Heavy traffic

Consider the $\mathrm{M}(\lambda ) / \mathrm{M}(\mu ) / 1$ queue with $\rho = \lambda / \mu < 1$. Let $Q_{\rho }$ be a random variable with the equilibrium queue distribution, and show that $(1-\rho ) Q_{\rho }$ converges in distribution as $\rho \uparrow 1$, the limit distribution being exponential with parameter 1.