Asymptotic Waiting Time Analysis of a Finite-Source M/M/1 Retrial Queueing System

Abstract

The aim of the paper is to derive the distribution of the number of retrial of the tagged request and as a consequence to present the waiting time analysis of a finite-source M/M/1 retrial queueing system by using the method of asymptotic analysis under the condition of the unlimited growing number of sources. As a result of the investigation, it is shown that the asymptotic distribution of the number of retrials of the tagged customer in the orbit is geometric with given parameter, and the waiting time of the tagged customer has a generalized exponential distribution. For the considered retrial queuing system numerical and simulation software packages are also developed. With the help of several sample examples the accuracy and range of applicability of the asymptotic results in prelimit situation are illustrated showing the effectiveness of the proposed approximation.

Waiting Time Asymptotic Analysis of a M/M/1 Retrial Queueing System Under Two Types of Limiting Condition

“…Also, for that purpose Kolmogorov distances between distributions were found: Let us denote the prelimit probability distribution of the number of returns of the tagged request to the orbit by π(n). We obtain it by numerical methods similar as it is shown in [27]. Using the law of total probability, π(n) can be written in the following form:…”

“…We find exact values of P k (i) and Π k (0, i) by solving (2) and ( 28) for some given parameters λ, μ, σ using numerical methods. Then we substitute found P k (i) and Π k (0, i) values into (27) and get the probability π (0). Solving ( 29) by numerical methods for some given parameters λ, μ, σ we similarly find Π k (n, i) for n = 1, 2, 3, ... (Table 2).…”

“…The waiting time distribution, the time a request spends in the orbit, is very complicated problem in the retrial queue system theory. Different approaches to the investigation of the waiting time can be found in Artalejo, Chakravarthy, Lopez-Herrero [1], Artalejo, Gomez-Corral [4], Choi, Chang [6], Falin, Artalejo [8], Gharbi, Dutheillet [14], Sudyko, Nazarov, Sztrik [27], Tóth, Bérczes, Sztrik [28], Zhang, Feng, Wang [29] for finite retrial group of sources and Chakravarthy, Dudin [7], Falin, Fricker [9], Falin [1,10,12], Gomez-Corral, Ramalhoto [15], Neuts [20], Nobel, Tijms [25], Lee, Kim, Kim [18] for infinite number of sources.…”

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

“…The aim of the present paper is to investigate such systems which has the above mention properties, that is finite-source, retrial, collisions, and non-reliability of the server. The present model is a generalization of the systems treated in Kvach and Nazarov (2015), Nazarov et al (2014), Nazarov and Sudyko (2010), Nazarov and Moiseeva (2006) and Sudyko et al (2018) and it is the natural continuation of the paper in which the asymptotic distribution of the number of customers in the system has been investigated.…”