## 4 Developing a Proxy Measure for Dynamically Determining Toll Plaza Capacity
\[A\left(t\right)-D\left(t\right)=V\left(t\right) \tag{1}\]  
\[TT\left(t\right)=\alpha+\beta V\left(t\right) \tag{2}\]  
\[\dot{TT}\left(t\right)=\beta[\dot{A}\left(t\right)-\dot{D}\left(t\right)] \tag{3}\]  
\[\dot{TT}\left(t\right)=\beta[\left(\dot{A}_{C}\left(t\right)-\dot{D}_{C}\left( t\right)\right)+\left(\dot{A}_{F}\left(t\right)-\dot{D}_{F}\left(t\right))] \tag{4}\]  
The estimated \(V\left(t\right)\) from 14 April and 15 May 2014, are used to evaluate their relationship with \(TT\left(t\right)\) (see Figure 4c). Compared with Figure 2f, \(V\left(t\right)\) displays a strong linear relationship with \(TT\left(t\right)\). \(TT_{F}\) shown on the \(y\)-axis of Figure 4c is the freely flowing travel time and \(V_{F}\) is the vehicle accumulation within the segment that can maintain the free flowing speed. When \(V\left(t\right)\) exceeds \(V_{F}\), the \(TT\left(t\right)\) begins to linearly increase. The slope of the line essentially indicates the changes in \(TT\left(t\right)\) resulting from the difference in \(\dot{A}\left(t\right)\) and \(\dot{D}\left(t\right)\) (see Equation (3)). The operator needs to carefully determine the number of cash and FasTrak lanes, since \(V\left(t\right)\) can increase (see Equation (4)) whenever the demand for FasTrak or cash lanes exceed the capacity of FasTrak or cash lanes. As noted earlier, toll staffing at the RSRB was increased on \(1\) May to better accommodate the cash lane demand, and a marked increase in travel time was observed on two of the seventeen days evaluated following the changes. This increase in travel time was greater than the typical delay observed prior to the changes and was accompanied by an increase in \(V\left(t\right)\). The increase  
Figure 4: (**a**) Segment travel times reported by INRIX. (**b**) Input–output diagram using the traffic count data from the toll plaza and segment travel times. (**c**) Relationship between cumulative count of vehicles and segment travel times from before and after period (the numbers in parentheses are \(t\)-value of parameters). in \(V\left(t\right)\) could have been the result of an increase in FasTrak demand each day coupled with a decrease in FasTrak capacity, as some of the FasTrak capacity was allocated to the cash lane capacity. However, this could not be confirmed with the data available from the RSRB. If this marked increase in travel times (at the RSRB) observed in two days were, indeed, a result of having fewer number of FasTrak lanes, the operator could evaluate the impact of converting existing cash lanes to FasTrak lanes using Equation (4). Discharge rate reported from the toll plazas (see Figure 1a) indicated that a FasTrak lane can serve about 1200-1300 vehicles per hour (vph) and a cash lane can serve about 360 vph while the toll plaza remains an active bottleneck [15; 16; 17]: The presence of a queue upstream of the toll plaza and downstream remaining freely flowing were visually confirmed by the toll plaza officers in estimating the FasTrak and cash lane capacity. The operator can increase the combined FasTrak lane capacity by an increment of 1200-1300 vph while decreasing the combined cash lane capacity by 360 vph in Equation (4) to estimate \(V\left(t\right)\) and the resulting \(TT\left(t\right)\) (see Figure 4c). Since \(V\left(t\right)\) is a proxy measure, and toll plaza capacity for cash and FasTrak changes in discrete increments, accurately estimating FasTrak and cash lane capacity is not a requirement for using proposed method. Suppose the changes in \(TT\left(t\right)\) and \(\Delta TT\left(t\right)\) were observed over some time \(T_{d}\) (see Equation (5)), and the relationship between \(V\left(t\right)\) and \(TT\left(t\right)\) shown in the figure holds. \(V\left(t\right)\) would have been changed by \(\Delta V\left(t\right)\) over \(T_{d}\). When \(T_{d}\) is set to one hour, then \(\Delta\)\(TT\) is the increase in travel time expected to accompany the difference in \(\dot{A}\left(t\right)\) and \(\dot{D}\left(t\right)\) that is sustained for an hour.