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

We study extreme values in certain fork-join queueing networks: consider $N$ identical queues with a common arrival process and independent service processes. All arrival and service processes are deterministic with random perturbations following Brownian motions. We prove that as $N\rightarrow \infty$, the scaled maximum of $N$ steady-state queue lengths converges in distribution to a normally distributed random variable. We also explore repercussions of this result for original equipment manufacturers (OEMs) that assemble a large number of components, each produced using specialized equipment, into complex systems. Component production capacity is subject to fluctuations, causing a high risk of shortages of at least one component, which in turn results in costly system production delays. OEMs hedge this risk by investing in a combination of excess production capacity and component inventories. We formulate a stylized model of the OEM that enables us to study the resulting trade-off between shortage risk, inventory costs, and capacity costs. Our asymptotic extreme value results translate into various asymptotically exact methods for cost-optimal inventory and capacity decisions, some of which are in closed form. Numerical results indicate that our results are asymptotically exact, while for transient times they depend on model parameters.