Extreme values for the waiting time in large fork-join queues
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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→∞. The maximum steady-state waiting time in this queueing system scales around 1/γlog N, where γ is determined by the cumulant generating function Λ of the service distribution and solves the Cramér-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value 1/γlog N is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation σ_A/√(Λ'(γ)γ). 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.
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