Vtyjkyuk

I have a continuous process for which I want to compute the probability that it did not fail until time $T$.

I tried the reliability / survival function:
$$ beginalign
R(T) = exp(-int_t=0^T h(t) dt)
endalign $$

I do not have a failure rate $h(t)$, but a failure probability given by a continuous but not necessarily monotonic function $p_fail(t)$, which gives the probability that the process fails at any time $t$ (with $t in R, t ge 0$).

Here, $p_fail(t)$ does not have a unit, which messes up the whole calculation. Therefore, I think my approach is wrong.

How can I calculate the reliability of the process?

To illustrate my problem:

Consider the simple function $p_1,fail(t [s]) = frac0.013600s cdot t$, for $t$ in seconds. This (in my opinion) is equal to $p_2,fail(t [h]) = frac0.011h cdot t$, for $t$ in hours.

With $T = 1h = 3600s$ the integration gives different results, depending on the unit of time:

beginarray c
hline & int_t=0^T p(t) dt \
hline p_1 & 18.00 \
hline p_2 & 0.005 \
hline
endarray