Name for a simple generalization of an exponentially distributed random variable

Question

An exponentially distributed random variable $X$ with mean $\mu$ has a simple survival function: $S(k) := \Pr (X>k) = \exp(-k/\mu)$ for $k$ and $\mu$ non-negative. Consider generalizing to $S(k) := \Pr (X>k) = \left(1 + c (\exp(k/\mu)-1)\right)^{-1}$ for $c>0$. This is a valid survival function of a non-negative random variable $X$ which reduces to the exponential survival function when $c=1$. This has to have been studied, but I can't find it. Does anyone know the name of this generalized exponential random variable?