Systems consisting of many queues in series have been considered by Glynn and Whitt (1991) and Baccelli, Borovkov and Mairesse (2000). We extend their results to apply to situations where the queues have finite capacity and so various types of «blocking» can occur. The models correspond to max-plus type recursions, of simple form but in infinitely many dimensions; they are related to «percolation» problems of finding paths of maximum weight through a two-dimensional lattice with random weights at the vertices. Topics treated include: laws of large numbers for the speed of customers progressing through the system; stationary behaviour for systems with external arrival processes; functional laws of large numbers describing the behaviour of the «front of the wave» progressing through a system which starts empty; stochastic orderings for waiting times of customers at successive queues. Several open problems are noted.