On the distributions of infinite server queues with batch arrivals
On the distributions of infinite server queues with batch arrivals
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DOI：
10.1007/s11134-019-09603-4
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发表时间：
2019-04-01
期刊：
影响因子：
1.2
通讯作者：
Pender, Jamol
中科院分区：
文献类型：
--
作者：
Daw, Andrew;Pender, Jamol
具有多个实体同时到达的排队是排队理论中最古老的模型之一，并且通常被称为批量（或在某些情况下，批量）到达排队系统。在这项工作中，我们研究了批量到达无限服务器队列的效果。我们假设到达时期发生根据泊松过程，与治疗的平稳和非平稳的到达率。我们考虑指数和一般分布的服务持续时间，我们分析固定和随机到达批量大小。除了推导出瞬态的平均值，方差和时刻生成函数随时间变化的到达率，我们还发现，稳态分布的队列是相当于比例的泊松随机变量的总和，其服务分布的顺序统计量成比例的速率。我们这样做，通过查看批到达系统作为一个相关的子队列的集合。此外，我们调查的限制行为的过程中，通过批量缩放的队列和通过流体和扩散的到达率的限制。在我们的分析过程中，我们的模型和调和数，广义厄米分布，截断多项式之间的重要联系。
Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as batch (or, in some cases, bulk) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.