Passage-time generating functions for continuous-time finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 414-426 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Weighted Sum ◽  
Continuous Time Markov Chain ◽  
Finite Markov Chains

Let S denote a subset of the states of a finite continuous-time Markov chain and let Y(a) denote the time that elapses until a weighted sum of a time units have been spent in S. Formulae are derived for the generating functions of Y(a) and of Y(a + b) – Y(b).

Passage-time generating functions for continuous-time finite Markov chains

1968 ◽  
Vol 5 (02) ◽  
pp. 414-426 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Weighted Sum ◽  
Continuous Time Markov Chain ◽  
Finite Markov Chains

Let S denote a subset of the states of a finite continuous-time Markov chain and let Y(a) denote the time that elapses until a weighted sum of a time units have been spent in S. Formulae are derived for the generating functions of Y(a) and of Y(a + b) – Y(b).

Some passage-time generating functions for discrete-time and continuous-time finite Markov chains

1967 ◽  
Vol 4 (03) ◽  
pp. 496-507 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Finite Markov Chain ◽  
Finite Markov Chains

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Yk denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Yk , of Yj + k — Yj and of Yj + k — Yj in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.

Some passage-time generating functions for discrete-time and continuous-time finite Markov chains

1967 ◽  
Vol 4 (3) ◽  
pp. 496-507 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Finite Markov Chain ◽  
Finite Markov Chains

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Yk denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Yk, of Yj + k — Yj and of Yj + k — Yj in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.

Lumpability and marginalisability for continuous-time Markov chains

1993 ◽  
Vol 30 (3) ◽  
pp. 518-528 ◽  
Author(s):  
Frank Ball ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Ion Channel ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Continuous Time Markov Chains ◽  
Probabilistic Proof ◽  
Necessary And Sufficient ◽  
Mutually Independent

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

1998 ◽  
Vol 35 (3) ◽  
pp. 545-556 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Hazard Rate ◽  
First Passage Time ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Reversed Hazard Rate ◽  
The Right ◽  
Birth Death

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

1989 ◽  
Vol 26 (3) ◽  
pp. 643-648 ◽  
Author(s):  
A. I. Zeifman
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

A Note on Approximating Mean Occupation Times of Continuous-Time Markov Chains

1988 ◽  
Vol 2 (2) ◽  
pp. 267-268
Author(s):  
Sheldon M. Ross
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Random Variables ◽  
Continuous Time Markov Chain ◽  
Occupation Times ◽  
Continuous Time Markov Chains ◽  
Exponential Random Variables

In [1] an approach to approximate the transition probabilities and mean occupation times of a continuous-time Markov chain is presented. For the chain under consideration, let Pij(t) and Tij(t) denote respectively the probability that it is in state j at time t, and the total time spent in j by time t, in both cases conditional on the chain starting in state i. Also, let Y1,…, Yn be independent exponential random variables each with rate λ = n/t, which are also independent of the Markov chain.

On a Markov chain approach for the study of reliability structures

1996 ◽  
Vol 33 (02) ◽  
pp. 357-367 ◽  
Author(s):  
M. V. Koutras
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Functions ◽  
General Framework ◽  
Reliability Evaluation ◽  
Markov Chain Approach ◽  
Markov Chain Imbedding ◽  
Finite Markov Chains

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

On a Markov chain approach for the study of reliability structures

1996 ◽  
Vol 33 (2) ◽  
pp. 357-367 ◽  
Author(s):  
M. V. Koutras
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Functions ◽  
General Framework ◽  
Reliability Evaluation ◽  
Markov Chain Approach ◽  
Markov Chain Imbedding ◽  
Finite Markov Chains

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

Time to Stationarity for a Continuous-Time Markov Chain

1991 ◽  
Vol 5 (1) ◽  
pp. 61-76 ◽  
Author(s):  
James Allen Fill
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Applied Mathematics ◽  
Continuous Time Markov Chain ◽  
Mixing Rates

Separation is one measure of distance from stationarity for Markov chains. Strong stationary times provide bounds on separation and so aid in the analysis of mixing rates. The precise connection between separation and strong stationary times was drawn by Aldous and Diaconis (1987) (Advances in Applied Mathematics8: 69−97) for discrete time chains. We develop the corresponding foundational theory for continuous time chains; several new and interesting mathematical issues arise.

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