Stochastic Analysis of an M/G/1 Retrial Queue with FCFS

Boualem, Mohamed; Cherfaoui, Mouloud; Djellab, Natalia; Aı̈ssani, Djamil

doi:10.1007/978-3-319-22476-3_8

Cited by 4 publications

(3 citation statements)

References 18 publications

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“…Additionally, it has been employed in studies of partially ordered spaces [35] and applied probability [47]. Several previous works have also applied the stochastic comparison method to queueing models, including [1,2,[7][8][9][10][11][12][13][14][15][16]34].…”

Section: Introductionmentioning

confidence: 99%

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

HABLAL,

TOUCHE,

ALEM

et al. 2023

Hacettepe Journal of Mathematics and Statistics

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Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

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Section: Introductionmentioning

confidence: 99%

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

HABLAL,

TOUCHE,

ALEM

et al. 2023

Hacettepe Journal of Mathematics and Statistics

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show abstract

“…Stochastic bounds take place in various retrial models. More precisely, in the literature of queues, many researchers have undertaken to examine such a question in the same spirit, among others, we cite: Boualem [5], Boualem et al [6,7,8,9,10,11,12,13], Khalil and Falin [18], Liang [19], Liang and Kulkarni [20], Shin [25] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls

2019

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In this article we analyze the M/G/1 retrial queue with two-way communication and n types of outgoing calls from a stochastic comparison viewpoint. The main idea is that given a complex Markov chain that cannot be analyzed numerically, we propose to bound it by a new Markov chain, which is easier to solve by using a stochastic comparison approach. Particularly, we study the monotonicity of the transition operator of the embedded Markov chain relative to the stochastic and convex orderings. Bounds are also obtained for the stationary distribution of the embedded Markov chain at departure epochs. Additionally, the performance measures of the considered system can be estimated by those of an M/M/1 retrial queue with two-way communication and n types of outgoing calls when the service time distribution is NBUE (respectively, NWUE). Finally, we test numerically the accuracy of the proposed bounds.

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“…Sakurai and Phung-Duc [18] used a mathematical method based on generating function approach to obtain explicit expressions for the joint stationary distribution of the number of calls in the orbit and the state of the server of a two way communication retrial queues with multiple types of outgoing calls whose durations follow distinct exponential distributions. Ouazine and Abbas [17] proposed a numerical approach based on Taylor series expansion with a statistical aspect for analyzing the stationary performances of the M1, M2/G1, G2/1 retrial queues with two way communication with one type of outgoing calls, while our work is focused on the application of stochastic comparisons of Markov chains, in order to derive bounds for the performance indices of the same model based on results achieved by Boualem [5], Boualem et al [6,7,8,9,10,11,12,13], Khalil et Falin [14], Liang [15], Liang and Kulkarni [16] and Stoyan [21].…”

Section: Introductionmentioning

confidence: 99%

Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication

Alem¹,

Boualem²,

Aı̈ssani³

2018

HJMS

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The main goal in the present paper is to provide a technique that considers the stochastic comparison approach for investigating monotonicity and comparability of an M1, M2/G1, G2/1 retrial queues with two way communication. This approach is developed for comparing a non Markov process to Markov process with many possible stochastic orderings. Particularly, we show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and convex ordering, as well as the comparability of two transition operators. Bounds are also obtained for the stationary distribution of the number of customers at departure epochs. Additionally, the performance measures of the system considered can be estimated by those of an M1, M2/M1, M2/1 retrial queue with two way communication when the service time distribution is NBUE (respectively NWUE). Finally, we validate stochastic comparison results by presenting a numerical example illustrating the interest of the approach.

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Stochastic Analysis of an M/G/1 Retrial Queue with FCFS

Cited by 4 publications

References 18 publications

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls

Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication

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