Asymptotic Properties of the Hazard Function and the Mean Residual Life Function

The memoryless or non-aging property of systems is of special relevance in reliability theory, which implies that the hazard function is constant in time, and the corresponding mean residual life function takes a reciprocal value. The only known continuous distribution with that property is the exponential distribution. However, many other distributions exist whose asymptotic behavior of underlying hazard functions approaches a constant, while the mean residual life function approaches a reciprocally constant value. Here we provide an analysis which enables us to study a class of distributions that asymptotically approach the memoryless property, and which include gamma, Erlangian, exponential resilience, exponential geometric, hyper exponential, logistic exponential and the inverse Gaussian distribution.