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

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

In this example, we study the effect of congestion levels on the performance of 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 demand multiplier *z* increases.

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

$$\text{Improvement} = \left( U \left[ \ddot{T} \ddot{T}\_{\text{toll-free}} \right] - U \left[ \ddot{T} \ddot{T}\_{\text{case}} \right] \right) / \left( U \left[ \ddot{T} \ddot{T}\_{\text{toll-free}} \right] - U \left[ \ddot{T} \ddot{T}\_{\text{SS-SD}} \right] \right) \times 100\text{\%} \tag{67}$$

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 perfor-

mance of the SN-MCP toll scheme decreases.

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

**Table 1.** *Link parameters.*

**66**

**Link Free-flow travel time**

**Design capacity**

11 3 2000 0.95

**Degradation parameter** *θ<sup>a</sup>*

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

 3 2000 0.95 12 3 1000 0.95 3 2000 0.95 13 3 2600 0.95 3 2000 0.95 14 3 2000 0.95 4 4.5 1800 0.95 15 3 1400 0.95 5 7.5 1200 0.95 16 3 2000 0.95 3 1000 0.95 17 3 800 0.95 3 2000 0.95 18 3 2000 0.95 3 1800 0.95 19 3 2000 0.95 3 1800 0.95 20 3 4000 0.95 10 4.5 1800 0.95 21 3 4000 0.95

**Link Free-flow travel time**

**Design capacity**

**Degradation parameter** *θ<sup>a</sup>*


#### **Table 2.**

*Comparison of system performance under different modeling scenarios and OD demand multipliers.*

#### **Figure 4.**

*Improvement in system performance under different modeling scenarios and OD demand multipliers.*

From the figure above, the improvement in the expected total perceived travel time obtained by the SS-DD, DS-DD, and DS-SD tolls is lower than that obtained by the SS-SD tolls. Besides, the gap between the expected total perceived travel time under the SS-SD tolls and other toll schemes increases as *z* increases. When traffic is light, all toll schemes achieve similar system performances, revealing that other toll schemes do not lose too much accuracy by ignoring the stochasticity of the traffic network. However, when traffic is heavy, the differences between them become pronounced. Furthermore, for the DS-SD tolls, neglecting the stochastic link capacity makes the system performance decrease rapidly. This indicates that the toll scheme is more sensitive to the stochasticity of link capacity.

actual effect of other toll schemes, which ignore the effect of stochastic travel

*Improvement in system performance under different modeling scenarios and* VoR *levels.*

*VoR U T*~*T*~

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

*Comparison of system performance under different modeling scenarios and* VoR *levels.*

**Toll free SS-SD SS-DD DS-DD DS-SD**

0.0068 179,233 177,335 177,384 177,413 177,510 0.0085 179,291 177,394 177,445 177,491 177,620 0.0104 179,344 177,468 177,525 177,568 177,716 0.0129 179,432 177,560 177,623 177,668 177,844 0.0165 179,550 177,688 177,757 177,801 177,996

From the above analysis, it can be concluded that the discrepancies of these simplifications depend on both the congestion and *VoR* levels. Capturing the effect of stochastic capacity degradation and stochastic travel demand is critically

**6.3 Analysis of the essentiality of incorporating the travelers' perception error**

The traffic network shown in **Figure 2** is again adopted in examining the PRSN-MCP model. By comparing the difference of the expected total perceived travel time achieved by the RSN-MCP tolls (expressed by *U TT* ½ � SS�SD ) and the PRSN-MCP tolls (denoted by *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD ), we examine the effect of incorporating the traveler's perception error into the RSN-MCP tolls. In this example, both stochastic fluctuations in supply (or link capacity) and demand are considered for both toll schemes. **Figure 6** illustrates the influence of various combinations of travel demand level and *VoR* level on the difference of the expected total perceived travel time achieved by the RSN-MCP tolls and the PRSN-MCP tolls. Based on the survey results of [24], it is reasonable to assume that all the travelers are risk-averse under an uncertain environment. Therefore, we test the *VoR* level from 0.0068 to 0.0165, and the OD demand multiplier *z* from 0.8 to 1 with an interval of 0.05. From **Figure 6**, it is clear that *U TT* ½ �� SS�SD *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD increases as the demand level *z* increases. This implies that the consideration of travelers' perception error in the RSN-MCP tolls may have a more significant impact on system performance

demand and link capacity.

important.

**69**

**Figure 5.**

**Table 3.**

#### *6.2.2 Effect of the VoR on the performance of different PRSN-MCP toll schemes*

By assuming the levels of congestion and *VMRw* are fixed (i.e., *z* = 1, *VMRw* ¼ 1*:*5), the effect of the *VoR* on the expected total perceived travel time under different toll schemes is examined in this section. In **Table 3**, the expected total perceived travel time at different *VoR* levels under the toll free, SS-SD, SS-DD, DS-DD, and DS-SD of the PRSN-MCP toll schemes are compared. The expected total perceived travel time increases with an increase in the level of the *VoR*. This is logical: when *VoR* increases, travelers need to budget a large buffer time to improve their travel time reliability.

Based on Eq. (67) and **Table 3**, we can obtain the percentage improvements in the expected total perceived travel time, as shown in **Figure 5**. It can be seen that the discrepancies between the performance of the SS-SD toll and that of other toll schemes become conspicuously larger as the *VoR* increases. This implies that the higher the travel time reliability that travelers are concerned with, the worse the


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

#### **Table 3.**

From the figure above, the improvement in the expected total perceived travel time obtained by the SS-DD, DS-DD, and DS-SD tolls is lower than that obtained by the SS-SD tolls. Besides, the gap between the expected total perceived travel time under the SS-SD tolls and other toll schemes increases as *z* increases. When traffic is light, all toll schemes achieve similar system performances, revealing that other toll schemes do not lose too much accuracy by ignoring the stochasticity of the traffic network. However, when traffic is heavy, the differences between them become pronounced. Furthermore, for the DS-SD tolls, neglecting the stochastic link capacity makes the system performance decrease rapidly. This indicates that the toll

*Improvement in system performance under different modeling scenarios and OD demand multipliers.*

scheme is more sensitive to the stochasticity of link capacity.

**Demand multiplier (***z***)** *U T*~*T*~

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

**Toll free SS-SD SS-DD DS-DD DS-SD**

0.8 132,261 129,158 129,171 129,184 129,300 0.85 142,651 139,878 139,908 139,928 140,084 0.9 153,870 151,438 151,474 151,502 151,695 0.95 166,113 163,979 164,033 164,072 164,283 1 179,550 177,688 177,757 177,801 177,996

*Comparison of system performance under different modeling scenarios and OD demand multipliers.*

their travel time reliability.

**68**

**Table 2.**

**Figure 4.**

*6.2.2 Effect of the VoR on the performance of different PRSN-MCP toll schemes*

1*:*5), the effect of the *VoR* on the expected total perceived travel time under different toll schemes is examined in this section. In **Table 3**, the expected total perceived travel time at different *VoR* levels under the toll free, SS-SD, SS-DD, DS-DD, and DS-SD of the PRSN-MCP toll schemes are compared. The expected total perceived travel time increases with an increase in the level of the *VoR*. This is logical: when *VoR* increases, travelers need to budget a large buffer time to improve

By assuming the levels of congestion and *VMRw* are fixed (i.e., *z* = 1, *VMRw* ¼

Based on Eq. (67) and **Table 3**, we can obtain the percentage improvements in the expected total perceived travel time, as shown in **Figure 5**. It can be seen that the discrepancies between the performance of the SS-SD toll and that of other toll schemes become conspicuously larger as the *VoR* increases. This implies that the higher the travel time reliability that travelers are concerned with, the worse the

*Comparison of system performance under different modeling scenarios and* VoR *levels.*

actual effect of other toll schemes, which ignore the effect of stochastic travel demand and link capacity.

From the above analysis, it can be concluded that the discrepancies of these simplifications depend on both the congestion and *VoR* levels. Capturing the effect of stochastic capacity degradation and stochastic travel demand is critically important.

#### **6.3 Analysis of the essentiality of incorporating the travelers' perception error**

The traffic network shown in **Figure 2** is again adopted in examining the PRSN-MCP model. By comparing the difference of the expected total perceived travel time achieved by the RSN-MCP tolls (expressed by *U TT* ½ � SS�SD ) and the PRSN-MCP tolls (denoted by *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD ), we examine the effect of incorporating the traveler's perception error into the RSN-MCP tolls. In this example, both stochastic fluctuations in supply (or link capacity) and demand are considered for both toll schemes. **Figure 6** illustrates the influence of various combinations of travel demand level and *VoR* level on the difference of the expected total perceived travel time achieved by the RSN-MCP tolls and the PRSN-MCP tolls. Based on the survey results of [24], it is reasonable to assume that all the travelers are risk-averse under an uncertain environment. Therefore, we test the *VoR* level from 0.0068 to 0.0165, and the OD demand multiplier *z* from 0.8 to 1 with an interval of 0.05. From **Figure 6**, it is clear that *U TT* ½ �� SS�SD *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD increases as the demand level *z* increases. This implies that the consideration of travelers' perception error in the RSN-MCP tolls may have a more significant impact on system performance

network (see **Figure 7**), which consists of 24 nodes and 76 links. The link design capacity and link free-flow travel time can be found in [25]. Degradation parameter *θ<sup>a</sup>* for each link is 0.95. In this example, we also assume *VMRw* ¼ 1*:*5, and the perception error distribution of a unit travel time follows *N*ð Þ 0*:*1, 0*:*2 . Forty-two OD pairs are considered in the Sioux Falls network and the mean of the lognormal demand for all OD pairs is given in **Table 4**. The stopping tolerance criterion is set

at 0.001. Convergence is achieved in 48 iterations as depicted in **Figure 8**.

method in reducing traffic congestion.

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

**Table 4.**

**Figure 8.**

**71**

*Convergence of the MSA for the Sioux Falls network.*

In this example, we compare two scenarios. One is the toll free case, and the other one is the PRSN-MCP toll scheme. **Table 5** presents the link toll under the PRSN-MCP scenario. By levying these tolls on each link, the network becomes smooth and efficient. At the equilibrium state, the expected total perceived travel time achieved by the toll free case and PRSN-MCP toll scheme is 345,749 and 324,636, respectively. Therefore, the proposed PRSN-MCP model is an efficient

**O/D 4 5 6 10 14 19 22** 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800

22 800 800 800 800 800 800

*Means of the stochastic demand for all OD pairs in the Sioux Falls network.*

#### **Figure 6.**

*Difference of the expected total perceived travel time between PRSN-MCP and RSN-MCP under different OD demand multiplier* z *and* VoR *levels.*

under heavier congestion levels. On the other hand, we can see that *U TT* ½ �� SS�SD *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD is increasing while the *VoR* level increases. This is to be expected, since a higher travel time reliability requires a larger time buffer. Therefore, ignoring the travelers' perception error may significantly reduce the performance of the RSN-MCP tolls, especially when the network congestion level is heavy and travelers require a higher travel time reliability level.

#### **6.4 Application to the Sioux Falls network in the PRSN-MCP (SS-SD) case**

The final example illustrates the calculation of the PRSN-MCP (SS-SD) toll in a larger network. This example network is the well-known medium-scale Sioux Falls

**Figure 7.** *Sioux Falls network.*
