Analysis of a single server batch arrival retrial queueing system with modified vacations and N-policy

Abstract

In this paper, a batch arrival single server retrial queue with modified vacations under N -policy is considered. If an arriving batch of customers finds the server busy or on vacation, then the entire batch joins the orbit in order to seek the service again. Otherwise, one customer from the arriving batch receives the service, while the rest joins the orbit. The customers in the orbit will try for service one by one when the server is idle with a classical retrial policy with the retrial rate ‘jv ’, where ‘j ’ is the size of the orbit. At a service completion epoch, if the number of customers in the orbit is zero, then the server leaves for a secondary job (vacation) of random length. At a vacation completion epoch, if the orbit size is at least N , then the server remains in the system to render service for the primary customers or orbital customers. On the other hand, if the number of customers in the orbit is less than ‘N ’ at a vacation completion epoch, the server avails multiple vacations subject to maximum ‘M ’ repeated vacations. After availing ‘M ’ consecutive vacations, the server returns to the system to render service irrespective of the orbit size. The model is studied using supplementary variable technique. For the proposed queueing system, the probability generating function of the steady state queue size distribution at an arbitrary time is obtained. Various performance measures are derived. A cost model for the queueing system is developed. Numerical illustration is provided.