The age-of-information ordering conjecture for finite-buffer systems

Consider the systems $\mathcal{P}_n$ and $\mathcal{B}_n$ for $n=1,2,\ldots$, each driven by a Poisson arrival process of rate $\lambda$. Let $\alpha_{\mathcal{I}}$ denote the stationary age of information for system $\mathcal{I}$, and write $\alpha_{\mathcal{I}}<_{\mathrm{st}}\alpha_{\mathcal{J}}$ for stochastic ordering.

Best-systems conjecture. Regardless of the message-size distribution,

$$\alpha_{\mathcal{I}}<_{\mathrm{st}}\alpha_{\mathcal{J}}$$

for every $\mathcal{I}\in\{\mathcal{P}_1,\mathcal{B}_1,\mathcal{P}_2,\mathcal{B}_2\}$ and every $\mathcal{J}\in\{\mathcal{P}_n,\mathcal{B}_n:n\ge3\}$.

The conjecture proposes that systems with one or two buffer positions dominate all corresponding systems with at least three positions in stochastic age comparison. The source states that no proof of the optimal system is known and supplies no resolution of this ordering.