ORSIS Conference, Jerusalem Mountains, Israel May 13, 2007

1. ORSIS Conference,Jerusalem Mountains, IsraelMay 13, 2007 Transient Fluid Solutions andQueueing Networks withInfinite Virtual Queues • Yoni Nazarathy • Gideon Weiss • University of Haifa

2. 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

3. 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

4. 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

5. AN EXAMPLE: A Push-Pull Queueing System (Weiss, Kopzon 2002,2006) Server 1 Server 2 PUSH PULL PROBABLYNOT WITH THESE POLICIES? Low variance of the departure process? PULL PUSH Require: “Inherently Unstable” “Inherently Stable” For Both Cases,Positive Recurrent Policies Exist Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

6. OUR MODEL: • MCQN+IVQ Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

7. 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

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

9. Static Fluid Formulation Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

10. is the average depletion of queue k per one unit of work on class k’. The input-output matrix (Harrison) Given, the rates assumptions , 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

11. - 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 (and much older ideas). Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

12. Rate Stable Controls Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

13. 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. • , so there is always some allocation. “Energy” Minimization • Lyapunov function: • Find allocation that reduces it as fast as possible: The Resulting Policy Choose: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

14. Rate Stability Theorem • Given a MCQN with IVQs defined with nominal production rates that are given by a feasible static allocation. The non-processor splitting, no-preemption Maximum Pressure Policy is stable for any primitive sequences that satisfy the rates assumptions in the following two senses: (1) – Rate Stability for infinite time horizon: (2) –Given a sequence for Finite time horizon, T: Where satisfies: Proof is an adaptation of Dai and Lin’s 2005, Theorem 2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

15. Work in progress…. How fast does the queue (virtual queue) sizes grow? How do simpler policies (randomized), that follow the static fluid equations compare? General Applications… Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

16. Possible Applications of the Theorem Steady State Systems Systems with Time Varying Parameters Tracking Transient Fluid Solutions of a MCQN Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

17. Transient Fluid Solutions Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

18. Server 2 Server 1 3 2 1 Example Network • Stacked Queue level representation: Attempt to minimize: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

19. Minimization of Corresponds to: Minimizing inventory costs. Minimizing the total job waiting time.(truncated to time horizon). Maximizing the total time from job completion to the time horizon. (maximizing “useful life”) Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

20. 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

21. Fluid solution • SCLP – Bellman, Anderson, Pullan, Weiss. • Piecewise linear solution. • Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss). • The Optimal Solution: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

22. Structure of the optimal solution – comparison to LBFS • Last Buffer First Server (LBFS): Improve: Don’t wait with the emptying of buffer 1 until 3 is empty… The optimal solution: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

23. Fluid TrackingPolicy Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

24. Near Optimal Control over a Finite Time Horizon • Approximation Approach: • 1) Approximate the problem using a fluid system. • 2) Solve the fluid system (SCLP). • 3) Track the fluid solution on-line (Using MCQN+IVQs). • 4) Under proper scaling, the approach is asymptotically optimal. Solution is intractable Finite Horizon Control of MCQN Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

25. 4 Time Intervals For each time interval, set a MCQN with Infinite Virtual Queues. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

26. 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. Let be an objective value for any general policy then: Using our maximum pressure based fluid tracking policy: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

28. Work in progress…. Continued…. How fast does the queue (virtual queue) sizes grow? How fast is convergence stated in the asymptotic optimality theorem??? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

30. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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