Should availability (calculated using mean times to fail/repair) correspond to average proportion of units available over time? If I have a mean time to repair (MTTR) of 3 days and a mean time between failures (MTBF) of 9 days, and so an availability of MTBF/(MTTR + MTBF) = 9/12= 75%, can I expect that, on average, 75% of my machines will be available per day, lets say I have a large number machines?  Do any assumptions need to be upheld in order this to occur?
 A: From Barlow and Proschan (1981):
Let $T_i$ be the duration of the $i$th functioning period and $D_i$ be the down time for the $i$th repair.  Let the sequence $\{T_i+D_i\}$ be mutually independent; $T_i$ and $D_i$ are not assumed independent.
Let $F$ be the common distribution of $T_i$, $H$ the common distribution of $T_i+D_i$, and 
$M_H(t) = \sum_{i=1}^\infty H^{(i)}(t)$ be the renewal function corresponding to $H$.
Then by the Key Renewal Theorem
$$A = \lim_{t\rightarrow\infty} A(t) = \frac{E[T]} { E[T] + E[D]}$$
when the distribution $H$ is non-lattice.
So, the equation you gave holds in the limit, with no assumptions other than that the expectations exist and the mild assumptions given above.
Now, if you have a system with $n$ machines following the same assumptions, and assuming that the machines are independent of each other, then the availability in the limit as $t\rightarrow\infty$ of each machine is also $A$.  
Let $X_i(t)$ be equal to 1 if the $i$th machine is operating at time $t$ and 0 if it is not.  Then the availability of $n$ machines at a given time $t$ is given by
$$A_1(t)+\cdots + A_n(t) = E[X_1(t)]+\cdots+E[X_n(t)]= E[X_1(t) + \cdots X_n(t)].$$
So, as $t\rightarrow\infty$ we must have a limiting availability of $nA$ machines.  This holds as time increases, rather than as the number of machines increases.  
It seems intuitively reasonable that as the number of machines increases, that the average availability at a given time will go to something, but I'm not sure that it would have to be the same quantity.
The dedication in this book is very funny --- I had forgotten it.
R.E. Barlow and F. Proschan (1981) Statistical Theory of Reliability and Life Testing.  TO BEGIN WITH.
