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 are equal to

\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

^ 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.
^ 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.
^ Harrison, Peter G.; Patel, Naresh M., Performance Modelling of Communication Networks and Computer Architectures, p. 336, ISBN 0-201-54419-9

[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

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v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job next Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean-field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
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