A queueing system under LCFS discipline with Markovian arrival process and general times of searching for service

We consider a single server queueing system with repeated attempts in which customers arrive according a Markov Arrival Process (MAP) and with a LCFS PR discipline. The service times are independent and have a common general distribution. After service completion time the server initiates his search time with an arbitrary distribution function. We consider two cases where the maximum number of repeated customers waiting in the orbit to seek service again is limited by r(r < ∞) or can be unlimited (r = ∞). We derive the steady state probabilities of the embedded Markov chain at service completion times of the process and also the steady state probabilities of the underlying Markov linear process.