= Solution
Use the lower limit $s=t+r_0$ for the inner integral:
$$
P(r>r_0)=\int_0^\infty\frac{e^{-t/\bar t}}{\bar t}
\left[\int_{t+r_0}^\infty\frac{e^{-s/\bar s}}{\bar s}ds\right]dt
=\frac{e^{-r_0/\bar s}}{\bar t}\int_0^\infty e^{-t(1/\bar s+1/\bar t)}dt.
$$
Hence
$$
\boxed{P(r>r_0)=\frac{\bar s}{\bar s+\bar t}e^{-r_0/\bar s},\qquad r_0>0.}
$$
The prefactor is the <probability> that interruption occurs; the remaining exponential is its conditional positive tail. This is the <positive difference of independent exponential waiting times>. The exponent is dimensionless because both $r_0$ and $\bar s$ are measured in minutes.
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