14 th INFORMS Applied Probability Conference, Eindhoven July 9, 2007

1. 14th INFORMS Applied Probability Conference,EindhovenJuly 9, 2007 Transient Fluid Solutions andQueueing Networks withInfinite Virtual Queues • Yoni Nazarathy • Gideon Weiss • University of Haifa

2. Overview: • MCQN model • Transient Fluid Solutions • Infinite Virtual Queues • Near Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

3. 6 1 2 3 5 4 Multi-Class Queueing Networks (Harrison 1988, Dai 1995,…) Queues/Classes Initial Queue Levels Routing Processes Resources Processing Durations Network Dynamics Resource Allocation (Scheduling) Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

4. Overview: • MCQN model • Transient Fluid Solutions • Infinite Virtual Queues • Near Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

5. Server 2 Server 1 3 2 1 Example Network Attempt to minimize: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

6. Fluid formulation Server 2 Server 1 3 2 s.t. 1 This is a Separated Continuous Linear Program (SCLP) Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

7. Fluid solution • SCLP – Bellman, Anderson, Pullan, Weiss. • Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss). • The Optimal Solution: The solution is piece-wise linear with a finite number of “time intervals” Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

8. Overview: • MCQN model • Transient Fluid Solutions • Infinite Virtual Queues • Near Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

9. m m INTRODUCING: Infinite Virtual Queues NominalProductionRate Regular Queue Infinite Virtual Queue Relative Queue Length Example Realization Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

10. IVQ’s Make Controlled Queueing Network even more interesting… What does a “good” control achieve? The Network Some Resource Stable and Low Queue Sizes PUSH High Utilization of Resources PULL High and Balanced Throughput Low variance of the departure process To Push Or To Pull? That is the question… Fluid oriented Approach:Choose a “good” nominal production rate (α)… Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

11. 6 1 2 3 5 4 Extend the MCQN to MCQN + IVQ Queues/Classes Initial Queue Levels Routing Processes Resources Processing Durations Network Dynamics Resource Allocation (Scheduling) NominalProductionRates Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

12. Rates Assumptions of the Primitive Sequences Primitive Sequences: May also define: rates assumptions: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

13. is the average depletion of queue k per one unit of work on class k’. The input-output matrix (Harrison) A fluid view of the outcome of one unit of work on class k’: The input-output matrix: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

14. - MCQN model - Nominal Production rates for IVQs - Resource Utilization - Resource Allocation A feasible static allocationis the triplet , such that: The Static Equations Similarto ideas from Harrison 2002 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

15. Maximum Pressure Policies (Tassiulas, Stolyar, Dai & Lin) Intuitive Meaning of the Policy • Reminder: is the average depletion of queue k per one unit of work on class k’. • Treating Z and T as fluid and assuming continuity: Feasible Allocations • An allocation at time t: a feasible selection of values of • At any time t, A(t) is the set of available allocations. “Energy” Minimization • Lyapunov function: • Find allocation that reduces it as fast as possible: The Policy: Choose: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

16. Rate Stability Theorem • MCQN + IVQ, Non-Processor Splitting, No-Preemption • Nominal production rates given by a feasible static allocation. • Primitive Sequences satisfy rates assumptions. • Using Maximum Pressure, the network is stable as follows: (1) – Rate Stability for infinite time horizon: (2) –Given a sequence : Where satisfies: Proof is an adaptation of Dai and Lin’s 2005, Theorem 2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

17. Overview: • MCQN model • Transient Fluid Solutions • Infinite Virtual Queues • Near Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

18. Back to the example network: For each time interval, set a MCQN with Infinite Virtual Queues: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

19. Example realizations, N={1,10,100} • seed 1 seed 2 seed 3 seed 4 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

20. Asymptotic Optimality Theorem - Queue length process of finite horizon MCQN - Scaling: speeding up processing rates by N and setting initial conditions: - Value of optimal fluid solution. (1) Let be an objective value for any general policy then: (2) Using the maximum pressure based fluid tracking policy: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

21. How fast is the convergence that is stated in the asymptotic optimality theorem ??? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

22. Empirical Asymptotics N = 1 to 106 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

23. ThankYou Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007