标题： 马尔可夫调制无限服务器系统的精细大偏差渐近性 显示英文标题

标题： Refined large deviations asymptotics for Markov-modulated infinite-server systems

摘要： 许多与网络相关的设置可以通过马尔可夫调制无限服务器系统来建模。在这些模型中，顾客的到达率和服务率由一个马尔可夫背景过程调制，此外，有无穷多个服务器（因此该模型常被用作相应多服务器模型的替代）。除了概率生成函数的基础系统的（常微分或偏微分）方程组的结果和递归以获得所有矩外，马尔可夫调制无限服务器模型几乎不允许任何显式分析。因此，最近的研究努力追求在各种极限情况下的渐近分析，特别是中心极限情况（描述围绕平均行为的波动）和大偏差情况（关注稀有事件）。其中许多结果使用了系统中顾客数量服从具有随机参数的泊松分布这一性质。本文的目标是开发技术以准确地近似大偏差情况下的尾部概率。我们考虑了到达率因因子$N$而膨胀的尺度，并且我们感兴趣的是顾客数量超过给定水平$Na$的概率。早期贡献专注于这个超越概率的所谓$logarithmic$$asymptotics$（这些本质上是不精确的），而本文通过建立$exact$ $asymptotics$ 改进了这些结果。这些结果分为两步：首先表征泊松分布的随机参数的分布，然后利用这些知识识别精确的渐近性。文章以一组数值实验结束，评估了渐近结果的准确性。 显示英文摘要

摘要： Many networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers' arrival rates and service rates are modulated by a Markovian background process, additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obtain all moments. As a consequence, recent research efforts have pursued an asymptotic analysis in various limiting regimes, notably the central-limit regime (describing fluctuations around the average behavior) and the large-deviations regime (focusing on rare events). Many of these results use the property that the number of customers in the system obeys a Poisson distribution with a random parameter. The objective of this paper is to develop techniques to accurately approximate tail probabilities in the large-deviations regime. We consider the scaling in which the arrival rates are inflated by a factor $N$, and we are interested in the probability that the number of customers exceeds a given level $Na$. Where earlier contributions focused on so-called $logarithmic$ $asymptotics$ of this exceedance probability (which are inherently imprecise), the present paper improves upon those results in that $exact$ $asymptotics$ are established. These are found in two steps: first the distribution of the random parameter of the Poisson distribution is characterized, and then this knowledge is used to identify the exact asymptotics. The paper is concluded by a set of numerical experiments, in which the accuracy of the asymptotic results is assessed.