Schol, Dennis

Vlasiou, Maria

Zwart, Bert

English

In this paper, we study the tail behavior of $\max_{i\leq N}\sup_{s&gt;0}\left(W_i(s)+W_A(s)-\beta s\right)$ as $N\to\infty$, with $(W_i,i\leq N)$ 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 $\frac{\sigma^2}{2\beta}\log N$. Here, we analyze the rare-event that this random variable reaches the value $(\frac{\sigma^2}{2\beta}+a)\log N$, with $a&gt;0$. 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 $a^{\star}$; namely, $0a^{\star}$. 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 $a=a^{\star}$

University of Twente Research Information

Tail Asymptotics for the Delay in a Brownian Fork-Join Queue

https://ris.utwente.nl/ws/files/284381322/2208.04796v1.pdf

Tail Asymptotics for the Delay in a Brownian Fork-Join Queue

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University of Twente Research Information