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Estimating probability distributions of dynamic queues

机译：估计动态队列的概率分布

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摘要

Queues are often associated with uncertainty or unreliability, which can arise from chance or climatic events, phase changes in system behaviour, or inherent randomness. Knowing the probability distribution of the number of customers in a queue is important for estimating the risk of stress or disruption to routine services and upstream blocking, potentially leading to exceeding critical limits, gridlock or incidents. The present paper focuses on time-varying queues produced by transient oversaturation during demand peaks where there is randomness in arrivals and service. The objective is to present practical methods for estimating a probability distribution from knowledge of the mean, variance and utilisation (degree of saturation) of a queue available from computationally efficient, if approximate, time-dependent calculation. This is made possible by a novel expression for time-dependent queue variance. The queue processes considered are those commonly used to represent isolated priority (M/M/1) and signal-like (M/D/1) systems, plus some statistical variations within the common Pollaczek-Khinchin framework. Results are verified by comparison with Markov simulation based on recurrence relations.
机译：队列通常与不确定性或不可靠性有关，不确定性或不可靠性可能由偶然或气候事件，系统行为的相变或固有的随机性引起。了解队列中客户数量的概率分布对于估计压力或中断常规服务和上游阻塞（可能导致超出关键限制，僵局或事件）的风险非常重要。本文着重研究在到达和服务中存在随机性的需求高峰期间，瞬态过饱和导致的时变队列。目的是提出一种实用的方法，用于根据对队列的均值，方差和利用率（饱和度）的了解来估计概率分布，该队列可从计算效率高的，近似的，与时间有关的计算中获得。对于与时间有关的队列差异，可以通过新颖的表达式来实现。所考虑的队列处理是通常用于表示隔离优先级（M / M / 1）和类信号（M / D / 1）系统的队列处理，以及常见Pollaczek-Khinchin框架内的一些统计变化。通过与基于递归关系的马尔可夫仿真比较，对结果进行了验证。

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