**5.2 Calculation of PRSN-MCP**

In order to illustrate the importance of incorporating both stochastic supply and demand into the proposed PRSN-MCP model, the calculation of PRSN-MCP can be separated into four scenarios based on (1) network uncertainty caused by the stochasticity of travel demand; and (2) network uncertainty induced by the stochastic supply (link capacity). Case A is the most complete situation in which both stochastic link capacity and travel demand are considered. In contrast to Case A, which describes the "true" behaviors of travelers, Case D is the simplest case, neglecting the stochasticity of traffic network. Case B and C ignore, respectively, the effect of stochastic demand and link capacity.

Differentiating Eq. (50) with respect to the mean link flow *va* yields

*cn*

<sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�2*<sup>n</sup> <sup>a</sup>* � �

<sup>1</sup> � *<sup>θ</sup>*1�*<sup>n</sup> <sup>a</sup>* � �

*<sup>a</sup>*ð Þ <sup>1</sup> � *<sup>θ</sup><sup>a</sup>* ð Þ <sup>1</sup> � *<sup>n</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *vn*

*<sup>a</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup><sup>a</sup>* ð Þ <sup>1</sup> � <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>*

<sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�2*<sup>n</sup> <sup>a</sup>* � �

*Cn a* � �

Differentiating Eq. (54) with respect to the mean link flow *va* we have, upon

*<sup>a</sup> va* þ ð Þ *n* þ 2 *βt*

By substituting Eqs. (20), (21), (51), (53), and (55) into Eq. (47), the value of

In Case C, only the effect of stochastic travel demand is captured in modeling travelers' route choice decision process. The effect of stochastic link capacity is

� � and *E* 1*=C*<sup>2</sup>*<sup>n</sup>*

*a*

*<sup>a</sup>* , respectively. Consequently, the mean and variance of *Ta* are given by Eqs. (28) and (29), respectively. Similar to Case B, we need to recalculate

*aTa*

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup> t* 0

*<sup>v</sup><sup>n</sup>*þ<sup>1</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ*<sup>n</sup> a*

0 *a cn a*

� � � �

¼ *t* 0 *a v*2 *<sup>a</sup>* þ *βt* 0 *a*

0 *a*

*cn*

*a*

� �*=∂va*, respectively.

*aE V*½ �þ*<sup>a</sup> βt*

0 *aE V<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

� � � �

" #<sup>2</sup> 8

*cn*

*<sup>a</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup><sup>a</sup>* ð Þ <sup>1</sup> � <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>*

" #<sup>2</sup> 8

*a* � � (51)

� �

9 = ;

9 = ; (52)

9 = ; (53)

*v<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup>*

(54)

*vn*þ<sup>1</sup> *<sup>a</sup>* (55)

*<sup>a</sup>* and

(56)

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

*cn*

*cn*

<sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>* � �

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

� � are simplified to 1*=c<sup>n</sup>*

*Cn a*

� �

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

<sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>* � �

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

*<sup>∂</sup>E TT* ½ � *∂va*

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

*Var TT* ½ �¼ *E TT*<sup>2</sup> � � � ð Þ *E TT* ½ � <sup>2</sup>

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup> <sup>β</sup><sup>t</sup>* 0 *a* � �<sup>2</sup> *Var V<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup> <sup>β</sup><sup>t</sup>* 0 *a* � �<sup>2</sup>

¼ ð Þ 2*n* þ 2 *βt*

0

*<sup>∂</sup>E V*ð Þ*<sup>a</sup>* <sup>2</sup>

*∂va*

ignored in this case. Therefore, *E* 1*=C<sup>n</sup>*

*<sup>∂</sup>E V*½ � *aTa <sup>=</sup>∂va*, *<sup>∂</sup>Var TT* ½ �*=∂va*, and *<sup>∂</sup>E V*<sup>2</sup>

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup> t* 0 *<sup>a</sup> va* <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

*E TT* ½ �¼ *<sup>E</sup>* <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

The expected total travel time is given by

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaTa* h i <sup>¼</sup> <sup>X</sup>

*aE V*½ � *<sup>a</sup>* <sup>2</sup> <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

*Ta* h i

PRSN-MCP in case of SS-DD can be determined.

8 < :

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

With Eq. (26) we have

*Ta* h i <sup>¼</sup> *<sup>t</sup>*

*<sup>∂</sup>Var TT* ½ � *∂va*

*E V*ð Þ*<sup>a</sup>* <sup>2</sup>

simplifying

1*=c*<sup>2</sup>*<sup>n</sup>*

**63**

¼ *t* 0 *<sup>a</sup>* þ *βt* 0 *a*

The variance of the total travel time is described by

� � ( )

*v*<sup>2</sup>*n*þ<sup>2</sup> *<sup>a</sup>*

0 *a* � �<sup>2</sup>

� � *Var C<sup>n</sup> a*

*c*2*n*

Differentiating Eq. (52) with respect to the mean link flow yields

< :

*c*2*n*

< :

*v*<sup>2</sup>*n*þ<sup>1</sup> *<sup>a</sup>*

0 *aE V<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

¼ 2 � *t* 0

*5.2.3 Case C: deterministic supply, stochastic demand (DS-SD)*

#### *5.2.1 Case a: stochastic supply, stochastic demand (SS-SD)*

To begin, we discuss the most complete and realistic case in which the travelers consider both stochastic fluctuations in supply (or link capacity) and demand in their route choice decision-making process. As of now, we have already obtained the values of *<sup>∂</sup>E TT* ½ �*=∂va*, *E T*½ � *<sup>a</sup>* ,*Var T*½ � *<sup>a</sup>* and *<sup>∂</sup>Var TT* ½ �*=∂va*. The only value left unknown is *∂E V*<sup>2</sup> *aTa* � �*=∂va*. With Eq. (26) we can obtain

$$\begin{split} E\left[ (V\_{a})^{2} T\_{a} \right] &= t\_{a}^{0} E[V\_{a}]^{2} + \beta t\_{a}^{0} E[V\_{a}^{n+2}] E\left[ \frac{\mathbf{1}}{C\_{a}^{n}} \right] \\ &= t\_{a}^{0} v\_{a}^{2} \mathcal{y}\_{a}^{2} + \beta t\_{a}^{0} \frac{(\mathbf{1} - \theta\_{a}^{1-n})}{\tilde{c}\_{a}^{n} (\mathbf{1} - \theta\_{a}) (\mathbf{1} - n)} \left( v\_{a}^{n+2} \mathcal{y}\_{a}^{n+3n+2} \right) \end{split} \tag{48}$$

Differentiating Eq. (48) with respect to the mean link flow *va* and performing some simple algebraic operations we have

$$\frac{\partial E\left[\left(V\_{a}\right)^{2}T\_{a}\right]}{\partial v\_{a}} = 2 \cdot t\_{a}^{0} v\_{a} \mathbf{y}\_{a}^{2} - t\_{a}^{0} \cdot \text{VMR} + \beta t\_{a}^{0} \frac{\left(\mathbf{1} - \theta\_{a}^{1-n}\right)}{\mathbf{c}\_{a}^{n} (\mathbf{1} - \theta\_{a})(\mathbf{1} - n)}\tag{49}$$

$$\left[ (n+2)v\_{a}^{n+1} \mathbf{y}\_{a}^{n+3n+2} - \frac{n^{2} + 3n + 2}{2} \cdot \text{VMR} \cdot v\_{a}^{n} \mathbf{y}\_{a}^{n+3n} \right]$$

Substituting Eqs. (30), (31), (34), (38), and (49) into Eq. (47), we can obtain the value of PRSN-MCP in case of SS-SD.

#### *5.2.2 Case B: stochastic supply, deterministic demand (SS-DD)*

In Case B, the effect of stochastic demand is neglected; only the effect of stochastic link capacity is considered in modeling the travelers' route choice decisionmaking process. Thus, the mean and variance of *Ta* are given by Eqs. (20) and (21), respectively. To calculate the value of PRSN-MCP in case of stochastic supply and deterministic demand, we need to recalculate *<sup>∂</sup>E V*½ � *aTa <sup>=</sup>∂va*, *<sup>∂</sup>Var TT* ½ �*=∂va*, and*∂E V*<sup>2</sup> *aTa* � �*=∂va*, respectively.

The expected total travel time can be simplified to

$$\begin{split} E[TT] &= E\left[\sum\_{a \in A} V\_a T\_a \right] = \sum\_{a \in A} \left\{ t\_a^0 E[V\_a] + \beta t\_a^0 E\left[V\_a^{n+1}\right] E\left[\frac{1}{C\_a^n}\right] \right\} \\ &= \sum\_{a \in A} \left\{ t\_a^0 v\_a + \beta t\_a^0 \frac{\left(1 - \theta\_a^{1-n}\right)}{\overline{c}\_a^n (1 - \theta\_a)(1 - n)} v\_a^{n+1} \right\} \end{split} \tag{50}$$

**5.2 Calculation of PRSN-MCP**

unknown is *∂E V*<sup>2</sup>

*E V*ð Þ*<sup>a</sup>* <sup>2</sup>

*<sup>∂</sup>E V*ð Þ*<sup>a</sup>* <sup>2</sup>

and*∂E V*<sup>2</sup>

**62**

*aTa*

*E TT* ½ �¼ *<sup>E</sup>* <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

*∂va*

the effect of stochastic demand and link capacity.

*aTa*

*Ta* h i

¼ *t* 0

¼ *t* 0 *a v*2 *ay*2 *<sup>a</sup>* þ *βt* 0 *a*

some simple algebraic operations we have

¼ 2 � *t* 0 *<sup>a</sup> vay*<sup>2</sup> *<sup>a</sup>* � *t* 0

the value of PRSN-MCP in case of SS-SD.

� �*=∂va*, respectively.

*Ta* h i

*5.2.1 Case a: stochastic supply, stochastic demand (SS-SD)*

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

In order to illustrate the importance of incorporating both stochastic supply and demand into the proposed PRSN-MCP model, the calculation of PRSN-MCP can be separated into four scenarios based on (1) network uncertainty caused by the stochasticity of travel demand; and (2) network uncertainty induced by the stochastic supply (link capacity). Case A is the most complete situation in which both stochastic link capacity and travel demand are considered. In contrast to Case A, which describes the "true" behaviors of travelers, Case D is the simplest case, neglecting the stochasticity of traffic network. Case B and C ignore, respectively,

To begin, we discuss the most complete and realistic case in which the travelers consider both stochastic fluctuations in supply (or link capacity) and demand in their route choice decision-making process. As of now, we have already obtained the values of *<sup>∂</sup>E TT* ½ �*=∂va*, *E T*½ � *<sup>a</sup>* ,*Var T*½ � *<sup>a</sup>* and *<sup>∂</sup>Var TT* ½ �*=∂va*. The only value left

> <sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>* � �

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

0 *a*

� �

*<sup>a</sup>* � *<sup>n</sup>*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

Substituting Eqs. (30), (31), (34), (38), and (49) into Eq. (47), we can obtain

In Case B, the effect of stochastic demand is neglected; only the effect of stochastic link capacity is considered in modeling the travelers' route choice decisionmaking process. Thus, the mean and variance of *Ta* are given by Eqs. (20) and (21), respectively. To calculate the value of PRSN-MCP in case of stochastic supply and deterministic demand, we need to recalculate *<sup>∂</sup>E V*½ � *aTa <sup>=</sup>∂va*, *<sup>∂</sup>Var TT* ½ �*=∂va*,

*cn*

Differentiating Eq. (48) with respect to the mean link flow *va* and performing

*<sup>a</sup>* � *VMR* þ *βt*

*Cn a* � �

> <sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>* � �

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

<sup>2</sup> � *VMR* � *<sup>v</sup><sup>n</sup>*

*vn*þ<sup>2</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ3*n*þ<sup>2</sup> *a* � �

> *ayn*<sup>2</sup>þ3*<sup>n</sup> a*

> > *Cn a*

(48)

(49)

� �*=∂va*. With Eq. (26) we can obtain

0 *aE V<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

*cn*

*aE V*½ � *<sup>a</sup>* <sup>2</sup> <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *<sup>v</sup><sup>n</sup>*þ<sup>1</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ3*n*þ<sup>2</sup>

*5.2.2 Case B: stochastic supply, deterministic demand (SS-DD)*

The expected total travel time can be simplified to

<sup>¼</sup> <sup>X</sup>

*cn*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup> t* 0

<sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>* � �

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

*aE V*½ �þ*<sup>a</sup> βt*

0 *aE V<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

*v<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>* ( ) (50)

� � � �

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>VaTa* h i

> *<sup>a</sup>* <sup>∈</sup> *<sup>A</sup> t* 0 *<sup>a</sup> va* þ *βt* 0 *a*

Differentiating Eq. (50) with respect to the mean link flow *va* yields

$$\frac{\partial E[TT]}{\partial v\_a} = t\_a^0 + \beta t\_a^0 \frac{\left(\mathbf{1} - \theta\_a^{1-n}\right)}{\overline{c}\_a^n (\mathbf{1} - \theta\_a)(\mathbf{1} - n)} \left[ (n+1)v\_a^n \right] \tag{51}$$

The variance of the total travel time is described by

$$\begin{split} Var[TT] &= E\left[ TT^2 \right] - \left( E[TT] \right)^2 \\ &= \sum\_{a \in A} \left\{ \left( \beta t\_a^0 \right)^2 \frac{Var\left[V\_a^{n+1}\right]}{Var\left[\mathcal{C}\_a^n\right]} \right\} \\ &= \sum\_{a \in A} \left\{ \left( \beta t\_a^0 \right)^2 v\_a^{2n+2} \left\{ \frac{\left(1 - \theta\_a^{1-2n}\right)}{\tilde{\varepsilon}\_a^{2n} (1 - \theta\_a)(1 - 2n)} - \left[ \frac{\left(1 - \theta\_a^{1-n}\right)}{\tilde{\varepsilon}\_a^n (1 - \theta\_a)(1 - n)} \right]^2 \right\} \right\} \end{split} \tag{52}$$

Differentiating Eq. (52) with respect to the mean link flow yields

$$\frac{\partial Var[TT]}{\partial v\_{a}} = (2n+2) \left( \beta t\_{a}^{0} \right)^{2} v\_{a}^{2n+1} \left\{ \frac{\left( \mathbf{1} - \theta\_{a}^{1-2n} \right)}{\overline{c}\_{a}^{2n} (\mathbf{1} - \theta\_{a}) (\mathbf{1} - 2n)} - \left[ \frac{\left( \mathbf{1} - \theta\_{a}^{1-n} \right)}{\overline{c}\_{a}^{n} (\mathbf{1} - \theta\_{a}) (\mathbf{1} - n)} \right]^{2} \right\} \tag{53}$$

With Eq. (26) we have

$$E\left[\left(V\_{a}\right)^{2}T\_{a}\right] = t\_{a}^{0}E\left[V\_{a}\right]^{2} + \beta t\_{a}^{0}E\left[V\_{a}^{n+2}\right]E\left[\frac{\mathbf{1}}{\mathcal{C}\_{a}^{n}}\right] = t\_{a}^{0}v\_{a}^{2} + \beta t\_{a}^{0}\frac{\left(\mathbf{1}-\theta\_{a}^{1-n}\right)}{\overline{c}\_{a}^{n}(\mathbf{1}-\theta\_{a})(\mathbf{1}-n)}v\_{a}^{n+2} \tag{54}$$

Differentiating Eq. (54) with respect to the mean link flow *va* we have, upon simplifying

$$\frac{\partial E\left[\left(V\_{a}\right)^{2}T\_{a}\right]}{\partial v\_{a}} = 2 \cdot t\_{a}^{0}v\_{a} + (n+2)\beta t\_{a}^{0} \frac{\left(1-\theta\_{a}^{1-n}\right)}{\overline{c}\_{a}^{n}(1-\theta\_{a})(1-n)}v\_{a}^{n+1} \tag{55}$$

By substituting Eqs. (20), (21), (51), (53), and (55) into Eq. (47), the value of PRSN-MCP in case of SS-DD can be determined.

#### *5.2.3 Case C: deterministic supply, stochastic demand (DS-SD)*

In Case C, only the effect of stochastic travel demand is captured in modeling travelers' route choice decision process. The effect of stochastic link capacity is ignored in this case. Therefore, *E* 1*=C<sup>n</sup> a* � � and *E* 1*=C*<sup>2</sup>*<sup>n</sup> a* � � are simplified to 1*=c<sup>n</sup> <sup>a</sup>* and 1*=c*<sup>2</sup>*<sup>n</sup> <sup>a</sup>* , respectively. Consequently, the mean and variance of *Ta* are given by Eqs. (28) and (29), respectively. Similar to Case B, we need to recalculate *<sup>∂</sup>E V*½ � *aTa <sup>=</sup>∂va*, *<sup>∂</sup>Var TT* ½ �*=∂va*, and *<sup>∂</sup>E V*<sup>2</sup> *aTa* � �*=∂va*, respectively.

The expected total travel time is given by

$$\begin{split} E[TT] &= E\left[\sum\_{a \in A} V\_a T\_a \right] = \sum\_{a \in A} \left\{ t\_a^0 E[V\_a] + \beta t\_a^0 E\left[V\_a^{n+1}\right] E\left[\frac{1}{C\_a^n}\right] \right\} \\ &= \sum\_{a \in A} \left\{ t\_a^0 v\_a + \frac{\beta t\_a^0}{\overline{c}\_a^n} \left( v\_a^{n+1} \mathbf{y}\_a^{n^2+n} \right) \right\} \end{split} \tag{56}$$

Differentiating Eq. (56) with respect to the mean link flow *va* yields

$$\frac{\partial E[TT]}{\partial v\_{a}} = t\_{a}^{0} + \frac{\beta t\_{a}^{0}}{\overline{c}\_{a}^{n}} \left[ \frac{n \nu\_{a}^{n-1} (1 - \nu\_{a}^{2})}{2 \nu\_{a}^{2}} + 1 \right] \left[ (n+1) \nu\_{a} \nu\_{a}^{n^{2}+n} \right] \tag{57}$$

The expected total travel time can be simplified to

0

*aE V*½ � *<sup>a</sup>* <sup>2</sup> <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

0

Consequently, we have, upon simplifying

*nβt* 0 *a cn a vn a* � �

0 *<sup>a</sup> vn*þ<sup>1</sup> *<sup>a</sup> cn a*

*<sup>a</sup>* <sup>þ</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>β</sup><sup>t</sup>*

� *E T*½ �¼ *<sup>a</sup>* ð Þ� 2*va* � 1 *t*

� �

0 *aE V<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup> t* 0

> 0 *a cn a vn a*

*aE V*½ �þ*<sup>a</sup> βt*

� *t* 0 *<sup>a</sup>* <sup>þ</sup> *<sup>β</sup><sup>t</sup>* 0 *a cn a vn a*

*Cn a* � �

0

By substituting Eqs. (63) and (65) into Eq. (47), the value of PRSN-MCP in case

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

**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)

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.

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

function with the following parameters:*β* ¼ 0*:*15, *n* ¼ 4, which is, *Ta* ¼ *t*

<sup>þ</sup> *VoR* � *<sup>ϖ</sup>*<sup>2</sup> ð Þ� <sup>2</sup>*va* � <sup>1</sup> *<sup>t</sup>*

� � (62)

0 *aE V<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

� �

¼ *t* 0 *a v*2 *<sup>a</sup>* <sup>þ</sup> *<sup>β</sup><sup>t</sup>* 0 *<sup>a</sup> v<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup> cn a*

*<sup>a</sup>* � ½ � ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � <sup>1</sup> *<sup>β</sup><sup>t</sup>*

0

� � � �

*Cn a*

<sup>¼</sup> *<sup>n</sup>β<sup>t</sup>* 0 *a cn a vn*

> 0 *a vn a cn a*

� �

*<sup>a</sup>* � ½ � ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � <sup>1</sup> *<sup>β</sup><sup>t</sup>*

0 *<sup>a</sup>* ð1 þ 0*:*15

*<sup>a</sup>* (63)

(64)

(65)

0 *a vn a cn a*

(66)

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaTa* h i <sup>¼</sup> <sup>X</sup>

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup> t* 0 *<sup>a</sup> va* <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

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

� *E T*½ �¼ *<sup>a</sup> t*

From Eq. (26) we can obtain

*Ta* h i <sup>¼</sup> *<sup>t</sup>*

> *Ta* h i

of DS-DD can be expressed as follows:

*E TT* ½ �¼ *<sup>E</sup>* <sup>X</sup>

Then we have

*<sup>∂</sup>E TT* ½ � *∂va*

*E V*ð Þ*<sup>a</sup>* <sup>2</sup>

*<sup>∂</sup>E V*ð Þ*<sup>a</sup>* <sup>2</sup>

PRSN � MCP ¼ ð Þ 1 þ *χ*

**6. Numerical examples**

successive averages (MSA).

ð Þ *Va=Ca* <sup>4</sup>

**65**

*∂va*

<sup>¼</sup> <sup>X</sup>

The variance of the total travel time is expressed as

$$\begin{split} \operatorname{Var}[\operatorname{TT}] &= \operatorname{E}[\operatorname{TT}^{2}] - \left(\operatorname{E}[\operatorname{TT}]\right)^{2} \\ &= \sum\_{a \in A} \left\{ \left(\mathbf{t}\_{a}^{0}\right)^{2} \cdot \operatorname{Var}[V\_{a}] + \left(\frac{\theta \mathbf{t}\_{a}^{0}}{\overline{\mathbf{c}\_{a}^{a}}}\right)^{2} \operatorname{Var}[V\_{a}^{n+1}] + \frac{2\beta \left(\mathbf{t}\_{a}^{0}\right)^{2}}{\overline{\mathbf{c}\_{a}^{a}}} \left\{ E\left[V\_{a}^{n+2}\right] - E\left[V\_{a}^{n+1}\right]E\left[V\_{a}\right] \right\} \right\} \\ &= \sum\_{a \in A} \left\{ \left(\mathbf{t}\_{a}^{0}\right)^{2} \cdot \operatorname{VAR} \cdot \nu\_{a} + \left(\frac{\beta \mathbf{t}\_{a}^{0}}{\overline{\mathbf{c}\_{a}^{a}}}\right)^{2} \left\{ \nu\_{a}^{2a} \mathbf{y}\_{a}^{4a^{2}+6n+2} - \left(\nu\_{a}^{n+1} \mathbf{y}\_{a}^{n+a}\right)^{2} \right\} + \frac{2\beta \left(\mathbf{t}\_{a}^{0}\right)^{2}}{\overline{\mathbf{c}\_{a}^{a}}} \nu\_{a}^{n+2} \mathbf{y}\_{a}^{n+a} \left(\mathbf{y}\_{a}^{2n+2} - 1\right) \right\} \end{split} \tag{58}$$

Differentiating Eq. (58) with respect to the mean link flow yields

$$\begin{split} \frac{\partial Var[TT]}{\partial v\_{a}} &= \left(t\_{a}^{0}\right)^{2} \cdot VMR + \left(\frac{\beta t\_{a}^{0}}{\tilde{c}\_{a}^{0}}\right)^{2} \left\{ \begin{aligned} &\left\{\nu\_{a}^{2}\nu\_{a}\frac{4\pi^{2}+6\pi}{a}[(2n+2)\nu\_{a} - (2n^{2}+n-1)\cdot VMR] \right\} \\ &- \left\{\nu\_{a}^{2}\frac{2\nu\_{a}^{2}\pi^{2}+2n-2}{\tilde{c}\_{a}^{0}}[(2n+2)\nu\_{a} - (n^{2}-n-2)\cdot VMR] \right\} \end{aligned} \right\} \\ &+ \frac{2\rho\{t\_{a}^{0}\}^{2}}{\tilde{c}\_{a}^{0}} \left\{ \begin{aligned} &\left\{\nu\_{a}^{2}\nu\_{a}^{n^{2}+3n} \left[(n+2)\nu\_{a} - \frac{(n^{2}+n-2)}{2}\cdot VMR\right] \right\} \\ &- \left\{\nu\_{a}^{2}\nu\_{a}^{n^{2}+n-2} \left[(n+2)\nu\_{a} - \frac{(n^{2}-n-4)}{2}\cdot VMR\right] \right\} \end{aligned} \right\} \end{split} \tag{59}$$

With Eq. (26) we have

$$E\left[\left(V\_{a}\right)^{2}T\_{a}\right] = t\_{a}^{0}E\left[V\_{a}\right]^{2} + \beta t\_{a}^{0}E\left[V\_{a}^{n+2}\right]E\left[\frac{\mathbf{1}}{\mathbf{C}\_{a}^{n}}\right] = t\_{a}^{0}v\_{a}^{2}\mathbf{y}\_{a}^{2} + \frac{\beta t\_{a}^{0}}{\overline{c}\_{a}^{n}}\left(\nu\_{a}^{n+2}\mathbf{y}\_{a}^{n+3n+2}\right) \tag{60}$$

Differentiating Eq. (60) with respect to the mean link flow *va* and performing some simple algebraic operations, we have

$$\begin{split} \frac{\partial E\left[\left(V\_{a}\right)^{2}T\_{a}\right]}{\partial v\_{a}} &= 2 \cdot t\_{a}^{0} v\_{a} \mathbf{y}\_{a}^{2} - t\_{a}^{0} \cdot \text{VMR} \\ &+ \frac{\beta t\_{a}^{0}}{\overline{c}\_{a}^{n}} \left[ (n+2) \boldsymbol{\nu}\_{a}^{n+1} \boldsymbol{\nu}\_{a}^{n+3n+2} - \frac{n^{2} + 3n + 2}{2} \cdot \text{VMR} \cdot \boldsymbol{\nu}\_{a}^{n} \boldsymbol{\nu}\_{a}^{n+3n} \right] \end{split} \tag{61}$$

Thus the value of PRSN-MCP in case of DS-SD can be determined by substituting Eqs. (28), (29), (57), (59), and (61) into Eq. (47).

#### *5.2.4 Case D: Deterministic supply, deterministic demand (DS-DD)*

Case D degenerates into the MCP model in a deterministic traffic network, in which neither the stochastic link capacity nor stochastic travel demand is considered in travelers' route choice decision making. In this case, the variance of both *Var TT* ½ � and *Var T*½ � is equal to zero, and *E T*½ �¼ *<sup>a</sup> t* 0 *<sup>a</sup>* þ *βt* 0 *a vn a=Cn <sup>a</sup>*. We only need to recalculate *<sup>∂</sup>E V*½ � *aTa <sup>=</sup>∂va*, and *<sup>∂</sup>E V*<sup>2</sup> *aTa* � �*=∂va*, respectively.

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

The expected total travel time can be simplified to

$$\begin{split} E[TT] &= E\left[\sum\_{a \in A} V\_a T\_a \right] = \sum\_{a \in A} \left\{ t\_a^0 E[V\_a] + \beta t\_a^0 E\left[V\_a^{n+1}\right] E\left[\frac{1}{C\_a^n}\right] \right\} \\ &= \sum\_{a \in A} \left\{ t\_a^0 v\_a + \frac{\beta t\_a^0 v\_a^{n+1}}{\overline{c}\_a^n} \right\} \end{split} \tag{62}$$

Then we have

Differentiating Eq. (56) with respect to the mean link flow *va* yields

*nvn*�<sup>1</sup> *<sup>a</sup>* <sup>1</sup> � *<sup>y</sup>*<sup>2</sup>

� �

2*β t* 0 *a* � �<sup>2</sup> *cn a*

*<sup>a</sup>* � *vn*þ<sup>1</sup> *<sup>a</sup> yn*<sup>2</sup>þ*<sup>n</sup>*

*<sup>a</sup>* � <sup>1</sup> � � ( )

� �<sup>2</sup> � �

� � ( )

2*y*<sup>2</sup> *a*

*a* � �

þ 1

*E V<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup>*

*a*

� � � *E V<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>*

þ 2*β t* 0 *a* � �<sup>2</sup> *cn a*

*<sup>a</sup> <sup>y</sup>*<sup>4</sup>*n*2þ6*<sup>n</sup> <sup>a</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � <sup>2</sup>*<sup>n</sup>* ½ � ð Þ� <sup>2</sup> <sup>þ</sup> *<sup>n</sup>* � <sup>1</sup> *VMR* n o

<sup>2</sup> � *VMR*

*<sup>a</sup> <sup>y</sup>*<sup>2</sup>*n*2þ2*n*�<sup>2</sup> *<sup>a</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � *<sup>n</sup>* ½ � ð Þ� <sup>2</sup> � *<sup>n</sup>* � <sup>2</sup> *VMR* n o

<sup>2</sup> � *VMR*

<sup>2</sup> � *VMR* � *<sup>v</sup><sup>n</sup>*

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *vayn*<sup>2</sup>þ*<sup>n</sup> a* h i

� �*E V*½ � *<sup>a</sup>*

*vn*þ<sup>2</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ*<sup>n</sup> <sup>a</sup> y*<sup>2</sup>*n*þ<sup>2</sup>

> 9 >>>=

> >>>;

*<sup>v</sup><sup>n</sup>*þ<sup>2</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ3*n*þ<sup>2</sup> *a* � �

> *<sup>a</sup> yn*<sup>2</sup>þ3*<sup>n</sup> a*

*<sup>a</sup>*. We only need to

(57)

(58)

(59)

(60)

(61)

9 >= >;

*<sup>∂</sup>E TT* ½ � *∂va*

*Var TT* ½ �¼ *E TT*<sup>2</sup> � � � ð Þ *E TT* ½ � <sup>2</sup>

� �<sup>2</sup> � *Var V*½ �þ*<sup>a</sup>*

� �<sup>2</sup> � *VMR* � *va* <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

� �<sup>2</sup> � *VMR* <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

8 >>><

>>>:

*aE V*½ � *<sup>a</sup>* <sup>2</sup> <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

some simple algebraic operations, we have

¼ 2 � *t* 0 *<sup>a</sup> vay*<sup>2</sup> *<sup>a</sup>* � *t* 0 *<sup>a</sup>* � *VMR*

> þ *βt* 0 *a cn a*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup> t* 0 *a*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup> t* 0 *a*

> ¼ *t* 0 *a*

> > þ 2*β t* 0 *a* � �<sup>2</sup> *cn a*

With Eq. (26) we have

¼ *t* 0

*Ta* h i

> *Ta* h i

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

*<sup>∂</sup>Var TT* ½ � *∂va*

*E V*ð Þ*<sup>a</sup>* <sup>2</sup>

*<sup>∂</sup>E V*ð Þ*<sup>a</sup>* <sup>2</sup>

**64**

*∂va*

¼ *t* 0 *<sup>a</sup>* <sup>þ</sup> *<sup>β</sup><sup>t</sup>* 0 *a cn a*

The variance of the total travel time is expressed as

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

0 *a cn a* � �<sup>2</sup>

> 0 *a cn a*

*vn*

� *vn*

0 *aE V<sup>n</sup>*þ<sup>2</sup> *<sup>a</sup>* � �*<sup>E</sup>* <sup>1</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *vn*þ<sup>1</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ3*n*þ<sup>2</sup>

*5.2.4 Case D: Deterministic supply, deterministic demand (DS-DD)*

ing Eqs. (28), (29), (57), (59), and (61) into Eq. (47).

*Var TT* ½ � and *Var T*½ � is equal to zero, and *E T*½ �¼ *<sup>a</sup> t*

recalculate *<sup>∂</sup>E V*½ � *aTa <sup>=</sup>∂va*, and *<sup>∂</sup>E V*<sup>2</sup>

� �<sup>2</sup> *<sup>v</sup>*<sup>2</sup>*<sup>n</sup>*

8 >< >:

*Var V<sup>n</sup>*þ<sup>1</sup> *<sup>a</sup>* � � <sup>þ</sup>

> *v*2*n <sup>a</sup> <sup>y</sup>*<sup>4</sup>*n*2þ6*n*þ<sup>2</sup>

Differentiating Eq. (58) with respect to the mean link flow yields

� *<sup>v</sup>*<sup>2</sup>*<sup>n</sup>*

*ayn*<sup>2</sup>þ3*<sup>n</sup> <sup>a</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � *<sup>n</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>n</sup>* � <sup>2</sup>

� � � �

*ayn*<sup>2</sup>þ*n*�<sup>2</sup> *<sup>a</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � *<sup>n</sup>*ð Þ <sup>2</sup> � *<sup>n</sup>* � <sup>4</sup>

*Cn a* � �

Differentiating Eq. (60) with respect to the mean link flow *va* and performing

� � � �

¼ *t* 0 *a v*2 *ay*2 *<sup>a</sup>* <sup>þ</sup> *<sup>β</sup><sup>t</sup>* 0 *a cn a*

*<sup>a</sup>* � *<sup>n</sup>*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

Thus the value of PRSN-MCP in case of DS-SD can be determined by substitut-

Case D degenerates into the MCP model in a deterministic traffic network, in which neither the stochastic link capacity nor stochastic travel demand is considered in travelers' route choice decision making. In this case, the variance of both

*aTa*

� �

0 *<sup>a</sup>* þ *βt* 0 *a vn a=Cn*

� �*=∂va*, respectively.

*βt* 0 *a cn a* � �<sup>2</sup>

$$\frac{\partial E[TT]}{\partial v\_{a}} - E[T\_{a}] = \left[t\_{a}^{0} + (n+1)\frac{\beta t\_{a}^{0}}{\overline{c}\_{a}^{n}}v\_{a}^{n}\right] - \left[t\_{a}^{0} + \frac{\beta t\_{a}^{0}}{\overline{c}\_{a}^{n}}v\_{a}^{n}\right] = \frac{n\beta t\_{a}^{0}}{\overline{c}\_{a}^{n}}v\_{a}^{n} \tag{63}$$

From Eq. (26) we can obtain

$$E\left[\left(V\_{a}\right)^{2}T\_{a}\right] = t\_{a}^{0}E[V\_{a}]^{2} + \beta t\_{a}^{0}E\left[V\_{a}^{n+2}\right]E\left[\frac{\mathbf{1}}{C\_{a}^{n}}\right] = t\_{a}^{0}v\_{a}^{2} + \frac{\beta t\_{a}^{0}v\_{a}^{n+2}}{\overline{c}\_{a}^{n}}\tag{64}$$

Consequently, we have, upon simplifying

$$\frac{\partial E\left[\left(V\_{a}\right)^{2}T\_{a}\right]}{\partial v\_{a}} - E[T\_{a}] = (2v\_{a} - 1) \cdot t\_{a}^{0} - \left[\left(n + 2\right)v\_{a} - 1\right] \frac{\beta t\_{a}^{0}v\_{a}^{n}}{\overline{c}\_{a}^{n}}\tag{65}$$

By substituting Eqs. (63) and (65) into Eq. (47), the value of PRSN-MCP in case of DS-DD can be expressed as follows:

$$\text{PRSN} - \text{MCP} = (\mathbf{1} + \boldsymbol{\chi}) \left( \frac{n\boldsymbol{\beta}t\_a^0}{\overline{\varepsilon}\_a^n} \boldsymbol{v}\_a^n \right) + \text{VoR} \cdot \boldsymbol{\sigma}^2 \left\{ (2\boldsymbol{v}\_a - \mathbf{1}) \cdot \boldsymbol{t}\_a^0 - [(n+2)\boldsymbol{v}\_a - \mathbf{1}] \frac{\boldsymbol{\beta}t\_a^0 \boldsymbol{v}\_a^n}{\overline{\varepsilon}\_a^n} \right\} \tag{66}$$
