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Tail asymptotics for M/M/c retrial queues with non-persistent customers

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Abstract

In this paper, we consider a variant of the classical M/M/c retrial queue, in which we allow non-persistent customers. When c > 1, this system does not have an explicit closed form solution for the joint stationary distribution of the number of retrial customers in the orbit and the number of busy servers. Our main focus is on the tail asymptotics for the joint probabilities. We first present a matrix-product solution for the joint stationary probability vectors, which is further simplified to a scalar-product form, according to matrix-analytic theory. We then apply the censoring technique, which has been proven an efficient approach for analyzing queueing systems including retrial queues, to obtain the censored equations and the Key Lemma. In terms of these results, we finally prove an exact tail asymptotic result for the stationary probabilities.

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Acknowledgments

This work was partially supported by a research grant from NSERC of Canada, National Natural Science Foundation of China Grant No. 10871020 and 973 Program Grant No. 2006CB805900, and a Summer Fellowship from the University of Northern Iowa, USA. The authors thank two anonymous referees for their suggestions of improving the quality of the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Northern Iowa, Cedar Falls, IA, 50614-0506, USA

    Bin Liu

  2. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 Zhongguancun East Road, 100190, Beijing, China

    Xi Wang

  3. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Yiqiang Q. Zhao

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  1. Bin Liu
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  2. Xi Wang
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  3. Yiqiang Q. Zhao
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Correspondence to Bin Liu.

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Liu, B., Wang, X. & Zhao, Y.Q. Tail asymptotics for M/M/c retrial queues with non-persistent customers. Oper Res Int J 12, 173–188 (2012). https://doi.org/10.1007/s12351-011-0106-6

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  • DOI: https://doi.org/10.1007/s12351-011-0106-6

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