An M/M/c queue with queueing-time dependent service rates

Bernardo D'Auria*, Ivo J.B.F. Adan, René Bekker, Vidyadhar Kulkarni

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

Recent studies indicate that in many situations service times are affected by the experienced queueing delay of the particular customer. This effect has been detected in different areas, such as health care, call centers and telecommunication networks. In this paper we present a methodology to analyze a model having this property. The specific model is an M/M/c queue in which any customer may be tagged at her arrival time if her queueing time will be above a certain fixed threshold. All tagged customers are then served at a given rate that may differ from the rate used for the non-tagged customers. We show how it is possible to model the virtual queueing time of this queueing system by a specific Markov chain. Then, solving the corresponding balance equations, we give a recursive solution to compute the stationary distribution, which involves a mixture of exponential terms. Using numerical experiments, we demonstrate that the differences in service rates can have a crucial impact on queueing time performance.

Original languageEnglish
Article number46
Pages (from-to)566-579
Number of pages14
JournalEuropean Journal of Operational Research
Volume299
Issue number2
Early online date21 Dec 2021
DOIs
Publication statusPublished - 1 Jun 2022

Bibliographical note

Funding Information:
The research of the first author was partially supported by Spain’s Ministry of Science and Innovation [Grants MTM2017-85618-P via FEDER funds and PID2020-116694GB-I00]

Publisher Copyright:
© 2021 The Author(s)

Funding

The research of the first author was partially supported by Spain’s Ministry of Science and Innovation [Grants MTM2017-85618-P via FEDER funds and PID2020-116694GB-I00]

FundersFunder number
Ministerio de Ciencia e InnovaciónMTM2017-85618-P, PID2020-116694GB-I00
Ministerio de Ciencia e Innovación

    Keywords

    • M/M/c queue
    • Queueing
    • Queueing-time dependency
    • Variable service rates

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