Performance Evaluation 70 (2013) 440–453 Contents lists available at SciVerse ScienceDirect Performance Evaluation journal homepage: www.elsevier.com/locate/peva Tandem queueing system with impatient customers as a model of call center with Interactive Voice Response Chesoong Kim a,∗ , Alexander Dudin b , Sergey Dudin b , Olga Dudina b a Sangji University, Wonju, Kangwon, 220-702, Republic of Korea b Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus article info Article history: Received 14 March 2012 Received in revised form 24 January 2013 Accepted 13 February 2013 Available online 24 February 2013 Keywords: Call center Interactive Voice Response Markovian arrival process Phase type service time distribution Impatient customers abstract A tandem queueing system with a Markovian Arrival Process (MAP ) useful in modeling a call center with Interactive Voice Response (IVR) is investigated. The first stage has a finite number of servers without buffer while the second stage of the tandem has a finite buffer and a finite number of servers. The service time at the first (second) stage has an exponential (phase type) distribution. A special approach for reducing the number of states of the stochastic process that describes the behavior of the system is used. The main performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution is derived. The numerical results are presented. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Call center is a centralized office used by companies for contact with its clients. To decrease load of operators in some call centers IVR (Interactive Voice Response) technology is used. IVR is a call center option that allows customers to interact with a company and serve their own inquiries via a telephone keypad without the involvement of an agent. Typically IVR is used in call centers of banks, insurance companies, travel agencies, mobile operators, etc. For example, if the subscriber of mobile network wants to know the balance of his(her) account, enable or disable the service, to find out some information about tariffs, he(she) can easily do it himself(herself) using IVR. If the customer cannot solve his(her) problem by using IVR and requires some assistance, he(she) can request to connect with an agent. In call centers with IVR the load of operator is lower, so the performance of call centers is higher. For modeling call centers, queueing theory is used. Adequate mathematical modeling call centers leads to a substantial increase of their economic efficiency, because exact prediction at the design stage can significantly reduce maintenance costs. Mathematical models allow to solve a problem of call center optimal design. For the references and the present state of art in investigation of call centers the reader is referred to survey [1], papers [2–5] and references therein. In [6], a call center with IVR is investigated. In this paper authors consider a two-stage queueing model with Poisson arrival process and exponential service time distribution at both stages with impatient customers and without balk of customers. Performance measures of this system are calculated approximately in an asymptotic Quality and Efficiency Driven (QED) regime. In our paper, we present an exact analytical analysis of the model without the restriction that the number of operators is large enough. We consider a tandem multi-server queueing system with a finite intermediate buffer and impatient ∗ Corresponding author. Tel.: +82 33 730 0464. E-mail addresses: dowoo@sangji.ac.kr (C. Kim), dudin@bsu.by (A. Dudin), dudin@madrid.com (S. Dudin), dudina_olga@email.com (O. Dudina). 0166-5316/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.peva.2013.02.001