论文标题
尾巴渐近学用于延迟布朗叉-join队列
Tail Asymptotics for the Delay in a Brownian Fork-Join Queue
论文作者
论文摘要
在本文中，我们研究了$ \ max_ {i \ leq n} \ sup_ {s> 0} \ left（w_i（s）+w_a（s）-βS-βS\ right）$ as $ n \ to \ infty $（w_i，w_i，w_i，i \ leq n）$ i.i.d.布朗尼动议和$ w_a $独立的布朗运动。这个随机变量可以看作是$ n $相互依赖的布朗人队列的最大变量，这又可以将其解释为布朗叉-join队列中的积压。在以前的工作中，我们已经表明，此随机变量围绕$ \ frac {σ^2} {2β} \ log n $。在这里，我们分析了此随机变量达到值$（\ frac {σ^2} {2β}+a）\ log n $的罕见事件，并带有$ a> 0 $。事实证明，其概率大致作为具有$ n $的电力法，指数取决于$ a $。但是，有三个制度，围绕一个关键点$ a^{\ star} $;也就是说，$ 0 <a <a^{\ star} $，$ a = a^{\ star} $，$ a> a> a^{\ star} $。后一种制度表现出一种渐近独立性的形式，而第一个制度揭示了高度不规则的行为，并在$ n $ suprema之间具有明显的依赖性结构，而在$ a = a^a^{\ star} $中的非平凡过渡。
In this paper, we study the tail behavior of $\max_{i\leq N}\sup_{s>0}\left(W_i(s)+W_A(s)-βs\right)$ 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 $\frac{σ^2}{2β}\log N$. Here, we analyze the rare-event that this random variable reaches the value $(\frac{σ^2}{2β}+a)\log N$, 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 $a^{\star}$; namely, $0<a<a^{\star}$, $a=a^{\star}$, and $a>a^{\star}$. 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 $a=a^{\star}$.
