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

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Overview:

14th INFORMS Applied Probability Conference,EindhovenJuly 9, 2007

Transient Fluid Solutions andQueueing Networks withInfinite Virtual Queues

- Yoni Nazarathy
- Gideon Weiss
- University of Haifa

Overview:

- MCQN model
- Transient Fluid Solutions
- Infinite Virtual Queues
- Near Optimal Finite Horizon Control

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

Overview:

- MCQN model
- Transient Fluid Solutions
- Infinite Virtual Queues
- Near Optimal Finite Horizon Control

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

Server 2

Server 1

3

2

1

Example NetworkAttempt to minimize:

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

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

Overview:

- MCQN model
- Transient Fluid Solutions
- Infinite Virtual Queues
- Near Optimal Finite Horizon Control

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

m

m

INTRODUCING: Infinite Virtual QueuesNominalProductionRate

Regular Queue

Infinite Virtual Queue

Relative Queue Length

Example Realization

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

6

1

2

3

5

4

Extend the MCQN to MCQN + IVQQueues/Classes

Initial Queue Levels

Routing Processes

Resources

Processing Durations

Network Dynamics

Resource Allocation (Scheduling)

NominalProductionRates

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

Rates Assumptions of the Primitive Sequences

Primitive Sequences:

May also define:

rates assumptions:

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

- 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

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

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

- MCQN model
- Transient Fluid Solutions
- Infinite Virtual Queues
- Near Optimal Finite Horizon Control

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

Back to the example network:

For each time interval, set a MCQN with Infinite Virtual Queues:

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

Example realizations, N={1,10,100}

- seed 1 seed 2 seed 3 seed 4

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

How fast is the convergence that is stated in the asymptotic optimality theorem ???

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

Empirical Asymptotics N = 1 to 106

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

ThankYou

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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