In an earlier paper, the authors posed a time-dependent traffic flow problem in which travellers go home from work on a long arterial street. Each traveller enters the street at a particular time and place, bound for some given destination as prescribed by a continuous trip density function, but the time at which his or her trip is completed depends on the traffic conditions en route; hence on the origins, destinations, and departure times of other motorists. A solution was obtained, but a solution that is valid only if no intersection becomes saturated. In this paper, the analysis is extended to cover cases where some key intersection does become saturated. The model is macroscopic and differs from alternative models of traffic flow in two ways. The first and more important is that the impact of the origin destination pattern on the traffic dynamics and vice versa are explicitly recognized and included in the model. This is particularly important when the impact of exit flows on downstream facilities is an issue and in applications involving route choice or elastic demand, since it allows choices based on the travel times that would be experienced by the travellers actually making the decisions. The second difference is that is is explicitly a queueing model: the basic assumption is htat intersections along the roadway have capacities and that when the capacity of some key intersection is exceeded by the arrival flow, the large queue that results will be the dominant element controlling subsequent operation of the system
