Let for $ 0\le\alpha\le 1 $ the quantity $ I_{\alpha}(x) : =\dfrac{\sharp\{n\le x,\log^{1-\alpha}n\le p_{n+1}-p_{n}\le\log^{1+\alpha}n\}}{\pi(x)} $ and $ J(\alpha)=\lim_{x\to\infty}I_{\alpha}(x) $.
Under Cramer's conjecture one has $J(1)=1 $ .
Is there a heuristics giving the value of $ \alpha_{0} $ such that $J(\alpha_{0})=1/2 $?
Conjecturally, the distribution of prime gaps (let's say we first divide them by $2$ since they are essentially all even) follows an approximate geometric distribution with scale $\lambda^{-1} = \log n$.
Thus I would expect the halfway point to occur in an interval of the form $ \alpha \log n \le p_{n+1} - p_n \le \beta \log n$, rather than a polylog law. In other words, we should expect $\lim_{x \to 0^+} J(x) = 1$, but $J(0) = 0$. It is likely that no such $\alpha_0$ exists.
Cramér's conjecture concerns the extreme values of the prime gap function: it does not predict that they occur with any regularity.