Asymptotic Analysis of Finite-Source M/GI/1 Retrial Queueing Systems with Collisions and Server Subject to Breakdowns and Repairs

Abstract

This paper deals with a retrial queuing system with a finite number of sources and collision of the customers, where the server is subject to random breakdowns and repairs depending on whether it is idle or busy. A significant difference of this system from the previous ones is that the service time is assumed to follow a general distribution while the server’s lifetime and repair time is supposed to be exponentially distributed. The considered system is investigated by the method of asymptotic analysis under the condition of an unlimited growing number of sources. As a result, it is proved that the limiting probability distribution of the number of customers in the system follows a Gaussian distribution with given parameters. The Gaussian approximation and the estimations obtained by stochastic simulations of the prelimit probability distribution are compared to each other and measured by the Kolmogorov distance. Several examples are treated and figures show the accuracy and area of applicability of the proposed asymptotic method.

Analysis of a Collision-Affected M/GI/1/ /N Retrial Queuing System Considering Negative Customers and Transmission Errors

“…For the purpose of calculating R Y (t), the first step to carry out is finding the transient probabilities of the new system by taking the Laplace transform of ( 21)- (23).…”

“…Bérczes [20] investigate the performance of an unreliable finite source retrial queue by software MOSEL-2. Nazarov [21][22][23] conduct an asymptotic analysis of unreliable finite source retrial queue with collisions. Although the introduction of finite sources brings the proposed model closer to reality, it also increases the complexity of the study.…”

“…Although the introduction of finite sources brings the proposed model closer to reality, it also increases the complexity of the study. So there are still very few related studies, and most of them used software, see [20] or other approximate methods, see [21][22][23] to study this kind of queuing model.…”

“…For all we know, not much work is found in analyzing the reliability of retrial queues with collisions and finite sources in the literature. Although [20][21][22][23] have conducted an analysis of unreliable finite source retrial queues with collisions by software and the asymptotic method, the reliability and availability have not been derived. This motivates us to deal with such a queuing model in this paper.…”