Parallel Scheduling of Multiclass M/M/M Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance

Univ. Pompeu Fabra, Economics and Business Working Paper No. 427

25 Pages Posted: 1 May 2000

See all articles by Kevin D. Glazebrook

Kevin D. Glazebrook

Newcastle University

José Niño Mora

Universitat Pompeu Fabra - Faculty of Economic and Business Sciences

Date Written: October 1999

Abstract

We address the problem of scheduling a multiclass "M/M/m" queue with Bernoulli feedback on "m" parallel servers to minimize time-average linear holding costs. We analyze the performance of a heuristic priority-index rule, which extends Klimov's optimal solution to the single-server case: servers select preemptively customers with larger Klimov indices. We present closed-form suboptimality bounds (approximate optimality) for Klimov's rule, which imply that its suboptimality gap is uniformly bounded above with respect to (i) external arrival rates, as long as they stay within system capacity; and (ii) the number of servers. It follows that its relative suboptimality gap vanishes in a heavy-traffic limit, as external arrival rates approach system capacity (heavy-traffic optimality). We obtain simpler expressions for the special no-feedback case, where the heuristic reduces to the classical "c mu" rule. Our analysis is based on comparing the expected cost of Klimov's rule to the value of a strong linear programming (LP) relaxation of the system's region of achievable performance of mean queue lengths. In order to obtain this relaxation, we derive and exploit a new set of work decomposition laws for the parallel-server system. We further report on the results of a computational study on the quality of the "c mu" rule for parallel scheduling.

JEL Classification: C60, C61

Suggested Citation

Glazebrook, Kevin D. and Niño Mora, José, Parallel Scheduling of Multiclass M/M/M Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance (October 1999). Univ. Pompeu Fabra, Economics and Business Working Paper No. 427, Available at SSRN: https://ssrn.com/abstract=224549 or http://dx.doi.org/10.2139/ssrn.224549

Kevin D. Glazebrook (Contact Author)

Newcastle University ( email )

5 Barrack Road
Devonshire Building
NE1 7RU Newcastle upon Tyne, NE1 7RU
United Kingdom

José Niño Mora

Universitat Pompeu Fabra - Faculty of Economic and Business Sciences ( email )

Ramon Trias Fargas 25-27
Barcelona, 08005
Spain
(34-3) 542 26 73 (Phone)
(34-3) 542 17 46 (Fax)

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