Читать книгу Reliability Analysis, Safety Assessment and Optimization - Enrico Zio - Страница 20

Let random variable T denote the time to failure. Then, the reliability function at time t can be expressed as the probability that the component does not fail at time t, that is,

(1.4)

Denote the cumulative distribution function (cdf) of T as F(t). The relationship between the cdf and the reliability is

(1.5)

Further, denote the probability density function (pdf) of failure time T as f(t). Then, equation (1.5) can be rewritten as

(1.6)

In all generality, the expected value or mean of the time to failure T is called the mean time to failure (MTTF), which is defined as

(1.7)

It is equivalent to

(1.8)

Another related concept is the mean time between failures (MTBF). MTBF is the average working time between two consecutive failures. The difference between MTBF and MTTF is that the former is used only in reference to a repairable item, while the latter is used for non-repairable items. However, MTBF is commonly used for both repairable and non-repairable items in practice.

The failure rate function or hazard rate function, denoted by h(t), is defined as the conditional probability of failure in the time interval [t, t+Δt] given that it has been working properly up to time t, which is given by

(1.9)

Furthermore, the cumulative failure rate function, or cumulative hazard function, denoted by H(t), is given by

(1.10)

Example 1.2 The failure time of a valve follows the exponential distribution with parameter h(t) (in arbitrary units of time-1). The value is new and functioning at time h(t). Calculate the reliability of the valve at time h(t) (in arbitrary units of time).

Solution

The pdf of the failure time of the valve is

The reliability function of the valve is given by

At time, the value of the reliability is