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Tail asymptotics for the delay in a Brownian fork-join queue

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  • Schol, Dennis
  • Vlasiou, Maria
  • Zwart, Bert

Abstract

We study the tail behavior of maxi≤Nsups>0Wi(s)+WA(s)−βs as N→∞, with (Wi,i≤N) i.i.d. Brownian motions and WA an independent Brownian motion. This random variable can be seen as the maximum of N mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around σ22βlogN. Here, we analyze the rare event that this random variable reaches the value (σ22β+a)logN, with a>0. It turns out that its probability behaves roughly as a power law with N, where the exponent depends on a. However, there are three regimes, around a critical point a⋆; namely, 0a⋆. The latter regime exhibits a form of asymptotic independence, while the first regime reveals highly irregular behavior with a clear dependence structure among the N suprema, with a nontrivial transition at a=a⋆.

Suggested Citation

  • Schol, Dennis & Vlasiou, Maria & Zwart, Bert, 2023. "Tail asymptotics for the delay in a Brownian fork-join queue," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 99-138.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:99-138
    DOI: 10.1016/j.spa.2023.06.013
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    References listed on IDEAS

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    1. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2015. "Extremes of vector-valued Gaussian processes: Exact asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4039-4065.
    2. Hongyuan Lu & Guodong Pang, 2017. "Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services," Queueing Systems: Theory and Applications, Springer, vol. 85(1), pages 67-115, February.
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