**3. Marginal cost pricing in a stochastic network (SN-MCP) with both supply and demand uncertainty**

#### **3.1 Analysis of SN-MCP**

In this section, we discuss the SN-MCP in the risk-neutral case. The MCP in the stochastic network aims to minimize the expected total travel time. Sumalee and Xu [18] investigated the relationship between the Stochastic Network-User Equilibrium (SN-UE) and Stochastic Network-System Optimal (SN-SO) models and established the first-marginal cost toll scheme for the SN model. They classified the marginal cost toll in the stochastic network into three forms. The first one is referred to as original marginal cost pricing, which takes the form of *E V*½ �� *<sup>a</sup> dt E V* ð Þ ½ � *<sup>a</sup> =dE V*½ � *<sup>a</sup>* ; the second one is referred to as average marginal cost pricing, which takes the form of *E V*½ �� *<sup>a</sup> dE T*½ � *<sup>a</sup>*ð Þ *Va =dE V*½ � *<sup>a</sup>* ; and the third one is referred to as Stochastic Network-Marginal Cost Pricing (SN-MCP), which takes the form of *∂E* P *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaTa* � �*=∂va* � *E T*½ � *<sup>a</sup>* . They further indicate that only the SN-MCP can make the traffic network achieve the optimal pattern.

Let, then, the real gap between the marginal social and marginal private costs in a stochastic network be represented by

$$\text{CSN} - \text{MCP} = \partial E \left[ \sum\_{a \in A} V\_a T\_a \right] / \partial v\_a - E[T\_a] = \partial E[TT] / \partial v\_a - E[T\_a] \tag{32}$$

#### **3.2 Calculation of SN-MCP**

In this study, we attempt to compute the value of SN-MCP in the case of both link capacity and demand variation. To achieve this goal, we need to calculate *∂E* P *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaTa* � �*=∂va* and *E T*½ � *<sup>a</sup>* , respectively. In considering the stochasticity of both link capacity and demand, *E T*½ � *<sup>a</sup>* should be determined by Eq. (30). The expected total travel time is expressed as

$$\begin{split} E[TT] &= E\left[\sum\_{a \in A} V\_a T\_a \right] = \sum\_{a \in A} \left\{ t\_a^0 E[V\_a] + \beta t\_a^0 E\left[V\_a^{n+1}\right] E\left[\frac{1}{T\_a^0}\right] \right\} \\ &= \sum\_{a \in A} \left\{ t\_a^0 v\_a + \beta t\_a^0 \frac{\left(1 - \theta\_a^{1-n}\right)}{\overline{c}\_a^n (1 - \theta\_a)(1 - n)} \left(v\_a^{n+1} v\_a^{n^2 + n}\right) \right\} \end{split} \tag{33}$$

Differentiating Eq. (33) with respect to the mean link flow *va* yields

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

*2.3.3 Both link capacity and demand variation*

*E T*½ �¼ *<sup>a</sup> t*

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

*c*2*n*

**supply and demand uncertainty**

the traffic network achieve the optimal pattern.

*<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*

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

<sup>¼</sup> <sup>X</sup>

*cn*

Differentiating Eq. (33) with respect to the mean link flow *va* yields

a stochastic network be represented by

SN � MCP <sup>¼</sup> *<sup>∂</sup><sup>E</sup>* <sup>X</sup>

**3.2 Calculation of SN-MCP**

total travel time is expressed as

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

<sup>¼</sup> <sup>X</sup>

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

< : 0 *<sup>a</sup>* þ *βt* 0 *a*

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

� �

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

demand variation as follows:

0 *a*

**3.1 Analysis of SN-MCP**

*Var T*½ �¼ *<sup>a</sup> <sup>β</sup>*<sup>2</sup> *<sup>t</sup>*

*∂E* P

*∂E* P

**58**

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

From the above analysis and under the Assumption A4, we can easily derive the

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

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

� � " #<sup>2</sup> <sup>8</sup>

*vn ay<sup>n</sup>*2�*<sup>n</sup> a* � �

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

*<sup>=</sup>∂va* � *E T*½ �¼ *<sup>a</sup> <sup>∂</sup>E TT* ½ �*=∂va* � *E T*½ � *<sup>a</sup>* (32)

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

*cn*

(30)

(31)

9 = ;

*vn ayn*<sup>2</sup>�*<sup>n</sup> a*

mean and variance of the link travel time in the case of both link capacity and

*cn*

*v*2*n <sup>a</sup> <sup>y</sup>*<sup>4</sup>*n*2�2*<sup>n</sup> a* � �

**3. Marginal cost pricing in a stochastic network (SN-MCP) with both**

In this section, we discuss the SN-MCP in the risk-neutral case. The MCP in the stochastic network aims to minimize the expected total travel time. Sumalee and Xu [18] investigated the relationship between the Stochastic Network-User Equilibrium (SN-UE) and Stochastic Network-System Optimal (SN-SO) models and established the first-marginal cost toll scheme for the SN model. They classified the marginal cost toll in the stochastic network into three forms. The first one is referred to as original marginal cost pricing, which takes the form of *E V*½ �� *<sup>a</sup> dt E V* ð Þ ½ � *<sup>a</sup> =dE V*½ � *<sup>a</sup>* ; the second one is referred to as average marginal cost pricing, which takes the form of *E V*½ �� *<sup>a</sup> dE T*½ � *<sup>a</sup>*ð Þ *Va =dE V*½ � *<sup>a</sup>* ; and the third one is referred to as Stochastic Network-Marginal Cost Pricing (SN-MCP), which takes the form of

� �*=∂va* � *E T*½ � *<sup>a</sup>* . They further indicate that only the SN-MCP can make

Let, then, the real gap between the marginal social and marginal private costs in

In this study, we attempt to compute the value of SN-MCP in the case of both link capacity and demand variation. To achieve this goal, we need to calculate

� �*=∂va* and *E T*½ � *<sup>a</sup>* , respectively. In considering the stochasticity of both link capacity and demand, *E T*½ � *<sup>a</sup>* should be determined by Eq. (30). The expected

*aE V*½ �þ*<sup>a</sup> βt*

� � ( ) (33)

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

� � � �

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

*Cn a*

*<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*

$$\frac{\partial E[TT]}{\partial v\_{a}} = t\_{a}^{0} + \beta t\_{a}^{0} \frac{\left(\mathbf{1} - \theta\_{a}^{1-n}\right)}{\tilde{c}\_{a}^{n} (\mathbf{1} - \theta\_{a}) (\mathbf{1} - n)} \left[\frac{n v\_{a}^{n-1} (\mathbf{1} - \mathbf{y}\_{a}^{2})}{2 \mathbf{y}\_{a}^{2}} + \mathbf{1}\right] \left[(n+1) v\_{a} y\_{a}^{n^{2} + n}\right] \tag{34}$$

By substituting Eqs. (30) and (34) into Eq. (32), the value of SN-MCP in case of Stochastic Supply and Stochastic demand (SS-SD) can be determined as follows:

$$\begin{split} \text{SN}-\text{MCP} &= \frac{\partial E[TT]}{\partial v\_{a}} - E[T\_{a}] \\ &= \beta t\_{a}^{0} \frac{(1-\theta\_{a}^{1-n})}{\overline{c}\_{a}^{n}(1-\theta\_{a})(1-n)} v\_{a}^{n} \boldsymbol{\chi}\_{a}^{n^{2}-n} \left\{ \left[ (n+1)\boldsymbol{\upchi}\_{a}^{2n} + \frac{(n^{2}+n)}{2} \boldsymbol{\upchi}\_{a}^{2n-2} (1-\boldsymbol{\upupchi}\_{a}^{2}) \right] - 1 \right\} \end{split} \tag{35}$$

Note that if we neglect the degradation of link capacity, Eq. (35) degenerates into the classical SN-MCP model proposed by [18], which considers only the stochastic travel demand. Furthermore, they pointed out that the SN-MCP toll is guaranteed to be positive when *ya* ≤ 1*:*4. This conclusion is also applicable in the SN-MCP proposed in this section.
