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Fitting correlated arrival and service times and related queueing performance

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  • Peter Buchholz

    (TU Dortmund)

  • Jan Kriege

    (TU Dortmund)

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.

Suggested Citation

  • Peter Buchholz & Jan Kriege, 2017. "Fitting correlated arrival and service times and related queueing performance," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 337-359, April.
    • Handle: RePEc:spr:queues:v:85:y:2017:i:3:d:10.1007_s11134-017-9514-5
    DOI: 10.1007/s11134-017-9514-5
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    References listed on IDEAS

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    Cited by:

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    2. Sindhu S & Achyutha Krishnamoorthy & Dmitry Kozyrev, 2023. "A Two-Server Queue with Interdependence between Arrival and Service Processes," Mathematics, MDPI, vol. 11(22), pages 1-25, November.
    3. Yera, Yoel G. & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2019. "Fitting procedure for the two-state Batch Markov modulated Poisson process," European Journal of Operational Research, Elsevier, vol. 279(1), pages 79-92.
    4. Civelek, Ismail & Biller, Bahar & Scheller-Wolf, Alan, 2021. "Impact of dependence on single-server queueing systems," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1031-1045.
    5. Olga Dudina & Alexander Dudin, 2019. "Optimization of Queueing Model with Server Heating and Cooling," Mathematics, MDPI, vol. 7(9), pages 1-14, August.
    6. Chesoong Kim & Alexander Dudin & Olga Dudina & Valentina Klimenok, 2020. "Analysis of Queueing System with Non-Preemptive Time Limited Service and Impatient Customers," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 401-432, June.
    7. Nima Manafzadeh Dizbin & Barış Tan, 2019. "Modelling and analysis of the impact of correlated inter-event data on production control using Markovian arrival processes," Flexible Services and Manufacturing Journal, Springer, vol. 31(4), pages 1042-1076, December.
    8. Dudin, A.N. & Dudin, S.A. & Dudina, O.S. & Samouylov, K.E., 2018. "Analysis of queueing model with processor sharing discipline and customers impatience," Operations Research Perspectives, Elsevier, vol. 5(C), pages 245-255.

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