Processor-sharing and random-service queues with semi-markovian arrivals

We consider single-server queues with exponentially distributed service times in which the arribal process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type l customer who meets upon his arrival k customers present in the SMP/M/1/PS queue is identical with the waiting time distribution of a type l customer who meets upon his arrival k+1 customers present in the SMP/M/1/RS queue. The Laplace-Stieltjes transforms of the sojourn time distribution for an arbitrary customer in the SMP/M/1/PS queue and the waiting time distribution for an arbitrary customer in the SMP/M/1/RS queue are derived. We also consider a special case of the SMP in which the interarrival time distriburion is determined only by the type of the next arrival.Includes bibliographical reference