Not much to say here right now, except that I walways forget the name of the useful tool from queuing theory, Kingman’s approximation
\[ {\displaystyle \mathbb {E} (W_{q})\approx \left({\frac {\rho }{1-\rho }}\right)\left({\frac {c_{a}^{2}+c_{s}^{2}}{2}}\right)\tau } \] where τ is the mean service time (i.e. μ = 1/τ is the service rate), λ is the mean arrival rate, ρ = λ/μ is the utilization, ca is the coefficient of variation for arrivals (that is the standard deviation of arrival times divided by the mean arrival time) and cs is the coefficient of variation for service times.