Theory of Probability &Its Applications, Volume 66, Issue 1, Page 1-14, January 2021.
We consider a single-channel system with service vacations, a Poisson input flow, and an arbitrarily distributed service time. Interruptions in service can mean either a complete shutdown of the server for a random period of time or a transition to a different (nonstandard) regime---they can occur either at the end of busy periods when the system is operating in the standard regime, or at the end of vacations at which the system contains no customers. We assume that there is a random timeout before a possible vacation and that the vacation occurs at the end of this timeout if no customers were received by the system during the timeout. Otherwise, the vacation is canceled and the system resumes standard operations. We consider three regimes with different conditions regarding the presence of timeouts and the rules for resuming the standard regime. Under fairly general assumptions concerning distributions of timeout times, we obtain durations of vacations, and processes describing the performance of the system during interruptions, formulas for the distribution, and expectation of the number of customers in the system in the stationary regime. Corresponding examples are given. For a number of special cases our results coincide with those available in the literature.

中文翻译：

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假期队列 $M|G|1$ 与关闭时间

概率论及其应用，第 66 卷，第 1 期，第 1-14 页，2021 年 1 月。
我们考虑具有服务假期、泊松输入流和任意分布服务时间的单通道系统。服务中断可能意味着在随机时间段内完全关闭服务器或转换到不同的（非标准）状态——它们可能发生在系统在标准状态下运行的繁忙时段结束时，或在系统不包含客户的假期结束时。我们假设在可能的假期之前有一个随机超时，并且如果在超时期间系统没有收到客户，则假期发生在此超时结束时。否则，假期将被取消，系统将恢复标准操作。我们考虑了在超时的存在和恢复标准制度的规则方面具有不同条件的三种制度。在关于超时时间分布的相当一般的假设下，我们获得了休假持续时间、描述系统中断期间系统性能的过程、分布公式以及系统中稳定状态下客户数量的期望。给出了相应的例子。对于一些特殊情况，我们的结果与文献中的结果一致。