Abstract
The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system with collision of 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, the primary request generation rate, retrial rate tend to zero while service rate, failure rates, repair rate are fixed. As the result of the analysis it is shown that the steady-state probability distribution of the number of transitions/retrials of the customer into the orbit is geometric with a given parameter, and the normalized sojourn time of the customer in the system follows a generalized exponential distribution. It is also proved that the limiting distributions of the normalized sojourn time of the customer in the system and the normalized sojourn/waiting time of the customer in the orbit coincide. The novelty of this investigation is the introduction of failure and repair of the server. Approximations of prelimit distributions obtained with the help of stochastic simulation by asymptotic one are considered and several illustrative examples show the accuracy and range of applicability of the proposed asymptotic method.
Highlights
Retrial queues, that is queues with repeated attempts have been effectively used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, call centers, wireless communication systems, etc
We deal with the case when the server is down all sources continue generation of customers and send it to the server, customers may retry from the orbit to the server but all arriving customers immediately go into the orbit
The research has been conducted by method of asymptotic analysis under condition of unlimited growing number of sources
Summary
That is queues with repeated attempts have been effectively used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, call centers, wireless communication systems, etc. In many practical cases it is essential to take into account that the rate of generation of new primary calls decreases as the number of customers in the system increases This can be done by means of finite-source, or quasi-random input models. Recent results on retrial queues with collisions can be found in, for example Ali and Wei (2015), Balsamo et al (2013), Choi et al (1992), Gómez-Corral (2010), Kim (2010), Kumar et al (2010a, b), Nazarov et al (2018) and Peng et al (2014) It is well-known that the analysis of waiting/response time and the number of retrials of a customer is much more complicated than the distribution of number of customers in the system.
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