Given the probability of being in queue \eqref{ErlangC}, it is easy to derive the expected number of customers in queue \(L_w\) - see: \cite{allen1978queueing} page 276: \[\label{Numb}
L_w = \frac{\rho}{1-\rho} \cdot P_w\] so that, by Little law Wiki - Little Law, we derive the average time spent in queue by customers \(T_w\) - see: \cite{allen1978queueing} page 276: \[\label{Tim}
T_w = \frac{L_w}{\lambda} = \frac{\rho}{1-\rho} \cdot \frac{P_w}{\lambda}\] The average time \(T_w\) is a non-linear function of the traffic intensity \(a\), of the number of servers \(c\) and of their ratio \(\rho\). We stress here that, as the average server utilization \(\rho\) increases getting close to one, the behavior of \(T_w\) in dependence of it becomes strongly non-linear with fast divergence.
The heuristic explanation of this singularity is related to the stochastic nature of this problem. Under any value of the average server utilization, from time to time, a big upsurge of customers can temporarily arise, leading to the formation of a queue. If the average server utilization is low, however, the system reacts in a timely manner by temporarily raising to \(100\%\) the server utilization so that the queue is rapidly disposed of and the impact on the average waiting time is very low. If, on the other hand, the server utilization is already high on average, the time needed to dispose of a temporary large queue can be very long, with material impact on the average waiting time.