This paper is concerned with strong approximation in queueing networks. A model of a circuit-switched network with fixed routes is considered in the limiting regime where the link capacities and the offered traffic are increased at the same rate. The process of normalized queue lengths is shown to converge almost surely to a sliding mode solution of an ordinary differential equation. The solution is shown to possess a unique stable point. It is reached exponentially fast or in finite time, depending on the values of the parameters. This has implications on the settling time of the network. The technique is applicable to closed Jackson networks and their settling times. In contrast with other asymptotic results on queueing networks it does not make use of product form distributions and extends easily to non- Markovian models.