# Kingman's formula

In queueing theory, a discipline within the mathematical theory of probability, Kingman's formula also known as the VUT equation, is an approximation for the mean waiting time in a G/G/1 queue.[1] The formula is the product of three terms which depend on utilization (U), variability (V) and service time (T). It was first published by John Kingman in his 1961 paper The single server queue in heavy traffic.[2] It is known to be generally very accurate, especially for a system operating close to saturation.[3]

## Statement of formula

Kingman's approximation states

${\displaystyle \mathbb {E} (W_{q})\approx \left({\frac {\rho }{1-\rho }}\right)\left({\frac {c_{a}^{2}+c_{s}^{2}}{2}}\right)\tau }$

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.

## References

1. ^ Shanthikumar, J. G.; Ding, S.; Zhang, M. T. (2007). "Queueing Theory for Semiconductor Manufacturing Systems: A Survey and Open Problems". IEEE Transactions on Automation Science and Engineering. 4 (4): 513. doi:10.1109/TASE.2007.906348.
2. ^ Kingman, J. F. C.; Atiyah (October 1961). "The single server queue in heavy traffic". Mathematical Proceedings of the Cambridge Philosophical Society. 57 (4): 902. doi:10.1017/S0305004100036094. JSTOR 2984229.
3. ^ Harrison, Peter G.; Patel, Naresh M., Performance Modelling of Communication Networks and Computer Architectures, p. 336, ISBN 0-201-54419-9