Performance estimation of an email contact center by a finite source discrete time Geo/Geo/1 queue with disasters

Abstract

In this paper, we model and analyze an email contact center as a queuing system. This queue system is a finite source discrete time Geo/Geo/1/∞/N queue with disasters. The arrival process of ordinary customers (OCs) and disasters and service process are mutually independent and geometrically distributed. Disasters in this system cause the system to be empty at once. The main contribution of the paper is to model a clearing mechanism in a real finite source system with the aid of disasters. The system states are a Markov chain and we develop a numerical scheme to compute the steady state probabilities of the system. Some of the performance measures of the system are calculated. At the end numerical illustrations show how sensitive the system performance measures are when parameters of the system change.

Introduction

In recent years, great attention has been paid to discrete time systems because of their ample application in communications. A fundamental motivation for the study of discrete-time queues is that these models are appropriate for modeling computer and telecommunications systems since the basic units are digital. In this sense, we refer to the books by Bruneel and Kim (1993) and Woodward (1994). Besides the communications applications of discrete time systems, these systems can be used in many cases such as traffic transportation, production and other stochastic systems. In Takagi (1993), discrete-time Geo/G/1/K/N systems are analyzed. To the best knowledge of the authors, this book is the most comprehensive one on this special topic in the existing literature. The assumption of geometric arrival and especially service time is a simplification of the system. Some of discrete time works that benefit this simplification are Artalejo and Hernández-Lerma, 2003, Atencia and Moreno, 2004, Atencia and Moreno, 2005, Chaudhry et al., 2004 and Gao, Wittevrongel, and Bruneel (2004).

In many practical situations, it is important to take into account the fact that the rate of generation of new primary calls decreases as the number of customers in the system increases. This can be done with the help of finite source models. Finite source queuing models are systems in which there is a limited number of OCs who use the service offered by the system. Two widely known applications of these models are machine interference problems (Mittler & Kern, 1997) and computer-communications systems (see Alfa and Sapna Isotupa, 2004, Drekic and Grassmann, 2002, Falin and Artalejo, 1998 and Sztrik, 2001, for continuous time cases and Artalejo and Lopez-Herrero, 2007, Minh, 1977 and Mittler & Kern, 1997, for discrete time ones). For finite source queuing systems applications and bibliography of related papers see Sztrik (2001).

A disaster kills simultaneously all the customers in the system. Disaster has no effect on an empty system. Disaster is also called catastrophe, mass exodus (Jain & Sigman, 1996) or queue flushing (Chen & Renshaw, 1997). Queue models with disasters can be used to analyze computer networks with virus infections and breakdowns due to a reset order. Disasters attendance in a queuing system can also be considered as a type of clearing mechanism which removes all workload in the system whenever it occurs to the system. That is, the arrival of a disaster not only destroys all the unfinished work but also breaks down the server. We can think of a disaster as a server reset or unplug which causes all the jobs in the system to be lost. The clearing mechanism of disasters can be applied to computer server systems in the presence of a virus as a clearing operation of all stored messages present in the system.

The presence of disasters in queuing systems was introduced by Jain and Sigman (1996). In the literature related to continuous time queues, numerous papers (Artalejo and GQomez-Corral, 1999, Chen and Renshaw, 1997, Jain and Sigman, 1996, Li and Lin, 2006, Towsley and Tripathi, 1991 and Wang, Liu, & Li, 2007) have recently appeared in which a disaster removes all the work present in the system. Disasters were applied to different single-server queues (Artalejo and GQomez-Corral, 1999, Chen and Renshaw, 1997 and Jain & Sigman, 1996) recently investigate an M/G/1 retrial queuing system with disasters and unreliable server. Li and Lin (2006) analyze an M/G/1 processor-sharing queue with disasters by means of extending the supplementary variable method. Nevertheless these studies only focused on the continuous time cases. As far as we know, Atencia and Moreno (2004) for the first time extend this topic to the discrete-time systems. Based on our best knowledge, the existing literature on discrete-time finite source queues does not cover the contribution of promoting disasters in the system. Our work differentiates itself by modeling a clearing mechanism in a queue whose arrivals come from a finite population. Existing literature on finite population discrete time queuing systems does not address this topic. We use this new model to approximate analysis an email contact center described in detail in next section.

Later, initially in Section 2, an email contact center is described as a queuing system. A description of the system model, assumptions and parameters is presented. In Section 3, steady state probabilities of the system are obtained in a efficient numerical approach and in Section 4, some system performance measures are computed. In Section 5, some numerical examples are given to illustrate the analysis and how sensitive the system performance measures are when parameters of the system change.