**2.1 Notations and assumptions**

Consider a strongly connected network *G* ¼ ð Þ *N*, *A* , where *N* is the set of nodes and *A* is the set of links in the network. Let *W* represent the set of OD pairs in the network and the set of routes between OD pair *w* ∈*W* be denoted by *Rw*. Random variables are expressed in capital letters and lower-case letters are used for mean values of random variables or deterministic variables.


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

derive the analytical function of SN-MCP for a risk-neutral case and RSN-MCP for a risk-based case in a stochastic network with both supply and demand uncertainty, respectively. In Section 5, the analysis for the PRSN-MCP is then described under different simplifications of network uncertainties. In Section 6, numerical examples with respect to a small-scale network and a medium-scale network (Sioux Falls network) are undertaken to demonstrate the effects of the proposed models. The final section contains some concluding remarks and recommends further research. The flow chart of the process applied in this chapter is presented in **Figure 1**.

Consider a strongly connected network *G* ¼ ð Þ *N*, *A* , where *N* is the set of nodes and *A* is the set of links in the network. Let *W* represent the set of OD pairs in the network and the set of routes between OD pair *w* ∈*W* be denoted by *Rw*. Random variables are expressed in capital letters and lower-case letters are used for mean

> *r* � �

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

**2. Framework of stochastic network model**

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

values of random variables or deterministic variables.

*<sup>q</sup>* variance of travel demand between OD pair *w* ∈*W*

**<sup>f</sup>** column vector of mean route flow, where **<sup>f</sup>** <sup>¼</sup> *<sup>f</sup> <sup>w</sup>*

**v** column vector of mean link flow, where **v** ¼ f g *va*

*TT* total travel time of the system, where *TT* <sup>¼</sup> <sup>P</sup>

*<sup>a</sup>*,*<sup>r</sup>* link-path incidence parameter; 1 if link *a* on path *r*, zero otherwise

*VoR* relative weight assigned to the travel time budget, that is, value of reliability

*VMRw* variance-to-mean ratio (*VMR*) of the random travel demand

*Q<sup>w</sup>* travel demand between OD pair *w* ∈*W q<sup>w</sup>* mean travel demand between OD pair *w* ∈*W*

*<sup>r</sup>* route flow on path *r* ∈*Rw*

*Va* traffic flow on link *a*∈ *A va* mean traffic flow on link *a* ∈ *A*

*<sup>r</sup>* travel time on path *r* ∈*Rw*

*Ta* travel time on link *a ta* mean travel time on link *a*

*<sup>r</sup>* mean travel time on path *r* ∈ *Rw ε<sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* variance of travel time on path *r* ∈*Rw*

*<sup>t</sup>* variance of travel time on link *a*∈ *A*

*<sup>r</sup>* perceived travel time on path *r* ∈*Rw*

*T*~ *<sup>a</sup>* perceived travel time on link *a*

*<sup>r</sup>* mean perceived travel time on path *r* ∈*Rw* ~*ε<sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* variance of perceived travel time on path *r* ∈*Rw*

*<sup>r</sup>* mean traffic flow on path *r* ∈ *Rw*

*<sup>f</sup>* variance of traffic flow on path *r* ∈*Rw*

*<sup>v</sup>* variance of traffic flow on link *a*∈ *A*

**2.1 Notations and assumptions**

*εw*

*F <sup>w</sup>*

*f w*

*ε<sup>w</sup>*,*<sup>r</sup>*

*ε a*

*δ w*

*T <sup>w</sup>*

*t w*

*ε a*

*T*~ *w*

~*t w*

**52**


Before proceeding with the analysis, some assumptions are made to allow for the closed-form formulation/calculation of the PRSN-MCP model.

A1. The travel demand *Q<sup>w</sup>* between each OD pair is assumed to be an independent random variable with a mean of *q<sup>w</sup>*and variance of *ε<sup>w</sup> <sup>q</sup>* , while *VMRw* is the variance-to-mean ratio (*VMR*) of the random travel demand in which *VMRw* ¼ *εw <sup>q</sup> =qw*. Stochastic demand is further assumed to follow a lognormal distribution, which is a nonnegative, asymmetrical distribution. This has been adopted in the literature as a more realistic approximation of the stochastic travel demand, as opposed to the more commonly used normal distribution [18, 23].

A2. The route flow *F <sup>w</sup> <sup>r</sup>* , and link flow *Va* are also assumed to be independent random variables that follow the same statistical distribution as OD demand. The *VMRs* of route flows are equal to those of the corresponding OD demand.

A3. The *VMRs* of travel demand are assumed to be the same for all OD pairs in order to derive the closed-form formulation of the PRSN-MCP model.

A4. The capacity degradation random variable *Ca* is independent of the traffic flow *va* on it and follows a uniform distribution with the design capacity *ca*of the link as its upper bound and the worst-degraded capacity as its lower bound (the lower bound would be a fraction *θ<sup>a</sup>* of the design capacity).

#### **2.2 VI formulation for different stochastic network models**

#### *2.2.1 Stochastic network-system optimal (SN-SO) formulation*

According to the Assumption A1 and A2, the OD travel demand, route flow *F <sup>w</sup> r* , and link flow *Va* are random variables, which consequently induce the random route/link travel times. As such, we have the following flow conservation relationships among them

$$Q^w = \sum\_{r \in \mathbb{R}\_w} F\_r^w, w \in W \tag{1}$$

$$W\_{\mathfrak{a}} = \sum\_{w \in W} \sum\_{r \in R\_w} \delta\_{a,r}^w F\_r^w, \forall a \in A \tag{2}$$

$$F\_r^w \ge 0, w \in \mathcal{W}, r \in \mathcal{R}\_w \tag{3}$$

where Eq. (1) is the travel demand conservation constraint, Eq. (2) is a definitional constraint that sums up all route flows that pass through a given link *a*, and Eq. (3) is a non-negativity constraint on the route flows. Let *<sup>Δ</sup>* <sup>¼</sup> *<sup>δ</sup><sup>w</sup> a*,*r* � � denote the route-link incidence matrix, *δ<sup>w</sup> <sup>a</sup>*,*<sup>r</sup>* <sup>¼</sup> 1 if route *<sup>r</sup>* traverses link *<sup>a</sup>*, and *<sup>δ</sup> <sup>w</sup> <sup>a</sup>*,*<sup>r</sup>* ¼ 0 otherwise. Let *f <sup>w</sup> <sup>r</sup>* , *va* denote the mean route flow and link flow, respectively. From Eqs. (1) � (3), these route and link flows satisfy the following conservation conditions:

$$q^w = \sum\_{r \in \mathbb{R}\_w} f^w\_r, w \in W \tag{4}$$

where *ε*

described by

variance as shown below:

*<sup>r</sup>* <sup>¼</sup> *E T <sup>w</sup> r*

<sup>¼</sup> <sup>X</sup>

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>δ<sup>w</sup>*

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

where <sup>∇</sup>**v***U TT*<sup>∗</sup> ½ �¼ *<sup>∂</sup><sup>E</sup>* <sup>P</sup>

*2.2.3 Perceived RSN-SO formulation*

vation conditions:

*bw*

*<sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* is the variance of route travel time, which represents the travel time

*<sup>t</sup>* be the

(13)

*a*

*<sup>t</sup>* , *a*∈ *A* (11)

*<sup>t</sup>* , *w* ∈*W*,*r*∈*Rw* (12)

X

*b w*

*<sup>t</sup>* be the variance of perceived link travel

*<sup>t</sup>* , *a*∈ *A* (16)

h i <sup>þ</sup> *VoR* � <sup>~</sup>*ε<sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* , *<sup>w</sup>* <sup>∈</sup>*W*,*r*∈*Rw* (15)

**<sup>v</sup>** � **<sup>v</sup>**<sup>∗</sup> ð Þ<sup>T</sup>∇**v***U TT*<sup>∗</sup> ½ �≥<sup>0</sup> (14)

*<sup>a</sup>* <sup>þ</sup> *VoR* � *<sup>∂</sup>Var* <sup>P</sup>

� �.

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>δ<sup>w</sup> a*,*rε<sup>a</sup> t*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>V* <sup>∗</sup> *<sup>a</sup> T*<sup>∗</sup> *a* � �*=∂v* <sup>∗</sup>

*<sup>r</sup>* is described as

*<sup>r</sup>* is the

reliability (TTR) of that route, is the route travel time, and *VoR* is the relative weight assigned to the TTR, that is, value of reliability. Similarly, let *ε <sup>a</sup>*

*ba* <sup>¼</sup> *E T*½ �þ*<sup>a</sup> VoR* � *<sup>ε</sup> <sup>a</sup>*

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>δ<sup>w</sup> <sup>a</sup>*,*rε <sup>a</sup>*

*<sup>ε</sup>w*,*<sup>r</sup> <sup>t</sup>* <sup>¼</sup> <sup>X</sup>

*<sup>a</sup>*,*rba*, *w* ∈*W*,*r*∈*Rw*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>V* <sup>∗</sup> *<sup>a</sup> T* <sup>∗</sup> *a* � �*=∂v* <sup>∗</sup>

choice process. The perceived TTB associated with route *r*, ~

~ *ba* <sup>¼</sup> *<sup>E</sup> <sup>T</sup>*<sup>~</sup> *<sup>a</sup>*

where <sup>~</sup>*ε<sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* is the variance of the perceived route travel time, and *<sup>T</sup>*~*<sup>w</sup>*

time, and *T*~ *<sup>a</sup>* be the perceived link travel time. The perceived TTB associated with

� � <sup>þ</sup> *VoR* � <sup>~</sup>*<sup>ε</sup> <sup>a</sup>*

Based on the assumption of independent travel time on each link, we can infer the following relationship between variances of perceived route travel time and

~ *b w <sup>r</sup>* <sup>¼</sup> *<sup>E</sup> <sup>T</sup>*~*<sup>w</sup> r*

perceived route travel time. Similarly, let ~*ε<sup>a</sup>*

*ba* can be described by

perceived link travel time as follows:

link *a*, ~

**55**

� � <sup>þ</sup> *VoR* � *<sup>ε</sup><sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* <sup>¼</sup> <sup>X</sup>

link-based RSN-SO model can be expressed as

Based on the assumption of independent travel time on each link, we can infer the following relationship between route travel time variance and link travel time

From Eqs. (10) � (12), the TTB of route and link satisfy the following conser-

Let *U TT* ½ �¼ *E TT* ½ �þ *VoR* � *Var TT* ½ �. With Eq. (13), the VI formulation for the

In the previous subsections, we consider that travelers can always choose the route with the minimum TTB; the resulting model is called a deterministic traffic assignment model. The main assumption underlying this kind of model is that travelers have full information about travel conditions, that is, they have perfect information about travel time and its variability. In this subsection, we relax this unreasonable assumption and include travelers' perception errors in their route

*<sup>a</sup>*,*rE T*½ �þ*<sup>a</sup> VoR* �

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>δ<sup>w</sup>*

variance of the link travel time, the TTB associated with link *a*, *ba*, which can be

$$\forall v\_d = \sum\_{w \in W} \sum\_{r \in R\_w} \delta\_{d,r}^w f\_r^w, \forall a \in A \tag{5}$$

$$f\_r^w \ge 0, w \in \mathcal{W}, r \in \mathcal{R}\_w \tag{6}$$

Let *ε w*,*r <sup>f</sup>* , *ε<sup>a</sup> <sup>v</sup>* be the variance of route flow and link flow, respectively. Then from the Assumption A1 and A2, we have

$$\sum\_{r \in R\_w} \varepsilon\_f^{w,r} = \sum\_{r \in R\_w} f\_r^w \text{VMR}\_w = q^w \text{VMR}\_w = \varepsilon\_q^w, w \in W \tag{7}$$

$$\begin{split} \varepsilon\_{v}^{t} &= \sum\_{w \in W} \sum\_{r \in R\_{w}} \left( \delta\_{a,r}^{w} \right)^{2} \varepsilon\_{f}^{w,r} = \sum\_{w \in W} \sum\_{r \in R\_{w}} \delta\_{a,r}^{w} \varepsilon\_{f}^{w,r} \\ &= \sum\_{w \in W} \sum\_{r \in R\_{w}} \delta\_{a,r}^{w} f\_{r}^{w} \text{VMR}\_{w} \end{split} \tag{8}$$

From Eqs. (7) and (8), we know that the variances of both route flow and link flow can be determined by the means of route flows. Furthermore, the route and link flow distribution can be derived through known travel demand distributions. Next, we discuss the VI formulation for the SN-SO model. In this section, we consider all the travelers to be risk-neutral. That is, travelers are not sensitive to the travel time variations and they do not need to budget the safety margin for their trips. The system optimal assignment under the stochastic network (SN-SO) aims to minimize the expected total travel time. The VI formulation for the SN-SO model can be obtained by finding **v**<sup>∗</sup> ∈ Ω**<sup>v</sup>** such that for any **v**∈ Ω**v**,

$$(\mathbf{v} - \mathbf{v}^\*)^T \nabla\_\mathbf{v} E[TT^\*] \ge \mathbf{0} \tag{9}$$

where <sup>∇</sup>**v***E TT*<sup>∗</sup> ½ �¼ *<sup>∂</sup><sup>E</sup>* <sup>P</sup> *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>V* <sup>∗</sup> *<sup>a</sup> T*<sup>∗</sup> *a* � �*=∂v* <sup>∗</sup> *a* � �, <sup>Ω</sup>**<sup>v</sup>** <sup>¼</sup> **<sup>v</sup>**j**<sup>v</sup>** <sup>¼</sup> *<sup>Δ</sup>***f**,**<sup>f</sup>** <sup>≥</sup> 0; *qw* <sup>¼</sup> <sup>P</sup> *r*∈ *Rw* n *f w <sup>r</sup>* , *w* ∈ *W*g. **v** and **f** are the column vector of mean link and route flow, respectively. *Ta* represents the travel time on link *a*. *TT* is the total travel time of the system, where *TT* <sup>¼</sup> <sup>P</sup> *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaTa*.

#### *2.2.2 Risk-based SN-SO (RSN-SO) formulation*

Up to this point, we have presented the risk-neutral case. However, several empirical studies reveal that travel time reliability plays an important role in the traveler's route choice decision process [1–3]. In this section, we consider the riskbased (averse or prone) case in which travelers are assumed to consider both the mean travel time and travel time variability in their route decision-making process. Researchers have used the Travel Time Budget (TTB) to represent travelers' risk-based travel behavior. Mathematically, the TTB associated with route *r*, *b<sup>w</sup> <sup>r</sup>* , is expressed as

$$b\_r^w = E\left[T\_r^w\right] + \text{VaR} \cdot \varepsilon\_t^{w,r}, w \in W, r \in R\_w \tag{10}$$

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

and Eq. (3) is a non-negativity constraint on the route flows. Let *<sup>Δ</sup>* <sup>¼</sup> *<sup>δ</sup><sup>w</sup>*

*<sup>q</sup><sup>w</sup>* <sup>¼</sup> <sup>X</sup>

*w* ∈*W*

*f w*

*r*∈*Rw f w*

> *δw a*,*r* � �<sup>2</sup> *ε w*,*r <sup>f</sup>* <sup>¼</sup> <sup>X</sup>

> > *<sup>r</sup> VMRw*

From Eqs. (7) and (8), we know that the variances of both route flow and link flow can be determined by the means of route flows. Furthermore, the route and link flow distribution can be derived through known travel demand distributions. Next, we discuss the VI formulation for the SN-SO model. In this section, we consider all the travelers to be risk-neutral. That is, travelers are not sensitive to the travel time variations and they do not need to budget the safety margin for their trips. The system optimal assignment under the stochastic network (SN-SO) aims to minimize the expected total travel time. The VI formulation for the SN-SO model

*a*

*<sup>r</sup>* , *w* ∈ *W*g. **v** and **f** are the column vector of mean link and route flow, respectively. *Ta* represents the travel time on link *a*. *TT* is the total travel time of the

Up to this point, we have presented the risk-neutral case. However, several empirical studies reveal that travel time reliability plays an important role in the traveler's route choice decision process [1–3]. In this section, we consider the riskbased (averse or prone) case in which travelers are assumed to consider both the mean travel time and travel time variability in their route decision-making process. Researchers have used the Travel Time Budget (TTB) to represent travelers' risk-based travel behavior. Mathematically, the TTB associated with route

� �, <sup>Ω</sup>**<sup>v</sup>** <sup>¼</sup> **<sup>v</sup>**j**<sup>v</sup>** <sup>¼</sup> *<sup>Δ</sup>***f**,**<sup>f</sup>** <sup>≥</sup> 0; *qw* <sup>¼</sup> <sup>P</sup>

*va* <sup>¼</sup> <sup>X</sup>

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

From Eqs. (1) � (3), these route and link flows satisfy the following conservation

*r*∈*Rw f w*

X *r*∈*Rw δw <sup>a</sup>*,*<sup>r</sup> <sup>f</sup> <sup>w</sup>*

*<sup>a</sup>*,*<sup>r</sup>* <sup>¼</sup> 1 if route *<sup>r</sup>* traverses link *<sup>a</sup>*, and *<sup>δ</sup> <sup>w</sup>*

*<sup>r</sup>* , *w* ∈*W* (4)

*<sup>r</sup>* ≥0, *w* ∈ *W*,*r*∈*Rw* (6)

X *r* ∈*Rw δw <sup>a</sup>*,*<sup>r</sup>ε w*,*r f*

**<sup>v</sup>** � **<sup>v</sup>**<sup>∗</sup> ð Þ<sup>T</sup>∇**v***E TT*<sup>∗</sup> ½ �≥<sup>0</sup> (9)

n

� � <sup>þ</sup> *VoR* � *<sup>ε</sup><sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* , *<sup>w</sup>* <sup>∈</sup>*W*,*r*<sup>∈</sup> *Rw* (10)

*<sup>r</sup>* , ∀*a*∈ *A* (5)

*<sup>q</sup>* , *w* ∈*W* (7)

(8)

*r*∈ *Rw*

*<sup>r</sup>* , *va* denote the mean route flow and link flow, respectively.

*<sup>v</sup>* be the variance of route flow and link flow, respectively. Then from

*<sup>r</sup> VMRw* <sup>¼</sup> *<sup>q</sup>wVMRw* <sup>¼</sup> *<sup>ε</sup><sup>w</sup>*

*w* ∈*W*

the route-link incidence matrix, *δ<sup>w</sup>*

the Assumption A1 and A2, we have

<sup>¼</sup> <sup>X</sup>

where <sup>∇</sup>**v***E TT*<sup>∗</sup> ½ �¼ *<sup>∂</sup><sup>E</sup>* <sup>P</sup>

system, where *TT* <sup>¼</sup> <sup>P</sup>

*<sup>r</sup>* , is expressed as

*f w*

*r*, *b<sup>w</sup>*

**54**

*w* ∈*W*

*w* ∈*W*

X *r*∈*Rw*

X *r*∈*Rw δw <sup>a</sup>*,*<sup>r</sup> <sup>f</sup> <sup>w</sup>*

can be obtained by finding **v**<sup>∗</sup> ∈ Ω**<sup>v</sup>** such that for any **v**∈ Ω**v**,

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

*2.2.2 Risk-based SN-SO (RSN-SO) formulation*

*bw*

*<sup>r</sup>* <sup>¼</sup> *E T <sup>w</sup> r*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>V* <sup>∗</sup> *<sup>a</sup> T*<sup>∗</sup> *a* � �*=∂v* <sup>∗</sup>

X *r*∈*Rw ε w*,*r <sup>f</sup>* <sup>¼</sup> <sup>X</sup>

*εa <sup>v</sup>* <sup>¼</sup> <sup>X</sup>

otherwise. Let *f <sup>w</sup>*

conditions:

Let *ε w*,*r <sup>f</sup>* , *ε<sup>a</sup>*

*a*,*r* � � denote

*<sup>a</sup>*,*<sup>r</sup>* ¼ 0

where *ε <sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* is the variance of route travel time, which represents the travel time reliability (TTR) of that route, is the route travel time, and *VoR* is the relative weight assigned to the TTR, that is, value of reliability. Similarly, let *ε <sup>a</sup> <sup>t</sup>* be the variance of the link travel time, the TTB associated with link *a*, *ba*, which can be described by

$$b\_{a} = E[T\_{a}] + \text{VaR} \cdot \varepsilon\_{t}^{a}, a \in \mathcal{A} \tag{11}$$

Based on the assumption of independent travel time on each link, we can infer the following relationship between route travel time variance and link travel time variance as shown below:

$$
\varepsilon\_t^{w,r} = \sum\_{a \in A} \delta\_{a,r}^w \varepsilon\_t^a, w \in W, r \in R\_w \tag{12}
$$

From Eqs. (10) � (12), the TTB of route and link satisfy the following conservation conditions:

$$\begin{split} b\_r^w &= E\left[T\_r^w\right] + VaR \cdot \varepsilon\_t^{w,r} = \sum\_{a \in A} \delta\_{a,r}^w E[T\_a] + VaR \cdot \sum\_{a \in A} \delta\_{a,r}^w \varepsilon\_t^a \\ &= \sum\_{a \in A} \delta\_{a,r}^w b\_a, w \in W, r \in R\_w \end{split} \tag{13}$$

Let *U TT* ½ �¼ *E TT* ½ �þ *VoR* � *Var TT* ½ �. With Eq. (13), the VI formulation for the link-based RSN-SO model can be expressed as

$$(\mathbf{v} - \mathbf{v}^\*)^T \nabla\_\mathbf{v} U[TT^\*] \ge \mathbf{0} \tag{14}$$

$$\text{where } \nabla\_{\mathbf{v}} U[TT^\*] = \left\{ \partial \mathbb{E} \left[ \sum\_{a \in A} V\_a^\* T\_a^\* \right] / \partial \boldsymbol{v}\_a^\* + \text{VaR} \cdot \partial \text{Var} \left[ \sum\_{a \in A} V\_a^\* T\_a^\* \right] / \partial \boldsymbol{v}\_a^\* \right\}.$$

#### *2.2.3 Perceived RSN-SO formulation*

In the previous subsections, we consider that travelers can always choose the route with the minimum TTB; the resulting model is called a deterministic traffic assignment model. The main assumption underlying this kind of model is that travelers have full information about travel conditions, that is, they have perfect information about travel time and its variability. In this subsection, we relax this unreasonable assumption and include travelers' perception errors in their route choice process. The perceived TTB associated with route *r*, ~ *b w <sup>r</sup>* is described as

$$
\tilde{\boldsymbol{b}}\_r^w = \boldsymbol{E} \left[ \tilde{\boldsymbol{T}}\_r^w \right] + \text{VoR} \cdot \tilde{\boldsymbol{\varepsilon}}\_t^{w,r}, w \in W, r \in \boldsymbol{R}\_w \tag{15}
$$

where <sup>~</sup>*ε<sup>w</sup>*,*<sup>r</sup> <sup>t</sup>* is the variance of the perceived route travel time, and *<sup>T</sup>*~*<sup>w</sup> <sup>r</sup>* is the perceived route travel time. Similarly, let ~*ε<sup>a</sup> <sup>t</sup>* be the variance of perceived link travel time, and *T*~ *<sup>a</sup>* be the perceived link travel time. The perceived TTB associated with link *a*, ~ *ba* can be described by

$$
\tilde{b}\_a = E\left[\tilde{T}\_a\right] + \text{Vol} \cdot \tilde{\varepsilon}\_t^a, a \in A \tag{16}
$$

Based on the assumption of independent travel time on each link, we can infer the following relationship between variances of perceived route travel time and perceived link travel time as follows:

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

$$\tilde{\epsilon}\_t^{w,r} = \sum\_{a \in A} \delta\_{a,r}^w \tilde{\epsilon}\_t^a, w \in W, r \in R\_w \tag{17}$$

*2.3.2 Demand fluctuation*

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

is given below

lognormal distribution

*w* ∈*W*

X *r*∈*Rw*

Let *ya* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � <sup>¼</sup> exp *<sup>n</sup>μ<sup>a</sup>*

*Var V<sup>n</sup> a* � � <sup>¼</sup> *E V*<sup>2</sup>*<sup>n</sup>*

of the link travel time as follows:

1 þ *VMR=va*

derivations according to [18], we can obtain

*<sup>v</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> *<sup>σ</sup><sup>a</sup> v* � �<sup>2</sup> *=*2 � � <sup>¼</sup> *<sup>v</sup><sup>n</sup>*

*Var T*½ �¼ *<sup>a</sup>*

*a* � � � *E V<sup>n</sup>*

*E T*½ �¼ *<sup>a</sup> t*

*β*<sup>2</sup> *t* 0 *a* � �<sup>2</sup> *C*<sup>2</sup>*<sup>n</sup> a*

0 *<sup>a</sup>* <sup>þ</sup> *<sup>β</sup><sup>t</sup>* 0 *a Cn a vn ay<sup>n</sup>*2�*<sup>n</sup> a*

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

*δw a*,*r* � �<sup>2</sup> *ε w*,*r*

*<sup>v</sup>* <sup>¼</sup> ln ð Þ� *va* <sup>1</sup>

variable exist and are given as follows:

where *μ<sup>a</sup>*

*εa <sup>v</sup>* <sup>¼</sup> <sup>X</sup>

**57**

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

Another main source of travel time variability, to be discussed in this section, is

<sup>2</sup>*<sup>π</sup>* <sup>p</sup> exp �ð Þ ln *<sup>x</sup>* � *<sup>μ</sup>*

where *x* is the random variable, *μ* and *σ* are the distribution parameters, and the

the Assumption A1 and A2, with lognormal OD demand, the link flows also follow a

*<sup>v</sup>* , *σ<sup>a</sup> v*

*v*

X *w* ∈*W*

*a*

*a* � � � � <sup>2</sup> <sup>¼</sup> *<sup>v</sup>*<sup>2</sup>*<sup>n</sup>*

Using the BPR function of link travel time, we can derive the mean and variance

<sup>p</sup> . Then, by using Eqs. (23) � (25) and performing some

and variance of link flow on link *a*∈ *A*. All of the moments of a lognormal random

*E X<sup>s</sup>* ½ �¼ exp *<sup>s</sup><sup>μ</sup>* <sup>þ</sup> *<sup>s</sup>*

where *E X<sup>s</sup>* ½ � is the *<sup>s</sup>*th moment of *<sup>X</sup>*. With Eq. (8) and A3, we have

*<sup>f</sup>* ¼ *VMR* �

2*σ*<sup>2</sup> !

� �<sup>2</sup> <sup>¼</sup> ln 1 <sup>þ</sup> *<sup>ε</sup> <sup>a</sup>*

2 *σ*2

> X *r*∈*Rw δw <sup>a</sup>*,*<sup>r</sup> <sup>f</sup> <sup>w</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *VMR=va* � � p *<sup>n</sup>*2�*<sup>n</sup>*

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

2

� �, ∀*a*∈ *A* (23)

*=*2 � � (24)

*v* ð Þ *va* <sup>2</sup> � �. *va*, *<sup>ε</sup> <sup>a</sup>*

, ∀*x*>0 (22)

� 1 � �. Based on

*<sup>v</sup>* are the mean

*<sup>r</sup>* ¼ *VMR* � *va*, *a*∈ *A*

*<sup>a</sup>* (26)

*<sup>a</sup>* (27)

<sup>¼</sup> *vn ay<sup>n</sup>*2�*<sup>n</sup>*

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

� � (28)

*ay<sup>n</sup>*2�*<sup>n</sup> a* � �<sup>2</sup> � � (29)

(25)

the stochastic travel demand. Several types of probability distributions of OD travel demand have been adopted by researchers to simulate the travel demand fluctuation, such as normal distribution [12], lognormal distribution [23], and Poisson distribution [11]. As indicated in Assumption A1, we use the lognormal distribution in this study, which is more realistic than the commonly adopted normal distribution. The probability density function of the lognormal distribution

*f x*ð Þ¼ <sup>j</sup>*μ*, *<sup>σ</sup>* <sup>1</sup>

*xσ* ffiffiffiffiffi

mean and variance of *<sup>x</sup>* are *E x*½ �¼ *<sup>e</sup>μ*þ*σ*2*=*<sup>2</sup> and Var½ �¼ *<sup>x</sup> <sup>e</sup>*<sup>2</sup>*μ*þ*σ*2*=*<sup>2</sup> *<sup>e</sup>σ*<sup>2</sup>

<sup>2</sup> ln 1 <sup>þ</sup> *<sup>ε</sup> <sup>a</sup>*

*Va* � *LN <sup>μ</sup><sup>a</sup>*

*v* ð Þ *va* <sup>2</sup> � �, *<sup>σ</sup><sup>a</sup>*

From Eqs. (15) � (17), the perceived TTB of the route and link satisfy the following conservation conditions

$$\begin{split} \ddot{\boldsymbol{b}}\_{r}^{w} &= \boldsymbol{E} \left[ \ddot{\boldsymbol{T}}\_{r}^{w} \right] + \text{VaR} \cdot \ddot{\boldsymbol{\varepsilon}}\_{t}^{w,r} = \sum\_{a \in A} \boldsymbol{\delta}\_{a,r}^{w} \boldsymbol{E} \left[ \ddot{\boldsymbol{T}}\_{a} \right] + \text{VaR} \cdot \sum\_{a \in A} \boldsymbol{\delta}\_{a,r}^{w} \ddot{\boldsymbol{\varepsilon}}\_{t}^{a} \\ &= \sum\_{a \in A} \boldsymbol{\delta}\_{a,r}^{w} \ddot{\boldsymbol{b}}\_{a}, w \in \boldsymbol{\mathcal{W}}, r \in \boldsymbol{\mathcal{R}}\_{w} \end{split} \tag{18}$$

Let *T*~*T*~ represent the total perceived travel time of the system, where *<sup>T</sup>*~*T*<sup>~</sup> <sup>¼</sup> <sup>P</sup> *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaT*<sup>~</sup> *<sup>a</sup>*, and let *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> � � <sup>¼</sup> *<sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � � <sup>þ</sup> *VoR* � *Var <sup>T</sup>*~*T*<sup>~</sup> � �. With Eq. (18), the VI formulation for the link-based perceived RSN-SO model can be expressed as

$$(\mathbf{v} - \mathbf{v}^\*)^T \nabla\_\mathbf{v} U \left[ \tilde{T} \tilde{T}^\* \right] \ge \mathbf{0} \tag{19}$$

$$\text{where } \nabla\_{\mathbf{v}} U\left[\tilde{T}\tilde{T}^\*\right] = \left\{\partial\!\!E\left[\sum\_{a\in A} V\_a^\* \tilde{T}\_a^\*\right] / \partial v\_a^\* + \text{VaR}\cdot\partial\!\!Var\left[\sum\_{a\in A} V\_a^\* \tilde{T}\_a^\*\right] / \partial v\_a^\*\right\}.$$

#### **2.3 Stochastic travel times under different sources of uncertainty**

Next, we will review the commonly adopted stochastic network models and their associated corresponding derivations of stochastic travel time in the literature in order to clarify the derivation of our proposed modeling approach.

The link travel time function is assumed to be the Bureau of Public Roads (BPR) function, *Ta* ¼ *t* 0 *<sup>a</sup>* <sup>1</sup> <sup>þ</sup> *<sup>β</sup>*ð Þ *Va=Ca <sup>n</sup>* ð Þ, <sup>∀</sup>*a*<sup>∈</sup> *<sup>A</sup>*, where *Ta*, *<sup>t</sup>* 0 *<sup>a</sup>* ,*Ca*,*Va* are the travel time, free-flow travel time, capacity, and traffic flow on link *a*. *β* and *n* are the deterministic parameters.

#### *2.3.1 Capacity degradation*

As has been discussed in Section 1, link capacities are subject to stochastic degradations to different degrees in the forms of traffic incidents, traffic management and control, work zones, and others. These constitute one of the main sources of travel time variability. To model the characteristics of stochastic link capacity degradation, [14] proposed the Probabilistic User Equilibrium (PUE) model. By assuming the capacity degradation random variable is independent of the traffic flow on it and follows a uniform distribution with the design capacity of the link as its upper bound and the worst-degraded capacity as its lower bound (the lower bound to be a fraction of the design capacity), they derived the mean and variance of *Ta* as follows:

$$E[T\_a] = t\_a^0 + \beta t\_a^0 \nu\_a^n \frac{\left(\mathbf{1} - \theta\_a^{1-n}\right)}{\overline{c}\_a^n (\mathbf{1} - \theta\_a)(\mathbf{1} - n)}\tag{20}$$

$$Var[T\_a] = \beta^2 \left(t\_a^0\right)^2 \nu\_a^{2n} \left\{ \frac{\left(\mathbf{1} - \theta\_a^{1-2n}\right)}{\tilde{c}\_a^{2n} (\mathbf{1} - \theta\_a)(\mathbf{1} - 2n)} - \left[\frac{\left(\mathbf{1} - \theta\_a^{1-n}\right)}{\tilde{c}\_a^n (\mathbf{1} - \theta\_a)(\mathbf{1} - n)}\right]^2 \right\} \tag{21}$$

They further indicated that the uniform distribution assumption can be relaxed with respect to other probability distributions via the Mellin transform technique [14].

#### *2.3.2 Demand fluctuation*

<sup>~</sup>*εw*,*<sup>r</sup> <sup>t</sup>* <sup>¼</sup> <sup>X</sup>

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

<sup>þ</sup> *VoR* � <sup>~</sup>*εw*,*<sup>r</sup> <sup>t</sup>* <sup>¼</sup> <sup>X</sup>

<sup>¼</sup> *<sup>∂</sup><sup>E</sup>* <sup>P</sup>

*ba*, *w* ∈*W*,*r*∈*Rw*

following conservation conditions

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>δ<sup>w</sup> a*,*r* ~

~ *b w <sup>r</sup>* <sup>¼</sup> *<sup>E</sup> <sup>T</sup>*<sup>~</sup> *<sup>w</sup> r* h i

*<sup>T</sup>*~*T*<sup>~</sup> <sup>¼</sup> <sup>P</sup>

<sup>¼</sup> <sup>X</sup>

where <sup>∇</sup>**v***<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> <sup>∗</sup> h i

function, *Ta* ¼ *t*

istic parameters.

of *Ta* as follows:

**56**

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

0 *a* � �<sup>2</sup> *v*2*n a*

*2.3.1 Capacity degradation*

0

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>δ<sup>w</sup> <sup>a</sup>*,*r*~*ε <sup>a</sup>*

From Eqs. (15) � (17), the perceived TTB of the route and link satisfy the

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>δ<sup>w</sup>*

Let *T*~*T*~ represent the total perceived travel time of the system, where

formulation for the link-based perceived RSN-SO model can be expressed as

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>V* <sup>∗</sup> *<sup>a</sup> <sup>T</sup>*<sup>~</sup> <sup>∗</sup> *a*

**2.3 Stochastic travel times under different sources of uncertainty**

in order to clarify the derivation of our proposed modeling approach.

*<sup>a</sup>* <sup>1</sup> <sup>þ</sup> *<sup>β</sup>*ð Þ *Va=Ca <sup>n</sup>* ð Þ, <sup>∀</sup>*a*<sup>∈</sup> *<sup>A</sup>*, where *Ta*, *<sup>t</sup>*

*E T*½ �¼ *<sup>a</sup> t*

*c*2*n*

< : 0 *<sup>a</sup>* þ *βt* 0 *a vn a*

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

h i

**<sup>v</sup>** � **<sup>v</sup>**<sup>∗</sup> ð Þ<sup>T</sup>∇**v***<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> <sup>∗</sup> h i

Next, we will review the commonly adopted stochastic network models and their associated corresponding derivations of stochastic travel time in the literature

The link travel time function is assumed to be the Bureau of Public Roads (BPR)

free-flow travel time, capacity, and traffic flow on link *a*. *β* and *n* are the determin-

As has been discussed in Section 1, link capacities are subject to stochastic degradations to different degrees in the forms of traffic incidents, traffic management and control, work zones, and others. These constitute one of the main sources of travel time variability. To model the characteristics of stochastic link capacity degradation, [14] proposed the Probabilistic User Equilibrium (PUE) model. By assuming the capacity degradation random variable is independent of the traffic flow on it and follows a uniform distribution with the design capacity of the link as its upper bound and the worst-degraded capacity as its lower bound (the lower bound to be a fraction of the design capacity), they derived the mean and variance

*<sup>a</sup>*,*rE <sup>T</sup>*<sup>~</sup> *<sup>a</sup>*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaT*<sup>~</sup> *<sup>a</sup>*, and let *<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> � � <sup>¼</sup> *<sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � � <sup>þ</sup> *VoR* � *Var <sup>T</sup>*~*T*<sup>~</sup> � �. With Eq. (18), the VI

*=∂v* <sup>∗</sup>

*<sup>a</sup>* <sup>þ</sup> *VoR* � *<sup>∂</sup>Var* <sup>P</sup>

n o

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*

*cn*

� �

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

*cn*

They further indicated that the uniform distribution assumption can be relaxed with respect to other probability distributions via the Mellin transform technique [14].

*<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

� � <sup>þ</sup> *VoR* �

*<sup>t</sup>* , *w* ∈*W*,*r*∈*Rw* (17)

X

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>δ<sup>w</sup> a*,*r*~*ε<sup>a</sup> t*

≥ 0 (19)

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>V* <sup>∗</sup> *<sup>a</sup> <sup>T</sup>*<sup>~</sup> <sup>∗</sup> *a*

*<sup>a</sup>* ,*Ca*,*Va* are the travel time,

h i

(18)

*=∂v* <sup>∗</sup> *a*

(20)

(21)

9 = ; .

Another main source of travel time variability, to be discussed in this section, is the stochastic travel demand. Several types of probability distributions of OD travel demand have been adopted by researchers to simulate the travel demand fluctuation, such as normal distribution [12], lognormal distribution [23], and Poisson distribution [11]. As indicated in Assumption A1, we use the lognormal distribution in this study, which is more realistic than the commonly adopted normal distribution. The probability density function of the lognormal distribution is given below

$$f(\mathbf{x}|\mu, \sigma) = \frac{1}{\varkappa \sigma \sqrt{2\pi}} \exp\left(\frac{-\left(\ln \pi - \mu\right)^2}{2\sigma^2}\right), \forall \mathbf{x} > \mathbf{0} \tag{22}$$

where *x* is the random variable, *μ* and *σ* are the distribution parameters, and the mean and variance of *<sup>x</sup>* are *E x*½ �¼ *<sup>e</sup>μ*þ*σ*2*=*<sup>2</sup> and Var½ �¼ *<sup>x</sup> <sup>e</sup>*<sup>2</sup>*μ*þ*σ*2*=*<sup>2</sup> *<sup>e</sup>σ*<sup>2</sup> � 1 � �. Based on the Assumption A1 and A2, with lognormal OD demand, the link flows also follow a lognormal distribution

$$V\_a \sim LN\left(\mu\_v^a, \sigma\_v^a\right), \forall a \in A \tag{23}$$

where *μ<sup>a</sup> <sup>v</sup>* <sup>¼</sup> ln ð Þ� *va* <sup>1</sup> <sup>2</sup> ln 1 <sup>þ</sup> *<sup>ε</sup> <sup>a</sup> v* ð Þ *va* <sup>2</sup> � �, *<sup>σ</sup><sup>a</sup> v* � �<sup>2</sup> <sup>¼</sup> ln 1 <sup>þ</sup> *<sup>ε</sup> <sup>a</sup> v* ð Þ *va* <sup>2</sup> � �. *va*, *<sup>ε</sup> <sup>a</sup> <sup>v</sup>* are the mean and variance of link flow on link *a*∈ *A*. All of the moments of a lognormal random variable exist and are given as follows:

$$E[X^{\circ}] = \exp\left(s\mu + s^2 \sigma^2 / 2\right) \tag{24}$$

where *E X<sup>s</sup>* ½ � is the *<sup>s</sup>*th moment of *<sup>X</sup>*. With Eq. (8) and A3, we have

$$\epsilon\_v^a = \sum\_{w \in \mathcal{W}} \sum\_{r \in R\_w} \left(\delta\_{a,r}^w\right)^2 \epsilon\_f^{w,r} = \text{VMR} \cdot \sum\_{w \in \mathcal{W}} \sum\_{r \in R\_w} \delta\_{a,r}^w f\_r^w = \text{VMR} \cdot v\_a, a \in A \tag{25}$$

Let *ya* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *VMR=va* <sup>p</sup> . Then, by using Eqs. (23) � (25) and performing some derivations according to [18], we can obtain

$$E\left[V\_{a}^{n}\right] = \exp\left(n\mu\_{v}^{a} + n^{2}\left(\sigma\_{v}^{a}\right)^{2}/2\right) = v\_{a}^{n}\left(\sqrt{1 + \text{VMR}/v\_{a}}\right)^{n^{2}-n} = v\_{a}^{n}y\_{a}^{n^{2}-n}\tag{26}$$

$$\operatorname{Var}\left[V\_{a}^{n}\right] = \operatorname{E}\left[V\_{a}^{2n}\right] - \left(\operatorname{E}\left[V\_{a}^{n}\right]\right)^{2} = \nu\_{a}^{2n}\mathcal{Y}\_{a}^{4n^{2}-2n} - \nu\_{a}^{2n}\mathcal{Y}\_{a}^{2n^{2}-2n} \tag{27}$$

Using the BPR function of link travel time, we can derive the mean and variance of the link travel time as follows:

$$E[T\_a] = t\_a^0 + \frac{\beta t\_a^0}{C\_a^n} \left( v\_a^n y\_a^{n^2 - n} \right) \tag{28}$$

$$\text{Var}[T\_{\boldsymbol{a}}] = \frac{\rho^2 \left(t\_{\boldsymbol{a}}^0\right)^2}{\mathbf{C}\_{\boldsymbol{a}}^{2n}} \left[\boldsymbol{v}\_{\boldsymbol{a}}^{2n} \boldsymbol{\upchi}\_{\boldsymbol{a}}^{4n^2 - 2n} - \left(\boldsymbol{v}\_{\boldsymbol{a}}^n \boldsymbol{\upchi}\_{\boldsymbol{a}}^{n^2 - n}\right)^2\right] \tag{29}$$

#### *2.3.3 Both link capacity and demand variation*

From the above analysis and under the Assumption A4, we can easily derive the mean and variance of the link travel time in the case of both link capacity and demand variation as follows:

$$E[T\_a] = t\_a^0 + \beta t\_a^0 \frac{\left(\mathbf{1} - \theta\_a^{1-n}\right)}{\overline{c}\_a^n (\mathbf{1} - \theta\_a)(\mathbf{1} - n)} \left(v\_a^n y\_a^{n^2 - n}\right) \tag{30}$$

*<sup>∂</sup>E TT* ½ � *∂va*

¼ *t* 0 *<sup>a</sup>* þ *βt* 0 *a*

SN � MCP <sup>¼</sup> *<sup>∂</sup>E TT* ½ �

¼ *βt* 0 *a* *∂va*

*cn*

SN-MCP proposed in this section.

**4.1 Analysis of risk-based SN-MCP**

**4.2 Calculation of RSN-MCP**

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

*t* 0 *a*

þ2*β t* 0 *a*

� *Var V*½ �þ*<sup>a</sup> βt*

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

*c n*

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

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

> 0 *a*

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

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

*c*2*n*

< :

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

> 8 >>>>><

> >>>>>:

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup> *a*∈ *A*

**59**

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

� �

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

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:

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

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

In the previous section, we know that the Stochastic Network-User Equilibrium (SN-UE) flow pattern can be driven toward a SN-SO flow pattern by charging a toll equal to the SN-MCP. Meanwhile, the expected total travel time can be minimized. In this section, we consider the risk-based (averse or prone) case. The objective function of the RSN-MCP model is to minimize the weighted sum of the mean and the variance of the total travel time, not simply to minimize the expected total

RSN � MCP <sup>¼</sup> f g *<sup>∂</sup>E TT* ½ �*=∂va* � *E T*½ � *<sup>a</sup>* <sup>þ</sup> *VoR* � f g *<sup>∂</sup>Var TT* ½ �*=∂va* � *Var T*½ � *<sup>a</sup>* (36)

In this section, we discuss the most complete and realistic situation in which travelers consider both stochastic fluctuations in supply (or link capacity) and demand in their route choice decision-making process. From Eqs. (32) and (36), we can see that the difference between SN-MCP and RSN-MCP is the term in the second parentheses of Eq. (36). This second term reflects the congestion toll on travel time reliability due to travelers' risk-based behavior. Let us now turn our attention to *<sup>∂</sup>Var TT* ½ �*=∂va*. The variance of the total travel time is described by

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

� � ( )

� �

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

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

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

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

� � *E Cn a*

" #<sup>2</sup> 8

� �*E V*½ � *<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> <sup>v</sup><sup>n</sup>*þ<sup>1</sup> *<sup>a</sup> <sup>y</sup><sup>n</sup>*2þ*<sup>n</sup>*

*a*

9 = ;

(37)

9 >>>>>=

>>>>>;

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

*c n*

*a* � �

þ 1

*a* þ

*<sup>n</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>n</sup>*

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

� � � � � <sup>1</sup> � �

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

*a*

(35)

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *vay<sup>n</sup>*2þ*<sup>n</sup> a* h i (34)

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

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

**4. Risk-based MCP (RSN-MCP) in a stochastic network**

travel time. Therefore, the RSN-MCP toll can be determined as

*cn*

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

� *E T*½ � *<sup>a</sup>*

<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*

$$Var[T\_a] = \beta^2 \left(t\_a^0\right)^2 \left\{ \frac{\left(\mathbf{1} - \theta\_a^{1-2n}\right)}{\tilde{c}\_a^{2n} (\mathbf{1} - \theta\_a)(\mathbf{1} - 2n)} \left(v\_a^{2n} y\_a^{4n^2 - 2n}\right) - \left[\frac{\left(\mathbf{1} - \theta\_a^{1-n}\right)}{\tilde{c}\_a^n (\mathbf{1} - \theta\_a)(\mathbf{1} - n)} \left(v\_a^n y\_a^{n^2 - n}\right)\right]^2 \right\} \tag{31}$$
