Abstract:
We consider a single-server discrete queueing system in which arrivals occur
according to a Markovian arrival process. Service is provided in groups of size no
more than M customers. The service times are assumed to follow a discrete
phase type distribution, whose representation may depend on the group size.
Under a probabilistic service rule, which depends on the number of customers
waiting in the queue, this system is studied as a Markov process. This type of
queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified)
matrix-geometric type. Efficient algorithmic procedures for the computation of
the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is
derived explicitly. Some numerical examples are presented.
