Large deviations and fast simulation in the presence of boundaries

• Let c(x) = inf {t > 0: Q(t) greater than or equal to x} be the time of first overflow of a queueing process 1001 over level x (the buffer size) and Z = P(T(X) less than or equal to T). Assuming that {Q(t)) is the reflected version of a Levy process {X(t)} or a Markov additive process, we study a variety of algorithms for estimating z by simulation when the event {tau(X) less than or equal to T} is rare, and analyse their performance. In particular, we exhibit an estimator using a filtered Monte Carlo argument which is logarithmically efficient whenever an efficient estimator for the probability of overflow within a busy cycle (i.e., for first passage probabilities for the unrestricted netput process) is available, thereby providing a way out of counterexamples in the literature on the scope of the large deviations approach to rare events simulation. We also add a counterexample of this type and give various theoretical results on asymptotic properties of Z=P(tau(x) less than or equal to T), both in the reflected Levy process setting and more generally for regenerative processes in a regime where T is so small that the exponential approximation for T(x) is not a priori valid.

Subject headings

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

Keyword

rare

queueing theory

local time

Levy process

importance sampling

filtered Monte Carlo

buffer overflow

exponential change of measure

event

reflection

regenerative process

saddlepoint

Publication and Content Type

art (subject category)

ref (subject category)