A finite capacity multi-server multi-queueing priority model with non-renewal input

We consider a non-preemptive head-of-the-line multi-server multi-queueing priority model with finite buffer capacity for each priority class. As an arrival process, a generalized Markovian arrival process with marked transitions is used. The service-time distribution is of phase-type and identical for the different priority classes. The model is described by a homogeneous continuous-time Markov chain (CTMC). From the steady-state distribution of the CTMC, which is calculated by matrix-geometric methods, we derive the steady-state distributions immediately after arrival instants of the different priority classes. Applying matrix-analytic methods, we calculate the Laplace–Stieltjes Transform of the actual waiting times for the different priority classes.