Asymptotic analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs

摘要

The aim of the present paper is to investigate a finite-source M/M/1 retrial queuing system with collision of the customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymptotic method is applied under the condition that the number of sources tends to infinity while the primary request generation rate, retrial rate tend to zero and service rate, failure rates, repair rate are fixed. It is proved that in steady state the limiting distribution of the centered and normalized number of customers in the system (orbit and service) follows a normal law with given parameters. The novelty of this investigation is the introduction of failure and repair of the service. Approximations of prelimiting distribution by asymptotic one are obtained and several illustrative examples show the accuracy and range of applicability of the proposed method.