TY - JOUR
T1 - Tail asymptotics for the delay in a Brownian fork-join queue
AU - Schol, Dennis
AU - Vlasiou, Maria
AU - Zwart, Bert
N1 - Funding Information:
This work is part of the research program Complexity in high-tech manufacturing, (partly) financed by the Dutch Research Council (NWO) through contract 438.16.121 .
Publisher Copyright:
© 2023 The Author(s)
PY - 2023/10
Y1 - 2023/10
N2 - 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⋆, 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⋆.
AB - 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⋆, 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⋆.
KW - Brownian queues
KW - Extreme-value theory
KW - Fork-join queues
KW - Tail asymptotics
UR - http://www.scopus.com/inward/record.url?scp=85165533964&partnerID=8YFLogxK
U2 - 10.1016/j.spa.2023.06.013
DO - 10.1016/j.spa.2023.06.013
M3 - Article
AN - SCOPUS:85165533964
SN - 0304-4149
VL - 164
SP - 99
EP - 138
JO - Stochastic processes and their applications
JF - Stochastic processes and their applications
ER -