1 research outputs found
Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues
In this paper we propose a new method for approximating the nonstationary
moment dynamics of one dimensional Markovian birth-death processes. By
expanding the transition probabilities of the Markov process in terms of
Poisson-Charlier polynomials, we are able to estimate any moment of the Markov
process even though the system of moment equations may not be closed. Using new
weighted discrete Sobolev spaces, we derive explicit error bounds of the
transition probabilities and new weak a priori estimates for approximating the
moments of the Markov processs using a truncated form of the expansion. Using
our error bounds and estimates, we are able to show that our approximations
converge to the true stochastic process as we add more terms to the expansion
and give explicit bounds on the truncation error. As a result, we are the first
paper in the queueing literature to provide error bounds and estimates on the
performance of a moment closure approximation. Lastly, we perform several
numerical experiments for some important models in the queueing theory
literature and show that our expansion techniques are accurate at estimating
the moment dynamics of these Markov process with only a few terms of the
expansion