Assuming that the conditions described in Useful Life Failure Period apply so that the failure rate is constant, the relationship between reliability, failure rate and time is given by the expression:

Where:

R(t)=Reliability, i.e., the probability that an item t will survive for time under the specified operating conditions.

e=The base of the natural logarithms (approximately 2.7183).

λ=The item failure rate under the specified operating conditions of temperature, stress, environment, etc.. It is constant for at least time t.

t=The time that the item is at risk under the specified operating conditions. This is sometimes called the mission time.

Failure causes are not always dependent upon time and may depend upon particular events, such as switching, handling, etc. In these cases, the relationship between reliability, failure probability and number of events is given by the expression:

Where:

R(N)=Reliability, i.e., the probability that the item survives N events under specified operating conditions.

ρE=The probability that an event will be defective under specified operating conditions.

N=The number of events.

When ρE <<1, ρE can be thought of as a failure rate or better still as a percent defective:

In general, the above assumption is valid for system prediction purposes and the following expression may be used:

Where:

ρE= The expected number of failures per event under specified operating conditions.

N=The number of events.

Consider now the case when the specified time interval, t (for which the reliability of an item is to be predicted), is made up of a number of different time intervals, ta, tb, tc, etc., each associated with different operating conditions. Then, from equation (2.1), the probability of failure in each time interval is given by:

Providing that R(ta), R(tb) and R(tc) can be considered independent of each other, they can be combined to R(t) give as follows:

Similarly, when the operational use of an item includes a number of independent events, the individual reliabilities given by equation (2.3) can be combined in a similar manner:

Where and are events occurring at different times:

M= The number of events associated with percent defective x.

N= The number of events associated with percent defective y.

Finally, time-based and event-based probabilities can also be combined together if the operational use of an item involves both: