Abstract:

This paper considers the M/G/1 queueing system under the delay Min(N,D)-policy. By using the renewal process theory, the total probability decomposition technique and the Laplace transform tool, we study the transient and equilibrium properties of the queue length from the beginning of the any initial state, and obtain both the recursion expressions of the Laplace transformation of the transient queue length distribution and the recursion expressions of the steady state queue length distribution. Meanwhile, we present the explicit expression of the additional queue-length distribution. Furthermore, we discuss some special cases, such as N \to \infty, or D \to \infty, or N=1 and P{Y=0}=1 or P{Y=0}=1, respectively.  Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters, and illustrate the important value of the expressions of the steady state queue length distribution in the system capacity optimum design.

Key words: M/G/1 queue, delay Min(N, D)-policy, total probability decomposition technique, Laplace transform, queue length distribution, system capacity optimum design