Estimation of average waiting time
If tasks arrive faster than they can be dealt with, average waiting time is unbounded. 

You can probably adapt the Pollaczek–Khinchine formula to give an analytic answer to your question: 

$$L = \rho + \frac{\rho^2 + \lambda^2 \operatorname{Var}(S)}{2(1-\rho)}$$

where


$L$ is the mean queue length;
$\lambda$ is the arrival rate of the Poisson process;
$1/\mu$ is the mean of the service time distribution $S$;
$\rho={\lambda \over \mu}$ is the utilization; and
$Var(S)$ is the variance of the service time distribution $S$.