Distributional Properties of Fluid Queues Busy Period and First Passage Times

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

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Ageing first-passage times of Markov processes: a matrix approach

Journal of Applied Probability ◽

10.2307/3215169 ◽

1997 ◽

Vol 34 (1) ◽

pp. 1-13 ◽

Cited By ~ 11

Author(s):

Haijun Li ◽

Moshe Shaked

Keyword(s):

Markov Process ◽

Markov Processes ◽

First Passage Time ◽

Passage Time ◽

Critical Value ◽

First Passage Times ◽

Matrix Approach ◽

First Passage ◽

Passage Times

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

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First-passage times with PFr densities

Journal of Applied Probability ◽

10.1017/s0021900200029119 ◽

1985 ◽

Vol 22 (01) ◽

pp. 185-196

Author(s):

David Assaf ◽

Moshe Shared ◽

J. George shanthikumar

Keyword(s):

First Passage Time ◽

Compound Poisson Process ◽

Passage Time ◽

First Passage Times ◽

Embedded Markov Chain ◽

First Passage ◽

Completely Monotone ◽

Totally Positive ◽

Finite State ◽

Passage Times

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PF r (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PF r first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

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Ageing first-passage times of Markov processes: a matrix approach

Journal of Applied Probability ◽

10.1017/s0021900200100646 ◽

1997 ◽

Vol 34 (01) ◽

pp. 1-13 ◽

Cited By ~ 3

Author(s):

Haijun Li ◽

Moshe Shaked

Keyword(s):

Markov Process ◽

Markov Processes ◽

First Passage Time ◽

Passage Time ◽

Critical Value ◽

First Passage Times ◽

Matrix Approach ◽

First Passage ◽

Passage Times

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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Spectral Theory for Skip-Free Markov Chains

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000098x ◽

1989 ◽

Vol 3 (1) ◽

pp. 77-88 ◽

Cited By ~ 9

Author(s):

Joseph Abate ◽

Ward Whitt

Keyword(s):

Markov Chains ◽

Spectral Theory ◽

First Passage Time ◽

Passage Time ◽

First Passage Times ◽

Simple Relationship ◽

First Passage ◽

Birth And Death Processes ◽

The Laplace Transform ◽

Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

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Probing Protein Shelf Lives from Inverse Mean First Passage Times

10.26434/chemrxiv.9529163 ◽

2019 ◽

Author(s):

Vishal Singh ◽

Parbati Biswas

Keyword(s):

Protein Aggregation ◽

Rate Constants ◽

Survival Probability ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

First Order Kinetics ◽

The Mean ◽

Passage Times ◽

Good Agreement

Protein aggregation is investigated theoretically via protein turnover, misfolding, aggregation and degradation. The Mean First Passage Time (MFPT) of aggregation is evaluated within the framework of Chemical Master Equation (CME) and pseudo first order kinetics with appropriate boundary conditions. The rate constants of aggregation of different proteins are calculated from the inverse MFPT, which show an excellent match with the experimentally reported rate constants and those extracted from the ThT/ThS fluorescence data. Protein aggregation is found to be practically independent of the number of contacts and the critical number of misfolded contacts. The age of appearance of aggregation-related diseases is obtained from the survival probability and the MFPT results, which matches with those reported in the literature. The calculated survival probability is in good agreement with the only available clinical data for Parkinson’s disease.<br>

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On Durbin's Series for the Density of First Passage Times

Journal of Applied Probability ◽

10.1017/s0021900200008263 ◽

2011 ◽

Vol 48 (03) ◽

pp. 713-722

Author(s):

P. Zipkin

Keyword(s):

Pointwise Convergence ◽

First Passage Time ◽

Convergent Series ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Substitution Method ◽

Curved Boundary ◽

Weiner Process ◽

Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

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On the moments of some first-passage times

Journal of Applied Probability ◽

10.1017/s0021900200037931 ◽

1985 ◽

Vol 22 (02) ◽

pp. 461-466

Author(s):

Valeri T. Stefanov

Keyword(s):

Continuous Function ◽

Exponential Type ◽

First Passage Time ◽

Sequential Estimation ◽

Passage Time ◽

Positive Continuous Function ◽

First Passage ◽

Finite Dimensional ◽

Independent Increments ◽

Passage Times

Let {Xt } t≧0 (t may be discrete or continuous) be a random process whose finite-dimensional distributions are of exponential type. The first-passage time inf{t:Xt ≧f(t)}, where f(t) is a positive, continuous function, such that f(t)= o(t) as t↑∞, is considered. The problem of finiteness of its moments is solved for both the case that {Xt } t≧0 has stationary independent increments as well as the case in which no assumptions are made about stationarity and independence for the increments of the process. Applications to sequential estimation are also given.

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Weak convergence and first passage times

Journal of Applied Probability ◽

10.1017/s0021900200048014 ◽

1975 ◽

Vol 12 (02) ◽

pp. 324-332

Author(s):

Allan Gut

Keyword(s):

First Passage Time ◽

Domain Of Attraction ◽

Passage Time ◽

Partial Sums ◽

First Passage ◽

Stable Law ◽

Common Distribution Function ◽

Functional Central Limit Theorems ◽

Functional Central Limit ◽

Passage Times

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp ], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

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