663 research outputs found
Central Limit Theorems and Large Deviations for Additive Functionals of Reflecting Diffusion Processes
This paper develops central limit theorems (CLT's) and large deviations
results for additive functionals associated with reflecting diffusions in which
the functional may include a term associated with the cumulative amount of
boundary reflection that has occurred. Extending the known central limit and
large deviations theory for Markov processes to include additive functionals
that incorporate boundary reflection is important in many applications settings
in which reflecting diffusions arise, including queueing theory and economics.
In particular, the paper establishes the partial differential equations that
must be solved in order to explicitly compute the mean and variance for the
CLT, as well as the associated rate function for the large deviations
principle
Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds
We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traf¿c limits for the distributions of the joint workload, waiting time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traf¿c limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale re¿ected Brownian motions in the non-negative orthant
The Evolution of a Spatial Stochastic Network
The asymptotic behavior of a stochastic network represented by a birth and
death processes of particles on a compact state space is analyzed. Births:
Particles are created at rate and their location is independent of
the current configuration. Deaths are due to negative particles arriving at
rate . The death of a particle occurs when a negative particle
arrives in its neighborhood and kills it. Several killing schemes are
considered. The arriving locations of positive and negative particles are
assumed to have the same distribution. By using a combination of monotonicity
properties and invariance relations it is shown that the configurations of
particles converge in distribution for several models. The problems of
uniqueness of invariant measures and of the existence of accumulation points
for the limiting configurations are also investigated. It is shown for several
natural models that if then the asymptotic configuration
has a finite number of points with probability 1. Examples with
and an infinite number of particles in the limit are also
presented
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