American Journal of Operations Research, 2013, 3, 570-580 Published Online November 2013 (http://www.scirp.org/journal/ajor) http://dx.doi.org/10.4236/ajor.2013.36054 Open Access AJOR A Two Stage Batch Arrival Queue with Reneging during Vacation and Breakdown Periods Monita Baruah 1* , Kailash C. Madan 2 , Tillal Eldabi 1 1 Brunel Business School, Brunel University, London, UK 2 College of Information Technology, Ahlia University, Manama, Bahrain Email: * Monita.Baruah@brunel.ac.uk, monitabrh@gmail.com, kmadan@ahlia.edu.bh, Tillal.Eldabi@brunel.ac.uk Received September 30, 2013; revised October 30, 2013; accepted November 7, 2013 Copyright © 2013 Monita Baruah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We study a two stage queuing model where the server provides two stages of service one by one in succession. We consider reneging to occur when the server is unavailable during the system breakdown or vacation periods. We concentrate on deriving the steady state solutions by using supplementary variable technique and calculate the mean queue length and mean waiting time. Further some special cases are also discussed and numerical examples are pre- sented. Keywords: Two-Stage Service; Batch Arrivals; Breakdowns; Reneging; Steady State Queue Size 1. Introduction Queues with impatient customers have attracted the at- tention of many researchers and we see significant con- tribution by numerous researchers in this area. One of the earliest works on balking and reneging was by Haight [1,2], Barrer [3], which was the first to introduce reneg- ing in which he studied deterministic reneging with sin- gle server markovian arrival and service rates. Another early work on markovian reneging with markovian and arrival and service pattern was by Ancker and Gafarian [4], Haghighi et al. [5] studied a markovian multiserver queuing system with balking and reneging. Consequently, we see a lot of developments in the study of queues with impatient customers in recent years. An M/G/1 queue with deterministic reneging was studied by Bae et al. [6]. We refer to some authors like Zhang et al. [7], El-Pau- omy [8], Altman and Yechiali [9,10], Kumar and Sharma [11] who studied queues with impatient customers in different contexts. Vacation queues are an important area in the literature of queuing theory. Since the past two decades it has emerged as an important area of study due to its various applicability in real life problems such as telecommuni- cation engineering, manufacturing and production indus- tries, computer and communication networks etc. A few of early works on queues with vacations are seen by au- thors like Levy and Yechailai [12], Doshi [13], Keilson and Servi [14]. A two stage batch arrival queuing system where customers receive a batch service in the first and individual service in the second stage was studied by Doshi [15] in the past. In recent years, extensive amount of work has been done on batch arrival queues with va- cations and breakdowns. We mention a few recent papers by Kumar and Arumuganathan [16], Choudhury, Tadj and Paul [17], Maraghi, Madan and Darby-Dowman [18], Khalaf, Madan and Lukas [19]. In this paper we consider a batch arrival queue where service is offered in two stages of service, one by one in succession. We extend and develop this model by adding new assumptions reneging and system breakdowns. Cus- tomers may renege (leave the queue after joining) during server breakdowns or during the time when the server takes vacation due to impatience. This is a very realistic assumption and often we come across such queuing si- tuations in the real world. 2. The Mathematical Model a) Customers arrive in batches following a compound Poisson Process with rate of arrival λ. Let ( ) d 1, 2, 3, i c t i λ =  be the first order probability that customers arrive at the system in batches of size i, at the system at a short interval of time ( ] , d x x t + , where * Corresponding author.