Definitions

Recommendations E.800 of the International Telecommunications Union (ITU-T) defines reliability as follows:

"The ability of an item to perform a required function
under given conditions for a given time interval."

Reliability R(t) is defined as the probability that the system continues to function throughout the interval (0,t). It is not necessary to (but it is often) assume that the system is functioning at time 0.

Let the random variable X be the time to failure of the system

R(t) = P(X > t) = 1- F(t)

When F(t) is the distribution function of system lifetime

Mean Time To system Failure



Where f(t) is the density function of a system lifetime

Reliability is often known as the Complementary distribution function or the Survivor function. Reliability is closely related but is distinct from availability. See the URL availability-modeling.com for more details on availability. The folowing simple example illustrates the difference between a reliability model and an availability model.

Consider a 2-processors system, Each processor has the failure rate . The repair rate is . For the availability analysis, the state space is shown as follows.



For reliability analysis, since we do not consider the repair once the system is in down state. All the down states will be considered as "absorbing" states, in a reliability model. Hence in our example we get

.

Note that the above Markov chain is a reliability model with repair since component repair is allowed if the system has not failed. It is also possible to construct a reliability model without repair.