We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined, and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regarding the busy period distributions including expectation and conditional expectation of busy periods are also explored.

中文翻译：

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具有一般状态相关控制的马尔可夫批量到达和批量服务队列

我们研究了一种改进的马尔可夫批量到达和批量服务队列，结合了一般的状态相关控制。首先研究停止的批量到达和批量服务队列，并检查和利用此停止队列与完整排队模型之间的关系。利用这种关系，研究了全排队过程的均衡行为，得到了均衡分布的概率生成函数。还检查了队列长度行为，并提供了队列长度分布的拉普拉斯变换。详细回答了有关点击次数和繁忙时段分布的重要问题，并介绍了这些分布的拉普拉斯变换。