Performance analysis of an M/G/1 queue with bi-level randomized (p, N1, N2)-policy

This paper proposes an M/G/1 queueing model with bi-level randomized (p, N-1, N-2)-policy. That is, after all of the customers in the system are served, the server is closed down immediately. If N-1(>= 1) customers are accumulated in the queue, the server is activated for service with probability p(0 <= p <= 1) or still left off with probability 1 - p. When the number of customers in the system becomes N-2(>= N-1), the server begins serving the waiting customers until the system becomes empty again. Using the total probability decomposition technique and the Laplace transform, we study the transient queue length distribution and obtain the expressions of the Laplace transform of the transient queue-length distribution with respect to time t. Then, employing L'Hospital's rule and some algebraic operations, the explicit recursive formulas of the steady-state queue-length distribution, which can be used to accurately evaluate the probabilities of queue length, are presented. Moreover, some other important queuing performance indices, such as the explicit expressions of its probability generating function of the steady-state queue-length distribution, the expected queue size and so on, are derived. Meanwhile, we investigate the system capacity optimization design by the steady-state queue-length distribution. Finally, an operating cost function is constructed, and by numerical calculation, we find the minimum of the long-run average cost rate and the optimal bi-level threshold policy (N*(1),N*(2)) that satisfies the average waiting time constraints.