Heavy-traffic Gamma limit conjecture for length-biased cycle time

Let $C_i$ denote the cycle time seen by an arbitrary customer or arrival at queue $i$, and let $\bm{C_i}$ denote its scaled length-biased version. Let $\alpha$ and $\mu$ be as defined above, and let $\delta$ be the quantity defined in Definition. Scaled cycle-time conjecture. As $\rho\uparrow1$, $(1-\rho)C_i$ converges in distribution to a random variable having a Gamma distribution with shape parameter $\alpha$ and rate parameter $\delta\mu$. This conjecture supplies the length-biased cycle-time limit used in the subsequent heavy-traffic approximations; the source gives no general proof beyond the stated conjectural framework.