1. Numerical Methods for Stochastic Networks Peter W. Glynn Institute for Computational and Mathematical Engineering Management Science and Engineering Stanford University Based on joint work with Jose Blanchet, Henry Lam, Denis Saure, and Assaf Zeevi Presented at Stochastic Networks Conference, Cambridge, UK March 23, 2010

2. Stochastic models: Descriptive Prescriptive Predictive

3. Today’s Talk: an LP based algorithm for computing the stationary distribution of RBM (Saure / Zeevi) a Lyapunov bound for stationary expectations (Zeevi) Rare-event simulation for many-server queues (Blanchet / Lam)

4. Computing Steady-State Distributions for Markov Chains

5. One Approach

7. An LP Alternative

9. Where does constraint come from? We assume that we can obtain a computable bound on i.e. This will come from Lyapunov bounds (later).

10. Application to RBM Reflected Brownian Motion (RBM): For some stochastic models, the LP algorithm is particularly natural and powerful

17. Theorem: This algorithm converges as, in the sense that as.

18. Numerical Results Smoothed marginal distribution estimates for the two-dimensional diffusion. The dotted line is computed via Monte Carlo simulation, and the solid line represents the algorithm estimates based on n = 50 and m = 4, incorporating smoothness constraints.

19. - valued Markov process with cadlag paths We say if there exists such that is a –local martingale for each

20. Computing Bounds on Stationary Expectations Main Theorem: Suppose is non–negative and satisfies Then,

21. Diffusion Upper Bound Suppose is and satisfies for, where and where. Then,

22. Many-Server Loss Systems

23. Many-Server Asymptotic Regime

24. Simplify our model (temporarily): using slotted time eliminate Markov modulation discrete service time distributions with finite support Consider equilibrium fraction of customers lost in the network.

25. Key Idea: Many server loss systems behave identically to infinite-server systems up to the time of the first loss.

26. Step 1:

27. Step 2:

28. Step 3:

29. Step 3:

30. Step 3:

31. Crude Monte Carlo : 3.7 days I.S. : A few seconds

32. Network Extension Estimate loss at a particular station If the most likely path to overflow a given station does not involve upstream stations, previous algorithm if efficient If an upstream station does hit its capacity constraint, we have “constrained Poisson statistics” that need to be sampled

33. Questions?