An M [X]/G/1 retrial queue with server breakdowns and constant rate of repeated attempts

Abstract

We consider an M [X] /G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically.

A batch arrival retrial queuing system for essential and optional services with server breakdown and Bernoulli vacation

“…According to the queuing and reliability point of view, the queue theorists are interested to develop the repairable service station. The worth nothing contributions in this area can be seen in the works of Ke (2005), Atencia et al (2008), Deka (2009), Falin (2010b), Jain and Bhargava (2010). They have obtained the joint distribution for the number of the customers in the queue in the retrial group.…”

Spectral properties of the M [X]/G/1 operator and its application

“…+ · · · |a 2,l (x)| + · · · + α γ + λ + α (c 1 + c 2 + c 3 + · · · ) ψ 1 E 1 ψ 2 k 21 4 h 62 + · · · |a 1,1 | + · · · + h ll + (c 1 + c 2 + c 3 + · · · ) h l+1l + (c 1 + c 2 + c 3 + · · · ) 2 h l+2l + (c 1 + c 2 + c 3 + · · · ) 3 h l+3l + (c 1 + c 2 + c 3 + · · · ) 4 h l+4l + · · · |a 1,l−1 | + · · · +v |E 1 ψ 2 k 11 + (c 1 + c 2 + c 3 + · · · ) E 1 ψ 2 k 21…”

“…Therefore, many researchers studied such queueing systems, see Falin and Templeton [6], Aissani [2], Atencia et al [5,4], Gupur [8], Gupur and Kasim [10], Jiang and Gupur [15], Lü and Gupur [17], Ismayil and Gupur [14], Zhang and Gupur [19], for instance. In 2008, Atencia et al [4] established the mathematical model of M [X ] /G/1 retrial queueing system with server breakdowns and constant rate of repeated attempts by using the supplementary variable technique and studied the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steadystate case under the following hypothesis: lim t→∞ p 0,n (t) = p 0,n , n ≥ 0, lim t→∞ p 1,n (x, t) = p 1,n (x), n ≥ 0, lim t→∞ p 2,n (x, y, t) = p 1,n (x, y), n ≥ 0.…”

State dependent arrival in bulk retrial queueing system with immediate Bernoulli feedback, multiple vacations and threshold

“…Such kind of retrial policy was first constructed by Fayolle [11], who investigated "M/M/1 retrial queue, where the queue will be formed by the retrial group of customers and request for service is possible only for customers at the head of the orbit queue after an exponentially distributed retrial time" with rate 'v'. Atencia et al [3] have analysed bulk retrial queue with constant retrial rate and server breakdowns. Recently Jailaxmi et al [14] examined M/G/1 retrial queue with modified vacations, collision and general retrial policy.…”