RELIABILITY:BOUNDS ON RELIABILITY BASED UPON FAILURE TIMES

BOUNDS ON RELIABILITY BASED UPON FAILURE TIMES

The previous sections considered systems that performed successfully during a designated period or failed during this same period. An alternative way of viewing systems is to view their performance as a function of time.

Consider a component (or system) and its associated random variable, the time to failure, T. Denote the cumulative distribution function of the time to failure of the component by F and its density function by f. In terms of the previous discussion, the random variables X and T are related in that X takes on the values

This function has a useful probabilistic interpretation, namely, r(t) dt represents the con- ditional probability that an object surviving to age t will fail in the interval [t, t + dt]. This function is sometimes called the hazard rate.

In many applications, there is every reason to believe that the failure rate tends to in- crease because of the inevitable deterioration that occurs. Such a failure rate that remains constant or increases with age is said to have an increasing failure rate (IFR).

In some applications, the failure rate tends to decrease. It would be expected to de- crease initially, for instance, for materials that exhibit the phenomenon of work hardening. Certain solid-state electronic devices are also believed to have a decreasing failure rate. Thus, a failure rate that remains constant or decreases with age is said to have a decreasing failure rate (DFR).

The failure rate possesses some interesting properties. The time to failure distribution is completely determined by the failure rate. In particular, it is easily shown that

Thus, an assumption made about the failure rate has direct implications on the time to failure distribution. As an example, consider a component whose failure distribution is given by the exponential distribution, i.e.,

Note that the exponential distribution has a constant failure rate and hence has both IFR and DFR. In fact, using the expression relating the time to failure distribution and the fail- ure rate, it is evident that a component having a constant failure rate must have a time to failure distribution that is exponential.

Bounds for IFR Distributions

Under either the IFR or DFR assumption, it is possible to obtain sharp bounds on the re- liability in terms of moments and percentiles: In particular, such bounds can be derived from statements based upon the mean time to failure. This fact is particularly important because many design engineers present specifications in terms of mean time to failure.

Because the exponential distribution with constant failure rate is the boundary distri- bution between IFR and DFR distributions, it provides natural bounds on the survival probability of IFR and DFR distributions. In particular, it can be shown that if all that is known about the failure distribution is that it is IFR and has mean JL, then the greatest lower bound on the reliability that can be given is and the inequality is sharp; i.e., the exponential distribution with mean JL attains the lower bound for t < JL, and the degenerate distribution concentrating at JL attains the lower bound for t > JL. This situation can be represented graphically as shown in Fig. 25.3.

where w depends on t and satisfies 1 - wJL = e-wt. It is important to note that the w in the term e-wt is a function of t, so that a different w must be found for each t. For fixed t and JL, this w is obtained by finding the intersection of the linear function (1 - wJL) and the ex- ponential function e-wt. It can be shown that for t > JL, such an intersection always exists.

Thus, R(t) for an IFR distribution with mean JL can be bounded above and below, as shown in Fig. 25.4. Note that the lower bound is the only one of consequence for t < JL, and that the upper bound is the only one of consequence for t > JL.

Increasing Failure Rate Average

Now that bounds on the reliability of a component have been obtained, what can be said about the preservation of monotone failure rate; i.e., what structures have the IFR prop- erty when their individual components have this property? Series structures of indepen- dent IFR (DFR) components are also IFR (DFR), k out of n structures consisting of n identical independent components, each having an IFR failure distribution, are also IFR; however, parallel structures of independent IFR components are not IFR unless they are composed of identical components. Thus, it is evident that, even for some simple systems, there may not be a preservation of the monotone failure rate.

Instead of using the failure rate as a means for characterizing the reliability,

is nondecreasing in t > 0. A similar definition is given for DFRA. It can be shown that a coherent system of independent components, each of which has an IFRA failure distribution, has a system failure distribution that is also IFRA.

As with IFR systems, there are bounds for IFRA systems. It can be easily shown that IFR distributions are also IFRA distributions (but not the reverse), and the same upper bound as given for IFR distributions is applicable here. A sharp lower bound for IFRA distributions with mean JL is given by

where b depends upon t and is defined by e-bt = b(JL - t).

As an example, a monotone system containing only independent components, each of which is exponential (thereby IFRA), is itself IFRA, and the aforementioned bounds are applicable. Furthermore, these bounds are dependent only upon the system mean time to failure.