10.8.1 Availability
Demonstration Plans
The availability tests are based on the assumption that a system
can be treated as being in one (and only one) of two states, “up” or “down.”
At t = 0 the system is up (state X) and operates until the first failure at T
= X_{1}; it is down for repairs during
the restore cycle Y_{1}. An
up/down cycle is complete by time X_{1} +
Y_{1}. The random variables,
X_{i} and Y_{i} are each assumed to be independent
and identically distributed with means E(X) and E(Y). The sequence of pairs
(X_{i}, Y_{i})
forms a two dimensional renewal process.
For this system, the availability,
A(t), equals the fraction of time the system is up during (0,
t).
The steady state availability is
Assume that E(X) and E(Y)
and, therefore, A are unknown. Hypothesize two values of A.

H_{o}: A =
A_{o}H_{1}:A = A_{1} where
A_{1} < A_{o} 
(10.101) 
On the basis of test or
field data, accept or reject the hypothesis Ho by
comparing the computed A to a critical value appropriate to the test type and
parameters.
It is assumed that both the up and
down times are gamma distributed in order to derive the relationships of each
test type. However, extremely useful results can be derived assuming the
exponential distribution in both cases; the exponential distribution is used
in the following examples.