Single-server queues under overdispersion in the heavy-traffic regime. Issue 1 (31st October 2020)

• Record Type: Journal Article
• Authors: Boxma, O.; Heemskerk, M.; Mandjes, M.
• Is Part Of: Stochastic models. Volume 37:Issue 1(2021)
• Page Start: 197
• Page End: 230
• Publication Date: 2020-10-31
• Subjects: Heavy traffic -- Markov modulation -- overdispersion -- random environment -- single-server queues; Primary: 60K25 -- Secondary: 60J60 -- 90B22; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/15326349.2020.1840394
• Languages: English
• ISSNs: 1532-6349

Abstract

Abstract: 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.