Refined large deviations asymptotics for Markovmodulated infiniteserver systems
Abstract
Many networkingrelated settings can be modeled by Markovmodulated infiniteserver 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 manyserver model). The Markovmodulated infiniteserver 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 centrallimit regime (describing fluctuations around the average behavior) and the largedeviations 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 largedeviations 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 socalled $logarithmic$ $asymptotics$ 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2016
 arXiv:
 arXiv:1608.04250
 Bibcode:
 2016arXiv160804250B
 Keywords:

 Mathematics  Probability;
 60K25;
 60K37;
 60F10;
 90B18