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Two phases queue systems with dependent phases service times via copula

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Abstract

A single server \({M/G_{1},G_{2}/1}\) queue with Poisson arrivals, two phases of general heterogeneous services that both phases are essential, with Bernoulli feedback and Bernoulli vacation is assumed. In the common queue models the service times of the phases are assumed to be independent. However, in this paper we assume that they are dependent random variables, considering that this dependency is one-way. It means that the second-phase service time does not affect the first-phase service time. Whereas, the first-phase service time affects the second-phase service time. For this model, the steady-state probability generating function of the queue size distribution is obtained. The Laplace–Stieltjes transform of the service times’ distributions and some important performance measures such as the mean of the queue size and the waiting time of a customer in the queue are obtained. Finally, numerical results via the Farlie–Gumbel–Morgenstern copula function are presented.

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  1. Department of Statistics, Faculty of Statistics, Mathematics and Computer, Allameh Tabataba’i University, Tehran, Iran

    Hossein Foroutan & Mohammad Reza Salehi Rad

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  1. Hossein Foroutan
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  2. Mohammad Reza Salehi Rad
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Correspondence to Mohammad Reza Salehi Rad.

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Foroutan, H., Salehi Rad, M.R. Two phases queue systems with dependent phases service times via copula. OPSEARCH 61, 189–204 (2024). https://doi.org/10.1007/s12597-023-00702-8

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