The three problems that follow were originally devised to test a computer program based on a related, but more powerful model than those discussed in class. The setting is a 10km long stretch of freeway beginning at x0=0 and ending at x3=10km, which is subdivided into three 3.33km segments: (x0,xl),(x1,x2),(x2,x3). A freeway flow q(x0,t) enters the section at x0 and a constant ramp flow qr=400veh/h enters at xI(x2 in Problem 3) without delay. (i.e. Since the ramp flow is not very large, the ramp has priority at the junction.) Each of the three segments has two lanes with a combined capacity of 4000veh/h and the flow-density relationship shown in Figure 1. The next downstream segment, (x3,x4), is a bottleneck with a capacity of =3200 veh/h. Problem 1: In Problem 1, which is designed to demonstrate the analysis techniques in the simplest setting possible, the arrival flow at x0 is q(x0,t)={3200veh/h2000veh/hift(0,15)minutesotherwise. These flows are increased by 400veh/h from the entrance ramp at x1 , and vehicles need 6 minutes to travel from x0 to x3 in the absence of congestion, so the "arrival" flow at x3=10 is q(x3,t)=3600veh/h2400veh/hotherwise.ift(6,21)minutes The actual flow at x3, however, cannot exceed the 3200veh/h capacity of segment (x3,x4), so the departure curve, D(x3,t), lies below the arrival curve and a queue exists from t=6 until t=28.5 minutes as shown in class handout. (All time coordinates will be given in minutes, often without showing the units, but hours will be used for most other purposes.) The maximum number of vehicles in the vertical stack queue is A(x3,21)D(x3, 21) =900800=100 vehicles and the delay is equal to the area of the shaded triangle (18.75 veh h). The shock wave analysis is also FCure2 Anabsisof Pmblem 1 shown in Figure 2. The number of vehicle hours of travel was ) Shoch Wew Technigus b) Cuniulatht Areival and Departurt determined by multiplying the area of each of the regions A1 through C in Figure 2a by its density, as shown in Table bellow. Comparing the two solutions, we see that the real queue in Figure 2 a reaches a maximum length of 2.5km or 180 vehicles at t=19.5 when the arrival flow at the queue end decreases, but the vertical stack queue in Figure 2b continues to grow until t=21 when the vehicle marking the flow change would reach the stack in the absence of roadway congestion. Even though the vertical stack continues to grow after the physical queue begins to shrink, its maximum size 100 vehicles at t=21 is smaller than the real queue's maximum because the stack does not include vehicles haven't joined the vertical stack yet. However, the two procedures yield the same estimate of total delay because whether one waits by traveling slowly in a horizontal queue or by descending through a vertical stack is irrelevant: all that matters is the time when one reaches x3, so is next in line for service - and that is the same in both models. Problem 2: In Problem 2, the arrival flow at x0 is considerably larger and has a slightly more complex pattern: q(x0,t)=3800veh/h2600veh/h2000veh/hift(0,15)minutesift(15,30)minutesotherwise The ramp flow entering at x1 remains 400veh/h at all times; it could also be time dependent, but that would make the calculations more complex without adding significantly to the problem's illustrative usefulness. With this arrival pattern, the capacity of the downstream section is exceeded not only at x3, but at x1 as well, so two queues form. Solve this problem using both the shock wave and cumulative arrival and departure technique. (Hint: in cumulative arrivaldeparture technique, you will need to draw two curves one for queue at x1 and one for the queue at x3). Problem 3: Problem 3 is exactly the same as Problem 2 except that the ramp is at x2 instead of x1. Solve this