The delay-information duality conjecture for lookahead queues

Let $p$ be fixed in $(0,1)$, let $w(\lambda)$ denote the size of the lookahead window, and let $C^*_{\Pi_{w(\lambda)}}(p,\lambda)$ be the best achievable delay under policies with lookahead window $w(\lambda)$. Delay-information duality conjecture. If

$$w(\lambda)\ll\log\frac{1}{1-\lambda}\quad\text{as }\lambda\to1,$$

then

$$\limsup_{\lambda\to1}C^*_{\Pi_{w(\lambda)}}(p,\lambda)=\infty.$$

Equivalently, delay collapse can occur only if $w(\lambda)=\Theta\!\left(\log\frac{1}{1-\lambda}\right)$. The preceding theorem shows that a window of order $\log\frac{1}{1-\lambda}$ achieves the optimal heavy-traffic delay limit, while this conjecture asserts that asymptotically smaller windows cannot keep the delay bounded.