Systems consisting of many queues in series have been considered by Glynn and Whitt (1991) and Baccelli, Borovkov and Mairesse (1999). I will discuss extensions of these models 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 a simple form but in infinitely many dimensions; they are related to problems of finding paths of maximum weight through a two-dimensional lattice with random weights at the vertices. Topics of interest include: laws of large numbers for 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. There are several interesting open problems.