338 The UMaP Journal 22.3 (2001) By far, the most common approach was to determine that the flow q, or flux, is a function of the density p of cars on a highway and the average speed v of those cars: q= pu. Successful approaches identified the following characteris tics of the basic traffic flow problem When the vehicle density on the highway is 0, the flow is also 0 As density increases, the flow also increases(up to a point) When the density reaches its maximum, or jam density Po, the flow must be 0 Therefore, the flow initially increases, as density does, until it reaches some maximum value. Further increase in the density, up to the jam density, results in a reduction of the flow At this point, many teams either derived from first principles or used one of the many resources available on traffic modeling(such as garber and Hoel [ 1999) to find a relationship between the density and the average speed three of the common macroscopic models were a linear model developed by greenshield U=0 9=p00 a fluid-flow model developed by greenberg q=puo log a higher-order model developed by Jayakrishnan q=p(1-2 Po where vo represents the speed that a vehicle would travel in the absence of other traffic(the speed limit). By taking the derivative of the flow equation with respect to speed(or density), teams then found the optimal speed (or density)to maximize flow. Many teams took the optimal flow from one of the macroscopic approaches nd used it as the basis for a larger model. One of the more common models was simulation to determine evacuation times under a variety of scenarios In order to make it beyond the Successful Participant category, teams had to find a way realistically to regulate traffic density to meet these optimality conditions. Many teams did this by stipulating that ramp metering systems (long term)or staggered evacuations(short term) could be used to control traffic density338 The UMAP Journal 22.3 (2001) By far, the most common approach was to determine that the flow q, or flux, is a function of the density ρ of cars on a highway and the average speed v of those cars: q = ρv. Successful approaches identified the following characteris￾tics of the basic traffic flow problem: • When the vehicle density on the highway is 0, the flow is also 0. • As density increases, the flow also increases (up to a point). • When the density reaches its maximum, orjam density ρ0, the flow must be 0. • Therefore, the flow initially increases, as density does, until it reaches some maximum value. Further increase in the density, up to the jam density, results in a reduction of the flow. At this point, many teams either derived from first principles or used one of the many resources available on traffic modeling (such as Garber and Hoel [1999]) to find a relationship between the density and the average speed. Three of the common macroscopic models were: • a linear model developed by Greenshield: v = v0  1 − ρ ρ0  , so q = ρv0  1 − ρ ρ0  ; • a fluid-flow model developed by Greenberg: v = v0 log ρ ρ0 , so q = ρv0 log ρ ρ0 ; or • a higher-order model developed by Jayakrishnan: v = v0  1 − ρ ρ0 a , so q = ρv0  1 − ρ ρ0 a , where v0 represents the speed that a vehicle would travel in the absence of other traffic (the speed limit). By taking the derivative of the flow equation with respect to speed (or density), teams then found the optimal speed (or density) to maximize flow. Many teams took the optimal flow from one of the macroscopic approaches and used it as the basis for a larger model. One of the more common models was simulation, to determine evacuation times under a variety of scenarios. In order to make it beyond the Successful Participant category, teams had to find a way realistically to regulate traffic density to meet these optimality conditions. Many teams did this by stipulating that ramp metering systems (long term) or staggered evacuations (short term) could be used to control traffic density