Conjecture on optimal allocation under probabilistic access

Let $p$ be the access failure probability for each node, let $\alpha$ be the allocation parameter, and let $\mu_s(\alpha)$ denote the average service rate under probabilistic access. Probabilistic-access conjecture. When $p$ is large, $\mu_s(1)$ is more likely to be the maximum; when $p$ is small, the maximum of $\mu_s(\alpha)$ is not attained at $1$, and the optimal $\alpha$ is decreasing with $p$. The preceding bounds identify regions where $\mu_s(1)$ is larger or smaller than $\mu_s(\alpha)$, but leave an intermediate region undecided.