A Bathtub Model of Downtown Traffic Congestion ( 8 page pdf, Richard Arnott, ACCESS, University of California, Jun. 10, 2015)
Today we review an analysis of the degree of congestion as a function of the timing of commuting and how applying a congestion charge can alter the flow sufficiently to allow free flowing traffic. This approach is called the bathtub model as it resembles the flow of water through a drain- as the water depth builds up the drainage rate increases up to a critical point when it decreases. In the same way, traffic reaches full congestion and then the flow slows down. Applying higher tolls as the traffic begins to reach the critical point reduces the traffic density and allows the flow to increase again.
“the dynamics of rush-hour traffic have the same properties as water flowing into and out of a hypothetical bathtub….. The height of water in the tub represents traffic density. The rate at which water drains increases with the height of the water until it reaches a critical height. … corresponds to the density of downtown traffic at which traffic jams start to become common. Above this level, traffic jams become more severe and the exit stream slows.”
"Commuters arriving exactly on time experience no schedule delay but a large travel time cost since they travel when traffic is most congested. In contrast, commuters departing at the beginning of the rush hour experience little congestion and therefore shorter travel times, but they arrive at work considerably before their work start time, experiencing high schedule delay cost.”
“ideal congestion pricing, with its no-toll equilibrium rush-hour traffic dynamics. Equilibriumis achieved when no commuter can reduce her trip price by altering her departure time. The optimum time pattern of departures minimizes total trip costs.“
“These optimal conditions can be achieved by charging a time-varying congestion toll set so that each commuter pays for the external cost her trip imposes on others. Each commuter’s trip price then equals the social cost of her trip. Since commuters respond to the toll by altering their departure times so that the equal trip-price condition (now including the toll) continues to be satisfied, the social cost of each trip is the same, which is the defining feature of the optimum scenario.”