The explicit heavy-traffic limit conjecture with mobility

Assume $\theta>0$. Let $\boldsymbol{X}=(X_1,X_2)$ be the stationary queue-length vector, let $\rho$ be the traffic intensity, and define

$$\boldsymbol{\xi}^*=\left(\frac{\rho_1}{1-\rho_1},1\right).$$

Explicit heavy-traffic limit conjecture. As $\rho\uparrow1$,

$$\frac{\boldsymbol{X}(\infty)}{-\log(1-\rho)}\Longrightarrow\frac{1}{1-\log(1-\rho_1)}\,\boldsymbol{\xi}^*.$$

This conjecture gives the explicit limit suggested by the conjectured empty-system exponential rate and would identify the exact logarithmic heavy-traffic asymptotics. The source presents it as a conjecture; the preceding theorem proves only tightness and an upper bound on accumulation points.