A queueing model with bmap arrivals and heterogeneous phase type group services

Abstract

We consider a queueing system in which the arrivals occur according to batch Markovian arrival process. There is a single server in the system. The server offers services in batches of varying sizes from 1 to b, where b is pre-determined finite positive integer. Upon completion of a service, the server will become idle if there is no customer waiting in the queue of infinite size. Otherwise, the server will offer services to the waiting customers by picking the minimum of b and the number waiting from the head of the queue. Thus, the services are offered for batches of size r varying from 1 to b. Assuming that the service times follow a phase type distribution with representation depending on r we analyze the queueing system in the steady-state using matrix-analytic methods. We report some interesting illustrative numerical examples point out that the type of the arrival (in other words, the variability in the inter-arrival times) and the type of batch size distribution have a significant effect on the some performance measures.