考虑基于 Min $(N,D)$-策略的M/G/1排队系统, 运用全概率分解技术和拉普拉斯变换工具, 获得了队长瞬态分布的拉普拉斯变换的递推表达式和稳态队长分布的递推表达式, 进一步通过数值实例, 讨论了稳态队长分布对系统参数的敏感性, 并阐述了稳态队长分布的表达式在系统容量优化设计中的重要价值. 最后, 建立了费用模型, 对最优 Min $(N,D)$-策略与单一的最优$N$-策略和单一的最优$D$-策略进行了比较分析.

In this paper, we consider the M/G/1 queueing system under the Min $(N,D)$-policy. By using the total probability decomposition technique and the Laplace transform tool, we 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. Furthermore, 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 design. Finally, we compare the optimal Min $(N,D)$-policy with the single optimal $N$-policy and the single optimal $D$-policy under a constructed cost model.