Flexible Group Service MAP/ PH/ 1 Queueing Model with Working Breakdown, Repair and Balking

In reality, there are many uses of queues where services are provided in groups and these type of queues are widely studied in the literature. In this paper we examine a particular queueing model, wherein the services are provided in groups ranging from 1 to a pre defined constant, denoted as K, and the arrival follows a Markovian arrival process. The service time of each individual customer follows phase type distribution. The maximum of each customers 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 K, the server won't begin the subsequent service until the system's customer size reaches K or a randomly assigned admission period expires, whichever happens first. The phase type representation of the service times depends on the group's size. Anytime a server breakdowns and it will not proceed to repair, instead it will serve the affected customer group at a slower pace. After that specific customer group's service is finished, the server will immediately undergo repair to fix any issues. The process of repair and breakdown occurs at exponential rate. When the server breakdowns, the customer might balk. The Markov chain's stability condition is determined and stationary probability vector is computed. Formulas for the primary system performance measures are given. Numerical and graphical representations of the proposed model are illustrated.