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Heavy-Traffic Limits for Nearly Deterministic Queues

Published online by Cambridge University Press:  14 July 2016

Karl Sigman*
Affiliation:
Columbia University
Ward Whitt*
Affiliation:
Columbia University
*
Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA.
Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA.
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Abstract

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We establish heavy-traffic limits for nearly deterministic queues, such as the G/D/n many-server queue. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn /D/1 model, where the Gn interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the Gn /D/1 model and the generalization to Gn /Gn /1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρn )√n → β as n → ∞ and (ii) when (1 − ρn )n → β as n → ∞, where ρn is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2011 

References

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