Statistics Seminar: Yoni Nazarathy -- Scaling limits of cyclically varying birth-death processes

Yoni Nazarathy Applied Mathematics, Faculty of Engineering & Industrial Sciences
Swinburne University of Technology 

Location: Carslaw 173 

Time: 2pm Friday, July 29, 2011 

Scaling limits of cyclically varying birth-death processes 

Abstract: Fluid limits of stochastic queueing systems have received considerable
attention in recent years.  The general idea is to scale space, time and/or system
parameters as to obtain a simpler, yet accurate description of the system.  A basic
example is the single server queue with time speeded up and space scaled down at the
same rate.  A second well known example is the Markovian infinite server queue with the
arrival rate speeded up and space scaled down at the same rate.  Such scalings and their
network generalizations are often useful for obtaining stability conditions and
approximating optimal control policies.  In this talk we consider birth-death processes
with general transition rates and obtain an asymptotic scaling result, generalizing the
Markovian single server and infinite server cases.  We apply our results to the
steady-state analysis of queueing systems with cyclic or time varying behaviour.
Examples are systems governed by deterministic cycles, queues with hysteresis control
and queues with Markov-modulated arrival or service rates.  The unifying property of
such systems, is that if they are properly scaled, the resulting trajectories follow a
cyclic or piece-wise deterministic behaviour which is determined by the asymptotic
scaling.  This yields simple a approximation for the stationary distribution which is
shown to be asymptotically exact.  Joint work with Matthieu Jonckheere.