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Fall 2023

CE 521: Transportation Systems Analysis

Problem Set 4:

Transportation Supply Models & Traffic Assignment

Issue date: Monday, October 16

Due date: Monday, October 30

1. A USC game at the Coliseum expects a full capacity. The game starts at 6pm. The incoming traffic before the game is to be routed thru two bottlenecks. Bottleneck 1 consists of two lanes and bottleneck 2 consists of three lanes. The max flow capacity of each lane is 1500 vehicles/hr. How should the traffic be routed thru the two bottlenecks so that the total waiting time of all vehicles is minimized? Assume that (i) the average occupancy of each vehicle is two; and (ii) α = 8$/hr, β = 3.5$/hr, and γ = 13.5$/hr.

Recall that for a single bottleneck problem with total demand N and bottleneck ca-pacity µ, the equilibrium arrival rates are given by A1 = α/α−β µ and A2 = α/α+γ µ. Fur-thermore, the start/end of congestion period are tstart = τ − γ/β+γ N/µ and tend = τ + β/β+γ N/µ respectively, and peak queue length occurs at τ , which also coincides with the time when the arrival rate switches from A1 to A2.

2. Consider the network and the link costs shown in the top of Figure 5.9 in [2]. Note that the demand from node 1 to node 4 is d1−4 = 1000. Compute the uncongested link flows under stochastic assignment using the logit path choice model with parameter θ = 1, according to the following two methods:

(a) Closed form expression of the logit path choice model; and

(b) Monte Carlo technique described in [2, Section 5.3.1]. Use at least five samples. The evrnd command in MATLAB might be helpful for this.

Compare the link flows obtained from the above two methods.

3. Revisit Q3 from HW3. Write computer code to find social equilibrium for the setup in (a) and (b).

4. Write computer code to find the (deterministic) user equilibrium and social equilibrium for the following setup: c1(f1) = 10(1 + 5 (f1/2)4), and c2(f2) = 9 + 2f2.

5. Write computer code to find the stochastic user equilibrium for the setup in Q4, for the logit path choice model with θ = 1, and verify that the path choice under logit model is consistent with the computed equilibrium flows.

References

[1] K. A. Small and E. T. Verhoef, The Economics of Urban Transportation. Routledge, 2 ed., 2007.

[2] E. Cascetta, Transportation Systems Analysis. Springer, 2009.