I am thinking about the consequences of adding prediction intervals and the consequence it has on the resulting interval. For example, I am considering when to expect the sum of two such intervals to be sub-additive or sup-additive, which I believe relates to the concavity/convexity of the quantile function or cumulative distribution function (depending on what you want to start with).
A good case study is the gamma distribution because it characterizes many waiting time distributions such as the exponential distribution, Erlang distribution, and others. Waiting times happen to be one of the kinds of things that I model in practice.
The quantile function of a gamma distribution is purportedly
$$Q_X(p) = \begin{cases} - \infty & p=0 \\ \frac{\gamma^{-1}(a, \Gamma(a)p)}{b} & p > 0 \end{cases}$$
where $$\gamma^{-1}$$ is the inverse of the lower incomplete gamma function.
I can see from the plot of the CDF under different parameters that it is not always convex or concave.
Under what parameters should I expect concavity/convexity of the quantile function?
The quantile function is the inverse of the c.d.f. (For functions whose c.d.f. is constant on some interval, it's more complicated than that. And this one is constant on the interval $(-\infty,0],$ but that need not concern us because that intersects the support of the distribution at just one point.)
The quantile function $p\mapsto x$ is concave at values of $p$ corresponding to values of $x$ where the c.d.f. is convex.
The c.d.f. is convex on intervals where its second derivative is positive. Those are intervals where the first derivative of the gamma density function is positive. That density is $$ x \mapsto \text{constant} \times x^{k-1} e^{-\text{constant} \cdot x}. $$ I don't know whether you intended that exponent that is $-\text{constant} \cdot x$ to be $-\theta x$ or $-x/\theta,$ and it seems to me you should have informed us of that as well. But ways of parameterizing the gamma density are used, so I would say which one I'm using.
If $k>1,$ then the function $x \mapsto \text{constant} \times x^{k-1} e^{-\text{constant} \cdot x}$ is $0$ when $x=0$ and positive when $x>0,$ so it has to increase and then later decrease, approaching $0$ as $x\to+\infty.$ So if $k>1,$ then on some interval from $0$ up to some positive number, the density is increasing, so to the c.d.f. is convex. Above that positive number it is concave.
It is good to remember that although convexity of a function on an interval follows from the second derivative of the function being positive on that interval, nonetheless the standard definition of convexity is that the chords of the graph lie above the graph.