International Journal of Scientific and Research Publications, Volume 3, Issue 11, November 2013 1 ISSN 2250-3153 www.ijsrp.org Optimal Strategy Analysis of an N-policy M/E k /1 Queueing System with Server Breakdowns and Multiple Vacations P.Jayachitra*, Dr.A.James Albert** * P.hD Research Scholar ,Mathematics, Karpagam University ** Dean, Mathematics, Karpagam University Abstract- This paper studies the optimal control of an N-policy M/E k /1 queueing system with server breakdowns and multiple vacations. The server is turned on when N units are accumulated in the system. The server is turned off and takes a vacation with exponential random length whenever the system is empty. If the number of units waiting in the system at any vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and find atleast N units in the system, it immediately starts to serve the waiting units. It is assumed that the server breakdown according to poisson process and the repair time has an exponential distribution. We derive the distribution of the system size and employ the probability generating function to obtain the mean queue length. It is proved that the service station is busy in the steady state is equal to the traffic intensity. The total expected cost function per unit time is developed to determine the optimal operating policy at minimum cost. This paper provides the minimum expected cost and the optimal operating policy based on numerical values of the system parameters. Sensitivity analysis is also provided. Index Terms- : M/E k /1 queueing system, multiple vacations, N-policy, probability generating function, Server breakdowns. I. INTRODUCTION his paper considers the modeling of a production system at which arrivals of production order follow a poisson process at a rate λ. The production times of the orders are made up of k independent and identically distributed exponential random variables with mean 1/kμ which yields an Erlang type k distribution. The system operation starts (turned on) only when N orders have accumulated and is shut down (turned off) when no orders are present. When the server is working he may meet unpredictable breakdown but it is immediately repaired. When the system is turned off, the server leaves the system for a random period of time called vacation. On returning to the system if the server finds less than N units in the system immediately he takes another vacation. This production system can be modeled by an M/E k /1 queueing system with server breakdowns and multiple vacations under N- policy. The concept of the N policy, was first introduced by Yadin and Naor [6]. Past work regarding queueing systems under the N policy may be divided into two categories: (i) cases with server’s vacations and (ii) cases with server’s breakdowns. For ca ses with server breakdowns, Wang [2] first proposed a management policy for Markovian queueing systems under the N policy with server breakdowns. Wang [4] and Wang et al. [5] extended the model proposed by Wang [2] to M/Ek/1 and M/H2/1 queueing systems respectively. Vasantha Kumar and Chandan [7] presented the optimal strategy analysis of two phase M/Ek/1 queueing system with server breakdowns and gating. Also they obtained the total expected cost function for the system and determine the optimal value of the control parameter N. Existing research works, including those mentioned above, have never covered cases involving both server breakdowns and vacations. Queueing models with server breakdowns and vacations accommodate the real-world situations more closely. The purpose of this paper is threefold. 1. The steady state equations are established to get the steady state probability distribution and to show that it generalizes the previous results. 2. We formulate the system total expected cost in order to determine the optimal operating N-policy numerically at the minimum cost for various values of system parameters while maintaining the minimal service quantity. 3. We perform a sensitivity analysis II. MODEL DESCRIPTION For the purpose of analytical investigation, we consider the model with the following assumptions: 1. The arrival is poisson process with parameter λ and with service times according to an Erlang distribution with mean 1/µ and stage parameter k. The Erlang type k distribution is made up of k independent and identical exponential stages, each with mean 1/kμ. A customer goes into the first stage of the service (say stage k) then progresses through the remaining stages and must T