Tail Asymptotics for Cumulative Processes Sampled at Heavy-Tailed Random Times with Applications to Queueing Models in Markovian Environments

This paper studies the tail asymptotics of a cumulative process $\{B(t); t \ge 0\}$ sampled at heavy-tailed random times $T$, where $T$ has a dominant impact on the asymptotic behavior of $\PP(B(T) > x)$. We establish several sufficient conditions for the asymptotic equality $\PP(B(T) > bx) \sim \PP(M(T) > bx) \sim \PP(T>x)$ as $x \to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certain positive constant. We also apply the obtained results to the subexponential asymptotics of the loss probability of a single-server finite-buffer queue with an on/off arrival process in a Markovian environment.