On the moments of Markov renewal processes

Recently Kshirsagar and Gupta [5] obtained expressions for the asymptotic values of the first two moments of a Markov renewal process. The method they employed involved formal inversion of matrices of Laplace-Stieltjes transforms. Their method also required the imposition of a non-singularity condition. In this paper we derive the asymptotic values using known renewal theoretic results. This method of approach utilises the fundamental matrix of the imbedded ergodic Markov chain and the theory of generalised matrix inverses. Although our results differ in form from those obtained by Kshirsagar and Gupta [5] we show that they reduce to their results under the added non-singularity condition. As a by-product of the derivation we find explicit expressions for the moments of the first passage time distributions in the associated semi-Markov process, generalising the results of Kemeny and Snell [4] obtained for Markov chains.

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Some Results in Markov Renewal Processes

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319690202 ◽

1969 ◽

Vol 18 (2) ◽

pp. 61-72 ◽

Cited By ~ 3

Author(s):

A.M. Kshirsagar ◽

Y. P. Gupta

Keyword(s):

Generating Function ◽

Renewal Process ◽

First Passage Time ◽

Probability Generating Function ◽

Passage Time ◽

Markov Renewal Process ◽

Renewal Processes ◽

First Passage ◽

Markov Renewal Processes ◽

Cumulative Processes

The following results are obtained in this paper: (1) The probability generating function of the simultaneous distribution of all the Nj( t)'s, where Nj( t) represents the number of times the jth state ( j = 1, 2, ... , m) is visited in time t, in a Markov Renewal Process ; (2) the covariance between Nj( t) and Nk( t); (3) the probability generating function and moments of Nj( t)'s in a General Markov Renewal Process i.e., a Markov Renewal Process with a random origin; (4) Cumulative processes associated with a Markov Renewal Process along with its first passage time and (5) Equilibrium Markov Renewal Processes.

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New better than used in expectation processes

Journal of Applied Probability ◽

10.1017/s0021900200106679 ◽

1992 ◽

Vol 29 (01) ◽

pp. 116-128 ◽

Cited By ~ 1

Author(s):

C. Y. Teresa Lam

Keyword(s):

Sufficient Conditions ◽

Markov Renewal Process ◽

Queueing Models ◽

Renewal Processes ◽

First Passage ◽

Markov Renewal Processes ◽

New Better Than Used ◽

Repair Models ◽

Better Than ◽

Passage Times

In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.

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On the level crossing of multi-dimensional delayed renewal processes

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953397000415 ◽

1997 ◽

Vol 10 (4) ◽

pp. 355-361 ◽

Cited By ~ 20

Author(s):

Jewgeni H. Dshalalow

Keyword(s):

Renewal Process ◽

Stochastic Models ◽

First Passage Time ◽

Passage Time ◽

The Other ◽

Renewal Processes ◽

Active Components ◽

Level Crossing ◽

First Passage ◽

Informal Discussion

The paper studies the behavior of an (l+3)th-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the crossing level and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented.

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A multivariate reward process defined on a semi-Markov process and its first-passage-time distributions

Journal of Applied Probability ◽

10.2307/3214872 ◽

1991 ◽

Vol 28 (2) ◽

pp. 360-373 ◽

Cited By ~ 22

Author(s):

Yasushi Masuda ◽

Ushio Sumita

Keyword(s):

Asymptotic Behavior ◽

Markov Process ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Renewal Equations ◽

Supplementary Variables ◽

Semi Markov Process ◽

Reward Process

A multivariate reward process defined on a semi-Markov process is studied. Transform results for the distributions of the multivariate reward and related processes are derived through the method of supplementary variables and the Markov renewal equations. These transform results enable the asymptotic behavior to be analyzed. A class of first-passage time distributions of the multivariate reward processes is also investigated.

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A multivariate reward process defined on a semi-Markov process and its first-passage-time distributions

Journal of Applied Probability ◽

10.1017/s0021900200039747 ◽

1991 ◽

Vol 28 (02) ◽

pp. 360-373 ◽

Cited By ~ 9

Author(s):

Yasushi Masuda ◽

Ushio Sumita

Keyword(s):

Asymptotic Behavior ◽

Markov Process ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Renewal Equations ◽

Supplementary Variables ◽

Semi Markov Process ◽

Reward Process

A multivariate reward process defined on a semi-Markov process is studied. Transform results for the distributions of the multivariate reward and related processes are derived through the method of supplementary variables and the Markov renewal equations. These transform results enable the asymptotic behavior to be analyzed. A class of first-passage time distributions of the multivariate reward processes is also investigated.

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Time dependent analysis of multivariate marked renewal processes

Journal of Applied Probability ◽

10.1017/s0021900200018866 ◽

2001 ◽

Vol 38 (03) ◽

pp. 707-721 ◽

Cited By ~ 2

Author(s):

Jewgeni H. Dshalalow

Keyword(s):

Poisson Process ◽

Point Process ◽

Stochastic Models ◽

First Passage Time ◽

Compound Poisson Process ◽

Passage Time ◽

Time Dependent ◽

Renewal Processes ◽

First Passage ◽

Time Dependent Analysis

The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.

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Time dependent analysis of multivariate marked renewal processes

Journal of Applied Probability ◽

10.1239/jap/1005091034 ◽

2001 ◽

Vol 38 (3) ◽

pp. 707-721 ◽

Cited By ~ 4

Author(s):

Jewgeni H. Dshalalow

Keyword(s):

Poisson Process ◽

Point Process ◽

Stochastic Models ◽

First Passage Time ◽

Compound Poisson Process ◽

Passage Time ◽

Time Dependent ◽

Renewal Processes ◽

First Passage ◽

Time Dependent Analysis

The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.

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New better than used in expectation processes

Journal of Applied Probability ◽

10.2307/3214796 ◽

1992 ◽

Vol 29 (1) ◽

pp. 116-128 ◽

Cited By ~ 4

Author(s):

C. Y. Teresa Lam

Keyword(s):

Sufficient Conditions ◽

Markov Renewal Process ◽

Queueing Models ◽

Renewal Processes ◽

First Passage ◽

Markov Renewal Processes ◽

New Better Than Used ◽

Repair Models ◽

Better Than ◽

Passage Times

In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.

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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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