Sampling at a Random Time with a Heavy-Tailed Distribution

Let Sn = 1 + + n be a sum of i.i.d. non-negative random variables, S0 = 0. We study the asymptotic behaviour of the probability PfX(T ) > ng, n!1, where X(t) = maxfn 0 : Sn tg, t 0, is the corre- sponding renewal process. The stopping time T has a heavy-tailed distribution and is independent of X(t). We treat two dierent approaches to the study: via the law of large numbers and by using the large deviation techniques. The rst approach is applied to the case when T has a heavier tail than exp( p 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( p x). As a corollary, the distribu- tional 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 gener- ally, if a stable queueing system satises 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.