论文标题
重尾叉-Join队列中的最大等待时间
Maximum waiting time in heavy-tailed fork-join queues
论文作者
论文摘要
在本文中，我们在$ n $ -server fork-join队列中研究了最大等待时间$ \ max_ {i \ leq n} w_i（\ cdot）$，带有重尾服务为$ n \ to \ infty $。服务时间是两个随机变量的乘积。一个随机变量具有定期变化的尾巴概率，并且在所有$ n $服务器中都是相同的，并且一个随机变量是Weibull分布式的，并且独立并且在所有服务器之间分布相同。该设置的物理解释是，如果作业的尺寸较大，那么所有子任务的尺寸都很大，而Weibull-Distribed部分描述了一些可变性。我们证明，在时间和空间缩放之后，最大等待时间过程在$ d [0，t] $中收敛到具有负漂移的极端过程的至上。时间和空间缩放是$ \ tilde {l}（b_n）b_n^{\fracβ{（（β-1）}} $，其中$β$是定期变化的分布中的形状参数，$ \ tilde {l}（x）$是一个缓慢的vary is $ n pec n os n $ n of n $ ge q ege（ $ \ max_ {i \ leq n} a_i/b_n \ oftset {\ mathbb {p}}} {\ longrightArrow} 1 $，as $ n \ to \ infty $，其中$ a_i $是i.i.d. i.i.d. \ weibull-distribull-distribed distribed随机变量。最后，我们证明了稳态收敛。
In this paper, we study the maximum waiting time $\max_{i\leq N}W_i(\cdot)$ in an $N$-server fork-join queue with heavy-tailed services as $N\to\infty$. The service times are the product of two random variables. One random variable has a regularly varying tail probability and is the same among all $N$ servers, and one random variable is Weibull distributed and is independent and identically distributed among all servers. This setup has the physical interpretation that if a job has a large size, then all the subtasks have large sizes, with some variability described by the Weibull-distributed part. We prove that after a temporal and spatial scaling, the maximum waiting time process converges in $D[0,T]$ to the supremum of an extremal process with negative drift. The temporal and spatial scaling are of order $\tilde{L}(b_N)b_N^{\fracβ{(β-1)}}$, where $β$ is the shape parameter in the regularly varying distribution, $\tilde{L}(x)$ is a slowly varying function, and $(b_N,N\geq 1)$ is a sequence for which holds that $\max_{i\leq N}A_i/b_N\overset{\mathbb{P}}{\longrightarrow}1$, as $N\to\infty$, where $A_i$ are i.i.d.\ Weibull-distributed random variables. Finally, we prove steady-state convergence.
