**6. Numerical examples**

The purpose of the numerical examples is to illustrate: (1) the effect of the *VMR* on the performance of the SN-MCP model; (2) the effect of both the demand and supply uncertainties on the performance of the PRSN-MCP model; (3) the importance of incorporating the travelers' perception error in the RSN-MCP model; and (4) the application of the proposed PRSN-MCP model in a medium-scale traffic network. The proposed models in this chapter can be solved by the method of successive averages (MSA).

#### **6.1 Effect of the VMR on the performance of SN-MCP toll scheme**

**Figure 2** shows a network consisting of 14 nodes and 21 directed links. There are two OD pairs, one is from node 1 to 12, and the other one is from node 1 to14. The link travel time function is assumed to be the Bureau of Public Roads (BPR) function with the following parameters:*β* ¼ 0*:*15, *n* ¼ 4, which is, *Ta* ¼ *t* 0 *<sup>a</sup>* ð1 þ 0*:*15 ð Þ *Va=Ca* <sup>4</sup> Þ, ∀*a*∈ *A*. The free-flow travel time, design capacity, and degradation parameter for each link are given in **Table 1**. In order to test the effects of different demand levels, the potential mean total demand for OD pair 1 and 2 is set as *<sup>q</sup>*<sup>1</sup> <sup>¼</sup> <sup>3800</sup>*<sup>z</sup>* and *<sup>q</sup>*<sup>2</sup> <sup>¼</sup> <sup>4200</sup>*z*, respectively. In 0 <sup>≤</sup>*z*≤1, *<sup>z</sup>* is the OD demand multiplier.

**Figure 2.** *Traffic network.*


**6.2 Importance of incorporating supply and demand uncertainty**

*6.2.1 Effect of congestion on the performance of different PRSN-MCP toll schemes.*

*Difference of the expected total perceived travel time between toll free case and SN-MCP under different OD*

stochasticity of the network, we compare the expected total perceived travel time under the four PRSN-MCP scenarios discussed in Section 5.2. These four scenarios are analyzed under different congestion levels (the OD demand multiplier *z* increases from 0.8 to 1 by interval 0.05). As a reminder, all the four scenarios consider the travelers' perception error, with the following differences: Case A is the most complete and realistic representation of the actual traffic flow as both stochastic fluctuations in supply (or link capacity) and demand are incorporated. In comparison, Case B and C are "incomplete cases," because they neglect certain aspects of the stochastic network. Case D is the classical MCP model in a deterministic traffic network.

simulated. To demonstrate the effects of neglecting certain aspects of the

We also use the traffic network shown in **Figure 2** in the following test, in which both supply and travel demand uncertainty and travelers' perception errors will be

In this example, we study the effect of congestion levels on the performance of

**Figure 4** demonstrates the percentage improvements in the expected total perceived travel time related to **Table 2**. The "Improvement" in **Figure 4** is, in this case, the percentage of improvement in the expected total perceived travel time

Improvement <sup>¼</sup> *<sup>U</sup> <sup>T</sup>*~*T*~toll�free � *<sup>U</sup> <sup>T</sup>*~*T*~case *<sup>=</sup> <sup>U</sup> <sup>T</sup>*~*T*~toll�free � *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD � 100%

(67)

different toll schemes with fixed *VoR* (i.e., *VoR* = 0.0165) and *VMRw* (i.e., *VMRw* ¼ 1*:*5). Furthermore, we assume the perception error distribution of unit travel time follows *N*ð Þ 0*:*1, 0*:*2 . **Table 2** displays the expected total perceived travel time at different congestion levels under the toll free, SS-SD, SS-DD, DS-DD, and DS-SD of the PRSN-MCP toll schemes. The results show that the expected total perceived travel time of the toll free and = other toll schemes increases as the

from the toll free case compared to the SS-SD tolls case, that is,

demand multiplier *z* increases.

**67**

**Figure 3.**

*demand multiplier* z *and* VMR *levels.*

*Reliability-Based Marginal Cost Pricing Problem DOI: http://dx.doi.org/10.5772/intechopen.92844*

#### **Table 1.**

*Link parameters.*

For the first example, we examine the effect of *VMR* on the performance of the SN-MCP model proposed in Section 3. All travelers are assumed to be risk-neutral (i.e., *VoR* = 0). In addition, travelers' perception errors are not considered in the first example. The relationship between the expected total perceived travel time, OD demand level, and *VMR* level under the toll free case and the SN-MCP toll scheme are shown in **Figure 3**. It can be observed that the difference of the expected total perceived travel time (i.e.,*U TT*toll free ½ �� *U TT* ½ � SN�MCP ) between these two scenarios decreases with the OD demand and *VMR* levels. For example, if the demand multiplier *z* is 0.8 and *VMR* level is 10, *U TT*toll free ½ �� *U TT* ½ � SN�MCP is more than 2900. However, when the demand multiplier *z* increases to 1 and *VMR* level increases to 50, *U TT*toll free ½ �� *U TT* ½ � SN�MCP is less than 1633. Remember that *VMRw* is the variance-to-mean ratio (*VMR*) of random travel demand. This indicates that along with the increase of travel demand variance and congestion level, the performance of the SN-MCP toll scheme decreases.

*Reliability-Based Marginal Cost Pricing Problem DOI: http://dx.doi.org/10.5772/intechopen.92844*

#### **Figure 3.**

*Difference of the expected total perceived travel time between toll free case and SN-MCP under different OD demand multiplier* z *and* VMR *levels.*
