Not much to say right now, except that I always forget the name of the useful tool from queuing theory, Kingman’s approximation for waiting time.

\[ \mathbb {E} (W_{q})\approx \left({\frac {\rho }{1-\rho }}\right)\left({\frac {c_{a}^{2}+c_{s}^{2}}{2}}\right)\tau \] 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, \(c_a\) is the coefficient of variation for arrivals (that is the standard deviation of arrival times divided by the mean arrival time) and \(c_s\) is the coefficient of variation for service times.

## References

Glasserman, Paul, Philip Heidelberger, Perwez Shahabuddin, and Tim Zajic. 1999. “Multilevel Splitting for Estimating Rare Event Probabilities.”

*Operations Research*47 (4): 585–600.Kingman, J. F. C. 1961. “The Single Server Queue in Heavy Traffic.”

*Mathematical Proceedings of the Cambridge Philosophical Society*57 (4): 902–4.Vázquez, Alexei, João Gama Oliveira, Zoltán Dezsö, Kwang-Il Goh, Imre Kondor, and Albert-László Barabási. 2006. “Modeling Bursts and Heavy Tails in Human Dynamics.”

*Physical Review E*73 (3): 036127.
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