5,639 research outputs found
Efficient Reactive Brownian Dynamics
We develop a Split Reactive Brownian Dynamics (SRBD) algorithm for particle
simulations of reaction-diffusion systems based on the Doi or volume reactivity
model, in which pairs of particles react with a specified Poisson rate if they
are closer than a chosen reactive distance. In our Doi model, we ensure that
the microscopic reaction rules for various association and disassociation
reactions are consistent with detailed balance (time reversibility) at
thermodynamic equilibrium. The SRBD algorithm uses Strang splitting in time to
separate reaction and diffusion, and solves both the diffusion-only and
reaction-only subproblems exactly, even at high packing densities. To
efficiently process reactions without uncontrolled approximations, SRBD employs
an event-driven algorithm that processes reactions in a time-ordered sequence
over the duration of the time step. A grid of cells with size larger than all
of the reactive distances is used to schedule and process the reactions, but
unlike traditional grid-based methods such as Reaction-Diffusion Master
Equation (RDME) algorithms, the results of SRBD are statistically independent
of the size of the grid used to accelerate the processing of reactions. We use
the SRBD algorithm to compute the effective macroscopic reaction rate for both
reaction- and diffusion-limited irreversible association in three dimensions.
We also study long-time tails in the time correlation functions for reversible
association at thermodynamic equilibrium. Finally, we compare different
particle and continuum methods on a model exhibiting a Turing-like instability
and pattern formation. We find that for models in which particles diffuse off
lattice, such as the Doi model, reactions lead to a spurious enhancement of the
effective diffusion coefficients.Comment: To appear in J. Chem. Phy
Diffusion Models for Double-ended Queues with Renewal Arrival Processes
We study a double-ended queue where buyers and sellers arrive to conduct
trades. When there is a pair of buyer and seller in the system, they
immediately transact a trade and leave. Thus there cannot be non-zero number of
buyers and sellers simultaneously in the system. We assume that sellers and
buyers arrive at the system according to independent renewal processes, and
they would leave the system after independent exponential patience times. We
establish fluid and diffusion approximations for the queue length process under
a suitable asymptotic regime. The fluid limit is the solution of an ordinary
differential equation, and the diffusion limit is a time-inhomogeneous
asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis
is also developed, and the diffusion limit in the stronger heavy traffic regime
is a time-homogeneous asymmetric O-U process. The limiting distributions of
both diffusion limits are obtained. We also show the interchange of the heavy
traffic and steady state limits
Some aspects of queueing and storage processes : a thesis in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University
In this study the nature of systems consisting of a single queue are first considered. Attention is then drawn to an analogy between such systems and storage systems.
A development of the single queue viz queues with feedback is considered after first considering feedback processes in general. The behaviour of queues, some with feedback loops, combined into networks is then considered. Finally, the application of such networks to the analysis of interconnected reservoir systems is considered and the conclusion drawn that such analytic methods complement the more recently developed mathematical programming methods by providing analytic solutions for
sub systems behaviour and thus guiding the development of a system model
Load Balancing in the Non-Degenerate Slowdown Regime
We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as
the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a
classical load balancing policy for queueing systems with multiple parallel
servers. Parallel server queueing systems are regularly analysed and
dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling
regime. However, when jobs must be dispatched to a server upon arrival, we
advocate the Non-Degenerate Slowdown regime (NDS) to compare different
load-balancing rules.
In this paper we identify novel diffusion approximation and timescale
separation that provides insights into the performance of JSQ. We calculate the
price of irrevocably dispatching jobs to servers and prove this to within 15%
(in the NDS regime) of the rules that may manoeuvre jobs between servers. We
also compare ours results for the JSQ policy with the NDS approximations of
many modern load balancing policies such as Idle-Queue-First and
Power-of--choices policies which act as low information proxies for the JSQ
policy. Our analysis leads us to construct new rules that have identical
performance to JSQ but require less communication overhead than
power-of-2-choices.Comment: Revised journal submission versio
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