Sampling at subexponential times, with queueing applications
S. Asmussen, C. Klüppelberg and K. Sigman
Department of Mathematical Statistics,
Lund Institute of Technology,
Lund University,
1998
ISSN 02811944
ISRN LUNFD6/NFMS3190SE

Abstract:

We study the tail asymptotics of the r.v. X(T) where {X(t)} is a stochastic
process with a linear drift and satisfying some regularity conditions like
a central limit theorem and a large deviations principle, and T is an independent
r.v. with a subexponential distribution. We find that the tail of X(T) is
sensitive to whether or not T has a heavier or lighter tail than a Weibull
distribution with tail e^{sqrt(x)} . This leads to two distinct
cases, heavytailed and moderately heavytailed, but also some results for
the classical lighttailed case are given. The results are applied via
distributional Little's law to establish tail asymptotics for steadystate
queue length in GI/GI/1 queues with subexponential service times. Further
applications are given for queues with vacations, and M/G/1 busy periods.


Key words:

busy period, independent sampling, Laplace's method, large deviations, Little's
law, Markov additive process, Poisson process, random walk, regular variation,
subexponential distribution, vacation model, Weibull distribution