Processes with new better than used first-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.

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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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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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The first-passage times of phase semi-Markov processes

Statistics & Probability Letters ◽

10.1016/j.spl.2011.08.021 ◽

2012 ◽

Vol 82 (1) ◽

pp. 40-48 ◽

Cited By ~ 4

Author(s):

Xuan Zhang ◽

Zhenting Hou

Keyword(s):

Markov Processes ◽

First Passage Times ◽

First Passage ◽

Semi Markov Processes ◽

Passage Times

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On ageing properties of first-passage times of increasing Markov processes

Advances in Applied Probability ◽

10.1017/s0001867800011484 ◽

2002 ◽

Vol 34 (01) ◽

pp. 241-259

Author(s):

Félix Belzunce ◽

Eva-María Ortega ◽

José M. Ruiz

Keyword(s):

Markov Chain ◽

Markov Processes ◽

Continuous Time ◽

Transition Matrix ◽

Sufficient Conditions ◽

First Passage Times ◽

Generator Matrix ◽

First Passage ◽

Finite Case ◽

Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

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On ageing properties of first-passage times of increasing Markov processes

Advances in Applied Probability ◽

10.1239/aap/1019160959 ◽

2002 ◽

Vol 34 (1) ◽

pp. 241-259 ◽

Cited By ~ 1

Author(s):

Félix Belzunce ◽

Eva-María Ortega ◽

José M. Ruiz

Keyword(s):

Markov Chain ◽

Markov Processes ◽

Continuous Time ◽

Transition Matrix ◽

Sufficient Conditions ◽

First Passage Times ◽

Generator Matrix ◽

First Passage ◽

Finite Case ◽

Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

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First passage times and lumpability of semi-Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200041462 ◽

1988 ◽

Vol 25 (04) ◽

pp. 675-687 ◽

Cited By ~ 2

Author(s):

Ushio Sumita ◽

Maria Rieders

Keyword(s):

Markov Processes ◽

First Passage Times ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

First Passage ◽

The Laplace Transform ◽

First Exit Times ◽

Semi Markov Processes ◽

Necessary And Sufficient ◽

Passage Times

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

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

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000005x ◽

1987 ◽

Vol 1 (3) ◽

pp. 279-291 ◽

Cited By ~ 16

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.

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First passage times and lumpability of semi-Markov processes

Journal of Applied Probability ◽

10.2307/3214288 ◽

1988 ◽

Vol 25 (4) ◽

pp. 675-687 ◽

Cited By ~ 4

Author(s):

Ushio Sumita ◽

Maria Rieders

Keyword(s):

Markov Processes ◽

First Passage Times ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

First Passage ◽

The Laplace Transform ◽

First Exit Times ◽

Semi Markov Processes ◽

Necessary And Sufficient ◽

Passage Times

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

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A class of correlated cumulative shock models

Advances in Applied Probability ◽

10.2307/1427145 ◽

1985 ◽

Vol 17 (2) ◽

pp. 347-366 ◽

Cited By ~ 48

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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