International Journal of Computer Applications (0975 – 8887) Volume 141 – No.7, May 2016 1 Analysis of M X /M/1/MWV/BD Queuing Systems K. Julia Rose Mary, PhD Asso. Prof. of Mathematics Nirmala College for Women Coimbatore-18 J. Maria Remona M.Sc Mathematics Nirmala College for Women Coimbatore-18 R. Rajalakshmi M.Sc Mathematics Nirmala College for Women Coimbatore-18 ABSTRACT In this paper, the batch arrival M X /M/1 queuing system along with server breakdowns and multiple working vacations is analyzed under exponential distribution. For this model Stochastic Decomposition is obtained and particular cases are evaluated. Further numerical illustration is also given to justify the validity of the model. Keywords Batch Arrival, Multiple Working Vacations, Breakdowns, Probability Generating Function(PGF), Stochastic Decomposition. 1. INTRODUCTION A classical queuing system may be described as one having a service facility at which units of some customer arrive for service and whenever there are more units in the system than the service facility can handle simultaneously, a queue or waiting line is developed. The waiting units take their turn for service according to as pre assigned rule and after service they leave the system. The study of classical queuing models are made by Saaty (1961), Gross & Harris (1985) and Medhi (2006). The batch arrival is described as the flow of arrivals in batches. Gaver (1959) introduced bulk arrival queues, where the arrivals could be in batch. Choudhury and Templeton (1983) and Medhi (1984) discuss the subject at great length. In N-policy the server does not start his service until there are N-customers in the queue. This policy is introduced by Yadin and Naor (1963) and is designed to minimize server switch over’s and to avoid excessive frequent use of setups. Lee and Srinivasan (1989), Lee et al., (1994 and 1995) studied the behavioral characteristics of batch arrival queues with N- policy and server vacations. Lee et al., (1994) successively combined the batch arrival queues with N-policy. Queuing systems with server classical vacations are characterized by the fact that the idle time of the server may be used for other secondary jobs. Allowing server to take vacation make queuing models more realistic and flexible in studying real world queuing situations. Applications arise naturally in call centers with multi task employees, maintenance activity, production and quality control problems etc.,. In N-policy queuing models, with server vacation, as soon as the system empties, the server leaves the system for a vacation of random length. When the server returns from the vacation and finds N or more customers, he immediately starts his service. Otherwise he takes repeated number of vacations until he finds N or more customers. This policy is called a Multiple Vacations Policy. Most of the classical queuing systems the server may fail and can be repaired. The performance of the system may be affected heavily by these breakdowns and limited repair capacity. Queuing systems with such unreliable stations are the topics of worth investigating from the performance prediction point of view. As a result of breakdowns, service facility becomes inoperative and the units demanding service can be served only when it is restored to operative state. Wang (1995) first proposed Markovian queuing system with removable service station. Ke J.C (2003) considered the control policy for batch arrival M X /M/1 queuing system under N-policy in which the server is characterized by breakdowns and multiple vacations. In working vacation queues, the server works at a lower service rate rather than completely stopping service during the vacation period. At the vacation termination epochs, if there are customers in the system, the server will start a new regular busy period. Otherwise, he takes another working vacation which follows multiple working vacations policy. In 2002, Servi and Finn, introduced a class of semi vacation policies, in which servers work at a lower rate rather than completely stopping primary service during vacation. Such a vacation is called working vacation (WV). Tian et al., (2008), Li and Tian (2007), Xu et al., (2009) considered M/M/1 queue with different working vacation policies. Xu et al., (2009) studied the results of Liu et al., (2007) to bulk input model M X /M/1/MWV. They have formulated the model as two dimensional Markovian chain and obtained the PGF of the stationary queue length and its stochastic decomposition result using the matrix analysis method. Their concept is motivated to combine the batch arrival queues under server breakdowns and multiple working vacations. In this paper, with the help of available literature a batch arrival M X /M/1 queuing system along with server breakdowns and multiple working vacations is analyzed under exponential distribution. The probability generating function (PGF) of the system size is obtained through the Chapman-Kolmogorov balanced equations satisfied by the steady state system size probabilities. The PGF is presented in closed form so that various performance measures are calculated easily. With the aid of PGF stochastic decomposition is obtained. Further particular cases are evaluated and sensitivity analysis is discussed. 2. MODEL DISCRIPTION Consider a batch arrival M X /M/1 queue in which, the arrival stream forms a Poisson process and the actual number of customers in any arriving module is a random variable X, which may take on any positive integral value k(<∞) with probability g k . If k is the arrival rate of a Poisson process of batches of size k then g k =λ k /λ, k=1,2,3,… where λ is the composite arrival rate of all batches equal to    1 i i  . This total process, which arises from the overlap of the set of Poisson processes with rates { k , k=1,2,…} is a compound Poisson process. Let X(z),E(X) and E(X 2 ) denote the PGF , first and second moments of random variable X.