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

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  • Kevin D. Glazebrook

    () (Department of Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK)

  • José Niño-Mora

    () (Department of Economics and Business, Universitat Pompeu Fabra, E-08005, Barcelona, Spain)

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.

Suggested Citation

  • Kevin D. Glazebrook & José Niño-Mora, 2001. "Parallel Scheduling of Multiclass M/M/m Queues: Approximate and Heavy-Traffic Optimization of Achievable Performance," Operations Research, INFORMS, vol. 49(4), pages 609-623, August.
  • Handle: RePEc:inm:oropre:v:49:y:2001:i:4:p:609-623
    DOI: 10.1287/opre.49.4.609.11225
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    File URL: http://dx.doi.org/10.1287/opre.49.4.609.11225
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    References listed on IDEAS

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    1. Paul J. Burke, 1956. "The Output of a Queuing System," Operations Research, INFORMS, vol. 4(6), pages 699-704, December.
    2. A. Federgruen & H. Groenevelt, 1988. "Characterization and Optimization of Achievable Performance in General Queueing Systems," Operations Research, INFORMS, vol. 36(5), pages 733-741, October.
    3. GLAZEBROOK, Kevin D. & NINO-MORA, José, 1999. "A linear programming approach to stability, optimisation and perfomance analysis for Markovian multiclass queueing networks," CORE Discussion Papers RP 1446, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. E. G. Coffman & I. Mitrani, 1980. "A Characterization of Waiting Time Performance Realizable by Single-Server Queues," Operations Research, INFORMS, vol. 28(3-part-ii), pages 810-821, June.
    5. Dimitris Bertsimas & José Niño-Mora, 1996. "Conservation Laws, Extended Polymatroids and Multiarmed Bandit Problems; A Polyhedral Approach to Indexable Systems," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 257-306, May.
    6. Dimitris Bertsimas & José Niño-Mora, 1999. "Optimization of Multiclass Queueing Networks with Changeover Times Via the Achievable Region Approach: Part I, The Single-Station Case," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 306-330, May.
    7. K.D. Glazebrook & R. Garbe, 1999. "Almost optimal policies for stochastic systemswhich almost satisfy conservation laws," Annals of Operations Research, Springer, vol. 92(0), pages 19-43, January.
    8. Dimitris Bertsimas & José Niño-Mora, 1999. "Optimization of Multiclass Queueing Networks with Changeover Times Via the Achievable Region Approach: Part II, The Multi-Station Case," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 331-361, May.
    9. Gideon Weiss, 1992. "Turnpike Optimality of Smith's Rule in Parallel Machines Stochastic Scheduling," Mathematics of Operations Research, INFORMS, vol. 17(2), pages 255-270, May.
    10. K.D. Glazebrook & J. Niño‐Mora, 1999. "A linear programming approach to stability, optimisationand performance analysis for Markovian multiclassqueueing networks," Annals of Operations Research, Springer, vol. 92(0), pages 1-18, January.
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    Citations

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    Cited by:

    1. José Niño-Mora, 2006. "Restless Bandit Marginal Productivity Indices, Diminishing Returns, and Optimal Control of Make-to-Order/Make-to-Stock M/G/1 Queues," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 50-84, February.
    2. Terry James & Kevin Glazebrook & Kyle Lin, 2016. "Developing Effective Service Policies for Multiclass Queues with Abandonment: Asymptotic Optimality and Approximate Policy Improvement," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 251-264, May.
    3. Esther Frostig & Gideon Weiss, 2016. "Four proofs of Gittins’ multiarmed bandit theorem," Annals of Operations Research, Springer, vol. 241(1), pages 127-165, June.
    4. R. T. Dunn & K. D. Glazebrook, 2004. "Discounted Multiarmed Bandit Problems on a Collection of Machines with Varying Speeds," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 266-279, May.
    5. Mor Armony & Amy R. Ward, 2010. "Fair Dynamic Routing in Large-Scale Heterogeneous-Server Systems," Operations Research, INFORMS, vol. 58(3), pages 624-637, June.
    6. Avishai Mandelbaum & Alexander L. Stolyar, 2004. "Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cμ-Rule," Operations Research, INFORMS, vol. 52(6), pages 836-855, December.
    7. K. D. Glazebrook & C. Kirkbride & J. Ouenniche, 2009. "Index Policies for the Admission Control and Routing of Impatient Customers to Heterogeneous Service Stations," Operations Research, INFORMS, vol. 57(4), pages 975-989, August.

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