Heavy-traffic waiting-time distribution conjecture for roving server networks

Let $W_i$ be the waiting time of a type-$i$ customer, and let $\mathcal{W}^{\textit{fluid}}_i$ be the standardised waiting time of a type-$i$ particle in the fluid model. Let $\Gamma$ be independent of $\mathcal{W}^{\textit{fluid}}_i$, with Gamma parameters $\alpha+1$ and $\delta\mu$. Waiting-time conjecture. As $\rho\uparrow1$, the scaled waiting time $(1-\rho)W_i$ converges in distribution to the product of these two independent random variables:

$$(1-\rho)W_i\xrightarrow{d}\Gamma\times\mathcal{W}^{\textit{fluid}}_i.$$

The assertion applies for $i=1,\dots,N$ and $k=i-N,\dots,i-1$. It follows the heavy-traffic averaging principle and is not proved in the supplied text for general arrivals.