SOME RESULTS ON A NEW CLASS OF SHOCK MODELS

2010 ◽  
Vol 27 (04) ◽  
pp. 503-515
Author(s):  
ALAGAR RANGAN ◽  
AYŞE TANSU
Keyword(s):  
Failure Time ◽  
System Failure ◽  
First Passage ◽  
New Class ◽  
Optimal Replacement ◽  
Random Threshold ◽  
Shock Models ◽  
Special Cases ◽  
Replacement Problem ◽  
First Passage Problem

Traditional shock models view system failure time as a first passage problem. Yeh Lam proposed a new class of models called δ-shock models in which failure was dependent on the frequency of shocks. The present work generalizes Yeh Lam's results for renewal shock arrivals and random threshold. Several special cases and an optimal replacement problem are also discussed.

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.

Moments of the first-passage time of a Wiener process with drift between two elastic barriers

1995 ◽  
Vol 32 (4) ◽  
pp. 1007-1013 ◽  
Author(s):  
Marco Dominé
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
Special Cases ◽  
Backward Equation ◽  
The One ◽  
First Passage Problem ◽  
Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

An Integral Equation Method for the First-Passage Problem in Random Vibration

1984 ◽  
Vol 51 (3) ◽  
pp. 674-679 ◽  
Author(s):  
P. H. Madsen ◽  
S. Krenk
Keyword(s):  
Integral Equation ◽  
Random Vibration ◽  
Integral Equation Method ◽  
Nonstationary Process ◽  
Alternative Methods ◽  
Approximation Formula ◽  
First Passage ◽  
Special Cases ◽  
Nonstationary Stochastic Process ◽  
First Passage Problem

The first-passage problem for a nonstationary stochastic process is formulated as an integral identity, which produces known bounds and series expansions as special cases, while approximation of the kernel leads to an integral equation for the first-passage probability density function. An accurate, explicit approximation formula for the kernel is derived, and the influence of uni or multi modal frequency content of the process is investigated. Numerical results provide comparisons with simulation results and alternative methods for narrow band processes, and also the case of a multimodal, nonstationary process is dealt with.

A class of correlated cumulative shock models

1985 ◽  
Vol 17 (02) ◽  
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.

scholarly journals Optimal replacement under a minimal repair strategy—a general failure model

1983 ◽  
Vol 15 (1) ◽  
pp. 198-211 ◽  
Author(s):  
Terje Aven
Keyword(s):  
Stopping Time ◽  
System Failure ◽  
Failure Model ◽  
Minimal Repair ◽  
Discounted Cost ◽  
Repair Strategy ◽  
Optimal Replacement ◽  
Special Cases ◽  
Replacement Model ◽  
Available Information

In this paper we generalize the minimal repair replacement model introduced by Barlow and Hunter (1960). We assume that there is available information about the underlying condition of the system, for instance through measurements of wear characteristics and damage inflicted on the system. We assume furthermore that the system failure rate and the expected cost of a repair/replacement at any point of time are adapted to this information. At time t = 0 a new system is installed. At a stopping time T, based on the information about the condition of the system, the system is replaced by a new and identical one, and the process is repeated. Failures that occur before replacement are rectified through minimal repair. We assume that a minimal repair changes neither the age of the system nor the information about the condition of the system. The problem is to find a T which minimizes the total expected discounted cost. Under appropriate conditions an optimal T is found. Some generalizations and special cases are given.

scholarly journals On the HNBUE property in a class of correlated cumulative shock models

1995 ◽  
Vol 27 (4) ◽  
pp. 1186-1188 ◽  
Author(s):  
Rafael Pérez-Ocón ◽  
M. Luz Gámiz-Pérez
Keyword(s):  
Failure Time ◽  
System Failure ◽  
Shock Model ◽  
Correct Proof ◽  
Shock Models

Conditions for a correlated cumulative shock model under which the system failure time is HNBUE are given. It is shown that the proof of a theorem given by Sumita and Shanthikumar (1985) relative to this property is not correct and a correct proof of the theorem is given.

Pure jump shock models in reliability

1986 ◽  
Vol 18 (02) ◽  
pp. 423-440 ◽  
Author(s):  
James W. Drosen
Keyword(s):  
Markov Process ◽  
Failure Rate ◽  
Failure Time ◽  
Jump Markov Process ◽  
Damage Level ◽  
Random Threshold ◽  
Shock Models ◽  
Environmental Shocks

There are many examples of a device suffering damage from random environmental shocks. We model the damage level of such a device as a pure jump Markov process, where the incremental damage caused by a shock depends both on the magnitude of the shock and on the damage level just before the shock. We also look at the time until failure of the device, which occurs when the damage level exceeds a random threshold. The distribution of the failure time and the failure rate are examined, and conditions for the failure rate to be increasing or to have an increasing average are found.

scholarly journals Optimal replacement under a minimal repair strategy—a general failure model

1983 ◽  
Vol 15 (01) ◽  
pp. 198-211
Author(s):  
Terje Aven
Keyword(s):  
Stopping Time ◽  
System Failure ◽  
Failure Model ◽  
Minimal Repair ◽  
Discounted Cost ◽  
Repair Strategy ◽  
Optimal Replacement ◽  
Special Cases ◽  
Replacement Model ◽  
Available Information

In this paper we generalize the minimal repair replacement model introduced by Barlow and Hunter (1960). We assume that there is available information about the underlying condition of the system, for instance through measurements of wear characteristics and damage inflicted on the system. We assume furthermore that the system failure rate and the expected cost of a repair/replacement at any point of time are adapted to this information. At time t = 0 a new system is installed. At a stopping time T, based on the information about the condition of the system, the system is replaced by a new and identical one, and the process is repeated. Failures that occur before replacement are rectified through minimal repair. We assume that a minimal repair changes neither the age of the system nor the information about the condition of the system. The problem is to find a T which minimizes the total expected discounted cost. Under appropriate conditions an optimal T is found. Some generalizations and special cases are given.

Distribution properties of the system failure time in a general shock model

1984 ◽  
Vol 16 (2) ◽  
pp. 363-377 ◽  
Author(s):  
J. G. Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Failure Time ◽  
Sufficient Conditions ◽  
System Failure ◽  
Shock Model ◽  
Completely Monotone ◽  
Shock Models ◽  
Subsequent Interval ◽  
New Better Than Used ◽  
Better Than

In this paper we study some distribution properties of the system failure time in general shock models associated with correlated renewal sequences (Xn, Yn) . Two models, depending on whether the magnitude of the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length of the subsequent interval to the next shock, are considered. Sufficient conditions under which the system failure time is completely monotone, new better than used, new better than used in expectation, and harmonic new better than used in expectation are given for these two models.

Moments of the first-passage time of a Wiener process with drift between two elastic barriers

1995 ◽  
Vol 32 (04) ◽  
pp. 1007-1013 ◽  
Author(s):  
Marco Dominé
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
Special Cases ◽  
Backward Equation ◽  
The One ◽  
First Passage Problem ◽  
Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

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