Bottlenecks and the evening commute.

Motion

During the evening commute there are often bottlenecks as people try to get home on fixe routes with finite capacity. Vickrey's "Congestion Theory and Transport Investment" (1969) decribes the problem of commuters trying to pass the bottleneck. A recent paper, "The evening commute with cars and transit: Duality results and user equilibrium for the combined morning and evening peaks" by Eric Gonzales and Carlos Daganzo tackles the commute problem looking at both the evening and morning commute, since mode travel decisions are often made based upon the travel needs for the whole day. 

The paper then considers both the morning and evening peaks together for a single mode bottleneck (all cars) with identical travelers that share the same wished times. For a schedule penalty function of the morning departure and evening arrival times that is positive definite and has certain properties, a user equilibrium is shown to exist in which commuters travel in the same order in both peaks. The result is used to illustrate the user equilibrium for two cases: (i) commuters have decoupled schedule preferences in the morning and evening and (ii) commuters must work a fixed shift length but have flexibility when to start. Finally, a special case is considered with cars and transit: commuters have the same wished order in the morning and evening peaks. Commuters must use the same mode in both directions, and the complete user equilibrium solution reveals the number of commuters using cars and transit and the period in the middle of each rush when transit is used.

The whole paper can be found here