A Preemptive Priority Queue with Accessible Batch Service and Heterogeneous Arrivals and with Vacation

In this paper a single server preemptive priority queuing system consisting of two types of customer units with Poisson arrival, exponential service time distribution and exponential vacation is analyzed. Two type of customer units of which higher priority units of unlimited Poisson input are served by bulk service rule and limited Poisson input of low priority units served singly. The bulk service rule admits in batches for service only if the batch size is not less than ‘a’ and not more than ‘b’ units. Such that the arriving units can enter service without affecting the service time. If the size of the batch being served is less than ‘a’ then the server turns to do service for the lower priority units with single service. Whenever the number of customers in the high priority queue is less than the quorum and there is no customers in low priority queue then the server takes an exponential vacation. Vacation is analyzed in two models as Model I and Model II. In Model I, the server avails multiple vacations and for Model II, only single vacation is allowed. The steady state probability vectors of the number of customers in the queue are obtained by the modified geometric method. The stability condition and mean queue length of customers are obtained. Numerical results are also presented. AMS (2000) Subject Classification Number: 60k25 and 65k30.