Mean normalized delay profile convergence for Oldest-Useful and Random-Useful

Consider a symmetric network with $N$ nodes and the Oldest-Useful (OU) or Random-Useful (RU) communication discipline. Under the transformation that maps availability levels to points $x\in[0,1]$ and rescales time so that the free-system expected sojourn time is $1$, let $R^N_D(x)$ denote the expected steady-state time for a packet to reach point $x$ under discipline $D$.

Delay-profile convergence conjecture. For $D=\mathrm{OU}$ or $D=\mathrm{RU}$, and for a given load $\lambda<1$, as $N\to\infty$,

$$R^N_D(x)\to R_D(x),\qquad 0\leqslant x\leqslant 1,$$

where $R_D$ is a continuous increasing function.

The profile describes packet-propagation delay throughout the network; its value at $x=1$ is the slowdown. The conjecture is supported by simulations, while convergence of the profiles for these disciplines remains unproved in the source.