • Sample path large deviations: possesses a good rate function : Efficient simulation for many-server queues

43. Logarithmic Estimate of Return Time • Bound the return time for many-server system in terms of the infinite-server queue where max of all residual service times at • Bounded service time: • consider blocks of where the service time is bounded in • Return time bounded by a geometric random variable independent of • Unbounded service time: • need to estimate the residual service time from previous block • Use Borell’s inequality to ensure significant probability of the Gaussian diffusion limit to stay in central region Efficient simulation for many-server queues

44. Other Extensions • Insurance portfolio problem: same algorithm with exponential tilting • Time-inhomogeneous arrivals: same algorithm • Markov modulation (on finite state-space): Sample Markov state ahead, then apply the same algorithm Efficient simulation for many-server queues