Let $\eta _n $ be the waiting time of the n-th 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 $\delta \to 0$, and $n\delta ^2 \to \infty $, where $\delta = {{({\bf M}\tau - {\bf M}\chi )} / {M\tau ;}}{\bf M}\tau $ and ${\bf M}\chi $ are the mean inter-arrival time and the service time, respectively.