This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously triggered random environments. Only in special cases explicit expressions can be obtained, e.g. when the random arrival and/or service rate can attain just finitely many values. While for more general model variants exact analysis is challenging, one ${\it can}$ derive limit theorems in the heavy-traffic regime. In some of our derivations we rely on evaluating the relevant Laplace transform in the heavy-traffic scaling using Taylor expansions, whereas other results are obtained by applying the continuous mapping theorem.

中文翻译：

---

大流量情况下过度分散的单服务器队列

本文讨论了过度分散下单服务器队列的队列长度过程的分析，即由到达过程馈送的队列，在给定时间窗口内到达数量的方差超过相应的平均值。考虑了几种变体，使用混合和马尔可夫调制等概念，从而产生具有内源触发或外源触发的随机环境的不同模型。仅在特殊情况下才能获得显式表达式，例如，当随机到达和/或服务率只能达到有限多个值时。虽然对于更一般的模型变体，精确分析具有挑战性，但可以推导出大流量情况下的极限定理。在我们的一些推导中，我们依赖于使用泰勒展开式评估重流量缩放中的相关拉普拉斯变换，