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 (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.

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.

Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (1) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie
Keyword(s):  
Recurrence Time ◽  
Renewal Processes ◽  
Convergence Properties ◽  
Renewal Theorem ◽  
Uniform Limit ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Positive Recurrent ◽  
Forward Recurrence Time ◽  
Initial Distributions

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (01) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie
Keyword(s):  
Recurrence Time ◽  
Renewal Processes ◽  
Convergence Properties ◽  
Renewal Theorem ◽  
Uniform Limit ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Positive Recurrent ◽  
Forward Recurrence Time ◽  
Initial Distributions

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

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.

Analysis of an M/G/1 queue with two types of impatient units

1995 ◽  
Vol 27 (03) ◽  
pp. 840-861 ◽  
Author(s):  
M. Martin ◽  
J. R. Artalejo
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Mathematical Analysis ◽  
Service System ◽  
Probability Distributions ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Stationary Probability ◽  
Markov Renewal Processes ◽  
Specific Performance

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later. We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.

Central limit theorems for multivariate semi-Markov sequences and processes, with applications

1999 ◽  
Vol 36 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Central Limit ◽  
Limit Theorems ◽  
Markov Models ◽  
Central Limit Theorems ◽  
Renewal Processes ◽  
Counting Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Processes

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

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.

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