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Product-form characterization for a two-dimensional reflecting random walk

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Abstract

We consider a two-dimensional skip-free reflecting random walk on the non-negative integers, which is referred to as a 2-d reflecting random walk. We give necessary and sufficient conditions for the stationary distribution to have a product-form. We also derive simpler sufficient conditions for the product-form for a restricted class of 2-d reflecting random walks. We apply these results and obtain a product-form approximation of the stationary distribution through a suitable modification of the parameters of the random walk.

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Acknowledgments

This paper was initiated while Masakiyo Miyazawa visited Guy Latouche at the Université Libre de Bruxelles. We thank the Ministère de la Communauté française de Belgique for its support through the ARC grant AUWB-08/13–ULB 5. We also thank Mr. Yusuke Fukui for numerical computations for the product-form and QBD approximations in Sect. 4. This research is supported in part by the Japan Society for the Promotion of Science under Grant No. 24310115.

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Authors and Affiliations

  1. Université Libre de Bruxelles, Brussels, Belgium

    Guy Latouche

  2. Tokyo University of Science, Noda, Japan

    Masakiyo Miyazawa

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  1. Guy Latouche
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  2. Masakiyo Miyazawa
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Correspondence to Guy Latouche.

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Latouche, G., Miyazawa, M. Product-form characterization for a two-dimensional reflecting random walk. Queueing Syst 77, 373–391 (2014). https://doi.org/10.1007/s11134-013-9381-7

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  • DOI: https://doi.org/10.1007/s11134-013-9381-7

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