Single-server queues under overdispersion in the heavy-traffic regime

This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. Several variants are considered, using concepts a...

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Bibliographic Details
Published in:Stochastic models 2020-10, Vol.37 (1), p.197-230
Main Authors: Boxma, O., Heemskerk, M., Mandjes, M.
Format: Article
Language:English
Subjects:
Heavy traffic
Laplace transforms
Markov modulation
overdispersion
Primary: 60K25
Queues
random environment
Secondary: 60J60
single-server queues
Theorems
Windows (intervals)
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Summary:This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously triggered random environments. Only in special cases explicit expressions can be obtained, e.g., when the random arrival and/or service rate can attain just finitely many values. While for more general model variants exact analysis is challenging, one can derive limit theorems in the heavy-traffic regime. In some of our derivations we rely on evaluating the relevant Laplace transform in the heavy-traffic scaling using Taylor expansions, whereas other results are obtained by applying the continuous mapping theorem.
ISSN:1532-6349
1532-4214
DOI:10.1080/15326349.2020.1840394