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

Let $W_{i_1,i_2,\dots,i_M}$ be the path time along the customer route $i_1,i_2,\dots,i_M$, and let $\mathcal{W}^{\textit{fluid}}_{i_1,i_2,\dots,i_M}$ be the corresponding standardised path time in the fluid model. Let $\Gamma$ have a Gamma distribution with parameters $\alpha+1$ and $\delta\mu$. Path-time conjecture. As $\rho\uparrow1$, the scaled path time satisfies

$$(1-\rho)W_{i_1,i_2,\dots,i_M}\xrightarrow{d}\Gamma\times\mathcal{W}^{\textit{fluid}}_{i_1,i_2,\dots,i_M}.$$

This applies for $i,k=1,\dots,N$; the distribution of the fluid factor is given by the path-time fluid analysis. It extends the heavy-traffic snapshot principle from queue lengths and waiting times to routed customer paths, while the general result remains conjectural.