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 New better than used in expectation processes

1992 ◽  
Vol 29 (1) ◽  
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.

Processes with new better than used first-passage times

1984 ◽  
Vol 16 (03) ◽  
pp. 667-686 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Point Process ◽  
Markov Processes ◽  
Probability Space ◽  
Sufficient Conditions ◽  
General Point ◽  
First Passage ◽  
Semi Markov Processes ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R 0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

Processes with new better than used first-passage times

1984 ◽  
Vol 16 (3) ◽  
pp. 667-686 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Point Process ◽  
Markov Processes ◽  
Probability Space ◽  
Sufficient Conditions ◽  
General Point ◽  
First Passage ◽  
Semi Markov Processes ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

First passage times for Markov renewal processes and applications

2000 ◽  
Vol 43 (12) ◽  
pp. 1238-1249 ◽  
Author(s):  
Guanghui Xu ◽  
Xueming Yuan ◽  
Quanlin Li
Keyword(s):  
First Passage Times ◽  
Renewal Processes ◽  
First Passage ◽  
Markov Renewal Processes ◽  
Passage Times

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.

Characterization of Some First Passage Times Using Log-Concavity and Log-Convexity as Aging Notions

1987 ◽  
Vol 1 (3) ◽  
pp. 279-291 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Branching Processes ◽  
Stochastic Ordering ◽  
Reliability Theory ◽  
First Passage Times ◽  
First Passage ◽  
Likelihood Ratio Ordering ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

An interpretation of log-concavity and log-convexity as aging notions is given in this paper. It imitates a stochastic ordering characterization of the NBU (new better than used) and the NWU (new worse than used) notions but stochastic ordering is now replaced by the likelihood ratio ordering. The new characterization of log-concavity and log-convexity sheds new light on these properties and enables one to obtain intuitively simple proofs of the log-convexity and log-concavity of some first passage times of interest in branching processes and in reliability theory.

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 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 class of correlated cumulative shock models

1985 ◽  
Vol 17 (2) ◽  
pp. 347-366 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Failure Time ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Random Variables ◽  
System Failure ◽  
Shock Models ◽  
The Times ◽  
New Better Than Used ◽  
Better Than

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn, Yn}∞n=0 of correlated random variables. The {Xn} denote the sizes of the shocks and the {Yn} denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.

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