Exact buffer overflow calculations for queues via martingales
Sören Asmussen, Manfred Jobmann and HansPeter Schwefel
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2000
ISSN 14039338

Abstract:

Let $\tau=\tau_n$ be the first time a queueing process like the queue length
or workload exceeds a level $n$. For

the M/M/1 queue length process, the mean $E\tau$ and the Laplace transform
$E e^{s\tau}$ is derived in

closed form using a martingale introduced in Kella \& Whitt (1992). For
workload processes and more general systems like MAP/PH/1, we use a Markov
additive extension given in Asmussen \& Kella (2000) to derive sets of
linear equations determining the same quantities. Numerical illustrations
are presented in the framework of M/M/1 and MMPP/M/1 with an application
to performance evaluation of telecommunication systems with longrange dependent
properties in the packet arrival process. Different approximations that are
obtained from asymptotic theory are compared with exact numerical results.



Key words:

Exponential martingale, extreme value theory, Lévy process, local
time, Markovmodulation, martingale, power tail, queue length, regenerative
process, Wald martingale