Optimal time-varying flows on congested networks
Operations Research
Extending and solving a multiperiod congested network flow model
Computers and Operations Research
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We consider the behaviour of a link exit-flow model that has been used to model link flows in dynamic traffic assignment (DTA) on networks. In particular, we investigate how the model behaves when time and space (the link length) discretised and the discretisation is varied. We present numerical examples based on various inflow patterns and exit-flow functions and draw conclusions for applications of the model in discrete space and in discrete or continuous time. If inflows are always less than capacity and the link is homogeneous, with no obstructions at the exit, then if the discretisation is refined to the continuous limit the model goes to the solution of the well-known LWR model. However, we observe, somewhat counter intuitively, that the usual continuous-time model does not give as good an approximation to the LWR solution as does the discrete-time model: for a best approximation, the discretisation of space and time should be synchronised. We also investigate the 'dip' in outflows, or 'jamming' of outflows, that the model display, if inflows are permitted to exceed a capacity limit (as they sometimes do in published applications of the model) and the exit-flow function has a downward sloping part (which has usually been assumed away in DTA applications). In that case, if the number of spatial segments is increased, or the number of time intervals is reduced, then any dip in outflows occurs sooner and is more pronounced, and leads to earlier jamming. In the continuous limit, the jam occurs at the link entrance, preventing inflows in excess of capacity.