A discrete-time retrial queueing model with one server

This paper presents a one-server queueing model with retrials in discrete-time. The number of primary jobs arriving in a time slot follows a general probability distribution and the different numbers of primary arrivals in consecutive time slots are mutually independent. Each job requires from the server a generally distributed number of slots for its service, and the service times of the different jobs are independent. Jobs arriving in a slot can start their service only at the beginning of the next slot. When upon arrival jobs find the server busy all incoming jobs are sent into orbit. When upon arrival in a slot jobs find the server idle, then one of the incoming jobs (randomly chosen) in that slot starts its service at the beginning of the next slot, whereas the other incoming jobs in that slot, if any, are sent into orbit. During each slot jobs in the orbit try to re-enter the system individually, independent of each other, with a given retrial probability.     

Transient solution for a finite capacity M/G a,b /1 queueing system with vacations to the server

In this paper we consider a single server queueing system with Poisson input, general service and a waiting room that allows only a maximum of b customers to wait at any time. A minimum of a customers are required to start a service and the server goes for a vacation whenever he finds less than a customers in the waiting room after a service. If the server returns from a vacation to find less than a customers waiting, he begins another vacation immediately. Using the theory of regenerative processes we derive expressions for the time dependent system size probabilities at arbitrary epochs.