Level crossings and stationary distributions for general dams

1980 ◽  
Vol 17 (1) ◽  
pp. 218-226 ◽  
Author(s):  
Michael Rubinovitch ◽  
J. W. Cohen
Keyword(s):  
Explicit Expression ◽  
Stationary Distribution ◽  
Basic Relation ◽  
Expected Number ◽  
Stationary Distributions ◽  
Level Crossings ◽  
Fixed Level

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

Level crossings and stationary distributions for general dams

1980 ◽  
Vol 17 (01) ◽  
pp. 218-226 ◽  
Author(s):  
Michael Rubinovitch ◽  
J. W. Cohen
Keyword(s):  
Explicit Expression ◽  
Stationary Distribution ◽  
Basic Relation ◽  
Expected Number ◽  
Stationary Distributions ◽  
Level Crossings ◽  
Fixed Level

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

scholarly journals Distributions stationnaires d'un système bonus–malus et probabilité de ruine

1988 ◽  
Vol 18 (1) ◽  
pp. 31-46 ◽  
Author(s):  
Par François Dufresne
Keyword(s):  
Theoretical Model ◽  
Stationary Distribution ◽  
Recursive Algorithm ◽  
Third Party ◽  
Ruin Probabilities ◽  
Stationary Distributions ◽  
Probability Of Ruin

AbstractIt is shown how the stationary distributions of a bonus–malus system can be computed recursively. It is further shown that there is an intrinsic relationship between such a stationary distribution and the probability of ruin in the risk-theoretical model. The recursive algorithm is applied to the Swiss bonus–malus system for automobile third-party liability and can be used to evaluate ruin probabilities.

scholarly journals Basing population genetic inferences and models of molecular evolution upon desired stationary distributions of DNA or protein sequences

2008 ◽  
Vol 363 (1512) ◽  
pp. 3931-3939 ◽  
Author(s):  
Sang Chul Choi ◽  
Benjamin D Redelings ◽  
Jeffrey L Thorne
Keyword(s):  
Molecular Evolution ◽  
Markov Model ◽  
Stationary Distribution ◽  
Population Genetic ◽  
Amino Acid Level ◽  
Protein Sequences ◽  
Relative Fitness ◽  
Evolutionary Models ◽  
Effective Population ◽  
Stationary Distributions

Models of molecular evolution tend to be overly simplistic caricatures of biology that are prone to assigning high probabilities to biologically implausible DNA or protein sequences. Here, we explore how to construct time-reversible evolutionary models that yield stationary distributions of sequences that match given target distributions. By adopting comparatively realistic target distributions, evolutionary models can be improved. Instead of focusing on estimating parameters, we concentrate on the population genetic implications of these models. Specifically, we obtain estimates of the product of effective population size and relative fitness difference of alleles. The approach is illustrated with two applications to protein-coding DNA. In the first, a codon-based evolutionary model yields a stationary distribution of sequences, which, when the sequences are translated, matches a variable-length Markov model trained on human proteins. In the second, we introduce an insertion–deletion model that describes selectively neutral evolutionary changes to DNA. We then show how to modify the neutral model so that its stationary distribution at the amino acid level can match a profile hidden Markov model, such as the one associated with the Pfam database.

Stationary distributions for dams with additive input and content-dependent release rate

1977 ◽  
Vol 9 (03) ◽  
pp. 645-663 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Stationary Distribution ◽  
Release Rate ◽  
Zero Set ◽  
Stationary Distributions ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Limiting Case ◽  
Finite Dam

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt } of an infinite dam whose cumulative input {At } is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = x α and {A t } is stable with index β ∊ (0, 1). In general if EAt , < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA 1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA 1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt } as t → ∞. For {At } stable with index β and r(x) = x α , α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt } in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.

scholarly journals On Truncation of the Matrix-Geometric Stationary Distributions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 798 ◽  
Author(s):  
Naumov ◽  
Gaidamaka ◽  
Samouylov
Keyword(s):  
Finite Number ◽  
Stationary Distribution ◽  
Queueing Systems ◽  
Death Process ◽  
Birth And Death Process ◽  
Loss System ◽  
Stationary Distributions ◽  
Open Network ◽  
The Matrix

In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed.

Quasi-stationary distributions and one-dimensional circuit-switched networks

1987 ◽  
Vol 24 (04) ◽  
pp. 965-977 ◽  
Author(s):  
Ilze Ziedins
Keyword(s):  
Communication Networks ◽  
Stationary Distribution ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Switched Networks ◽  
One Dimensional ◽  
Special Cases ◽  
Quasi Stationary Distribution

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

scholarly journals Level crossings of a random trigonometric polynomial with dependent coefficients

1995 ◽  
Vol 58 (1) ◽  
pp. 39-46
Author(s):  
K. Farahmand
Keyword(s):  
Trigonometric Polynomial ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Expected Number ◽  
Level Crossings ◽  
Random Trigonometric Polynomial ◽  
Normally Distributed

AbstractThis paper provides an asymptotic estimate for the expected number of K-level crossings of the random trigonometric polynomial g1 cos x + g2 cos 2x+ … + gn cos nx where gj (j = 1, 2, …, n) are dependent normally distributed random variables with mean zero and variance one. The two cases of ρjr, the correlation coeffiecient between the j-th and r-th coefficients, being either (i) constant, or (ii) ρ∣j−r∣ρ, j ≠ r, 0 < ρ < 1, are considered. It is shown that the previous result for ρjr = 0 still remains valid for both cases.

scholarly journals Level crossings and turning points of random hyperbolic polynomials

1999 ◽  
Vol 22 (3) ◽  
pp. 579-586
Author(s):  
K. Farahmand ◽  
P. Hannigan
Keyword(s):  
Asymptotic Estimate ◽  
Turning Points ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Level Crossings ◽  
Hyperbolic Polynomials ◽  
Normally Distributed

In this paper, we show that the asymptotic estimate for the expected number ofK-level crossings of a random hyperbolic polynomiala1sinhx+a2sinh2x+⋯+ansinhnx, whereaj(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one, is(1/π)logn. This result is true for allKindependent ofx, providedK≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomiala1coshx+a2cosh2x+⋯+ancoshnx, withaj(j=1,2,…,n)as before, is also(1/π)logn.

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