Understanding the shape of the mixture failure rate (with engineering and demographic applications)

Mixtures of distributions are usually effectively used for modeling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behavior as t tends to infty. Some demographic and engineering examples are considered. The corresponding inverse problem is discussed.

Shocks in homogeneous and heterogeneous populations

A system subject to a point process of shocks is considered. Shocks occur in accordance with a nonhomogeneous Poisson process. Different criterions of system failures are discussed in a homogeneous case. Two natural settings are analyzed. Heterogeneity is modeled by an unobserved univariate random variable (frailty). It is shown that reliability (safety) analysis for a heterogeneous case can differ dramatically from that for a homogeneous setting. A shock burn-in procedure for a heterogeneous population is described. The corresponding bounds for the failure rates are obtained.

Modeling failure (mortality) rate with a change point

Simple models for the failure (mortality) rate change point are considered. The relationship with the mean residual lifetime function change point problem is discussed. It is shown that when the change point is random, the observed failure (mortality) rate can be obtained via a specific mixture of lifetime distributions. The shape of the observed failure (mortality) rate is analyzed and the corresponding sim-ple but meaningful example is considered.

Mortality in varying environment

An impact of environment on mortality, similar to survival analysis, is often modeled by the proportional hazards model, which assumes the corresponding comparison with a baseline environment. This model describes the memory-less property, when the mortality rate at a given instant of time depends only on the environment at this in-stant of time and does not depend on the history. In the presence of degradation the assumption of this kind is usually unrealistic and history-dependent models should be considered. The simplest stochastic degradation model is the accelerated life model. We discuss these models for the cohort setting and apply the developed approach to the period setting for the case when environment (stress) is modeled by the functions with switching points (jumps in the level of the stress).

Aging: damage accumulation versus increasing mortality rate

If aging is understood as some process of damage accumulation, it does not necessarily lead to increasing mortality rates. Within the framework of a suggested generalization of the Strehler-Mildwan (1960) model, we show that even for models with monotonically increasing degradation, the mortality rate can still decrease. The de-cline in vitality and functions, as manifestation of aging, is modeled by the monotonically decreasing quality of life function. Using this function, the initial lifetime ran-dom variable with ultimately decreasing mortality rate is ‘weighted’ to result in a new random variable which is already characterized by the increasing rate.

On engineering reliability concepts and biological aging

Some stochastic approaches to biological aging modeling are studied. We assume that an organism acquires a random resource at birth. Death occurs when the accumulated dam-age (wear) exceeds this initial value, modeled by the discrete or continuous random vari-ables. Another source of death of an organism is also taken into account, when it occurs as a consequence of a shock or of a demand for energy, which is a generalization of the Strehler-Mildwan’s model (1960). Biological age based on the observed degradation is also defined. Finally, aging properties of repairable systems are discussed. We show that even in the case of imperfect repair, which is certainly the case for organisms, aging slows down with age and eventually can even fade out. This presents another possible explanation for the human mortality rate plateaus.mortality

Lifesaving increases life expectancy

The notion of repeated minimal repair is analyzed and applied to modeling the lifesaving procedure of organisms. Under certain assumptions the equivalence between demographic lifesaving model and reliability shock model is proved. Both of these models are based on the non-homogeneous Poisson processes of underlying potentially harmful events The lifesaving ratio for homogeneous and heterogeneous populations is defined. Some generalizations are discussed. Several simple examples are considered.

On systems with shared resources and optimal switching strategies

Simple series systems of identical components with spare parts are considered. It is shown that the cumulative distribution function of a system failure time tends to a step function as the number of components increases and resources can be shared. An example of ‘continuous resources’ is also described. The time-sharing strategy for standby systems is investigated. It is proved that an optimal rule for a system of standby components with increasing failure rates is the single switching performed at t/2 , where t is a mission time.

Stochastically ordered subpopulations and optimal burn-in procedure

Burn-in is a widely used engineering method which is adopted to eliminate defective items before they are shipped to customers or put into the field operation. In the studies of burn-in, the assumption of bathtub shaped failure rate function is usually employed and optimal burn-in procedures are investigated. In this paper, however, we assume that the population is composed of two ordered subpopulations and optimal burn-in procedures are studied in this context. Two types of risks are defined and an optimal burn-in procedure, which minimizes the weighted risks is studied. The joint optimal solutions for the optimal burn-in procedure, which minimizes the mean number of repairs during the field operation, are also investigated.