Spectral Theory for Skip-Free Markov Chains

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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First-passage times with PFr densities

Journal of Applied Probability ◽

10.2307/3213758 ◽

1985 ◽

Vol 22 (1) ◽

pp. 185-196 ◽

Cited By ~ 27

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 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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The Protein Hourglass: Exact First Passage Time Distributions for Protein Thresholds

10.1101/2020.08.14.251223 ◽

2020 ◽

Author(s):

Krishna Rijal ◽

Ashok Prasad ◽

Dibyendu Das

Keyword(s):

First Passage Time ◽

Self Regulation ◽

Threshold Value ◽

Passage Time ◽

First Passage Times ◽

Lambda Phage ◽

First Passage ◽

Analytical Expression ◽

Protein Levels ◽

Passage Times

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of E. coli infected with bacteriophage lambda. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on lambda-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

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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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Mean first-passage times of Brownian motion and related problems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1952.0051 ◽

1952 ◽

Vol 211 (1106) ◽

pp. 431-443 ◽

Cited By ~ 42

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Passage Time ◽

First Passage Times ◽

Analytical Relationship ◽

First Passage ◽

Special Boundaries ◽

Mean First Passage Times ◽

Special Cases ◽

Passage Times

The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.

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On passage and conditional passage times for Markov chains in continuous time

Journal of Applied Probability ◽

10.1017/s0021900200040924 ◽

1988 ◽

Vol 25 (02) ◽

pp. 279-290 ◽

Cited By ~ 1

Author(s):

Masaaki Kijima

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Continuous Time ◽

First Passage Time ◽

Passage Time ◽

Death Process ◽

First Passage ◽

Reversible Markov Chain ◽

Irreducible Markov Chain ◽

Passage Times

Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on . For k < i < j, let H = {k + 1, ···, j − 1} and let kTij ( jTik ) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to . These conditional passage times are studied through first-passage times of a modified chain HX(t) constructed by making the set of states absorbing. It will be shown that the densities of kTij and jTik for any birth-death process are unimodal and the modes kmij ( jmik ) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of kTij and jTik for a time-reversible Markov chain are presented. Symmetry among kTij, jTik , and is also discussed, where , and are conditional passage times of the reversed process of X(t).

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First-passage-time moments of Markov processes

Journal of Applied Probability ◽

10.2307/3213962 ◽

1985 ◽

Vol 22 (4) ◽

pp. 939-945 ◽

Cited By ~ 13

Author(s):

David D. Yao

Keyword(s):

Markov Chains ◽

Markov Processes ◽

Continuous Time ◽

Fundamental Matrix ◽

Generalized Inverse ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Continuous Time Markov Chains ◽

Passage Times

We consider the first-passage times of continuous-time Markov chains. Based on the approach of generalized inverse, moments of all orders are derived and expressed in simple, explicit forms in terms of the ‘fundamental matrix'. The formulas are new and are also efficient for computation.

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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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First-passage-time moments of Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200108186 ◽

1985 ◽

Vol 22 (04) ◽

pp. 939-945

Author(s):

David D. Yao

Keyword(s):

Markov Chains ◽

Markov Processes ◽

Continuous Time ◽

Fundamental Matrix ◽

Generalized Inverse ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Continuous Time Markov Chains ◽

Passage Times

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On limiting behavior of ordinary and conditional first-passage times for a class of birth-death processes

Journal of Applied Probability ◽

10.2307/3214074 ◽

1987 ◽

Vol 24 (1) ◽

pp. 235-240 ◽

Cited By ~ 4

Author(s):

U. Sumita

Keyword(s):

First Passage Time ◽

Arrival Rate ◽

Passage Time ◽

First Passage Times ◽

Death Process ◽

Service Rate ◽

First Passage ◽

Limiting Behavior ◽

Passage Times ◽

Birth Death

Let N(t) be a birth-death process on 𝒩= {0, 1, 2, ·· ·} governed by the transition rates λn > 0 (n ≧ 0) and μn > 0 (n ≧ 1) where λn → λ > 0 and μn → μ > 0 as n → ∞ and ρ = λ/μ. Let Tmn be the first-passage time of N(t) from m to n and define It is shown that, when converges in distribution to TBP(μ,λ) as n → ∞ where TΒΡ (μ,λ) is the server busy period of an M/M/1 queueing system with arrival rate μ and service rate λ. Correspondingly T0n/E[T0n] converges to 1 with probability 1 as n →∞. Of related interest is the conditional first-passage time mTrn of N(t) from r to n given no visit to m where m < r < n. As we shall see, the conditional first-passage time of N(t) can be viewed as an ordinary first-passage time of a modified birth-death process M(t) governed by where are generated from λn and μn. Furthermore it is shown that for and while for and This enables one to establish the relation between the limiting behavior of the ordinary first-passage times and that of the conditional first-passage times.

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