I am using the following probability distribution function defined for $x \in [0, \infty)$ with $\alpha>0$:
$$ f(x\mid\alpha)= \frac{\alpha}{(x+\alpha)^2}$$ the CDF is $$ F(x\mid\alpha)= \frac{x}{x+\alpha}$$
does this distribution have a name? Has it been studied?
I don't know if this distribution has a name, but it is closely related to a well-studied distribution. If $X$ is Pareto-distributed with scale $\alpha$ and shape $1$ (this is a bit confusing since $\alpha$ normally denotes the shape), then the random variable $X-\alpha$ has density $f(\cdot\mid\alpha)$ and CDF $F(\cdot\mid\alpha)$.