The work in this thesis is inspired by modeling delays in supply chains for high-tech manufacturers, such as ASML, Philips Healthcare, and Boeing; these supply chains are large. High-tech suppliers specialize in producing and delivering a specific component of the final product. We can see the process of the manufacturer ordering items to a supplier as a queue; the manufacturer is a customer in the store, and he is getting served by the supplier. The time it takes for an order to be completed, depends on the working speed of the supplier, and the number of orders that are already waiting. High-tech manufacturers have many suppliers at the same time, each supplier delivers a specific part of the final product; the manufacturer is therefore in many queues at the same time. Thus, the time it takes to get all the components for one final product is equal to the time it takes for the slowest supplier. The central objective in this thesis, is to model this time, or equivalently, the average number of orders waiting to be served by the slowest supplier. The queueing network which models this is called the fork-join queue. Analyzing the performance of the slowest supplier among many queues is not easy. One way to do this, is by looking at limiting behavior; we let the number of queues go to infinity. The resulting limits can be used to approximate the behavior of the slowest supplier in the finite situation. In this thesis, we explore different situations. We investigate the situation that the manufacturer and the suppliers behave in a nearly deterministic fashion, the situation that suppliers’ service times are random but without having extremely large service times, the situation that extremely large service times are likely, and we look at a continuous-time approximation. We derive different limiting results; we derive convergence results that describe how slow the slowest supplier will work on average, a result that describes the typical standard deviation, and a result that describes how likely it is that extreme events occur. Finally, we consider situation that suppliers already have items in stock, such that suppliers can deliver without the manufacturer having to wait, to reduce costs due to delay. Of course, having items in stock incurs costs as well, thus there is a trade-off to be made, we derive a decision rule that determines the optimal number of items in stock.