Initial and final behaviour of failure rate functions for mixtures and systems

2003 ◽  
Vol 40 (03) ◽  
pp. 721-740 ◽  
Author(s):  
Henry W. Block ◽  
Yulin Li ◽  
Thomas H. Savits
Keyword(s):  
Asymptotic Behaviour ◽  
Failure Rate ◽  
Rate Function ◽  
Coherent Systems ◽  
Failure Rate Function ◽  
Rate Functions ◽  
Limiting Behaviour

In this paper we consider the initial and asymptotic behaviour of the failure rate function resulting from mixtures of subpopulations and formation of coherent systems. In particular, it is shown that the failure rate of a mixture has the same limiting behaviour as the failure rate of the strongest subpopulation. A similar result holds for systems except the role of strongest subpopulation is replaced by strongest min path set.

Initial and final behaviour of failure rate functions for mixtures and systems

2003 ◽  
Vol 40 (3) ◽  
pp. 721-740 ◽  
Author(s):  
Henry W. Block ◽  
Yulin Li ◽  
Thomas H. Savits
Keyword(s):  
Asymptotic Behaviour ◽  
Failure Rate ◽  
Rate Function ◽  
Coherent Systems ◽  
Failure Rate Function ◽  
Rate Functions ◽  
Limiting Behaviour

In this paper we consider the initial and asymptotic behaviour of the failure rate function resulting from mixtures of subpopulations and formation of coherent systems. In particular, it is shown that the failure rate of a mixture has the same limiting behaviour as the failure rate of the strongest subpopulation. A similar result holds for systems except the role of strongest subpopulation is replaced by strongest min path set.

scholarly journals Determination of failure rate function on the basis оf the markovian process

2021 ◽  
pp. 79-87
Author(s):  
Iryna Bashkevych ◽  
◽  
Yurii Yevseichyk ◽  
Kostiantyn Medvediev ◽  
Leonid Yanchuk ◽  
...  
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Analytical Form ◽  
Rate Function ◽  
Markovian Process ◽  
Failure Rate Function ◽  
Repair Work ◽  
Rate Functions ◽  
Maintenance Work ◽  
The Cost

The life cycle of a construction (or its element) is considered as markovian process with discrete states and continuous time. Five operational states have been accepted, in which the construction may be. The corresponding system of differential equations is obtained for the case of a homogeneous markovian process with a constant conversion rate (Kolmogorov system). The method of uncertain coefficients is applied to solve the system of equations in analytical form. The obtained solutions make it possible to determine the probability of finding the construction in a particular state as well as the most likely transition time from one operational state to another. Security function defined as the probability of not finding the construction in its last (inoperable) state and the failure rate function. The graphs of the probability of finding a construction in each of the five states, reliability and failure rate functions are presented and investigated. The obtained analytical dependences make it possible to determine the longevity and residual life of the work both individual elements and structures as a whole and optimize scheduling for ongoing maintenance work, significantly improve the performance of the structure, reduce the cost of repair work and extend the life of the structure.

THE FAILURE RATE SHAPE FOR A MIXTURE OF TWO GAMMAS

2015 ◽  
Vol 29 (2) ◽  
pp. 253-264 ◽  
Author(s):  
Henry W. Block ◽  
Thomas H. Savits ◽  
Naftali A. Langberg
Keyword(s):  
Failure Rate ◽  
Rate Function ◽  
Shape Parameters ◽  
Failure Rate Function ◽  
Scale Parameters ◽  
Mixing Parameter ◽  
Gamma Density

In this paper, we continue our investigation of the shape of the failure rate of a mixture of two densities. In our recent paper, Block, Langberg and Savits [2], we introduced a variation of Glaser's method in which we emphasized the role of the mixing parameter q. There we determined all possible shapes of the failure rate function for a mixture of one exponential and one gamma density. Here we classify all possible shapes for a mixture of two gamma densities having shape parameters b, c>0 and scale parameters λ, μ>0.

Burn-in and the performance quality measures in continuous heterogeneous populations

2012 ◽  
Vol 226 (4) ◽  
pp. 417-425
Author(s):  
Ji Hwan Cha ◽  
Maxim Finkelstein
Keyword(s):  
Failure Rate ◽  
Performance Measures ◽  
Quality Measures ◽  
Rate Function ◽  
Heterogeneous Population ◽  
Failure Rate Function ◽  
Heterogeneous Populations ◽  
Performance Quality ◽  
Rate Functions ◽  
Overall Performance

Burn-in is a method used to eliminate initial failures in field use. To burn-in a component or system means to subject it to a period of use prior to the time when it is to actually be used. Under the assumption of decreasing or bathtub-shaped population failure rate functions, various problems of determining optimal burn-in have been intensively studied in the literature. In this paper, we assume that a population is composed of stochastically ordered subpopulations, described by their own performance quality measures and study optimal burn-in, which optimizes overall performance measures. It turns out that this setting can justify burn-in even when it is not necessary in the framework of conventional approaches. For instance, it could be reasonable to perform burn-in even when the failure rate function that describes a heterogeneous population of items increases and this is one of the main and important findings of the current study.

scholarly journals TOWARDS DURABILITY OF AUTOMOBILE ROADS

2019 ◽  
pp. 207-213
Author(s):  
R. P. Moiseenko ◽  
V. M. Efimenko
Keyword(s):  
Failure Rate ◽  
Objective Assessment ◽  
Simple Algorithm ◽  
Rate Function ◽  
Technical Condition ◽  
Failure Rate Function ◽  
Factors Affecting ◽  
The Road ◽  
Rate Functions ◽  
Probabilistic Nature

Evaluation of the durability of the road is a consistently relevant industry problem. The use of mathematical theory of reliability is due to the probabilistic nature of many factors affecting the duration of the effective service of road elements. The article presents the results of calculating the durability of road using the failure rate functions based on the exponential law, the Weibull law and the United law. The variants of calculation of durability considered in article, taking into account the called functions, provide rather objective assessment of indicators for highways with non-rigid type of road clothes. The calculation of the durability of the transport structure is made by a simple algorithm. The main difficulty in using this algorithm is the need for experimental determination of the parameters of the failure rate function, but this difficulty is typical for any statistical method. At the same time, among the factors affecting the variability of parameters should be identified geographical complex, individually characteristic of the region of study. Therefore, in the future, to determine the failure rate of the elements of the operated roads, it is necessary to create regional Bach data characterizing the technical condition of roads and their elements in accordance with industry regulators.

STOCHASTIC SURVIVAL MODELS WITH EVENTS TRIGGERED BY EXTERNAL SHOCKS

2012 ◽  
Vol 26 (2) ◽  
pp. 183-195 ◽  
Author(s):  
Ji Hwan Cha ◽  
Maxim Finkelstein
Keyword(s):  
Failure Rate ◽  
Rate Function ◽  
Poisson Processes ◽  
Survival Models ◽  
Failure Rate Function ◽  
Limiting Behavior ◽  
Wear Process ◽  
Shock Models ◽  
Nonhomogeneous Poisson Processes ◽  
Rate Functions

In most conventional settings, the events caused by an external shock are initiated at the moments of its occurrence. In this paper, we study the new classes of shock models: (i) When each shock from a nonhomogeneous Poisson processes can trigger a failure of a system not immediately, as in classical extreme shock models, but with delay of some random time. (ii) When each shock from a nonhomogeneous Poisson processes results not in an ‘immediate’ increment of wear, as in classical accumulated wear models, but triggers its own increasing wear process. The wear from different shocks is accumulated and the failure of a system occurs when it reaches a given boundary. We derive the corresponding survival and failure rate functions. Furthermore, we study the limiting behavior of the failure rate function where it is applicable.

REDEFINING FAILURE RATE FUNCTION FOR DISCRETE DISTRIBUTIONS

2002 ◽  
Vol 09 (03) ◽  
pp. 275-285 ◽  
Author(s):  
M. XIE ◽  
O. GAUDOIN ◽  
C. BRACQUEMOND
Keyword(s):  
Failure Rate ◽  
Rate Function ◽  
Monotonicity Property ◽  
Discrete Distributions ◽  
The Other ◽  
Continuous Case ◽  
Discrete Version ◽  
Series System ◽  
Failure Rate Function ◽  
Definition Of

For discrete distribution with reliability function R(k), k = 1, 2,…,[R(k - 1) - R(k)]/R(k - 1) has been used as the definition of the failure rate function in the literature. However, this is different from that of the continuous case. This discrete version has the interpretation of a probability while it is known that a failure rate is not a probability in the continuous case. This discrete failure rate is bounded, and hence cannot be convex, e.g., it cannot grow linearly. It is not additive for series system while the additivity for series system is a common understanding in practice. In the paper, another definition of discrete failure rate function as In[R(k - 1)/R(k)] is introduced, and the above-mentioned problems are resolved. On the other hand, it is shown that the two failure rate definitions have the same monotonicity property. That is, if one is increasing/decreasing, the other is also increasing/decreasing. For other aging concepts, the new failure rate definition is more appropriate. The failure rate functions according to this definition are given for a number of useful discrete reliability functions.

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