Transient Fluid Solutions and Queueing Networks with Infinite

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

Overview: ØMCQN model ØTransient Fluid Solutions ØInfinite Virtual Queues ØNear Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 1

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

Overview: ØMCQN model ØTransient Fluid Solutions ØInfinite Virtual Queues ØNear Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 3

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

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

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 sol cewis utio fini e line n is of “ te n a um r with tim e in ber a terv als ” pie Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 6

Overview: ØMCQN model ØTransient Fluid Solutions ØInfinite Virtual Queues ØNear Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 7

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

IVQ’s Make Controlled Queueing Network even more interesting… What does a “good” control achieve? Some Resource PUSH Sta ble a Qu eue nd Lo Siz w es The Network PULL Hig ha n Th d Ba rou l gh anced pu Lo t wv dep aria n art ure ce of pro the ces s on i t za es i l i t rc U u gh eso i H f. R o 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 9

Extend the MCQN to MCQN + IVQ Queues/Classes 6 Initial Queue Levels 1 Routing Processes 2 3 Resources 5 Processing Durations 4 Network Dynamics Resource Allocation (Scheduling) Nominal Productio n Rates Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 10

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

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: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 12

The Static Equations - MCQN model - Nominal Production rates for IVQs - Resource Utilization - Resource Allocation A feasible static allocation is the triplet , such that: Similar to ideas from Harrison 2002 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. “Energy” Minimization • Lyapunov function: • Find allocation that reduces it as fast as possible: The Policy: Choose: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 14

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 15

Overview: ØMCQN model ØTransient Fluid Solutions ØInfinite Virtual Queues ØNear Optimal Finite Horizon Control Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 16

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

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

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 19

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

Empirical Asymptotics N = 1 to 106 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 21

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