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

We study the tail behavior of max_i≤Nsup_s>0W_i(s)+W_A(s)−βs as N→∞, with (W_i,i≤N) i.i.d. Brownian motions and W_A 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^⋆.