Markov modulated fluid network process: Tail asymptotics of the stationary distribution

TL;DR

This paper investigates the tail behavior of the stationary distribution in Markov modulated fluid networks, providing bounds for tail decay rates in multiple directions using Dynkin's formula, advancing understanding of high-dimensional tail asymptotics.

Contribution

It introduces two approaches based on Dynkin's formula to derive bounds for tail decay rates in multidimensional Markov modulated fluid networks, addressing a complex high-dimensional problem.

Findings

Derived upper and lower bounds for tail decay rates

Applied approaches to multidimensional tail asymptotics

Enhanced understanding of high-dimensional tail behavior

Abstract

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotics behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on a well-known time-evolution formula of a Markov process, so-called Dynkin's formula, where a key ingredient is a suitable choice of test functions. Those results show how multidimensional tail asymptotics can be studied for the more than two-dimensional case, which is known as a hard problem.