Abstract: In this paper, we consider the M/G/1 queueing system with multiple server vacations and Min(N,D,V)-policy. By using the total probability decomposition technique and the Laplace transformation tool, the transient queue-length distribution and the steady queue-length distribution are discussed. Both the s of the Laplace transformation of the transient queue-length distribution and the recursive s of the steady queue-length distribution are obtained. Meanwhile, we present the stochastic decomposition result of the steady queue length and the explicit of the additional queue length distribution. Furthermore, some special cases,such as when N→∞ or D→∞, and when P{V=∞}=1 or P{V=0}=1, are discussed. Finally, the explicit of the long-run expected cost rate is derived under a given cost structure. And by through numerical calculation , we determine the optimal control policy (N*,D*) for minimizing the long-run expected cost per unit time as well as compare with the single optimal N*-policy and the single optimal D*-policy .

Key words: Multiple server vacation, Min(N,D,V)-policy, total probability decomposition technique, queue length distribution, optimal policy