  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.
