Heavy-traffic Gamma limit conjecture for total workload

Let $V$ be the total amount of work in the system at the beginning of a cycle, and let $r$, $\rho$, $\tilde{b}_i$, $\tilde{B}_i$, $\tilde{A}_i$, $\tau$, and $\beta$ have the meanings assigned in the model. Define

$$\sigma^2=\sum_{i=1}^N \hat\lambda_i\left(\operatorname{Var}[\widetilde{B}_i]+(\hat\lambda_i\tilde{b}_i)^2\operatorname{Var}[\hat A_i]\right),\qquad \alpha=\frac{2r\delta}{\sigma^2},\qquad \mu=\frac{2}{\sigma^2}.$$

Here $\delta$ is the quantity defined in Definition. Heavy-traffic workload conjecture. As $\rho\uparrow1$, $(1-\rho)V$ has a Gamma distribution with shape parameter $\alpha$ and rate parameter $mu$. This conjecture adapts the heavy-traffic workload conjecture of Olsen and Van der Mei to roving server networks; the corresponding result is known from the cited Poisson-arrival case, while the general renewal-arrival setting remains conjectural.