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.

Shock models with underlying birth process

1975 ◽  
Vol 12 (01) ◽  
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.

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.

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

1988 ◽  
Vol 20 (01) ◽  
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.

CLASSIFICATION OF MULTIVARIATE LIFE DISTRIBUTIONS BASED ON PARTIAL ORDERING

2002 ◽  
Vol 16 (1) ◽  
pp. 129-137 ◽  
Author(s):  
Dilip Roy
Keyword(s):  
Present Article ◽  
Failure Rate ◽  
Partial Ordering ◽  
Increasing Failure Rate ◽  
Multivariate Generalization ◽  
Partial Orderings ◽  
Life Distributions ◽  
New Better Than Used ◽  
Better Than

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

Failure distributions of shock models

1980 ◽  
Vol 17 (03) ◽  
pp. 745-752 ◽  
Author(s):  
Gary Gottlieb
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Shock Model ◽  
Life Distribution ◽  
Increasing Failure Rate ◽  
Shock Models ◽  
Damage Process

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

scholarly journals Reversed Preservation Properties for Series and Parallel Systems

2007 ◽  
Vol 44 (4) ◽  
pp. 928-937 ◽  
Author(s):  
Félix Belzunce ◽  
Helena Martínez-Puertas ◽  
José M. Ruiz
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Parallel Systems ◽  
Mean Residual Life ◽  
Convex Order ◽  
Increasing Failure Rate ◽  
Decreasing Failure Rate ◽  
New Better Than Used ◽  
Better Than

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

Failure distributions of shock models

1980 ◽  
Vol 17 (3) ◽  
pp. 745-752 ◽  
Author(s):  
Gary Gottlieb
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Shock Model ◽  
Life Distribution ◽  
Increasing Failure Rate ◽  
Shock Models ◽  
Damage Process

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

scholarly journals AGING FUNCTIONS AND MULTIVARIATE NOTIONS OF NBU AND IFR

2010 ◽  
Vol 24 (2) ◽  
pp. 263-278 ◽  
Author(s):  
Fabrizio Durante ◽  
Rachele Foschi ◽  
Fabio Spizzichino
Keyword(s):  
Failure Rate ◽  
Survival Function ◽  
Conditional Survival ◽  
Survival Functions ◽  
Increasing Failure Rate ◽  
Joint Survival Function ◽  
History Of ◽  
Aging Function ◽  
New Better Than Used ◽  
Better Than

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function $\overline{F}$. For such models, we study some properties of multivariate aging of $\overline{F}$ that are described by means of the multivariate aging function $B_{\overline{F}}$, which is a useful tool for describing the level curves of $\overline{F}$. Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals.

scholarly journals Reversed Preservation Properties for Series and Parallel Systems

2007 ◽  
Vol 44 (04) ◽  
pp. 928-937
Author(s):  
Félix Belzunce ◽  
Helena Martínez-Puertas ◽  
José M. Ruiz
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Parallel Systems ◽  
Mean Residual Life ◽  
Convex Order ◽  
Increasing Failure Rate ◽  
Decreasing Failure Rate ◽  
New Better Than Used ◽  
Better Than

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

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