Disquisition of a Retrial Queueing System with Batch Markovian Arrival Process, Nonidentical Service Devices and Phase-Type Distribution of Service Times

We study a retrial queueing system with N ranked heterogeneous service devices where processing times at each device follow a phase-type ($PH$) distribution with device-dependent parameters. Requests arrive according to a Batch Markovian Arrival Process ($BMAP$). The system uses a preemptive priority rule: idle devices with smaller serial numbers are preferred, and when a lower-numbered device completes service, the request being processed at the highest-numbered busy device is moved there and its service restarts. Requests that cannot be served immediately join an orbit of infinite capacity and retry after random time intervals. We describe the system dynamics by a multidimensional continuous-time Markov chain with a block upper-Hessenberg generator. A sufficient ergodicity condition for this Markov chain is derived. We present formulas for the key performance measures, including the mean orbit length, device utilizations, and the probability of immediate service. Numerical experiments show how the arrival rate and the coefficient of variation of processing times affect system performance. In particular, higher processing-time variability (hyperexponential case) in the considered example widens the stability region, while lower variability (Erlang case) narrows it, compared with exponential service. A supplementary study shows that higher arrival correlation under the chosen set of the system parameters amplifies orbit congestion and shifts utilization from the fastest server to the slower ones.