A distributional form of Little's law in heavy traffic (Q1196940)

Consider a system of single-server queues indexed by the traffic intensity \(\rho<1\), and let \(l(\rho)\), \(w(\rho)\) denote the queue length, resp. the waiting time in the steady state. The input is assumed to have a general form, stationary or nearly stationary in the sense of time averages. It is shown that \[ \sqrt{1-\rho}(l(\rho)-\lambda_ \rho w(\rho))\to_{\mathcal D}bN\sqrt M,\quad\rho\uparrow 1, \] where \(\lambda_ \rho\) is the arrival rate, \(b\) a constant, and \(N\) and \(M\) independent random variables such that \(N\) is standard normal and \(M\) standard exponential.