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Queues with time-dependent arrival rates. III — A mild rush hour

Published online by Cambridge University Press:  14 July 2016

G. F. Newell
G. F. Newell*
Affiliation:
Institute of Transportation and Traffic Engineering, University of California, Berkeley
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Abstract

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 3 , December 1968 , pp. 591 - 606
Copyright
Copyright © Applied Probability Trust 1968 

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References

[1] Newell, G. F. (1968) Queues with time-dependent arrival rates. I — The transition through saturation. J. Appl. Prob. 5, 436–451.Google Scholar
[2] Newell, G. F. (1968) Queues with time-dependent arrival rates. II — The maximum queue and the return to equilibrium. J. Appl. Prob. 5, 579–590.Google Scholar
[3] Newell, G. F. (1962) Asymptotic extreme value distribution for one-dimensional diffusion processes. J. Math. Mech. 11, 481–496.Google Scholar