Service Systems in Heavy Traffic

Let $\eta _n$ be the waiting time of the $n$ customer arriving at a service line. It is proved that under certain conditions the distribution of $\delta\eta _n$ tends to a negative exponential distribution as $\delta\to0$, and $n\delta^2\to\infty$, where $\delta={{({\mathbf M}\tau-{\mathbf M}\chi)}/{\mathbf M\tau;}}$ ${\mathbf M}\tau$ and ${\mathbf M}\chi$ are the mean inter-arrival time and the service time, respectively.