## 4 Developing a Proxy Measure for Dynamically Determining Toll Plaza Capacity
Dividing observed \(TT\left(t\right)\) by \(\beta\) would provide an estimate of the additional capacity that needs to be maintained to change \(\Delta TT\left(t\right)\) to 0 based on the proposed model (see Equation (6)). \[\Delta TT=\beta\cdot\Delta V=\beta[\dot{A}\left(t\right)-\dot{D} \left(t\right)]\cdot T_{d} \tag{5}\] \[\frac{\Delta TT}{\beta\cdot T_{d}}=\left[\dot{A}\left(t\right)- \dot{D}\left(t\right)\right] \tag{6}\]  
Travel times along a 2.97-mile segment leading to the BEN during the Friday evening peak period are shown in Figure 5a. The dotted black line represents the average Friday afternoon travel times observed during the before period. The solid black line represents the travel time observed on 9 May 2014. Recall that the facility has seventeen lanes, among which lanes 10-17 are open road tolling lanes, while a varying number of lanes among lanes 1-10 operated as cash lanes throughout the day. Staffing for the cash lanes was reduced from five lanes to three lanes for the Friday evening peak operation beginning on 1 May 2014: While \(\dot{D}_{C}\left(t\right)\) has been reduced, the number of open road tolling lanes remains the same--\([\dot{A}_{F}\left(t\right)-\dot{D}_{F}\left(t\right)]\) can be assumed to have remained constant. A marked increase in travel times accompanied the change in staffing and lane availability (see the travel on 9 May in Figure 5a), and complaints were filed by commuters. After an in-depth discussion among toll plaza stakeholders, it was agreed that the staffing of cash lanes at the BEN for the Friday evening period would be increased from three lanes to four lanes. The revised scheduled was implemented on 16 May 2014, and travel times on 16 May returned to a level similar to the initial condition. Figure 5b shows the relationship between \(V\left(t\right)\) and \(TT\left(t\right)\) observed at the BEN on 9 May 2014. Based on the observed relationship between \(V\left(t\right)\) and \(TT\left(t\right)\), and assuming that the demand on 16 May would be comparable to that of 9 May, the proposed model can be used to predict segment travel times in a scenario of an increased number of cash lanes. The cash lane capacity of the toll plaza is roughly estimated to be 360 vehicles per hour based on observations at the toll plaza. Since drivers using the cash lane can pay either with cash or via the electronic transponder, as was stated in the introduction, this observed cash lane capacity would have been affected by electronic transponder users. Assuming \(\dot{A}\left(t\right)\) would remain comparable to the previous Friday, the solid black line in Figure 5c is the estimated rescaled cumulative vehicle count passing the toll plaza under the increased capacity schedule, \(CS\left(t\right)\). \(D\left(t\right)\) (represented by the dotted black line in Figure 5c) depicts the rescaled departure curve at the toll plaza if staffing had not been increased. In the figure, \(A^{\prime}\left(t\right)\) is the rescaled \(A\left(t\right)\) shifted forward in travel time. Notice how the vertical separation between \(A^{\prime}\left(t\right)\) and \(CS\left(t\right)\) disappears the majority of the time, indicating that the traffic will remain mostly freely flowing in response to opening up an additional cash lane. The proposed model would predict delays where \(A^{\prime}\left(t\right)\) and \(CS\left(t\right)\) do not remain superimposed. The vertical separation is the excess vehicle accumulation based on the proxy measure. By adding this number to \(V_{F}\) (see Figure 4c) and using the relationship shown in Equation (2), the travel times during the Friday afternoon peak after the staffing increase were predicted--the parameters for Equation (2) were calibrated and are shown in Figure 5b. The predicted travel times are shown together with the observed travel times (see the light grey line labeled \"May 16\" and dotted grey line labeled \"Predicted\"). The predicted and observed travel times were remarkably comparable.