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A Product-Form Solution for a Two-Class \(Geo^{Geo}/D/1\) Queue with Random Routing and Randomly Alternating Service

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Performance Evaluation Methodologies and Tools (VALUETOOLS 2022)

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Abstract

We analyze a discrete-time queueing system, consisting of two queues and a single server. The server randomly distributes its time between the two queues. Service times of any customer of either queue are deterministically equal to 1 time slot. In general, the joint analysis of such a two-queue system turns out to be very hard. In this paper, we assume that the total number of arrivals into the system constitutes a series of i.i.d. random variables with common geometric distribution. Each arriving customer is routed probabilistically to a queue. By means of a state-of-the-art approach, we obtain a closed-form expression of the steady-state joint PGF of the number of customers present (“system contents”) in both queues, at the beginning of a random slot. We find that the joint PGF is of product form, which proves that the system contents in both queues are independent. We provide an additional intuitive stochastic explanation for this remarkable result. We discuss several model extensions using the stochastic analysis.

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Authors and Affiliations

  1. Department of Telecommunications and Information Processing (EA07), Ghent University - UGent, Sint-Pietersnieuwstraat 41, 9000, Gent, Belgium

    Arnaud Devos, Michiel De Muynck, Herwig Bruneel & Joris Walraevens

Authors
  1. Arnaud Devos
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  2. Michiel De Muynck
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  3. Herwig Bruneel
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  4. Joris Walraevens
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Correspondence to Arnaud Devos .

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Editors and Affiliations

  1. University of Iceland, Reykjavik, Iceland

    Esa Hyytiä

  2. Indian Institute of Technology Bombay, Mumbai, India

    Veeraruna Kavitha

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Devos, A., De Muynck, M., Bruneel, H., Walraevens, J. (2023). A Product-Form Solution for a Two-Class \(Geo^{Geo}/D/1\) Queue with Random Routing and Randomly Alternating Service. In: Hyytiä, E., Kavitha, V. (eds) Performance Evaluation Methodologies and Tools. VALUETOOLS 2022. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 482. Springer, Cham. https://doi.org/10.1007/978-3-031-31234-2_6

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  • DOI: https://doi.org/10.1007/978-3-031-31234-2_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-31233-5

  • Online ISBN: 978-3-031-31234-2

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