Analytic queueing network models constitute a flexible tool for the study of network flow. These aggregate models are simple to manipulate and their analytic aspect renders them suitable for use within an optimization framework. Analytic queueing network models often assume infinite capacity for all queues. For real systems this infinite capacity assumption does not hold, but is often maintained due to the difficulty of grasping the between-queue correlation structure present in finite capacity networks. This correlation structure helps explain bottleneck effects and spillbacks, the latter being of special interest in networks containing loops because they are a source of potential deadlock (i.e. gridlock). We present an analytic queueing network model which acknowledges the finite capacity of the different queues. The model is adapted for multiple server finite capacity queueing networks with an arbitrary topology and blocking-after-service. By explicitly modelling the blocking phase the model yields a description of the congestion effects. A decomposition method allowing the evaluation of the model is also described. The methods validation, by comparison to both pre-existing methods and simulation results, is presented, as well as its application to the study of patient flow in a network of operative and post-operative units of the Geneva University Hospital.