Extreme values for the waiting time in large fork-join queues

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 $$N\rightarrow \infty $$ N → ∞ . The maximum steady-state waiting time in this queueing system scales around $$\frac{1}{\gamma }\log N$$ 1 γ log N , where $$\gamma $$ γ is determined by the cumulant generating function $$\Lambda $$ Λ of the service times distribution and solves the Cramér–Lundberg equation with stochastic service times and deterministic interarrival times. This value $$\frac{1}{\gamma }\log N$$ 1 γ 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 $$\frac{\sigma _A}{\sqrt{\Lambda '(\gamma )\gamma }}$$ σ A Λ ′ ( γ ) γ . By using the distributional form of 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.