2022/08/09 by Dennis Schol, Schol, Dennis, Maria Vlasiou +3
Business, Management and Accounting · Decision Sciences · Mathematics · #60G15 #60G70 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2208.04796
openalex publication_date 2022/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the tail behavior of maxi≤ Nsups>0(Wi(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 (σ2)/(2β)log N. Here, we analyze the rare-event that this random variable reaches the value ((σ2)/(2β)+a)log N, 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⋆.