  High-tech systems are typically produced in two stages: 1) Production of
components using specialized equipment and staff; 2) System
assembly/integration. Component production capacity is subject to fluctuations,
causing a high risk of shortages of at least one component, which results in
costly delays. Companies hedge this risk by strategic investments in excess
production capacity and in buffer inventories of components. To optimize these,
it is crucial to characterize the relation between component shortage risk and
capacity and inventory investments. We suppose that component production
capacity and produce demand are normally distributed over finite time
intervals, and we accordingly model the production system as a symmetric
fork-join queueing network with $N$ statistically identical queues with a
common arrival process and independent service processes. Assuming a symmetric
cost structure, we subsequently apply extreme value theory to gain analytic
insights into this optimization problem. We derive several new results for this
queueing network, notably that the scaled maximum of $N$ steady-state queue
lengths converges in distribution to a Gaussian random variable. These results
translate into asymptotically optimal methods to dimension the system. Tests on
a range of problems reveal that these methods typically work well for systems
of moderate size.
