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We study extreme values in certain fork-join queueing networks: consider $N$identical queues with a common arrival process and independent serviceprocesses. All arrival and service processes are deterministic with randomperturbations following Brownian motions. We prove that as $N\rightarrow\infty$, the scaled maximum of $N$ steady-state queue lengths converges indistribution to a normally distributed random variable. We explorerepercussions of this result for original equipment manufacturers (OEMs) thatassemble a large number of components, each produced using specializedequipment, into complex systems. Component production capacity is subject tofluctuations, causing high risk of shortages of at least one component, whichresults in costly system production delays. OEMs hedge this risk by investingin a combination of excess production capacity and component inventories. Weformulate a stylized model of the OEM that enables us to study the resultingtrade-off between shortage risk, inventory costs, and capacity costs. Ourasymptotic extreme value results translate into asymptotically exact methodsfor cost-optimal inventory and capacity decisions, some of which are in closedform. We validate our asymptotic results with a set of detailed numericalexperiments. These experiments indicate that our results are asymptoticallyexact, while for transient times they depend on model parameters.

Extreme-value theory for large fork-join queues, with an application to high-tech supply chains