ANALYSIS OF SINGLE SERVER FIXED BATCH SERVICE QUEUEING SYSTEM UNDER MULTIPLE WORKING VACATIONS WITH NEGATIVE CUSTOMERS

Abstract

This study considered a single server batch service queueing system under multiple working vacations with batch size k the arrival rate 𝜆 follows a Poisson process, the service time follows an exponential distribution with parameter 𝜇. Further we assumed negative arrival occur during busy period at the rate of 𝜈 which follows a Poisson distribution and the length of time the server in vacation follows an exponential distribution with parameter 𝛼. It was assumed that the system initially contains 𝑘 customers when the server enters into the system and starts the service immediately with a batch of size 𝑘. After completion of a service, if the server finds less than 𝑘 customers in the queue, then the server goes for a multiple working vacations. During working vacation the server servers customers at slow rate than regular service rate (𝜂 < 𝜇). If there are 𝑘 or more customers in the queue then the first 𝑘 customers will be selected from the queue and service will be given as a batch of size k. That is if there are less than 𝑘 customers in the queue when the server returns from the vacation, the server immediately leaves for another vacation and so on until it finds at least 𝑘 customers in the queue. The recursive technique was employed to obtain the steady state probabilities. By using steady state probabilities important performance measures of the system were derived. The cost model for the system also formulated by using the performance measures of the system. The overall behaviour of the system was analyzed by varying the model parameters and MATLAB was used in this situation.