## 4 Developing a Proxy Measure for Dynamically Determining Toll Plaza Capacity  
Figure 4a shows the segment travel times reported on 14 April and 15 May 2014, along the RSRB. Using the traffic count data and segment travel times, an input-output diagram can be constructed as shown in Figure 4b. The solid line labeled \(D\left(t\right)\) denotes the cumulative count of vehicles leaving the toll plaza with respect to time. Then, using the average travel times, \(D\left(t\right)\) was shifted back in time to reconstruct the cumulative arrival curve at the start of the segment, \(A\left(t\right)\). \(V\left(t\right)\), the vertical difference between \(A\left(t\right)\) and \(D\left(t\right)\), is the estimated number of vehicles stored in the segment at time \(t\) (see Figure 4b and Equation (1)). When \(A\left(t\right)\) is shifted forward by the free flow travel, \(TT_{F}\) (see \(A^{\prime}\left(t\right)\) in Figure 4b), the horizontal distance between \(A^{\prime}\left(t\right)\) and \(D\left(t\right)\) is the delay, \(d\left(t\right)\). \(A^{\prime}\left(t\right)\) and \(D\left(t\right)\) will remain superimposed when traffic is freely flowing. The area under between \(A^{\prime}\left(t\right)\) and \(D\left(t\right)\) is the total delay. The changes in travel time that encompasses the queue near the upstream of the toll plaza were linearly approximated by the number of vehicles stored within the segment: An increase in queue length will change the portion of the trip that will be traveled by \(v_{f}\) (see Figure 2c,d) under dense traffic conditions. Equation (2) is used based on this assumption, and \(V(t)\) is the proxy measure that was estimated following the procedure described in the preceding paragraph. Equation (3) can be obtained by differentiating \(TT\left(t\right)\), \(D\left(t\right)\), and \(A\left(t\right)\) with respect to time. Note that the right side of equation consists of the difference between the rate of vehicles arriving at the cash lane, \(\dot{A}_{C}\left(t\right)\), and the FasTrak lane, \(\dot{A}_{F}\left(t\right)\) and the corresponding cash lane, \(\dot{D}_{C}\left(t\right)\), and FasTrak lane, \(\dot{D}_{F}\left(t\right)\), discharge rates. Even though the combined demand for cash lanes and FasTrak lanes is less than the combined capacity, when the capacity for the two kinds of lanes is not properly assigned, the right side of equation can rapidly increase, resulting in a marked increase in travel times. This is what happened at the BEN Bridge after the staffing changes were made on 1 May 2014, as explained in detail later in this section. Figure 3: (**a**) Travel time along the 2.87-mile segment near the RSRB (Richmond San Rafael Bridge) toll plaza observed on 15 May. (**b**) Snapshot of cumulative travel time at time 07:10. (**c**) Snapshot of cumulative travel time at time 07:40. (**d**) Changes in cumulative travel time from P to the toll booth. (**e**) Length of the queue. Recall that \(V\left(t\right)\) is the only proxy measure developed to evaluate the relationship among \(TT\left(t\right)\), \(\dot{A}\left(t\right)\), and \(\dot{D}\left(t\right)\). If the flow between upstream and downstream of the segment is conserved (_i.e._, no on- or off-ramps within the segment), \(V\left(t\right)\) shown in Equation (1) would be the number of vehicles stored in the segment. Although \(V\left(t\right)\) estimated in the manner described in the preceding paragraph can be biased depending on the locations of the ramps and the flow using them, it can still be used as a proxy measure as shown in Figure 4c.