Queueing Systems

, Volume 85, Issue 3–4, pp 337–359 | Cite as

Fitting correlated arrival and service times and related queueing performance

  • Peter Buchholz
  • Jan Kriege


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.


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

Mathematics Subject Classification

65C40 60K20 68M20 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Informatik IVTU DortmundDortmundGermany

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