A queueing system with markov arrival process and exceptional first service

We analyse a special kind of a MAP/G/l/r FCFS queue where the first customer who arrives in the system, at the beginning of a busy period, has a service time ruled by a different distribution function than customers getting service during the busy period. An arriving customer who finds the system completely full is lost. We derive the steady state probabilities of the Markov process underlying the considered queueing system.