On the number of customers served in the M/ G/1 retrial queue: first moments and maximum entropy approach

Abstract

In this paper we present general results on the number of customers, I, served during the busy period in an M/ G/1 retrial system. Its analysis in terms of Laplace transforms has been previously discussed in the literature. However, this solution presents important limitations in practice; in particular, the moments of I cannot be obtained by direct differentiation. We propose a direct method of computation for the second moment of I and also for the probability of k, k⩽4, customers being served in a busy period. Then, the maximum entropy principle approach is used to estimate the true distribution of I according to the available information. Scope and purpose We consider an M/ G/1 queue with retrials. Retrial queueing systems are characterized by the fact that, an arriving customer who finds the server busy is obliged to leave the service area and return later to repeat his request after some random time. We deal with I, the number of customers served during the busy period of a retrial queue, and obtain closed expressions for its main characteristics, which will be employed in order to estimate the true distribution of this random variable.