There is probably no function $f$ such that $f(A,X)=A^X$.

Question - why do you not want to assume the observations are independent?

Then since 15 seconds is very little compared to 5 hours, there is a very small probability that the user tries to access the system just as it is switching between up and down, so the probability of running into a system that is down is still about 1.59%...

If we just want a set that contains no largest cone (i.e., no smallest base) then we can take the nonzero degrees, i.e., those $\mathbf a$ with $\mathbf a>\mathbf 0$.

Reimann and Slaman show that there is a cone consisting of continuously random reals. However they then show that every non-$\Delta^1_1$ real is continuously random. So Kleene's $\mathcal O$ is a fairly natural base for such a cone.