Characterizations of generalized hyperexponential distribution functions

Botta, Robert F.; Harris, Carl M.; Marchal, William G.

doi:10.1080/15326348708807049

Cited by 110 publications

(34 citation statements)

References 12 publications

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“…The information (under the reversibility assumption) that we can take M = N is important since it reduces the size of a representation in terms of a simple Markov chain. Note that in the general case of real eigenvalues, one may have to take M > N, as it is shown by an example due to Botta et al [3]. Further, to estimate the minimal M seems a difficult task.…”

Section: Theorem 12 Under the Continuous Time Assumptions (B1) And mentioning

confidence: 98%

“…In the same spirit, note that due to the example of Botta et al [3] alluded to in the introduction, Theorems 1.1 and 1.2 are no longer true when the reversibility condition is replaced by the assumption that all the eigenvalues of P are nonnegative. It follows that there are such subMarkovian matrices P that cannot be approached by reversible subMarkovian matrices (but maybe this is true for real-diagonalizable subMarkovian matrices).…”

Section: Remark 44mentioning

confidence: 98%

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On absorption times and Dirichlet eigenvalues

Miclo

2010

ESAIM: PS

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Abstract. This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point) are nonnegative, we show that T is distributed as a mixture of sums of independent geometric laws whose parameters are successive Dirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law. This result leads to a probabilistic interpretation of the spectrum, in terms of strong random times and local equilibria through a simple intertwining relation. Next this study is extended to the continuous time framework, where geometric laws have to be replaced by exponential distributions having the (opposite) Dirichlet eigenvalues of the generator as parameters. Returning to the discrete time setting we consider the influence of negative eigenvalues which are given another probabilistic meaning. These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

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Section: Theorem 12 Under the Continuous Time Assumptions (B1) And mentioning

confidence: 98%

Section: Remark 44mentioning

confidence: 98%

On absorption times and Dirichlet eigenvalues

Miclo

2010

ESAIM: PS

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“…The GH family is dense in the set of all distributions on [0, ∞), and has coefficients of variation in (0, ∞) (Botta et al [26]). More properties are in Botta and Harris [27] and Harris et al [28].…”

Section: Generalized Hyper-exponential (Gh) Inter-arrival Timesmentioning

confidence: 99%

Alternative Analysis of Finite-Time Probability Distributions of Renewal Theory

Brill¹

2014

Prob. Eng. Inf. Sci.

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We introduce a level-crossing analysis of the finite time-t probability distributions of the excess life, age, total life, and related quantities of renewal processes. The technique embeds the renewal process as one cycle of a regenerative process with a barrier at level t, whose limiting probability density function leads directly to the time-t quantities. The new method connects the analysis of renewal processes with the analysis of a large class of stochastic models of Operations Research. Examples are given.

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“…Specifically, in the paper by Cao et al [2], the amount of rainfall in an area is modeled using a mixture of two exponential densities. A further exposition of generalized hyperexponential distribution functions may also be found in Botta et al [3]. The focus of this paper is on a subset of exponential probability densities, which will be henceforth referred to as hyperexponential densities.…”

Section: Introductionmentioning

confidence: 99%