The $MAP/M/N$ retrial queueing system with time-phased batch arrivals

We consider a retrial queueing system with batch arrival of customers. Unlike standard batch arrival, where a whole batch enters the system simultaneously, we assume that customers of a batch (session) arrive one by one in exponentially distributed time intervals. Service time is exponentially distributed. The batch arrival flow is MAP. The number of customers in a session is geometrically distributed. The number of sessions that can enter the system simultaneously is a control parameter. We analyze the joint probability distribution of the number of sessions and customers in the system using the techniques of multidimensional asymptotically quasi-Toeplitz Markov chains.