A brief review on retrial queue: Progress in 2010-2015

Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No. 4, 2016 © 2015 by the authors – Licensee IJASER- Under Creative Commons License 3.0 Research article

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A brief review on retrial queue: Progress in 2010-2015 Shekhar C1, Raina A.A2, Kumar A3 1,3 Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan 333 031 India 2 Department of Mathematical Sciences, BGSB University, Rajouri, J & K 185 234 (India). DOI: 10.6088/ijaser.05032 Abstract: The present survey is based on the principle of retrials in queueing theory i.e. the customers who repeatedly try to avail the services from the server, on finding dejection, because of the inactivity of the server to provide service at an instant of selection from the queue. A bunch of work has previously been done on this front from past since it is realistic phenomena of many service system. The current investigation aims to present a brief review of the major works done on retrial queueing systems in recent years. The bibliography consists of research articles which were published in journals of repute during the period 2010-2015. In addition to the above references, an exhaustive list of books and survey papers on retrial queueing systems is also prepared so as to provide a detailed enough catalog for further understanding and research in retrial queueing domain. We have classified the journal papers according to the year of publishing. The aim of the present study is to turn up at a broad enough for a collection of important results in the theory of retrial queueing systems and their applications in solving several realistic problems. The main contribution is the bibliography of recent papers and books collected from different databases and private sources. At last, we have arranged some of the references according to the classification based on analysis and solution technique. The author(s) hope that this survey paper could be of help to learners contemplating research on retrial queues. Keywords: Queueing theory; Retrial queue; Priorities; Communication; Computer networks; Maintenance policy.

1. Introduction 1.1 Queueing theory – An outline Since the past few decades, it has come to the fore that in most cases, it is the probabilistic models that score an edge over the deterministic models in terms of practical applicability. Most of the probabilistic models find their extensive applications in broader domains of common fields such as statistics, operations research and in non-trivial areas such as applied industrial research. One of the most predominant areas where probabilistic models have been exploited for the good is Queueing theory, a branch of applied probability theory which deals with an in-depth study of various service systems plagued by congestion. Broadly speaking, a queueing system involves a chain of events such as the arrival of customers for availing service, waiting for the provision of service in case provision of service is not immediate, and departure from the system after having received the service. Some of the crux questions that queueing theory tries to address right from the outset are finding the mean waiting time in the queue, the mean system response time, mean utilization time of service, etc and all of the findings are based upon existence of stochastic scenario, where the various times such as service time, inter-arrival time etc are all assumed to be random in nature. To cater to broad and detailed coverage of queueing theory, a list of introductory and some advanced textual references can be found in the bibliography. However, having said that, there are some very basic and foundational texts such as (Bocharov et al. (2004), Cooper (1981), Giambene ————————————— *Corresponding author (e-mail: [email protected]) Received on January, 2016; Published on August, 2016

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(2005), Gross and Harris (1998), Kleinrock (1975), Newell (1971) and Saaty (1961)) which can help beginner level researchers, to explore the field of Queueing theory in a streamlined manner.

1.1.1 Basic Queueing Structure

Figure 1: Basic Queue Structure

A basic queuing process comprises of several sub-processes such as: (1) Process of Arrival of customers or users (2) Order of access of service by users after joining the queue (3) Service availing process and departure thereafter from the student (a) Arrival denotes the number of customers or users on average who want to avail the service with stipulated time. (b) The size of the queue at any given moment is the number of customers present in the queue. Potential customers include people, inventory which is itself a work in progress, raw materials or any other entity which can be modeled to wait for availing some service. Also, note that size of a queue can be finite or infinite. (c) A server can be thought of as a service vendor cum processor which carries out the execution of some process for customers in waiting for service to be availed. The inception which brought queueing theory in limelight dates back to the publication of Erlang’s paper "The theory of probabilities and telephone conversations" in 1909 which was meant for study of telephonic traffic congestion. Erlang succinctly delineated the concept of statistical equilibrium which deals with constant service times and arrivals following a Poisson distribution. Queueing theory has found umpteen numbers of applications in a plethora of fields such as in banking industry, in booking offices, automating machines, telephonic networks, railways, elevators, travel and transportation industry etc. All these applications are associated with design, planning, and execution of service facilities to assemble randomly fluctuating demands for service so as to minimize the overcrowding and to strike a fine balance between trade and industry to ensure a balance between service cost and the cost associated with waiting time. The early fifties was one of the most fertile periods of improvement in queueing theory. Mathematicians of this time propelled developments in the field of queueing theory to an altogether a different level and established queueing theory as a remarkably fruitful field of mathematics which yielded applications, results, and methods in probability theory. Undoubtedly, to understand queueing theory requires a fair amount of background in statistics and probability theory. Although there exist a multitude of texts to establish a decent level of grounding in queueing theory, but we mention two very popular modern texts which are (Cohen (1969) and Cooper (1990)) besides which these are other classics such as (Takacs (1962), Trivedi (1982), Walrand (1988) and Wolff (1989)).

1.1.2 Areas of application Since the time, the first telephonic systems made their intervention in human lives; queueing theory has 325 Shekhar C et al., Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No.4, 2016

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been veritably providing the background research for understanding the working and functioning of these systems. These systems remained very much the talk of the town throughout the 1950s. The paradigm shift in applications and other applied research work pertaining to queueing theory started after the Second World War. It was during this time that another, a relatively new branch of applied probability theory, namely “reliability” too came in existence and active developments in this new branch too came into the light of mathematical literature. The oft-so used “machine interference model” (which is a special case of queueing and reliability), that is predominantly prevalent in today’s statistical parlance was also developed during this time. Another discovery that too came around during this time was that there was a much easier and reasonable way of understanding and formulating models of the reliability of seemingly complex systems and that was, to think in terms of queues (arrivals of breakdowns and repair services).There is significant enough intersection in the aforementioned areas of applied probability theory and more or less, the same statistical underpinnings, techniques, and procedures can be followed for gaining a deeper insight into those. With the advancement in the modeling of computer systems and rapid progress in the area of data transmission systems in the 1960s, the study of queues became easier and it started getting characterized by complex service systems, hence paving the way for the need to understand, analyze interconnected systems. Since then, there has been an incrementally speedy progress in this area and the number of industrial applications to has multiplied since the 1970s. Queueing networks have always been an essential component in the study of communication systems. The extensive induction of computers into these systems has paved the way for use of new results in queueing networks into the studies of the performance of large communication networks.

1.2 Retrial queue – An outline Queueing systems in which arriving customers who repeatedly try to avail the services from the server, on finding dejection, because of the inactivity of the server to provide service at any instant of selection from the queue are known as retrial queueing systems or queues with repeated attempts. The customer is in “orbit” whenever it is in between the retrials. The customers, in retrial queueing systems, may wait in the orbit and while being in the orbit, they can try as many times as they want, to avail service from the same server. The basic structure of a retrial queue is shown in Figure 2.

Figure 2: Basic Retrial Structure Immediate example to gain an intuition of working of retrial queueing is a case wherein, a person wishes to 326 Shekhar C et al., Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No.4, 2016

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make a phone call across a busy line of the telephone network. If the line is busy, then he has no option but to try repeatedly to reach to the point where his call actually gets passed through the networking line. Thus, there is a unique standout feature of retrial queueing systems and that is, a customer who arrives and enters into the service system, on finding that all service providing servers are busy, decides to depart from the service area but after some time comes back again to seek his demand for service. In a way, retrial queues can be spoken of as a network which has re-servicing capabilities after crowding beyond capacity. Thus, a retrial network has two major constituent nodes: the main node where blocking is possible and a delay node for repeated attempts. As far as other networks with blocking are concerned, there are too many logical difficulties involved which hinder their in-depth comprehension and extensive development. Nevertheless, the salient features of the theory of retrial queueing systems as an independent domain of queueing theory are quite clearly laid out. Broadly speaking from the point of view of various results that have been obtained so far, the prevalent and currently existing methods of analysis and the vivid areas of applications in which retrial queueing systems are used, retrial queues can be divided into following three major groups: (a) Uni-channel systems, (b) Multiple-channel systems, (c) Complex systems. Though standard queueing models are widely applicable but, because these models don’t take into account the happening of recurring retrials, therefore, standard queueing models cannot be thought of as a solution for a plethora of practically crux conundrums. To address this very issue, Retrial queues have been roped in to iron out this deficiency. Of late, there has been an unprecedented surge in the development of retrial queueing systems, since they are closer to how most of the queuing processes occur in daily lives of human beings. As far as relevant literature pertaining to retrial queueing systems is concerned, it is highly imperative to focus on specific results that appeared in the 1950s, and survey papers that were published by Yang and Templeton (1987), Falin (1990, 1986), Kulkarni and Liang (1997), Artalejo (1998, 1999a, 1999b, 2009, 2010), Artalejo and Falin (2002), Gomez–Corral (2006) and so on. Most recently Kim and Kim (2015) provided the detailed survey on retrial queueing systems. These papers constitute a very exhaustive source of references on this topic. For more detail on sources to study retrial queueing systems, the curious reader is advised to look up at the texts mentioned in section 4.

1.3 Examples The present section delineates some pertinent examples of systems which can be modeled as retrial queues: (a) Telephone systems: One of the frequently used ways to model the commonly and widely used telephone systems is to model them as retrial queues. Because of the recurring trials of a telephonic caller, who on finding his call in congestion network does not get the service, comes back and makes a call again, all this makes telephone systems a case of retrial queues. The above resemblance of the way in which telephone systems and retrial queues work provides enough considerations for focusing on the need of development of sound methods and ways to develop a theory of retrial queues for in-depth understanding 327 Shekhar C et al., Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No.4, 2016

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& modeling of customer behavior patterns in classical telephone systems. With recent advancements in telecommunication sector happening at an electric pace, all this has led to an ever-increasing need to develop and extend the retrial phenomenon. Also the fact that under the overload conditions, most telephone systems perform poorly, it becomes all the more important to take a step back and relook at the models that we develop while modeling the telephone systems on retrial theme. Hence, in the remainder of this sub-section on telephone systems, we mainly focus on the role of retrial queueing systems with respect to call centers and cellular mobile networks. The prime goal of any call center is to deliver quality service by means of a device, in our case that device being a telephone. As far as modeling a call center is concerned, thinking in terms of retrial queue can provide us a glaring qualitative insight. Also, when carefully researched upon the structure, planning, management and execution of a typical call center, it becomes immediately clear as to how modeling telephone systems as retrial queues best explains the customer behavior in such a scenario. Almost all the major players in the telecom industry use call centers as their prime way of communication and interaction with their customers. To get a perspective, a call center can be thought of as a basic queueing structure based on M/M/C queue (one of the most widely used queueing model). As performance metrics on which call centers are evaluated, quality of service and speed in times of network congestion are the two most prominent evaluation criteria’s, and it is here that efficient call handling mechanisms can improve the classical telephone systems. Thus, we realize that while modeling a telephone network or mobile cellular network, one cannot ignore the inherent existence of recurring calls. All the above-discussed factors vouch strongly enough in favor of a new queueing system which rightly is the retrial queueing system. (b) Computer networks: The underlying rules governing the operations of random access protocols in computer networks provide a sound enough background and motivation at the same, for design of communication protocols with new feature and that is allowing the protocols have to have retransmission control which is nothing but thinking of protocols to have retrial capabilities in the queueing domain. To understand it a bit more clearly, let us take an example of a communication line which has slotted time which is evenly shared among several stations. The slot duration equals the total time taken during the transmission of a single packet of data. A clash occurs whenever two or more stations are transmitting packets together. A clash will always result in damaged packets across the entire transmission line and these damaged packets of data must now be retransmitted. The immediate quick fix can be to allow the clashing stations to retransmit the damaged packets in the nearest available time slot but then such a fix is bound to run into a clash again, hence again leading to damaged packets and again a need for retransmission again. To avoid above mentioned complicated situations, we can allow each clashing station to introduce a random delay before the next challenge to transmit the packet. This lucid yet layman enough explanation justifies the need for incorporation of retrial feature in computer networks. (c) Priorities: In order to accommodate modeling of complex real life scenarios in which, in addition to usual queue, there is a need of a prioritized queue, then we go for an alternate implementation of retrial queueing system which is nothing but prioritized retrial queueing system. Prioritized retrial queueing systems are characterized by the prevalence of two types of customers: firstly, the primary customers who arrive as per independent streams and secondly the customers who previously were in the usual queue, making them the low priority entities in the new queue set up. In such a revised retrial queueing system, the primary customers are treated as a first preference and hence given higher priority and these customers 328 Shekhar C et al., Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No.4, 2016

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are queued first and served too according to some order of service. Due to the existence of priorities, any case of blocking between the first stream of customers and the second stream of customers will be dealt to accommodate interests of higher priority customer. Hence, in the case of blocking, it is the low priority customer who has to leave the service area and do the task of waiting until subsequent retrials happen to lead to an extension of waiting and service time for the low priority customer. As is evident now, higher priority customers in case of prioritized retrial queueing systems enjoy a non-answerable authority and priority over the low priority customers. The most commonly prevalent and succinct example to visualize the importance of such a prioritized retrial queueing system is a case of a large hospital where a higher order priority queue is always kept for emergency cases or for patients with special needs, even though a usual queue is always the order of the day to accommodate interests of the usual population arriving at the hospital for availing the treatment services.

1.3 Computational methods The following computational methods provide efficient implementation of retrial queueing models: (a) Direct truncation method (b) Generalized truncation method (c) Truncation method using level dependent quasi-birth-and-death process (LDQBD) (d) Matrix-geometric approximation. In this introductory section, the emphasis is on capturing the fundamentally predominant role of the retrial queueing theory in classical telephone systems, call centers, mobile telephone systems and local computer networks.

2. Classifications on Analysis and Solution Technique 2.1 Matrix Method Dudin et al.(2015a), Shin (2015), Dudin et al.(2015b), Vijaya and Soujanya (2015), Jain et al.(2015), Phung-Duc (2015), Kuo et al.(2014), Phung-Duc and Kawanishi (2014), Gomez-Corral and Garcia (2014), Rabia (2014), Kim et al. (2014), Shin and Moon (2013), Do et al.(2013), Kim et al.(2012), Kim and Kim (2012), Do (2011), Ke et al.(2011), Efrosinin and Winkler (2011), Wu et al.(2011), Gharbi and Dutheillet (2011), Kim et al.(2010a), Kim and Kim (2010), Kim et al.(2010b), Kim et al.(2010c), Kim et al.(2010d), Shin and Moon (2010), Do (2010a), Do (2010b), Do and Chakka (2010), Do (2010c), Lin and Ke (2011), Liu and Song (2012), Cordeiro and Kharoufeh (2012), Krishnamoorthy et al.(2012), Li et al.(2012), Kumar et al.(2013), Kim et al.(2015).

2.2 Generating function Dimitriou (2015), Gao and Wang (2014), Artalejo and Phung-Duc (2013), Wu and Lian (2013a), Wu and Lian (2013b), Dimitriou (2013), Wu et al.(2013), Deepak et al.(2013), Ammara et al. (2013), Arrar et al.(2012), Wang et al.(2011), Shin and Moon (2011), Wu and Yin (2011), Amador and Moreno (2011), Wu et al.(2011), Kumar et al.(2010), Falin (2010a), Dimitriou and Langaris (2010), Langaris and Dimitriou (2010), Falin (2010b), Choudhury et al. (2010), Atencia et al. (2010), Wang et al.(2011), Gao et al. (2012), Wang (2012), Choudhury and Deka (2013), Wang et al. (2013), Jain and Bhagat (2015), Jain et al. (2015). 329 Shekhar C et al., Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No.4, 2016

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2.3 Supplementary variable method Wu and Yin (2011), Gao (2015), Haridass and Arumuganathan (2015), Rajadurai et al.(2014), Gao et al.(2014), Gao and Wang (2014), Jailaxmi et al.(2014).

2.4 Recursively Shin and Moon (2014), Avrachenkov and Yechiali (2010), Dragieva (2013), Zhang and Wang (2013).

3. Classifications on Key Features The extensive research on retrial queue has been done in past due its advances in socioeconomic issues involved in day to day life and computational techniques. In table 1, some keynote works of pioneer researchers, mathematicians etc with different key features are tabulated to enrich the present literature. Authors

Key Features

Avrachenkov and Yechiali (2010)

Retrial networks; Mean value analysis; Fixed point approach.

Kim and Kim (2011)

M/G/1 retrial queue; Waiting time; Laplace-stieltjes transform.

Liu and Gao

Discrete-time queues; Retrial queues; Markov chain; Bernoulli feedback.

(2011)

Wu et al. (2011)

Retrial G-queue; Markovian arrival process (MAP); Phase-type service; Markovian random environment; Finite number of sources.

Zhang and Hou (2011)

Retrial queues; G-queues; Random vacations; Stability condition; Markov chain; Reliability.

Arivudainambi

and

Retrial queue; Two phases of heterogeneous service; Bernoulli feedback; Optional vacations.

Godhandaraman (2012) Artalejo

and

Lopez-Herrero

Retrial queue; BSDE approach; Limiting distribution; Waiting time.

(2012) Choudhury and Ke (2012)

Stationary system size distribution; Random breakdown; Delay time; Repair time; Retrial time; Reliability indices.

Khodadadi and Jolai (2012)

Retrial queue; Vacation; Fuzzy inference; Threshold policy; Threshold accepting; Simulation.

Jain and Bhagat (2012)

Finite retrial queue; Geometric arrivals; Finite capacity; Threshold recovery; Markovian system; Transient solution.

Dudina et al. (2013)

Call centers; Retrial customers; Markovian arrival process; Phase-type service time distribution.

Kim and Kim (2013)

M/PH/1 retrial queue; Waiting time distribution; Laplace–Stieltjes transform; Coupling method.

Wang and Zhang (2013)

Queueing; M/M/1 queue; Balking; Equilibrium strategies; Threshold strategies; Retrials.

Zhang and Wang (2013)

Busy period; Interruptions; Quasi-random input; Recovery times; Retrial queues.

Avrachenkov and Morozov (2014)

Retrial queue; GI/GI/c/K type queue; Constant retrial rate; Stability conditions; Regenerative approach.

Bhulai et al. (2014)

Monotonicity; Processor-sharing queue; Retrial queue.

Choudhury and Ke (2014)

Stationary system size distribution; Random breakdown; Delay time; Repair time; Retrial time and reliability indices.

Do et al. (2014)

Retrial queue; Working vacations; Negative customers.

Bouchentouf et al. (2015)

Queueing system; Call center, Retrial queue, Abandonment, Feedback.

Bagyam et al. (2015)

Bulk; Arrival; Retrial; Queueing model.

Choudhury et al. (2015)

Random breakdown; Reliability function; Repair time; Stationary queue size distribution; Second phase of service.

Kim and Kim (2015)

Retrial queue; Markov process; Stability; Lyapunov function.

Kim and Kim (2015)

Stability; Two-class two-server retrial queue; Markov process.

Table 1: Some advanced key features 330 Shekhar C et al., Int. Journal of Applied Sciences and Engineering Research, Vol. 5, No.4, 2016

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4. Conclusions The present survey paper discusses the areas of application from the point of view of retrial queueing systems. After a basic description of underlying queueing structure which every queueing theoretic model entails, a modern and very recent branch of the queueing system, namely the retrial queueing system has been brought to attention. The survey paper identifies the potential areas of application such as call centers, mobile cellular networks, computer networks where efficient retrial queueing systems can play a pivotal role in delivering quality service in quick time. Along with that, the survey paper has thrown light on some of the most popular and timeless classic texts in the field of queueing theory which early stage researchers in the field of operations research can explore for further fruitful research. For the descriptions of other practical applications, readers may look into the exhaustive list of references. The authors have tried their best to provide as exhaustive a list of texts, description, and other relevant concerned paraphernalia as concerned, and it is believed that this survey paper will be of immense help to learners who are eyeing on research in the domain of queueing systems with a specialized focus on retrial queueing systems.

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