A discrete queue with the Markovian arrival process and phase type primary and secondary services

A discrete time queueing model with Markovian arrival process in which jobs require primary and possibly secondary service, is analysed. Under the assumptions of infinite waiting room for the primary queue and no waiting room for the secondary queue the authors show that the steady state queue length density is of the matrix-geometric type. Stationary waiting time distribution of an arriving job is derived explicitly. Efficient algorithmic procedures for the computation of the rate matrix and the waiting time distribution are given.