We study asymptotics for the probability ${\bf P} (T > S_n)$, $n\to \infty$, where $S_n = \xi_1+\ldots + \xi_n$ is a sum of i.i.d. non-negative r.v.'s with a positive mean and a finite variance, and $T$ is an independent r.v. with a heavy-tailed distribution. The talk focuses mainly on the case of so-called ``moderately heavy tails'' when $T$ has a heavier tail than $\exp^{-\sqrt{x}}$. We treat two different approaches to the problem: via the normal approximation and by use of large deviation technics. As a corollary, the distributional Little's law allows us to get asymptotics for stationary queue length in FCFS queues with subexponential service times. The talk is based on a joint paper Dima Korshunov (Novossibirsk)