scholarly journals PRESERVATION OF SOME NEW PARTIAL ORDERINGS UNDER POISSON AND CUMULATIVE DAMAGE SHOCK MODELS

2002 ◽  
Vol 32 (1) ◽  
pp. 95-105
Author(s):  
S.E. Khider ◽  
A.-H.N. Ahmed ◽  
M.K. Mohamed
Keyword(s):  
Cumulative Damage ◽  
Shock Models ◽  
Partial Orderings

Partial orderings under cumulative damage shock models

1993 ◽  
Vol 25 (04) ◽  
pp. 939-946 ◽  
Author(s):  
Franco Pellerey
Keyword(s):  
Likelihood Ratio ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Discrete Distributions ◽  
Counting Processes ◽  
Likelihood Ratio Order ◽  
Common Shocks ◽  
Shock Models ◽  
Partial Orderings ◽  
The Mean

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

Partial orderings under cumulative damage shock models

1993 ◽  
Vol 25 (4) ◽  
pp. 939-946 ◽  
Author(s):  
Franco Pellerey
Keyword(s):  
Likelihood Ratio ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Discrete Distributions ◽  
Counting Processes ◽  
Likelihood Ratio Order ◽  
Common Shocks ◽  
Shock Models ◽  
Partial Orderings ◽  
The Mean

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

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.

A useful ageing property based on the Laplace transform

1983 ◽  
Vol 20 (3) ◽  
pp. 615-626 ◽  
Author(s):  
Bengt Klefsjö
Keyword(s):  
Laplace Transform ◽  
Damage Model ◽  
Cumulative Damage ◽  
Closure Properties ◽  
Shock Models ◽  
The Laplace Transform ◽  
Life Distributions ◽  
Cumulative Damage Model ◽  
Ageing Property

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.

scholarly journals On preservation of some partial orderings under shock models

1990 ◽  
Vol 22 (02) ◽  
pp. 508-509 ◽  
Author(s):  
Subhash C. Kochar
Keyword(s):  
Poisson Process ◽  
Homogeneous Poisson Process ◽  
Shock Models ◽  
Partial Orderings ◽  
Life Distributions

Singh and Jain (1989) have proved some preservation results for partial orderings of life distributions assuming that shocks occur according to a homogeneous Poisson process. It is shown that their results hold under less restrictive conditions.

A useful ageing property based on the Laplace transform

1983 ◽  
Vol 20 (03) ◽  
pp. 615-626 ◽  
Author(s):  
Bengt Klefsjö
Keyword(s):  
Laplace Transform ◽  
Damage Model ◽  
Cumulative Damage ◽  
Closure Properties ◽  
Shock Models ◽  
The Laplace Transform ◽  
Life Distributions ◽  
Cumulative Damage Model ◽  
Ageing Property

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.

Optimal replacement for self-repairing shock models by general failure rate

1984 ◽  
Vol 21 (1) ◽  
pp. 108-119 ◽  
Author(s):  
Gary Gottlieb ◽  
Benny Levikson
Keyword(s):  
Markov Process ◽  
Failure Rate ◽  
Lower Cost ◽  
Cumulative Damage ◽  
Control Limit ◽  
Replacement Policy ◽  
Time Of Arrival ◽  
Optimal Replacement ◽  
Shock Models ◽  
Optimal Replacement Policy

A device is subject to a series of shocks which cause damage and eventually failure will occur at the time of arrival of one of the shocks. In between the shocks, the device is partially repaired as the cumulative damage decreases as some Markov process. The device must be replaced upon failure at some cost but it can also be replaced before failure at a lower cost. We consider the general case where the failure rate need not be increasing and replacement can be made at any time. The form of the optimal replacement policy is found and fairly general conditions are given for which a control limit policy is optimal.

Optimal replacement with semi-Markov shock models

1976 ◽  
Vol 13 (01) ◽  
pp. 108-117 ◽  
Author(s):  
Richard M. Feldman
Keyword(s):  
Markov Process ◽  
Cumulative Damage ◽  
Control Limit ◽  
Damage Level ◽  
Fixed Level ◽  
Optimal Replacement ◽  
Shock Models ◽  
Semi Markov Process ◽  
Damage Process ◽  
Replacement Rule

Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused from all previous shocks. The purpose of this paper is to derive the optimal replacement rule for such a system whose cumulative damage process is a semi-Markov process. This allows for both the time between shocks and the damage due to the next shock to be dependent on the present cumulative damage level. Only policies within the class of control-limit policies will be considered; namely, policies with which no action is taken if the damage is below a fixed level, and a replacement is made if the damage is above that. An example will be given illustrating the use of the optimal replacement rule.

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.

Laplace ordering and its applications

1991 ◽  
Vol 28 (1) ◽  
pp. 116-130 ◽  
Author(s):  
Abdulhamid Alzaid ◽  
Jee Soo Kim ◽  
Frank Proschan
Keyword(s):  
Time Series ◽  
Operations Research ◽  
Laplace Transforms ◽  
Cumulative Damage ◽  
Coherent Systems ◽  
Damage Models ◽  
Shock Models ◽  
Life Distributions

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

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