在平稳状态下, Baba利用补充变量方法研究了多重休假的$M^X/G/1$排队, 但作者假定了休假时间和服务时间都有概率密度函数. 本文考虑推广的多重休假 $M^X/G/1$ 排队, 在假定休假时间和服务时间都是一般概率分布函数下, 我们研究了队长的瞬态和稳态性质. 通过引进``服务员忙期''和使用不同于Baba文 中使用的分析技术, 我们导出了在任意时刻 $t$ 瞬态队长分布的 $L$ 变换的递推表 达式和稳态队长分布的递推表达式, 以及平稳队长的随机分解. 特别地, 通过本文 可直接获得多重休假的 $M/G/1$ 与标准的 $M^X/G/1$ 排队系统相应的结果.

Under the steady state Baba used the supplementary variable to study the $M^X/G/1$ queue with multiple server vacations in which it was assumed that the vacation time and the service time have probability density functions. In this paper we consider the generalized $M^X/G/1$ queue with multiple server vacations. Assuming that the vacation time and the service time have general distribution functions, we study the transient and equilibrium properties of the queue length. By introducing the server busy period and using the different technique we derive the recursion expression of the L-transformation of the transient queue length distribution at time t, and also the expressions of the distribution and stochastic decomposition of the queue length at a random point in equilibrium. Especially we obtain some corresponding results for $M/G/1$ queue with server vacations and the $M^X/G/1$ queue with server vacations but no delay.