2023/09/15 by Schol, Dennis, Vlasiou, Maria, Zwart, Bert
#FOS: Computer and information sciences #FOS: Mathematics #Performance (cs.PF) #Probability (math.PR)
paper · doi:10.48550/arxiv.2309.08373
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 \frac1γlog 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 \frac1γlog N is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation (σA)/(√(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.