Study of Impact of Negative Arrivals in Single Server Fixed Batch Service Queueing System with Multiple Vacations

Consider a single server fixed batch service queueing system under multiple vacation with a possibility of negative arrival in which the arrival rate λ follows a Poisson process, the service time follows an exponential distribution with parameter μ. Further we assume a negative arrival rate occur at the rate of ν which follows a Poisson process and the length of time the server in vacation follows an 6968 G. Ayyappan, G. Devipriya and A. Muthu Ganapathi Subramanian exponential distribution with parameter α. Assume 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 he finds less than k customers in the queue, then the server goes for a multiple vacation of a length α. If there are more than k customers in the queue then the first k customers will be selected from the queue and service will be given as a batch. We are analyzing the possibility of negative arrival in this model. Negative customers have the effect of deleting some customer in the queue. In the simplest version, a negative arrival removes an ordinary positive customer or a random batch of positive customers according to some strategy. It is noted that the existence of a flow of negative arrivals provides a control mechanism to control excessive congestion at the queue and also assume that the negative customers only act when the server is busy . This model is completely solved by using the generating function technique. 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. Various particular cases of this model have been discussed.