Sampling at a Random Time with a Heavy-Tailed Distribution∗

Let Sn = ξ1 + · · · + ξn be a sum of i.i.d. non-negative random variables, S0 = 0. We study the asymptotic behaviour of the probability P{X(T ) > n}, n→∞, where X(t) = max{n ≥ 0 : Sn ≤ t}, t ≥ 0, is the corresponding renewal process. The stopping time T has a heavy-tailed distribution and is independent of X(t). We treat two different approaches to the study: via the law of large numbers and by using the large deviation techniques. The first approach is applied to the case when T has a heavier tail than exp(− √ x). The second one is mostly applied to the case of the so-called “moderately heavy tails” when T has a lighter tail than exp(− √ x). As a corollary, the distributional Little’s law allows us to obtain the tail asymptotics for a stationary queue length in a single server queue with subexponential service times. More generally, if a stable queueing system satisfies the distributional Little’s law and if a stationary sojourn time distribution of a “typical” customer is heavy-tailed and its asymptotics is known, then the results of this paper provide a way for obtaining the tail asymptotics for a stationary queue length.