We study the maximum remaining service time in infinite-server queues of type M|G|∞ (at a given time and in a stationary regime). The following cases for the arrival flow rate are considered: (1) time-independent, (2) given by a function of time, (3) given by a random process. As examples of service time distributions, we consider exponential, hyperexponential, Pareto, and uniform distributions. In the case of a constant rate, we study effects that arise when the average service time is infinite (for power-law distribution tails). We find the extremal index of the sequence of maximum remaining service times. The results are extended to queues of type M^X|G|∞, including those with dependent service times within a batch.

中文翻译：

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无限服务器队列中的最大剩余服务时间

我们研究型的无限服务器队列的最大剩余使用时间中号| G |∞（在给定时间和固定状态下）。考虑以下到达流量的情况：（1）与时间无关，（2）由时间函数给出，（3）由随机过程给出。作为服务时间分布的示例，我们考虑指数分布，超指数分布，帕累托分布和均匀分布。在恒定速率的情况下，我们研究了平均服务时间无限长（对于幂律分布尾部）产生的影响。我们找到最大剩余服务时间序列的极值索引。结果扩展到类型M ^X |的队列。G |∞，包括批次中具有相关服务时间的那些。