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Heavy-Traffic Limits for Nearly Deterministic Queues

Part of: Limit theorems Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Karl Sigman  and
Ward Whitt
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 G n /D/1 model, where the G n interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the G n /D/1 model and the generalization to G n /G n /1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρ n )√n → β as n → ∞ and (ii) when (1 − ρ n )n → β as n → ∞, where ρ n is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.

Keywords

Heavy trafficnearly deterministic queueErlang queuemany-server queuedeterministic service timeGaussian random walkcyclic thinningfunctional central limit theoreminvariance principle

MSC classification

Secondary: 60K25: Queueing theory 60F17: Functional limit theorems; invariance principles 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 657 - 678
Copyright
Copyright © Applied Probability Trust 2011 

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