Analysis of an MAP/PH/1 Queue with Flexible Group Service

Novel customers batch service discipline for a single server queue is introduced and analyzed. Service to customers is offered in batches of a certain size. If the number of customers in the system at a service completion moment is less than this size, the server does not start next service until the number of customers in the system will reach this size or a random limitation of the idle time of the server will expire, whichever occurs first. Customers arrive according to a Markovian Arrival Process. Individual customer's service time has phase type distribution. Service time of a batch is defined as the maximum of individual service times of customers which form a batch. Dynamics of such a system is described by a multi-dimensional Markov chain. Ergodicity condition for this Markov chain is derived, stationary probability distribution of the states is computed, formulas for the main performance measures of the system are attained. Laplace-Stieltjes transform of waiting time is obtained. Results are numerically illustrated.