Investigating the behaviour of a discrete retrial system

Abstract

Certain technological applications use queuing systems where the service time of entering entities cannot take any value, it can only be a multiple of a certain cycle-time. As examples of this, one can mention the landing of aeroplanes and the optical buffers of internet networks. Servicing an entering customer can be started immediately, or, if the server is busy, or there are waiting customers, the new customer joins a queue, moving along a closed path which can be completed within a fixed cycle-time of T units. Applications in digital technology induce the investigation of discrete systems. We give the mathematical description of systems serving two types of customers, where inter-arrival times follow a geometric distribution, and service times are distributed uniformly. A Markov-chain is defined and the generating functions of transition probabilities are calculated. The condition of ergodicity is established and the equilibrium distribution is given.