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.

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.

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).

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

Lumpability for non-irreducible finite markov chains

1984 ◽  
Vol 21 (03) ◽  
pp. 567-574 ◽  
Author(s):  
Atef M. Abdel-Moneim ◽  
Frederick W. Leysieffer
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
The State ◽  
Discrete Time Markov Chain ◽  
Finite Markov Chains

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.

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.

scholarly journals Normal distributions of finite Markov chains

2019 ◽  
Vol 29 (08) ◽  
pp. 1431-1449
Author(s):  
John Rhodes ◽  
Anne Schilling
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Planar Graphs ◽  
Finite Semigroup ◽  
Finite Markov Chain ◽  
Normal Distributions ◽  
Straight Line ◽  
Finite Markov Chains ◽  
The Right

We show that the stationary distribution of a finite Markov chain can be expressed as the sum of certain normal distributions. These normal distributions are associated to planar graphs consisting of a straight line with attached loops. The loops touch only at one vertex either of the straight line or of another attached loop. Our analysis is based on our previous work, which derives the stationary distribution of a finite Markov chain using semaphore codes on the Karnofsky–Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain.

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.

General transition probabilities for finite Markov chains

1964 ◽  
Vol 60 (1) ◽  
pp. 83-91 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Finite Number ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Initial State ◽  
Discrete Time Markov Chain ◽  
Finite Markov Chains

We consider a stationary discrete-time Markov chain with a finite number m of possible states which we designate by 1,…,m. We assume that at time t = 0 the process is in an initial state i with probability (i = 1,…, m) and such that and .

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