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Fitting correlated arrival and service times and related queueing performance

Queueing Systems volume 85, pages 337–359 (2017)Cite this article

Abstract

In this paper, we consider a queue where the inter-arrival times are correlated and, additionally, service times are also correlated with inter-arrival times. We show that the resulting model can be interpreted as an MMAP[K]/PH[K]/1 queue for which matrix geometric solution algorithms are available. The major result of this paper is the presentation of approaches to fit the parameters of the model, namely the MMAP, the PH distribution and the parameters introducing correlation between inter-arrival and service times, according to some trace of inter-arrival and corresponding service times. Two different algorithms are presented. The first algorithm is based on available methods to compute a MAP from the inter-arrival times and a PH distribution from the service times. Afterward, the correlation between inter-arrival and service times is integrated by solving a quadratic programming problem over some joint moments. The second algorithm is of the expectation maximization type and computes all parameters of the MAP and the PH distribution in an iterative way. It is shown that both algorithms yield sufficiently accurate results with an acceptable effort.

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Acknowledgements

We thank the anonymous reviewers and the editor for their very thorough reviews that helped us a lot to improve the paper.

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Authors and Affiliations

  1. Informatik IV, TU Dortmund, 44221, Dortmund, Germany

    Peter Buchholz & Jan Kriege

Authors
  1. Peter Buchholz
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  2. Jan Kriege
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Correspondence to Peter Buchholz.

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Buchholz, P., Kriege, J. Fitting correlated arrival and service times and related queueing performance. Queueing Syst 85, 337–359 (2017). https://doi.org/10.1007/s11134-017-9514-5

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  • DOI: https://doi.org/10.1007/s11134-017-9514-5

Keywords

  • Markovian arrival process
  • Marked Markovian arrival processes
  • Phase type distributions
  • Multi-class queues
  • Expectation maximization algorithm

Mathematics Subject Classification

  • 65C40
  • 60K20
  • 68M20