A queueing system with a batch Markovian arrival process and varying priorities

We consider herein a single-server queueing system with a finite buffer and a batch Markovian arrival process. Customers staying in the buffer may have a lower or higher priority. Immediately after arrival each of the customer is assigned the lowest priority and a timer is set for it, which is defined as a random variable distributed according to the phase law and having two absorbing states. After the timer enters one of the absorbing states, the customer may leave the system forever (get lost) or change its priority to the highest. When the timer enters another absorbing state, the customer is lost with some probability and the timer is set again with an additional probability. If a customer enters a completely full system, it is lost. Systems of this type can serve as mathematical models of many real-life medical care systems, contact centers, perishable food storage systems, etc. The operation of the system is described in terms of a multidimensional Markov chain, the stationary distribution and a number of performance characteristics of the system are calculated.