Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments

TL;DR

This paper establishes conditions under which the tail probabilities of a cumulative process sampled at heavy-tailed random times are asymptotically equivalent to the tail of the sampling time, with applications to queueing models in Markovian environments.

Contribution

It provides new sufficient conditions for tail asymptotics of cumulative processes at heavy-tailed times, enabling analysis of queueing systems in Markovian settings.

Findings

Derived asymptotic formulas for loss probabilities in finite-buffer queues.

Established conditions for tail equivalence of process and sampling time.

Applied results to queueing models with on/off arrivals in Markovian environments.

Abstract

This paper considers the tail asymptotics for a cumulative process $\{B(t); t \ge 0\}$ sampled at a heavy-tailed random time $T$. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality ${\sf P}(B(T) > bx) \sim {\sf P}(M(T) > bx) \sim {\sf P}(T>x)$ as $x \to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certain positive constant. The main results of this paper can be used to obtain the subexponential asymptotics for various queueing models in Markovian environments. As an example, using the main results, we derive subexponential asymptotic formulas for the loss probability of a single-server finite-buffer queue with an on/off arrival process in a Markovian environment.