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The Single Server Queue with Mixing Dependencies

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Abstract

We study a single server queue, where a certain type of dependence is introduced between the service times, or between the inter-arrival times, or both between the service times and the inter-arrival times. This dependence arises via mixing, i.e., a parameter pertaining to the distribution of the service times, or of the inter-arrival times, is itself considered to be a random variable. We give a duality result between such queueing models and the corresponding insurance risk models, for which the respective dependence structures have been studied before. For a number of examples we provide exact expressions for the waiting time distribution, and compare these to the ones for the standard M/M/1 queue. We also investigate the effect of dependence and derive first order asymptotics for some of the obtained waiting time tails. Finally, we discuss this dependence concept for the waiting time tail of the G/M/1 queue.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Youri Raaijmakers & Onno Boxma

  2. Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, CH-1015, Lausanne, Switzerland

    Hansjörg Albrecher

  3. Swiss Finance Institute, Lausanne, Switzerland

    Hansjörg Albrecher

Authors
  1. Youri Raaijmakers
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  2. Hansjörg Albrecher
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  3. Onno Boxma
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Corresponding author

Correspondence to Youri Raaijmakers.

Additional information

Hansjörg Albrecher acknowledges financial support from the Swiss National Science Foundation Project 200021_168993.

The research of Onno Boxma was partly funded by the NWO Gravitation Project NETWORKS, Grant Number 024.002.003

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Raaijmakers, Y., Albrecher, H. & Boxma, O. The Single Server Queue with Mixing Dependencies. Methodol Comput Appl Probab 21, 1023–1044 (2019). https://doi.org/10.1007/s11009-018-9683-7

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  • DOI: https://doi.org/10.1007/s11009-018-9683-7

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