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Waiting times in queueing networks with a single shared server

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Abstract

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper, we develop a new method to obtain exact expressions for the Laplace–Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now.

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Acknowledgments

The authors are grateful to Ivo Adan and Onno Boxma for providing valuable comments on earlier drafts of this paper. The research was done in the framework of the BSIK/BRICKS project, the European Network of Excellence Euro-NF, and of the project “Service Optimization and Quality” (SeQual), funded by the Dutch agency SenterNovem.

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Authors and Affiliations

  1. Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB , Eindhoven, The Netherlands

    M. A. A. Boon

  2. Department of Mathematics, Faculty of Sciences, VU University Amsterdam, De Boelelaan 1081a, 1081 HV , Amsterdam, The Netherlands

    R. D. van der Mei

  3. Center for Mathematics and Computer Science (CWI), Science Park 123, 1098 XG , Amsterdam, The Netherlands

    R. D. van der Mei

  4. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH , Amsterdam, The Netherlands

    E. M. M. Winands

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  1. M. A. A. Boon
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  2. R. D. van der Mei
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  3. E. M. M. Winands
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Correspondence to M. A. A. Boon.

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Boon, M.A.A., van der Mei, R.D. & Winands, E.M.M. Waiting times in queueing networks with a single shared server. Queueing Syst 74, 403–429 (2013). https://doi.org/10.1007/s11134-012-9334-6

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  • DOI: https://doi.org/10.1007/s11134-012-9334-6

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