The Annals of Applied Probability

Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy

S. L. Bell and R. J. Williams

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Abstract

This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean 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) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold control policy for use in the original parallel server system. We show that this policy is asymptotically optimal in the heavy traffic limit and the limiting cost is the same as the optimal cost in the Brownian control problem. The techniques developed here are expected to be useful for analyzing the performance of threshold-type policies in more complex multiserver systems.

Article information

Source
Ann. Appl. Probab., Volume 11, Number 3 (2001), 608-649.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015345343

Digital Object Identifier
doi:10.1214/aoap/1015345343

Mathematical Reviews number (MathSciNet)
MR1865018

Zentralblatt MATH identifier
1015.60080

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20] 90B35: Scheduling theory, deterministic [See also 68M20]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Queueing networks dynamic control resource pooling heavy traffic

Citation

Bell, S. L.; Williams, R. J. Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy. Ann. Appl. Probab. 11 (2001), no. 3, 608--649. doi:10.1214/aoap/1015345343. https://projecteuclid.org/euclid.aoap/1015345343


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