Some Results in Markov Renewal Processes

1969 ◽  
Vol 18 (2) ◽  
pp. 61-72 ◽  
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.

scholarly journals On the moments of Markov renewal processes

1969 ◽  
Vol 1 (02) ◽  
pp. 188-210 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
First Passage ◽  
Ergodic Markov Chain ◽  
Asymptotic Values ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Stieltjes Transforms

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.

scholarly journals On the moments of Markov renewal processes

1969 ◽  
Vol 1 (2) ◽  
pp. 188-210 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
First Passage ◽  
Ergodic Markov Chain ◽  
Asymptotic Values ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Stieltjes Transforms

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.

scholarly journals On the level crossing of multi-dimensional delayed renewal processes

1997 ◽  
Vol 10 (4) ◽  
pp. 355-361 ◽  
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.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (03) ◽  
pp. 752-756
Author(s):  
Per Kragh Andersen
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
The Moment

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

scholarly journals New better than used in expectation processes

1992 ◽  
Vol 29 (01) ◽  
pp. 116-128 ◽  
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.

scholarly journals First excess levels of vector processes

1994 ◽  
Vol 7 (3) ◽  
pp. 457-464 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Point Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Compound Process ◽  
First Time

This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e. reaches or exceeds) its associated level for the first time. The author targets a joint transformation of the first excess level, first passage time, and the index of the point process which labels the first passage time. The cases when both marks are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic models are discussed.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂ t , the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂ t , and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (4) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

Time dependent analysis of multivariate marked renewal processes

2001 ◽  
Vol 38 (03) ◽  
pp. 707-721 ◽  
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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