Abstract
In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.
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Kijima, M., Makimoto, N. Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues. Queueing Syst 11, 255–272 (1992). https://doi.org/10.1007/BF01164005
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DOI: https://doi.org/10.1007/BF01164005