Queue-Length Distribution for the Discriminatory Processor-Sharing Queue

In this paper, we study a multiple class discriminatory processor-sharing queue. The queue is assumed to have Poisson input and exponentially distributed service times. In this discipline there are K classes of customers. When there are n_i customers present in the system of class i(i = 1, …, K), each member of class j receives a fraction of the server's capacity given by α_j/∑_i=1^Kn_iα_i. Thus, associated with class i customers is a weight α_i which determines the level of service discrimination. For this problem, we find the moments of the queue-length distribution as a solution of linear simultaneous equations. We also prove a heavy traffic limit theorem for the joint queue-length distribution for this queue.