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 [Formula presented]logN. Here, we analyze the rare event that this random variable reaches the value ([Formula presented]+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, 0<a<a⋆, a=a⋆, and a>a⋆. 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⋆.

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Stochastic Processes and their Applications

Schol, D., Vlasiou, M., & Zwart, B. (2023). Tail asymptotics for the delay in a Brownian fork-join queue. Stochastic Processes and their Applications, 164, 99–138. doi:10.1016/j.spa.2023.06.013