1,812 research outputs found
The invariant measure of homogeneous Markov processes in the quarter-plane: Representation in geometric terms
We consider the invariant measure of a homogeneous continuous-time Markov process in the quarter-plane. The basic solutions of the global balance equation are the geometric distributions. We first show that the invariant measure can not be a finite linear combination of basic geometric distributions, unless it consists of a single basic geometric distribution. Second, we show that a countable linear combination of geometric terms can be an invariant measure only if it consists of pairwise-coupled terms. As a consequence, we obtain a complete characterization of all countable linear combinations of geometric distributions that may yield an invariant measure for a homogeneous continuous-time Markov process in the quarter-plane
Modeling with bivariate geometric distributions
This dissertation studied systems with several components which were subject to different types of failures. Systems with two components having frequency counts in the domain of positive integers, and the survival time of each component following geometric or mixture geometric distribution can be classified into this category. Examples of such systems include twin engines of an airplane and the paired organs in a human body. It was found that such a system, using conditional arguments, can be characterized as multivariate geometric distributions. It was proved that these characterizations of the geometric models can be achieved using conditional probabilities, conditional failure rates, or probability generating functions. These new models were fitted to real-life data using the maximum likelihood estimators, Bayes estimators, and method of moment estimators. The maximum likelihood estimators were obtained by solving score equations. Two methods of moments estimators were compared in each of the several bivariate geometric models using the estimated bias vectors and the estimated variance-covariance matrices. This comparison was done through a Monte-Carlo simulation for increasing sample sizes. The Chi-square goodness-of-fit tests were used to evaluate model performance
A Homotopy Algorithm for Approximating Geometric Distributions by Integrable Systems
In the geometric theory of nonlinear control systems, the notion of a
distribution and the dual notion of codistribution play a central
role. Many results in nonlinear control theory require certain
distributions to be integrable. Distributions (and codistributions)
are not generically integrable and, moreover, the integrability
property is not likely to persist under small perturbations of the
system. Therefore, it is natural to consider the problem of
approximating a given codistribution by an integrable codistribution,
and to determine to what extent such an approximation may be used for
obtaining approximate solutions to various problems in control
theory. In this note, we concentrate on the purely mathematical
problem of approximating a given codistribution by an integrable
codistribution. We present an algorithm for approximating an
m-dimensional nonintegrable codistribution by an integrable one using
a homotopy approach. The method yields an approximating codistribution
that agrees with the original codistribution on an m-dimensional
submanifold E_0 of R^n
Batch queues, reversibility and first-passage percolation
We consider a model of queues in discrete time, with batch services and
arrivals. The case where arrival and service batches both have Bernoulli
distributions corresponds to a discrete-time M/M/1 queue, and the case where
both have geometric distributions has also been previously studied. We describe
a common extension to a more general class where the batches are the product of
a Bernoulli and a geometric, and use reversibility arguments to prove versions
of Burke's theorem for these models. Extensions to models with continuous time
or continuous workload are also described. As an application, we show how these
results can be combined with methods of Seppalainen and O'Connell to provide
exact solutions for a new class of first-passage percolation problems.Comment: 16 pages. Mostly minor revisions; some new explanatory text added in
various places. Thanks to a referee for helpful comments and suggestion
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