4,224 research outputs found
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
We present a mathematical framework for constructing and analyzing parallel
algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting
algorithms have the capacity to simulate a wide range of spatio-temporal scales
in spatially distributed, non-equilibrium physiochemical processes with complex
chemistry and transport micro-mechanisms. The algorithms can be tailored to
specific hierarchical parallel architectures such as multi-core processors or
clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms
are controlled-error approximations of kinetic Monte Carlo algorithms,
departing from the predominant paradigm of creating parallel KMC algorithms
with exactly the same master equation as the serial one.
Our methodology relies on a spatial decomposition of the Markov operator
underlying the KMC algorithm into a hierarchy of operators corresponding to the
processors' structure in the parallel architecture. Based on this operator
decomposition, we formulate Fractional Step Approximation schemes by employing
the Trotter Theorem and its random variants; these schemes, (a) determine the
communication schedule} between processors, and (b) are run independently on
each processor through a serial KMC simulation, called a kernel, on each
fractional step time-window.
Furthermore, the proposed mathematical framework allows us to rigorously
justify the numerical and statistical consistency of the proposed algorithms,
showing the convergence of our approximating schemes to the original serial
KMC. The approach also provides a systematic evaluation of different processor
communicating schedules.Comment: 34 pages, 9 figure
Asymptotic Approximations for TCP Compound
In this paper, we derive an approximation for throughput of TCP Compound
connections under random losses. Throughput expressions for TCP Compound under
a deterministic loss model exist in the literature. These are obtained assuming
the window sizes are continuous, i.e., a fluid behaviour is assumed. We
validate this model theoretically. We show that under the deterministic loss
model, the TCP window evolution for TCP Compound is periodic and is independent
of the initial window size. We then consider the case when packets are lost
randomly and independently of each other. We discuss Markov chain models to
analyze performance of TCP in this scenario. We use insights from the
deterministic loss model to get an appropriate scaling for the window size
process and show that these scaled processes, indexed by p, the packet error
rate, converge to a limit Markov chain process as p goes to 0. We show the
existence and uniqueness of the stationary distribution for this limit process.
Using the stationary distribution for the limit process, we obtain
approximations for throughput, under random losses, for TCP Compound when
packet error rates are small. We compare our results with ns2 simulations which
show a good match.Comment: Longer version for NCC 201
Error bounds for last-column-block-augmented truncations of block-structured Markov chains
This paper discusses the error estimation of the last-column-block-augmented
northwest-corner truncation (LC-block-augmented truncation, for short) of
block-structured Markov chains (BSMCs) in continuous time. We first derive
upper bounds for the absolute difference between the time-averaged functionals
of a BSMC and its LC-block-augmented truncation, under the assumption that the
BSMC satisfies the general -modulated drift condition. We then establish
computable bounds for a special case where the BSMC is exponentially ergodic.
To derive such computable bounds for the general case, we propose a method that
reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to
level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the
properties of the bounds through the numerical results on an M/M/ retrial
queue, which is a representative example of LD-QBDs. Finally, we present
computable perturbation bounds for the stationary distribution vectors of
BSMCs.Comment: This version has fixed the bugs for the positions of Figures 1
through
Applying Mean-field Approximation to Continuous Time Markov Chains
The mean-field analysis technique is used to perform analysis of a systems with a large number of components to determine the emergent deterministic behaviour and how this behaviour modifies when its parameters are perturbed. The computer science performance modelling and analysis community has found the mean-field method useful for modelling large-scale computer and communication networks. Applying mean-field analysis from the computer science perspective requires the following major steps: (1) describing how the agents populations evolve by means of a system of differential equations, (2) finding the emergent
deterministic behaviour of the system by solving such differential equations, and (3) analysing properties of this behaviour either by relying on simulation or by using logics. Depending on the system under analysis, performing these steps may become challenging. Often, modifications
of the general idea are needed. In this tutorial we consider illustrating examples to discuss how the mean-field method is used in different application areas. Starting from the application of the classical technique,
moving to cases where additional steps have to be used, such as systems with local communication. Finally we illustrate the application of the simulation and
uid model checking analysis techniques
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