Poisson and delay-profile conjecture for the large reshuffling system

Fix a load $\lambda<1$ and, for each $N$, consider the reshuffling system in steady state. At a communication epoch, let the number of useful packets on a link be the relevant link-level random variable. Let $R^N_{\mathrm{RS}}(x)$ and $T^N_{\mathrm{RS}}(x)$ denote, respectively, the mean and random time for a packet to reach point $x\in[0,1]$ in the transformed reshuffling system, and let the slowdown be the mean normalized sojourn time at $x=1$.

Large reshuffling-system conjecture. As $N\to\infty$: (i) the number of useful packets on a link at a communication epoch converges in distribution to $\operatorname{Poisson}(\psi)$ for a fixed $\psi\geqslant0$; (ii) the numbers of useful packets at links at consecutive communication epochs are asymptotically independent; (iii) $\psi=-\log(1-\lambda)$; and (iv) the slowdown converges to

$$-\frac{\log(1-\lambda)}{\lambda},$$

while, for every $x\in[0,1]$,

$$R^N_{\mathrm{RS}}(x)\to\left[-\frac{\log(1-\lambda)}{\lambda}\right]x,$$ $$T^N_{\mathrm{RS}}(x)\Rightarrow\left[-\frac{\log(1-\lambda)}{\lambda}\right]x.$$

This conjecture predicts both the local useful-packet statistics and the macroscopic propagation delays in the reshuffling model. The reshuffling system is proved stable exactly when $\lambda<1$, but the asserted large-$N$ distributional and profile limits remain conjectural.