Heavy-traffic stable-limit conjecture for Hawkes/M/∞ queues with regularly varying service times

Let $N$ denote the number of customers in a Hawkes/M/$\infty$ queue, with ${\mathscr J}(u)=e^{-\mu u}$ and $h(u)=e^{-u}$ for $u\geqslant 0$. Assume that the service-time distribution $B$ is regularly varying with index $-\alpha$, where $\alpha\in(1,2)$. Let $S_\alpha(c,0,0)$ denote a stable random variable with parameters $\alpha,c,0,0$. Heavy-traffic stable-limit conjecture. As $\rho\uparrow 1$,

$$(1-\rho)^{\frac{2}{\alpha}}N\stackrel{d}{\longrightarrow}S_\alpha(c,0,0),$$

where

$$c=\frac{1}{\mu}\left(\frac{\Gamma(1-\alpha)\lambda_\infty\ell(\infty)}{\alpha r}\right)^{\frac{1}{\alpha}}.$$

The conjecture proposes a stable heavy-traffic limit for the queue length when service times have regularly varying tails with index between $-2$ and $-1$, extending the light-tailed heavy-traffic result. Its resolution is not indicated in the supplied text.