Solution of an M/G/1 queueing model with k sequential heterogeneous service steps and vacations using the Tauberian property

Ali Mohammadi and Mohammad Reza Salehi Rad

Communications in Statistics - Theory and Methods, 2021, vol. 50, issue 14, 3235-3248

Abstract: This article studies a transient M/G/1 queueing model with k sequential heterogeneous service steps and vacations. Entrants arrive according to a Poisson process, and each arrival is serviced sequentially and consistently in k phases in the order of arrival. The service times follow a general distribution function. Upon completion of the service, the server could take a vacation with the probability θ and remains in the system to provide service to other customers with the probability 1−θ. For this model, first, we obtain the Laplace transform of the probability generating functions (PGFs) of system size, and then we obtain the PGFs for a special case in the transient state, and the corresponding steady state results explicitly. Also, we derive the system performance measures such as the means system size, waiting time, and busy period in a closed form.

Date: 2021
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DOI: 10.1080/03610926.2019.1691232

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