1. STUDENTS PROBABILITY DAY Weizmann Institute of Science March 28, 2007 Yoni Nazarathy (Supervisor: Prof. Gideon Weiss) University of Haifa Yoni Nazarathy (Supervisor: Prof. Gideon Weiss) University of Haifa Queueing Networks with Infinite Virtual Queues An Example, An Application and a Fundamental Question

2. Yoni Nazarathy, University of Haifa, 2007 1 Multi-Class Queueing Networks (Harrison 1988, Dai 1995, … ) 12 6 54 3 Queues 1/4 3/4 Routes Initial Queue Levels Servers Processing Durations Resource Allocation (Scheduling) Network Dynamics

3. Yoni Nazarathy, University of Haifa, 2007 2 INTRODUCING: Infinite Virtual Queues Regular Queue Infinite Virtual Queue Example Realization   Relative Queue Length: Nominal Production Rate

4. Yoni Nazarathy, University of Haifa, 2007 3 MCQN+IVQ 12 6 54 3 Queues 1/4 3/4 Routes Initial Queue Levels Servers Processing Durations Resource Allocation (Scheduling) Network Dynamics Nominal Productio n Rates

5. Yoni Nazarathy, University of Haifa, 2007 4 An Example

6. Yoni Nazarathy, University of Haifa, 2007 5 A Push-Pull Queueing System (Weiss, Kopzon 2002,2006) Server 1Server 2 PUSH PULL PUSH Fluid Solution: or Require Full Utilization Require Rate Stability “Inherently Stable” “Inherently Unstable” Proportion of time server i allocates to “Pulling”

7. Yoni Nazarathy, University of Haifa, 2007 6 Maximum Pressure (Dai, Lin 2005) Max-Pressure is a rate stable policy (even when ρ=1). Push-Pull acts like a ρ=1 System. As Proven by Dai and Lin, Max-Pressure is rate stable. But for the Push-Pull system Max-Pressure is not Positive Recurrent: Queue on Server 1 Queue on Server 2

8. Yoni Nazarathy, University of Haifa, 2007 7 Positive Recurrent Policies Exist!!! Kopzon, Weiss 2002 Kopzon, Weiss 2006

9. Yoni Nazarathy, University of Haifa, 2007 8 An Application

10. Yoni Nazarathy, University of Haifa, 2007 9 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. 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 Weiss, Nazarathy 2007

11. Yoni Nazarathy, University of Haifa, 2007 10 Fluid formulation s.t. This is a Separated Continuous Linear Program (SCLP) Server 1Server 2 1 2 3

12. Yoni Nazarathy, University of Haifa, 2007 11 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: 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:

13. Yoni Nazarathy, University of Haifa, 2007 12 4 Time Intervals For each time interval, set a MCQN with Infinite Virtual Queues.

14. Yoni Nazarathy, University of Haifa, 2007 13 Maximum Pressure (Dai, Lin) is such a policy, even when ρ=1 Now Control the MCQN+IVQ Using a Rate Stable Policy

15. Yoni Nazarathy, University of Haifa, 2007 14 Example realizations, N={1,10,100} seed 1 seed 2 seed 3 seed 4

16. Yoni Nazarathy, University of Haifa, 2007 15 A Fundamental Question

17. Yoni Nazarathy, University of Haifa, 2007 16 Is there a characterization of MCQN+IVQs that allows: Full Utilization of all the servers that have an IVQ. Stability of all finite queues. Proportional equality among production streams. Is there a characterization of MCQN+IVQs that allows: Full Utilization of all the servers that have an IVQ. Stability of all finite queues. Proportional equality among production streams. ?

18. Yoni Nazarathy, University of Haifa, 2007 17 Thank You