We prove that the scaled maximum steady-state waiting time and the scaled maximum steady-state queue length among $N$ $GI/GI/1$-queues in the $N$-server fork-join queue, converge to a normally distributed random variable as $N\to\infty$. The maximum steady-state waiting time in this queueing system scales around $\frac{1}{\gamma}\log N$, where $\gamma$ is determined by the cumulant generating function $\Lambda$ of the service distribution and solves the Cram\'er-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value $\frac{1}{\gamma}\log N$ is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $\frac{\sigma_A}{\sqrt{\Lambda'(\gamma)\gamma}}$. By using distributional Little's law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers.

翻译：我们证明了在$N$服务器分叉-合并队列中，$N$个$GI/GI/1$队列的稳态最大等待时间与稳态最大队长经尺度变换后，当$N\to\infty$时收敛于正态分布随机变量。该排队系统中稳态最大等待时间尺度约为$\frac{1}{\gamma}\log N$，其中$\gamma$由服务分布的累积生成函数$\Lambda$决定，并满足具有随机服务时间与确定性到达间隔的Cramér-Lundberg方程。该值$\frac{1}{\gamma}\log N$在特定命中时刻达到。到达该命中时刻的到达次数满足中心极限定理，标准差为$\frac{\sigma_A}{\sqrt{\Lambda'(\gamma)\gamma}}$。通过应用分布形式的Little定律，我们可将该结论推广至最大队长。最后，我们将这些结论扩展至包含不同服务类别的分叉-合并队列。