Analysis of Single Server Queueing System With Batch Service Under Multiple Vacations With Loss And Feedback

Abstract

Consider a single server queueing system with foxed batch service under multiple vacations with loss and feedback in which the arrival rate ? follows a Poisson process and the service time follows an exponential distribution with parameter ?. Assume that the system initially contain k customers when the server enters the system and starts the service in batch. The concept of feedback is incorporated in this model (i.e) after completion of the service, if this batch of customers dissatisfied then this batch may join the queue with probability q and with probability (1-q) leaves the system. This q is called a feedback probability. After completion of the service if he finds more than k customers in the queue then the first k customers will be taken for service and service will be given as a batch of size k and if he finds less than k customers in the queue then he leaves for a multiple vacation of exponential length ?. The impatient behaviour of customer is also studied in this model (i.e) the arriving customer may join the queue with probability p when the server is busy or in vacation. This probability p is called loss probability. This model is completely solved by constructing the generating function and Rouche’s theorem is applied and we have derived the closed form solutions for probability of number of customers in the queue during the server busy and in vacation. Further we are providing the analytical solution for mean number of customers and variance of the system. Numerical studies have been done for analysis of mean and variance for various values of ?, µ, ?, p, q and k and also various particular cases of this model have been discussed.

Keywords : Single Server , Batch Service, Loss and Feedback,  Multiple vacations, Steady state distribution.