An approach to modelling time-varying ¯ows on congested networks Malachy Carey a, * , Eswaran Subrahmanian b a Faculty of Business and Management, University of Ulster, Newtownabbey, Northern Ireland, BT37 0QB, UK b Engineering Design Research Center, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburg, PA 15213, USA Received 13 August 1998; received in revised form 9 February 1999; accepted 9 April 1999 Abstract In mathematical programming models of time-varying ¯ows on trac networks (dynamic trac as- signment) a key component is the model of ¯ow behaviour within individual links. However, to maintain tractability in these models, time-varying link ¯ows tend to be modelled in very simple ways. Here we try to model link ¯ows more ¯exibly, so that the trip time of a vehicle on a link is in¯uenced by the ¯ow rate when the vehicle enters the link, the ¯ow rate when the vehicle exits from the link, and knock-on eects from trac ahead on the link. We concentrate on congestion along links, but the model can be extended, for example by dividing each link into a travel link followed by a queue `link'. We also concentrate on a system optimising model but outline how this can be extended to user equilibrium. We consider the properties of the model, and ®nd that the ®rst-in-®rst-out (FIFO) property of road trac holds unless there is a sharp increase in in¯ows to a link followed by a sharp decrease. We also investigate the ``holding back'' of ¯ows, a phenomenon associated with intertemporal network optimisation models in general. We apply the model to simple network examples. The model has the advantage of being linear and having a special structure which may be exploited to develop more ecient solution tech- niques. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic trac assignment; Time-varying ¯ows; Networks; Congestion; FIFO 1. Introduction Mathematical programming models of time-varying ¯ows on trac networks (dynamic trac assignment) consist basically of three parts: conservation of ¯ow, route choice criteria (usually user equilibrium or system optimum), and a model of ¯ow behaviour or propagation within Transportation Research Part B 34 (2000) 157±183 www.elsevier.com/locate/trb * Corresponding author. Tel.: +44-01232-366352; fax: +44-01232-366868. E-mail address: m.carey@ulst.ac.uk. 0191-2615/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII:S0191-2615(99)00019-3