1. 14 th INFORMS Applied Probability Conference, Eindhoven July 9, 2007 Yoni Nazarathy Gideon Weiss University of Haifa Yoni Nazarathy Gideon Weiss University of Haifa Transient Fluid Solutions and Queueing Networks with Infinite Virtual Queues

2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 1 Overview:  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control

3. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 2 Multi-Class Queueing Networks (Harrison 1988, Dai 1995, … ) 12 6 54 3 Queues/Classes Routing Processes Initial Queue Levels Resources Processing Durations Resource Allocation (Scheduling) Network Dynamics

4. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 3 Overview:  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control

5. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 4 Example Network Server 1Server 2 1 2 3 Attempt to minimize:

6. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 5 Fluid formulation s.t. This is a Separated Continuous Linear Program (SCLP) Server 1Server 2 1 2 3

7. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 6 Fluid solution SCLP – Bellman, Anderson, Pullan, Weiss. Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss). The Optimal 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”

8. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 7 Overview:  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control

9. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 8 INTRODUCING: Infinite Virtual Queues Regular Queue Infinite Virtual Queue Example Realization   Nominal Production Rate Relative Queue Length

10. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 9 IVQ ’ s Make Controlled Queueing Network even more interesting … Some Resource The Network PUSH PULL To Push Or To Pull? That is the question… High Utilization of Resources High and Balanced Throughput Stable and Low Queue Sizes Low variance of the departure process What does a “good” control achieve? Fluid oriented Approach: Choose a “good” nominal production rate (α)…

11. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 10 Extend the MCQN to MCQN + IVQ 12 6 54 3 Queues/Classes Routing Processes Initial Queue Levels Processing Durations Resource Allocation (Scheduling) Network Dynamics Nominal Productio n Rates Resources

12. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 11 Rates Assumptions of the Primitive Sequences Primitive Sequences: May also define: rates assumptions:

13. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 12 The input-output matrix (Harrison) A fluid view of the outcome of one unit of work on class k’: is the average depletion of queue k per one unit of work on class k’. The input-output matrix:

14. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 13 The Static Equations A feasible static allocation is the triplet, such that: - MCQN model - Nominal Production rates for IVQs - Resource Utilization - Resource Allocation Similar to ideas from Harrison 2002

15. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 14 Lyapunov function: Find allocation that reduces it as fast as possible: Maximum Pressure Policies (Tassiulas, Stolyar, Dai & Lin) 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: 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: An allocation at time t: a feasible selection of values of At any time t, A(t) is the set of available allocations. Intuitive Meaning of the Policy “Energy” Minimization The Policy: Choose: Feasible Allocations

16. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 15 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: Rate Stability Theorem (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.

17. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 16 Overview:  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control  MCQN model  Transient Fluid Solutions  Infinite Virtual Queues  Near Optimal Finite Horizon Control

18. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 17 Back to the example network: For each time interval, set a MCQN with Infinite Virtual Queues:

19. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 18 Example realizations, N={1,10,100} seed 1 seed 2 seed 3 seed 4

20. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 19 (1) Let be an objective value for any general policy then: - Scaling: speeding up processing rates by N and setting initial conditions: Asymptotic Optimality Theorem - Queue length process of finite horizon MCQN - Value of optimal fluid solution. (2) Using the maximum pressure based fluid tracking policy:

21. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 20 How fast is the convergence that is stated in the asymptotic optimality theorem ???

22. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 21 Empirical Asymptotics N = 1 to 10 6

23. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 22 Thank You