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Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

机译：多个适应性休假下具有批马尔可夫到达过程的队列的平稳分布和最优控制

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摘要

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.
机译：我们考虑了具有多个自适应休假策略的具有批处理马尔可夫到达过程（BMAP）和穷举服务规范的无限缓冲区单服务器队列。就是说，服务器一直服务到系统清空为止，然后该服务器随机抽取最大数目H个休假，直到他发现队列中至少有一个客户或服务器穷尽了所有休假为止。服务器采取的最大休假数H是离散的随机变量。我们在各个时期获得队列长度分布，例如服务完成/休假终止，到达前，任意，出发后和服务前。所提出的分析是基于使用矩阵分析过程来获得出发后时期的队列长度分布的。稍后，我们使用补充变量方法和简单的代数运算，使用出发后纪元的队列长度分布来获取其他纪元的队列长度分布。已经获得了一些重要的性能指标，例如平均队列长度和平均等待时间。作为我们模型的特殊情况，可以获取其他几个休假排队模型，例如，单休假，多休假模型和具有特别的第一个休假时间的队列。最后，考虑每单位时间的总预期成本函数，以最低成本确定局部最优的多重适应性休假策略。

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