Dynamic Scheduling of a Parallel Server System in Heavy
Traffic with Complete Resource Pooling:
Asymptotic Optimality of a
S. L. Bell and R. J. Williams
We consider a parallel server queueing system consisting of
a bank of buffers for holding incoming jobs
and a bank of flexible servers for processing
Incoming jobs are classified into one of several different
classes (or buffers).
Jobs within a class are processed on a first-in-first-out basis,
where the processing of a given job may be performed
by any server from a given (class-dependent) subset
of the bank of servers.
The random service time of a job may depend on both its class
and the server providing the service.
Each job departs the system after receiving service from one server.
The system manager seeks to minimize
holding costs by dynamically scheduling waiting jobs to available servers.
We consider a parameter regime in which the system satisfies
both a heavy traffic and a complete resource pooling condition.
Our cost function is an expected cumulative discounted cost of
holding jobs in the system, where the (undiscounted) cost per unit
time is a linear function of normalized (with heavy traffic scaling)
In a prior work,
the second author proposed
a continuous review
threshold control policy for use in such a parallel server system.
This policy was advanced as an ``interpretation" of
the analytic solution to an associated Brownian control problem (formal
heavy traffic diffusion approximation).
In this paper we show that the policy proposed previously is asymptotically
optimal in the heavy traffic limit and that the limiting cost is the same
as the optimal cost in the Brownian control problem.
Published in Electronic
J. of Probability, 10 (2005), 1044-1115.
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Last updated: July 27, 2004