# Queueing

## The mathematical field whose major result is enraging you about call centres

Not much to say here 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.

Glasserman, Paul, Philip Heidelberger, Perwez Shahabuddin, and Tim Zajic. 1999. “Multilevel Splitting for Estimating Rare Event Probabilities.” Operations Research 47 (4): 585–600. https://doi.org/10.1287/opre.47.4.585.

Kingman, J. F. C. 1961. “The Single Server Queue in Heavy Traffic.” Mathematical Proceedings of the Cambridge Philosophical Society 57 (4): 902–4. https://doi.org/10.1017/S0305004100036094.

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. https://doi.org/10.1103/PhysRevE.73.036127.