4 research outputs found
Tutte's invariant approach for Brownian motion reflected in the quadrant
We consider a Brownian motion with drift in the quarter plane with orthogonal
reflection on the axes. The Laplace transform of its stationary distribution
satisfies a functional equation, which is reminiscent from equations arising in
the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant
approach to this continuous setting, and we obtain an explicit formula for the
Laplace transform in terms of generalized Chebyshev polynomials.Comment: 14 pages, 3 figure
Steady-state simulation of reflected Brownian motion and related stochastic networks
This paper develops the first class of algorithms that enable unbiased
estimation of steady-state expectations for multidimensional reflected Brownian
motion. In order to explain our ideas, we first consider the case of compound
Poisson (possibly Markov modulated) input. In this case, we analyze the
complexity of our procedure as the dimension of the network increases and show
that, under certain assumptions, the algorithm has polynomial-expected
termination time. Our methodology includes procedures that are of interest
beyond steady-state simulation and reflected processes. For instance, we use
wavelets to construct a piecewise linear function that can be guaranteed to be
within distance (deterministic) in the uniform norm to Brownian
motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sensitivity analysis for diffusion processes constrained to an orthant
This paper studies diffusion processes constrained to the positive orthant
under infinitesimal changes in the drift. Our first main result states that any
constrained function and its (left) drift-derivative is the unique solution to
an augmented Skorohod problem. Our second main result uses this
characterization to establish a basic adjoint relationship for the stationary
distribution of the constrained diffusion process jointly with its
left-derivative process.Comment: Published in at http://dx.doi.org/10.1214/13-AAP967 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Many-server queues with customer abandonment: numerical analysis of their diffusion models
We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
first-in-first-out (FIFO) order and the customers waiting in queue may abandon
the system without service. Two diffusion models are proposed in this paper.
They differ in how the patience time distribution is built into them. The first
diffusion model uses the patience time density at zero and the second one uses
the entire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distribution of such
a diffusion process. A crucial part of the algorithm is to choose an
appropriate reference density. Using a conjecture on the tail behavior of a
limit queue length process, we propose a systematic approach to constructing a
reference density. With the proposed reference density, the algorithm is shown
to converge quickly in numerical experiments. These experiments also show that
the diffusion models are good approximations for many-server queues, sometimes
for queues with as few as twenty servers