From Did's comment following his reply, given a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$ for some function $r:[0,\infty) \to [0, \infty)$ every $t⩾0$.
Is there a name for such a distribution?
If $r$ is constant, then $U$ has an exponential distribution.
If $r$ is piecewise constant, what is the name of the distribution? "Piecewise exponential"?