Priority tandem queueing system with retrials and reservation of channels as a model of call center

https://doi.org/10.1016/j.cie.2016.03.012Get rights and content

Highlights

  • A tandem queue consisting of two multi-server stations is analyzed.

  • Retrials and impatience of customers are taken into account.

  • Ergodicity condition is derived.

  • Steady state distribution of the system states and performance measures are computed.

  • Numerical results showing significant effect of providing the priority are presented.

Abstract

We study a tandem queueing system consisting of two multi-server stations and finite intermediate buffer. Customers arrive at the first station of the tandem according to a Markovian Arrival Process. The first station does not have a buffer. Customers, who do not succeed to enter the service immediately upon arrival, retry for the service after a random amount of time. After service completion at the first station, a customer leaves the system permanently or moves for the service at the second station. A part of customers entering the second station have a priority over other customers. The priority is provided by means of reservation of a part of servers for the service of priority customers only. Usually, non-priority customers are not allowed to occupy the reserved servers. However, if the queue of non-priority customers in the intermediate buffer becomes larger than some preassigned threshold while there are free servers, a non-priority customer is picked-up for the service. Customers staying in the buffer are impatient. Priority and non-priority customers have different patience time and may leave the system or return to the first station if the patience time expires. The system is analyzed in steady state. A condition for existence of the stationary regime in the system is derived, the steady state distribution and various performance measures of the system are calculated, some illustrative numerical examples are discussed. A tandem queue under consideration is suitable, e.g., for modeling call centers with Interactive Voice Response Machines. Analysis presented in this paper was implemented in borders of the applied project funded by one of the banks.

Introduction

At the present days, call centers are complex socio-technical systems that play an increasingly important role in the modern world society. They are used to provide services in information and emergency centers, help-desks, tele-marketing, etc. To offer high quality services, call center managers and designers should consider the complex of the factors associated with arrivals of customers at random instants and a variety of customer requirements for quality of the service. Queueing models can be effectively used for call centers design and support of their management. The surveys of research works devoted to mathematical modeling of call centers can be seen in the papers by Aksin et al., 2007, Jouini et al., 2009, Jouini et al., 2010, Khudyakov et al., 2010 and references therein. In this paper, we consider a tandem queueing system that takes into account such important features of call centers as: possibility that a customer may need service from several sequentially located servers; the presence of multi-class customers with different Quality of Service (QoS) requirements; the impatience of customers and the retrial phenomena.

Tandem queueing systems can be used for modeling real-life queueing networks as well as for validation of general decomposition algorithms in networks. These systems have found great attention in the literature, since the practical importance and mathematical complexity of such systems make them attractive for researchers in the field of queuing theory and its applications, see, e.g., the papers by Balsamo et al., 2003, Perros, 1989.

Retrial queueing systems differ from the systems with waiting room or losses in the fact that a customer that does not succeed to get an access to the service facility immediately upon arrival neither enters a buffer, nor leaves the system permanently. He/she enters the so-called orbit (the virtual room for such customers) from which he/she makes repeated attempts to reach the service facility in a random amount of time. There are many publications in the field of queueing theory and telecommunications devoted to investigation of retrial queues. The state of the art in research in the field of retrial queueing systems is partially presented by Gomez-Corral, 2006, Artalejo and Gomez-Corral, 2008, Artalejo, 2010, Kim and Kim, 2016.

In the literature, there is only a relatively small number of publications dealing with tandem queues with retrials, although the effect of retrials is an integral part of many real telecommunication systems and ignorance of this effect may lead to significant errors in the design and evaluation of their performance. To the best of our knowledge, tandem queuing systems with retrials were considered only in the papers by Falin, 2013, Gomez-Corral and Martos, 2002, Kim et al., 2010a, Kim et al., 2010b, Klimenok and Savko, 2013, Moutzoukis and Langaris, 2013, Phung-Duc, 2012. The papers by Falin, 2013, Phung-Duc, 2012 deal with exponential retrial tandem queue consisting of two single-server stations without intermediate buffer. For this quite simple queue, the explicit formulas for steady-state probabilities are obtained. Moutzoukis and Langaris (2013) have considered the system with stationary Poisson input, single-server stations and constant intensity of repeated attempts from the orbit. In the work by Klimenok and Savko (2013), the tandem queue with stationary Poisson flow, two types of customers and reservation of channels for the priority customers at the second station is studied. Gomez-Corral and Martos (2002) deal with queueing system with Markovian Arrival Process (MAP), single-server stations and phase type service time distributions. In the paper by Kim et al. (2010b), more general model is under study. The arrival flow is a BMAP, the service time distribution at Station 1 is arbitrary. The paper by Kim et al. (2010a) is devoted to the tandem retrial queue with BMAP, single-server first station, an additional flow at multi-server second station and reservation of channels for the customers from this flow.

The phenomena of impatience of customers is an important feature of many telecommunication systems including call centers and contact centers. For references relating to application of queueing models with the impatience to analysis of call centers we refer the reader to the works by Dudin et al., 2013, Garnett et al., 2002, Jouini et al., 2010, Kim et al., 2013, Roubus and Jouini, 2013 and references therein. But we know only the paper by Klimenok and Savko (2013) where a tandem queue with retrials and the impatience of customers is dealt.

In the present paper, we consider a tandem consisting of two stations, Station 1 and Station 2. Station 1 is represented by a multi-server queue with retrials. A customer who finds all lines busy upon arrival joins the orbit and retries for the service after a random amount of time independently of other orbital customers. Such a behavior is typical for customers calling to a call center and receiving a signal that all trunk lines are busy.

All incoming calls at Station 1 (primary and repeated), can finish the service in the tandem system at this (first) stage or require an additional service at Station 2 (second stage). The first stage can be considered, e.g., as dial phase or IVR-Interactive Voice Response (for definition see the papers by Khudyakov et al. (2010) and by Dudin et al. (2013)). Unsatisfied clients are directed to the agents at the second stage to resolve their problems. Kim and Park (2010) propose also the following interpretation of two-stage service in a call center. All customers are initially handled by the agents at the first stage of call center (servers of Station 1) and a part of the customers can be satisfied with the service at this stage. However, the service of some customers cannot be completed by these agents due their limited authority and knowledge. These customers are transferred to experts at the second stage who can handle them.

Station 2 is represented by a multi-server queue with a finite buffer. We assume that all agents (servers of Station 2) are flexible enough to answer all requirements of service. But the customers to be served at Station 2 are divided into two different classes according to their ability to wait for the connection to the agent. We assume that the company that owns the call center provides preferences to high-valued clients who are more impatient while staying in the buffer. To prevent the loss of most of these clients, management of call center may decide that some group of agents will be reserved for service of high-valued (priority) clients only.

In this paper we study the operation of the system in steady state and discuss the question “how many agents should be reserved to provide the best QoS for various types of customers in the call center?”

The rest of the paper is organized as follows. In Section 2, the queueing system under consideration is defined. In Section 3, the multi-dimensional continuous time Markov chain, which describes the behavior of this system, is constructed. The generator of this chain is presented and the fact that this Markov chain belongs to the class of the asymptotically quasi-Toeplitz Markov chains is proved. In Section 4, the ergodicity condition of this Markov chain is derived. The algorithm for computing the stationary probabilities is outlined in Section 5. In Section 6, formulas for some key performance measures of the system are given. Numerical examples are presented and an optimization problem is formulated and solved in Section 7. Section 8 concludes the paper.

Section snippets

Model description

We consider a tandem queueing system consisting of two stations in series, Station 1 and Station 2. The structure of the system is presented in Fig. 1.

Station 1 is represented by the N-server retrial queue without a buffer. Station 2 is represented by (K+R)-server queue with a finite buffer of capacity M.

Customers arrive at Station 1 according the Markovian Arrival Process (MAP). The MAP is governed by some underlying process νt,t0, which is an irreducible continuous time Markov chain with

Paper contributions

Call center are quite popular subject of research and there is huge literature on this topic. So, it is necessary to stress distinguishing features of the considered model comparing to the existing in the literature. First of all, it should be mentioned that the majority of models in the literature assume that arrival flow is described by the stationary Poisson process, i.e., inter-arrival times are independent identically distributed random variables having exponential distribution. In this

Process of the system states

The process of the system states is described in terms of an irreducible continuous-time Markov chain ξt,t0, with state spaceX=X1X2X3whereX1=(i,n,r,m,ν),i0;n=0,,N;r=0,,K-1;m=0;ν=0,,W,X2=(i,n,r,m,ν),i0;n=0,,N;r=K,,K+R-1;m=0,,M;ν=0,,W,X3=(i,n,r,m,l,ν),i0;n=0,,N;r=K+R;m=0,,M;l=0,,m;ν=0,,W.The components of vectors representing the states from X have the following sense:

  • i is the number of customers in the orbit;

  • n is the number of busy servers at Station 1;

  • r is the number of busy

Ergodicity condition

In this section, we derive the sufficient condition for the stability (non-stability) of the queue under consideration that coincides with the condition of ergodicity (non-ergodicity) of the Markov chain ξt. The Markov chain ξt is ergodic if the limiting probabilities as t goes infinity do not depend on the initial distribution.

Theorem 1

(i) The Markov chain ξt,t0, is ergodic if the following inequality holds:λ+Nμ1p2(1-q1)r=KK+R-1xr,M+(1-q2)r=KK+R-1m=1Mxr,mmγ2+Nμ1(1-p0)(1-q1)l=0MxK+R,l(M)+(1-q2)m=1M

Stationary distribution

In the following we assume that ergodicity condition (8) holds. Denote the stationary probabilities of the states of the Markov chain ξt,t0, byp(i,n,r,ν)=limtPit=i,nt=n,rt=r,νt=ν,i0;n=0,,N;r=0,,K-1;ν=0,,W;p(i,n,r,m,ν)=limtPit=i,nt=n,rt=r,mt=m,νt=ν,i0;n=0,,N;r=K,,K+R-1;m=0,,M;ν=0,,W;p(i,n,m,m(2),ν)=limtPit=i,nt=n,rt=K+R,mt=m,mt(2)=m(2),νt=ν,i0;n=0,N;m(2)=0,m;ν=0,,W.

Let us enumerate the states of the chain ξt in the lexicographic order, and denote by pi the row vector of

Performance measures

In this section, we calculate a number of performance measures of the queue under consideration based on the steady state probabilities pi,i0.

  • Stationary distribution of the number of customers in the orbitpi=pie,i0.

  • Mean number of customers in the orbitLorbit=i=0ipie.

  • Stationary distribution of the number of busy servers at Station 1Nbusy(1)=P(IN+1e(K+RM+M)W)where P=i=0pi.

  • Mean number of busy servers at Station 1N(1)=Nbusy(1)diag{0,1,,N}e.

  • Intensity of output flow from Station 1μ¯(1)=μ1N

Numerical experiments

The purpose of this section is to give an example of numerical solution of optimization problem and to bring out some qualitative aspects of the queue under consideration. To this end, we present the results of three numerical experiments. In the first experiment, we show the dependence of the most important measures of the system, the loss probabilities of priority and non-priority customers, on the number of the reserved servers R and the threshold j. The goal of the second experiment is to

Conclusion

In this paper, a tandem queueing system with customers retrials and impatience is analyzed. The arrival flow is assumed to be a MAP what allows, in particular, to take into account correlation of inter-arrival times. Customers having higher impatience have a privilege in service at the second stage of a tandem. This privilege is ensured by means of providing non-preemptive priority in access to the service from a buffer and by means of reservation of a part of servers at the second stage

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2014K2A1B8048465) and by Belarusian Republican Foundation of Fundamental Research (Grant No. F15KOR-001).

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