Chapter 7, Problem 5P

a. Draw trajectories of approximately 8 to 11 vehicles moving on a one way 1-lane road with different time-varying speeds. b. Consider a time-space window in the time-space diagram of part (a). See below. Denote the number of vehicles passing BD, DC, AC, and AB respectively as n1, N2, N3, and n4. Write an equation relating n₁, N2, N3, and n4 to each other. What is the physical intuition of this equation? Please elaborate. X 4 X. n4 n3 с n2 X1 D B n1 t c. Using density (k) definition at time instances t₁ and t₂ and flow (q) definition at locations X1 and X2, rewrite equation of interest in part (b) to demonstrate KAB-KCD 9BD-9AC + t₂-t1 x2-x1 = 0. d. What will be the equation in part (c), in case of x2 → x1 and t₂ → t₁.

Consider a city center where the traffic conditions are described by a Macroscopic Fundamental Diagram (MFD) of network outflow (g- rate of trips finished) vs. accumulation (n- number of cars) with a trapezoidal shape, as shown in the figure below. The values of the parameters are: • • maximum trip completion rate gmax=100 [veh/min] critical accumulations ncr1=1000 [veh] and ncr2=1500 [veh] jam accumulation njam=4000 [veh]. gmax ncr1 ncr2 njam There are two types of demands in the morning peak hour (7-8am): trips generated from outside the city center with rate q1=80 [veh/min], and trips generated from within the city center with rate q2=50 [veh/min]. In addition, a perimeter traffic control, u, is available that only restricts vehicles entering the city from outside. If at 7am there are already no=500 [veh] in the city center: a. Write the dynamic equations (mass conservation equation) in a continuous form for the center of the city. b. Convert the continuous dynamic into the discrete…

Assume a car park facility where the arrival rate is 1 customer every minute, and the service process including pressing the button, taking the card, and waiting for the boom to rise leads to service rate of μ customer every minute. a. Assume the arrival and service processes are stochastic. Using any software (Excel, Matlab, or the one you prefer), plot average delay time (including service time) and average queue size (including the vehicle currently being served) for all combinations of λ = {1,2,3,..,10} and p = {0.1,0.3,0.5,0.7,0.9}. Specifically, we ask you to make 2 graphs (one for average delay and the other for average queue size), where the x-axes contains the different values for 1, and where you make one curve for each p. b. Assume the arrival process is stochastic but the service process is deterministic with rate μ. Using any software (Excel, Matlab, or the one you prefer), plot average delay time (including service time) and average queue size (including the vehicle…