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Abstract

There are many uses of queues, where services are provided in groups; these types of queues are widely studied in the literature. In this paper we examine a particular queueing model wherein the services are provided in groups and the group size may be less than or equal to the size initially fixed. The arrival follows a Markovian arrival process. The service time of each individual customer follows phase type distribution. The maximum of each customer’s individual service time within a group is defined as the group’s service time. At the service completion moment if there are fewer customers than the initially fixed size, the server won’t begin the subsequent service until the system’s customer size reaches the initially fixed size or a randomly assigned admission period expires, whichever happens first. The phase type representation of the service times depends on the group’s size. If there is no customer block in the waiting line after the server finishes serving, the server will leave for vacation. If any customer block arrives within the designated vacation period, the server immediately will start serving them at a slower pace than the regular pace. After a regular service each customer group has an option of receiving the optional service from the server. The Markov chain’s stability condition is determined and stationary probability vector is computed. Formulas for the primary system performance measures are given. Waiting time distribution of the model is derived. Numerical and graphical representations of the proposed model are illustrated.

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