Abstract— We consider a finite buffer queueing model with several key features of call centers, such as retrials, feedbacks, and impatience. In addition, because we do not completely understand the customer impatience behavior, we use a general distribution for the maximum waiting time before abandoning the call. We develop a QBD process with infinite state space for the queue in a call center situation. To solve for the stationary performance measures, we introduce an effective approximation method, and numerical examples have been presented to show the effectiveness of our method. Index Terms— Multi-server queues, call centers, retrials, feedbacks, QBD process. I. INTRODUCTION Queueing models are the main quantitative technique in evaluating the operating performance of call centers. There are three common characteristics in the customer’s (or caller’s) behavior: (1) a customer may try to call again if he or she gets a busy signal; (2) for a customer on hold, if his or her waiting time reaches a limit, he or she will hang up and leave; and (3) a customer may call again if his or her problems are not solved completely after a service (see [1], [14]). Therefore, we present a queueing model with customers’ retrials, feedbacks, and impatience. In addition, to realistically model call centers, we assume a finite buffer to hold the waiting customers. There are many works on queueing models for call centers due to the recent and rapid growth of this industry. For retrial queueing models, most existing studies are on queues with one or no waiting spot (see [10]). Another class of queueing models is the multi-server queue with both customer retrials and impatience. Most studies in this class focus on models with only one or two servers (see [2], [3], and [4]). Since the retrial models with many waiting spots and/or multiple servers usually require the infinite state space Quasi-Birth-and-Death (QBD) processes, it Manuscript received.. Zhe George Zhang and Peter Haug are with the Western Washington University, Bellingham, WA 98225 USA: 360-650-2867; fax: 360-650-4844; (e-mail: george.zhang@ wwu.edu). Yi-Jun Zhu and Ren-Xiang Zhu are with 1 Science Faculty, Jiangsu University Zhenjiang Jiangsu 212013, China (e-mail: yjzhu@ujs.edu.cn). is extremely difficult or even impossible to obtain the stationary performance measures of the systems. However, to quantitatively evaluate the performance of practical call centers, we need to consider these difficult models. There are mainly three types of methods to solve the QBDs for call center models. Type 1 is to formulate a QBD process with a special transition probability matrix structure where the entries become the same after a certain level (see [5]). For this kind of QBDs, a matrix geometric solution can be obtained in terms of a rate matrix which can be evaluated using a numerical method. Type 2 is to use the state space truncation to convert infinite state models to finite state ones which can be solved (see [6], [7]). Type 3 is to approximate the original infinite QBD model by another infinite one which is solvable (see [8], [9], and [10]). The model of this paper has not been studied via QBD approach in the past. We formulate the QBD process for a call center system with all its main features. To solve the QBD process, we proposed a method of Type 3 in which the original model is approximated by a simpler and solvable QBD process. Then, the stationary performance measures of our original model can be obtained via this easier-to-solve QBD process. Numerical examples have been used to show the effectiveness and efficiency of our method. The rest of the paper is organized as follows. In section 2, we formulate a QBD process model for the queueing system with the main features of call centers. In section 3, we present the approximation method to solve the QBD process model and give some useful performance measures. In section 4, we provide some computational results to discuss the effectiveness and the efficiency of the approximation method and conclude the paper with a summary. II. MODEL FORMULATION – A QBD PROCESS Consider a queueing system with a waiting and service area and a retrial area (see Figure 1). In the waiting and service area, there are s servers and s k − < ∞ waiting spots (or the system can hold a maximum of k waiting and in-service customers.) We assume that customers arrive to the system according to a Poisson process with rate λ , and the service time is exponentially distributed with rate μ . The service discipline is a “first-come-first-served” (FCFS) sequence. An after-service customer may enter the retrial area and call again for further service. This behavior is called the feedback. The feedback probability is β <1 and the probability of leaving the system then is ) 1 ( β β β − = . A Performance Analysis of Call Centers Based on a Multi-server Queue with Retrials, Feedbacks, and Impatience Yi-Jun Zhu, Ren-Xiang Zhu, Zhe George Zhang, Peter Haug IAENG International Journal of Applied Mathematics, 37:1, IJAM_37_1_10 ______________________________________________________________________________________ (Advance online publication: 15 August 2007)