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.

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.

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.

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

1984 ◽  
Vol 16 (02) ◽  
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 (X n, Y n) . Two models, depending on whether the magnitude of the nth shock X n is correlated to the length Y n 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.

The probability that the largest observation is censored

1993 ◽  
Vol 30 (03) ◽  
pp. 602-615 ◽  
Author(s):  
R. A. Maller ◽  
S. Zhou
Keyword(s):  
Survival Time ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Consistent Estimator ◽  
Survival Times ◽  
Practical Difficulty ◽  
Independent Censoring ◽  
The Mean ◽  
Necessary And Sufficient

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

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 Generalized Variability Orderings among Nonnegative Fuzzy Random Variables

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
S. Ramasubramanian ◽  
P. Mahendran
Keyword(s):  
Distribution Function ◽  
Integral Inequality ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Equivalent Conditions ◽  
New Better Than Used ◽  
Better Than ◽  
Fuzzy Random

The variability ordering for more and less variables of fuzzy random variables in terms of its distribution function is defined. A property of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) is derived as an application to the variability ordering of fuzzy random variables. The concept of generalized variability orderings of nonnegative fuzzy random variables representing lifetime of components is introduced. The<Pdomination is a generalized variability ordering. We proposed an integral inequality to the case of fuzzy random variables using<Pordering. The results included equivalent conditions which justify the generalized variability orderings.

scholarly journals The new better than used failure rate class of life distribution

1988 ◽  
Vol 20 (1) ◽  
pp. 237-240 ◽  
Author(s):  
A. M. Abouammoh ◽  
A. N. Ahmed
Keyword(s):  
Failure Rate ◽  
Life Distribution ◽  
Shock Models ◽  
Rate Class ◽  
Life Distributions ◽  
Dual Class ◽  
New Better Than Used ◽  
Ageing Distribution ◽  
Better Than

A new concept of ageing distribution, namely new better than used in failure rate (NBUFR), is introduced. Different properties of the NBUFR class and its dual class are presented. Its relations to other classes of life distributions are investigated. Finally, NBUFR survival under shock models is discussed.

Shock models with underlying birth process

1975 ◽  
Vol 12 (1) ◽  
pp. 18-28 ◽  
Author(s):  
M. S. A-Hameed ◽  
F. Proschan
Keyword(s):  
Failure Rate ◽  
Life Distribution ◽  
Birth Process ◽  
Increasing Failure Rate ◽  
Shock Models ◽  
Life Distributions ◽  
New Better Than Used ◽  
Better Than

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process: given k shocks have occurred in [0, t], the probability of a shock occurring in (t, t + Δ] is λ kλ (t)Δ + o (Δ). We show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the ‘new better than used' property, etc.) are obtained under appropriate assumptions on λ k, λ (t), and on the probability of surviving a given number of shocks.

The probability that the largest observation is censored

1993 ◽  
Vol 30 (3) ◽  
pp. 602-615 ◽  
Author(s):  
R. A. Maller ◽  
S. Zhou
Keyword(s):  
Survival Time ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Consistent Estimator ◽  
Survival Times ◽  
Practical Difficulty ◽  
Independent Censoring ◽  
The Mean ◽  
Necessary And Sufficient

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

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.

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