Strong approximation theorems for density dependent Markov chains
TLDR
A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form X N (t)=x 0 + ∑ 1 N lY 1 N ∫ t 0 f 1 (X N (s))ds where l ∈ Z t, the Y 1 are independent Poisson processes, and N is a parameter with a natural interpretation.About:
This article is published in Stochastic Processes and their Applications.The article was published on 1978-02-01 and is currently open access. It has received 596 citations till now.read more
Citations
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The chemical Langevin equation
TL;DR: In this article, it is shown that the chemical Langevin equation can be derived from the microphysical premise from which the chemical master equation is derived, which leads directly to an approximate time-evolution equation of the Langevin type.
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The power of two choices in randomized load balancing
TL;DR: This work uses a limiting, deterministic model representing the behavior as n/spl rarr//spl infin/ to approximate the behavior of finite systems and provides simulations that demonstrate that the method accurately predicts system behavior, even for relatively small systems.
Book
Logarithmic Combinatorial Structures: A Probabilistic Approach
TL;DR: In this article, the authors explain the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition.
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Stochastic theory of a fluid model of producers and consumers coupled by a buffer
TL;DR: In this article, the authors derived efficient computational procedures and numerically investigated the following fluid model which is of interest in manufacturing and communications: m producing machines supply a buffer, n consuming machines feed off it, each machine independently alternates between exponentially distributed random periods in the ‘in service' and 'failed' states.
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Extracting macroscopic dynamics: model problems and algorithms
TL;DR: A number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description are described.
References
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Book
A first course in stochastic processes
Samuel Karlin,Howard M. Taylor +1 more
TL;DR: In this paper, the Basic Limit Theorem of Markov Chains and its applications are discussed and examples of continuous time Markov chains are presented. But they do not cover the application of continuous-time Markov chain in matrix analysis.
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An Approximation of Partial Sums of Independent RV's, and the Sample DF. II
TL;DR: In this article, the authors introduced a new construction for the pair S¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ n�, T¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ n>>\s, and proved that if X>>\s has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then ¦S>>\s n� -T� n� nၡ 1/4(log n) 1/1(log log n)1/4) with probability one.
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Solutions of ordinary differential equations as limits of pure jump markov processes
TL;DR: In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development as discussed by the authors.
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Limit theorems for sequences of jump Markov processes approximating ordinary differential processes
TL;DR: In this paper, the authors gave conditions under which a sequence of jump Markov processes Xn (t) will converge to the solution X(t) of a system of first order ordinary differential equations, in the sense that for every δ > 0,