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Nonparametric Estimation for Multi-server Queues Based on the Number of Clients in the System

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Abstract

In this article, we introduce a nonparametric (or distribution-free) estimator for traffic intensity in multi-server queues, which has not yet been discussed in the literature. Because this is a very useful model with many potential practical applications, it is the main focus of this study. We compare the performance of a new nonparametric estimator for situations in which the use of Markovian multi-server queues (M/M/s queues in Kendall notation) is adequate or in which it is necessary to consider multi-server queues with general arrival and general service times. We show that, when the parametric Markovian assumptions of M/M/s queues are satisfied, the new estimator is not superior to the maximum likelihood estimator based on the Markovian assumption with respect to M/M/s queues. However, for situations in which the interarrival time distribution and/or the service time distribution cannot be considered exponential (that is, non-Markovian), the new nonparametric estimator is superior. All evaluations are carried out using Monte Carlo simulations. A detailed numerical example is presented to show the usefulness of the technique for practical applications.

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Acknowledgements

We would like to thank the Editor-in-Chief and the two referees for their detailed and insightful comments, which led to a much-improved manuscript.

Funding

VBQ and FRBC acknowledges CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, grants 160974/2020-8, 148989/2021-7, and 305442/2022-8) and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, grant CEX-PPM-00564-17), for partial financial support.

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Authors and Affiliations

  1. Departamento de Engenharia de Produção, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, MG, Brazil

    V. B. Quinino

  2. Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, MG, Brazil

    F. R. B. Cruz & R. C. Quinino

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  1. V. B. Quinino
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  2. F. R. B. Cruz
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  3. R. C. Quinino
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Contributions

All authors, VBQ, FRBC, and RCQ contributed equally to the design and implementation of the research, to the analysis of the results, and to the final writing of the manuscript.

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Correspondence to F. R. B. Cruz.

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Appendix A Script in R

Appendix A Script in R

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Quinino, V.B., Cruz, F.R.B. & Quinino, R.C. Nonparametric Estimation for Multi-server Queues Based on the Number of Clients in the System. Sankhya A 86, 494–529 (2024). https://doi.org/10.1007/s13171-023-00331-9

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