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

摘要

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 ofthis paper is to establish several sufficient conditions for the asymptoticequality ${\sf P}(B(T) > bx) \sim {\sf P}(M(T) > bx) \sim {\sf P}(T>x)$ as $x\to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certainpositive constant. The main results of this paper can be used to obtain thesubexponential asymptotics for various queueing models in Markovianenvironments. As an example, using the main results, we derive subexponentialasymptotic formulas for the loss probability of a single-server finite-bufferqueue with an on/off arrival process in a Markovian environment.