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.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (3) ◽  
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 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.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (2) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

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.

Duality results for block-structured transition matrices

1999 ◽  
Vol 36 (4) ◽  
pp. 1045-1057 ◽  
Author(s):  
Yiqiang Q. Zhao ◽  
Wei Li ◽  
Attahiru Sule Alfa
Keyword(s):  
Transition Matrix ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Transition Kernel ◽  
Transition Matrices ◽  
Duality Results ◽  
Markov Renewal Processes ◽  
The Matrix ◽  
Probabilistic Measures ◽  
Block Structured

In this paper, we consider a certain class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewal process possesses some block-structured property, including repeating rows. Duality conditions and properties are obtained on two probabilistic measures which often play a key role in the analysis and computations of such a block-structured process. The method used here unifies two different concepts of duality. Applications of duality are also provided, including a characteristic theorem concerning recurrence and transience of a transition matrix with repeating rows and a batch arrival queueing model.

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.

Exponential ergodicity in Markov renewal processes

1968 ◽  
Vol 5 (2) ◽  
pp. 387-400 ◽  
Author(s):  
Jozef L. Teugels
Keyword(s):  
Markov Renewal Process ◽  
Renewal Processes ◽  
Continuous Time Markov Chain ◽  
Uniform Estimates ◽  
Exponential Ergodicity ◽  
Markov Renewal Processes ◽  
Recurrent Case ◽  
Transient Case ◽  
Positive Recurrent ◽  
Semi Markov Processes

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.An application to the M/G/1 queue is included in the last section.

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.

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