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

Mirjam Meijer, Dennis Schol, Willem van Jaarsveld, Maria Vlasiou, Bert Zwart

Research output: Working paperProfessional

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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 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.
Original languageEnglish
Publication statusPublished - 19 May 2021

Publication series

PublisherCornell University


  • math.PR
  • math.OC


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