Retrial queueing systems pdf

Markovian queues in equilibrium. Intermediate queueing theory. The method of collective marks. Advanced material. A queueing theory primer; Bounds, inequalities and approximations. Priority queueing. Introduction to Queueing Systems with Telecommunication Applications. The book is composed of two main parts: mathematical background and queueing systems with applications. Moutzoukis and Langaris [ 88 ] derive the explicit results for the tandem model with constant retrial rate and blocking at the first server.

For complex retrial queueing networks, a practical approach is the fixed point approximation. For example, fixed point approximations are used to analyze some tandem models with retrials by Avrachenkov et al.

In fixed point approximation, the system is divided into multiple subsystems whose input parameters are unknown. Furthermore, these subsystems are assumed to be independent. The output of one subsystem is the input of another subsystem. After some iterative calculations, one will get a convergence determining all the unknown parameters. One drawback of this methodology is that a rigorous proof of the convergence and the accuracy of the approximation are not always presented.

Moreover, the approximation is basically valid only under the slow retrial regime. Numerical solutions for some simple tandem retrial models are presented in [ 74 , 73 , , 65 ]. Recently, Fiems and Phung-Duc [ 60 ] present a light-traffic analysis for finite-source retrial systems arising from CSMA protocols without collision.

These systems could be formulated by multidimensional Markov chains which are also used to represent retrial queueing networks. The analysis by Fiems and Phung-Duc [ 60 ] is based on series expansion subject to the arrival rate around the origin and is validated by simulation.

Thus, power series expansion may be useful for analyzing queueing networks with retrials. Orbital search mechanism for customers in the orbit is introduced by Artalejo et al. This mechanism is further extended in [ 49 , 40 , 84 ]. Chakravarthy et al. Recently, Dragieva and Phung-Duc [ , 48 ] propose a related model that the authors call the retrial queueing model with interaction between server and customers in the orbit. The main idea is that not only customers retry to capture an idle server incoming calls but the server also makes outgoing calls to retrial customers.

The distributions of the durations of incoming calls and outgoing calls are different. Recently, game theoretic analysis of queues has been attracted much attention. Some authors study game theoretic analysis for retrial models. In particular, a series of works by Economou and Kanta [ 51 , 53 ] provide detailed analysis for retrial models with constant retrial rate. To be more precise, models in Wang et al. Thus, the analyses of more general models might be promising future topics.

Most of researches assume the exponential retrial time. Only a few references are devoted to models with other retrial time distributions where each customer in the orbit acts independently of other customers [ 34 , , ].

In the current literature, there is not an exact analysis for this type of settings and only some approximations or simulations are presented. Thus, researches in this direction may be highly appreciated. In almost the work mentioned above, the main quantity of interest is the stationary queue length distribution. A few references are devoted to some new quantities of interest. In particular, channel idle period is analyzed by Artalejo and Gomez-Corral in [ 13 ]. Distributions of the successful and blocked events are studied in [ 5 , 6 , 7 ].

Maximum queue length in busy period is presented in [ 9 ] while that in a fixed time interval is analyzed by Gomez-Corral and Garcia [ 68 ]. Dragieva [ 46 , 47 ] studies the distribution of the number of retrials in model with finite source with arbitrary service time distribution. Gao et al. In this paper, we have surveyed the main theoretical results for retrial queueing models.

We have also investigated retrial queueing models arising from real applications such as call centers, random access protocols, cellular networks etc. We hope that this paper can be served as a basic reference for researchers who want to enter and deepen this field. Because the retrial queue literature is rich, we also refer to some earlier survey papers [ , 56 , 11 , 66 , 18 ] , two books [ 57 , 19 ] and the recent Special Issue [ 69 ]. Most of references in this paper is for continuous time retrial queues.

We refer to Nobel [ 94 ] for a survey on results of discrete time retrial queues. Sections 1 and 2 are partially based on the dissertation of the author [ ]. The author would like to thank two anonymous reviewers whose comments greatly helped to improve the presentation of the paper. I would like to thank the Editors of the book, especially Professor Shoji Kasahara for giving me an opportunity to write this survey.

The author would like to devote this paper to the memory of Professor Jesus Artalejo who was a leading researcher in the fields of Queueing Theory and Mathematical Biology publishing more than one hundred papers and was a co-author and friend of the author of this paper. A classified bibliography of research on retrial queues: Progress in Already have an account?

Login here. Don't have an account? Signup here. There are no comments yet. Tuan Phung-Duc 1 publication. Related Research. Alireza Sepas-Moghaddam , et al. Victor Selivanov , et al. Huawei Huang , et al. Kamal Ahmat , et al. Abolfazl Lavaei , et al. Barbara Franci , et al. Rami Akeela , et al. For an extensive comparison of standard and retrial queueing systems, the readers are referred to the paper of Artalejo and Falin [ 15 ].

However, from a management point of view we need to minimize the number of agents, due to the fact that the cost of a call center is mainly the human cost [ ]. In order to achieve the customer satisfaction under some constraint on the cost, we need some mathematical model to express the trade-off between the customer satisfaction and the human cost.

A queueing model is one of the most appropriate mathematical models for the design of call centers. In addition, in order to capture the retrial phenomenon as presented above, a retrial queueing model is expected to be more appropriate than the corresponding standard queueing model [ 62 , 81 , , 1 ]. See Figure 2 for a simple retrial queueing model for call centers. For a detailed explanation on call centers, we refer to the book by Stolletz [ ]. Numerical results by Phung-Duc et al.

This is an evidence for modelling call centers by retrial queues. Under light-tailed assumption of the service time distributions of incoming and outgoing calls, asymptotic analysis of the joint queue length is also presented in [ 22 ] using a simpler method in comparison with that of Kim et al.

Under some heavy tailed assumptions of the service times, the queue length asymptotics is presented in Shang et al. The book is intended for an audience ranging from advanced undergraduates to researchers interested not only in queueing theory, but also in applied probability, stochastic models of the operations research, and engineering.

The prerequisite is a graduate course in stochastic processes, and a positive attitude to the algorithmic probability.

It gives a good review of computational aspects of performance evaluation and matrix-analytic formalism. For those … will be encountered with the feature of retrials in some modeling process, the book will surely support their investigation. Skip to main content Skip to table of contents.

Advertisement Hide. Heavy traffic analysis of a polling model with retrials and glue periods. Accessible bibliography on retrial queues: progress in Mathe- matical and Computer Modelling, 51 , Retrial queueing systems. Analysis of multiserver queues with constant retrial rate.

A retrial system with two input streams and two orbit queues. Queueing Systems, 77, Retrial networks with finite buffers and their applications to internet data traffic. Probability in the Engineering and Informational Sciences, 22 4 , Stability analysis and simulation of N-class retrial system with constant retrial rates and Poisson inputs.

Asia Pacific Journal of Operational Research 31 2 , 18 pages. Sufficient stability conditions for multi-class constant retrial rate systems. Queueing Systems, 82, Vacation and polling models with retrials. Wolter Eds. Amsterdam: North Holland. Boundary value problems in queueing theory. Queueing Systems, 3, Kylstra Ed. Ams- terdam: North-Holland. Queueuing Systems, 14, Probability in the Engineering and Informational Sciences, 7, A queueing model with two types of retrial customers and paired services.

Annals of Operations Research, 1 , A two class retrial system with coupled orbit queues. Probability in the Engineering and Informational Sciences, 31 2 , A queueing system for modeling cooperative wireless networks with coupled relay nodes and synchronized packet arrivals. Performance Evaluation , European Journal of Opera- tional Research, , Retrial queues.

Chapman and Hall. On a multiclass batch arrival retrial queue. Advances in Applied Probability, 20, Single line queue with repeated demands. Queueing Systems, 6, Boundary Value Problems. Oxford: Pergamon Press. A survey of retrial queueing systems.

Expected waiting times in a multiclass batch arrival retrial queue.