**1. Introduction**

It is well known, due to stochastic variations in both supply and demand, that travel time almost always involves a measure of uncertainty. Recently, several empirical studies on the value of time and reliability revealed that travel time reliability plays an important role in the traveler's route choice decision-making process [1–3]. With these studies as a basis, the study of travel time variability (reliability) has gradually emerged as an important topic. In this context, travel time reliability pertains to the probability that a trip can be successfully completed within a specified time interval, reflecting the uncertainty in trip journey times [4, 5]. To model the characteristics of travel time reliability, the concept of TTB is commonly used. TTB is defined as the average travel time plus extra time (for a measure of the buffer time) such that the probability of completing the trip within the TTB is no less than a predefined reliability threshold α [6]. Earlier research applied the concept of effective travel time to capture the travel time reliability [7]. Recently, [6] further proposed a stochastic mean-excess traffic equilibrium model to represent both the reliability and unreliability aspects of travel time variability and travelers' route choice perception errors.

Generally speaking, uncertainties from both the demand and supply sides of a system directly lead to recurring variability and unreliability of travel times and

have an obvious impact on the traveler's route choice behavior. Supply-side sources refer to the capacity variations that can occur, due to several exogenous sources of uncertainty on the road sections or at-grade intersections concerned. These exogenous sources of uncertainty may take different forms, such as environmental conditions, traffic incidents, traffic management and control, work zones, and so on. Such stochastic link capacity degradations usually lead to non-recurrent congestion [8–10]. Demand-side sources are regarded as the travel demand fluctuations, which result from various endogenous sources. These endogenous sources can include temporal factors, special events, population characteristics, and traffic information among others. Travel demand variations usually lead to recurrent congestion [4, 11, 12].

Although considerable research exists on congestion pricing and travel time reliability, relatively little research combines the two, especially regarding travelers' risk attitudes and/or the valuation of reliable travel [19]. Some examples of research that do combine congestion pricing and travel time reliability are included here. Li et al. [20] proposed a reliability-based optimal toll design bi-level model. On the upper level, network performance is optimized from a road authority point of view including travel time reliability, while a dynamic user-equilibrium is achieved from the viewpoint of travelers on the lower level. Boyles et al. [19] proposed a first-best congestion pricing model considering network capacity uncertainty and user valuation of travel time reliability, while [18] investigated marginal cost pricing in a stochastic traffic network in which demand uncertainty is explicitly considered. By assuming that all travelers have complete information about the road traffic condition, [18] derived an analytical function of Stochastic Network-Marginal Cost Pricing (SN-MCP) for a risk-neutral case and risk-based SN-MCP (RSN-MCP) for a risk-based case under the assumptions of lognormal demand and constant VMR across all OD pairs. Gardner et al. [21] consider the uncertainty in long-term travel demand and in day-to-day network capacity, and discuss the benefit of responsive

In the above-mentioned studies, MCP is analysis in a stochastic network, which considers either link supply uncertainty (e.g., see [19]) or stochastic travel demand

researchers usually assume the commonly adopted Gumbel variate as the random error term and use the conventional logit-based Stochastic User Equilibrium (SUE) model. However, this approach may not reflect the travelers' perception of the random travel time exactly. Due to the variation of travel time, it is more rational to assume that the travelers' perception error is also dependent on the random perceived travel time [22]. Therefore, in order to explicitly consider both supply and demand aspects of a stochastic network and to reflect the travelers' perception error of the random travel time, this investigation extends [18] by (1) considering both the stochastic travel demand and link capacity degradation, and (2) incorporating

The remainder of the chapter is organized as follows. The next section introduces the assumptions used in the analysis and presents the variational inequality (VI) formulation for different stochastic models. It also discusses the stochastic travel times under different sources of uncertainty. Then, Section 3 and Section 4

(e.g., see [18]). In addition, to account for the travelers' perception error,

travelers' perception error into the first-best MCP analysis.

pricing and travel information.

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

**Figure 1.**

**51**

*Flow chart for the research process.*

Several stochastic traffic network (SN) modeling approaches have been proposed to represent such uncertainties. On the capacity side, [13] proposed a probabilistic approach using the concept of capacity reliability to model the uncertain characteristics of link capacities. Lo et al. [14] proposed the Probabilistic User Equilibrium (PUE) model, which takes the fact that the link capacities are subject to stochastic degradations into account. In subsequent research using the concept of Travel Time Budget (TTB), [10] further extended the PUE model to capture the route choice behaviors of travelers with heterogeneous risk aversions. On the demand side, [11] proposed a framework of the stochastic network model to represent the stochastic demand. Ref. [12] extended the TTB model and proposed a travel time reliability-based traffic assignment model to consider the effect of daily demand fluctuations. On both the demand and supply sides, [15] proposed a traffic assignment model, which considers the uncertainties of a traffic network due to adverse weather conditions. Sumalee et al. [16] proposed a stochastic network model with log-normal distributed origin–destination (OD) travel demands and link capacities. It should be noted that all of the above studies focused on the question of how to represent the travel time reliability in a traffic assignment model, but did not answer the question of how to improve the travel time reliability in a stochastic traffic network.

All the aforementioned studies discovered that travelers do indeed consider travel time variability as a risk in their route choice decisions. Nevertheless, the first-best marginal cost pricing (MCP) is commonly modeled via a deterministic approach, which assumes that both traffic supply and travel demand are known, and that the route travel times are deterministic [17]. Furthermore, travelers are assumed to know exactly the time on each available route and can always choose the least-cost routes for their trips. As indicated earlier, due to various sources of uncertainty coming from both supply-side and demand-side of road network, it is unreasonable to assume that travel times are deterministic and known perfectly by all the travelers. Though several traffic equilibrium models have been developed for environments characterized by uncertainty in the past decades, such models have not been adopted in the analysis of first-best MCP. Intuitively, the variability and unreliability of travel times caused by network uncertainties directly influence the traveler's route choice behavior, thereby negatively affecting the performance of MCP. However, there is little theoretical basis for this intuition. At least, it is not yet clear to what extent the stochastic demand and supply and the travelers' perception error affect the performance of MCP. In this context, the study of first-best MCP under an uncertain environment is a necessary and urgent theoretical task. In addition, this investigation is also practically relevant. As indicated by [18], the recent change in the Electronic Road Pricing (ERP) toll adjustment scheme in Singapore involves the consideration of the 85th-percetile traffic condition (speed) to reflect the variability of traffic conditions. This involves determining optimal tolls in a stochastic environment, where both demand and capacity are subject to uncertainty.

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

have an obvious impact on the traveler's route choice behavior. Supply-side sources refer to the capacity variations that can occur, due to several exogenous sources of uncertainty on the road sections or at-grade intersections concerned. These exogenous sources of uncertainty may take different forms, such as environmental conditions, traffic incidents, traffic management and control, work zones, and so on. Such stochastic link capacity degradations usually lead to non-recurrent congestion [8–10]. Demand-side sources are regarded as the travel demand fluctuations, which result from various endogenous sources. These endogenous sources can include temporal factors, special events, population characteristics, and traffic information among others. Travel demand variations usually lead to recurrent congestion [4, 11, 12]. Several stochastic traffic network (SN) modeling approaches have been proposed to represent such uncertainties. On the capacity side, [13] proposed a probabilistic approach using the concept of capacity reliability to model the uncertain characteristics of link capacities. Lo et al. [14] proposed the Probabilistic User Equilibrium (PUE) model, which takes the fact that the link capacities are subject to stochastic degradations into account. In subsequent research using the concept of Travel Time Budget (TTB), [10] further extended the PUE model to capture the route choice behaviors of travelers with heterogeneous risk aversions. On the demand side, [11] proposed a framework of the stochastic network model to represent the stochastic demand. Ref. [12] extended the TTB model and proposed a travel time reliability-based traffic assignment model to consider the effect of daily demand fluctuations. On both the demand and supply sides, [15] proposed a traffic assignment model, which considers the uncertainties of a traffic network due to adverse weather conditions. Sumalee et al. [16] proposed a stochastic network model with log-normal distributed origin–destination (OD) travel demands and link capacities. It should be noted that all of the above studies focused on the question of how to represent the travel time reliability in a traffic assignment model, but did not answer the question of how to improve the travel time reliability

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

All the aforementioned studies discovered that travelers do indeed consider travel time variability as a risk in their route choice decisions. Nevertheless, the first-best marginal cost pricing (MCP) is commonly modeled via a deterministic approach, which assumes that both traffic supply and travel demand are known, and that the route travel times are deterministic [17]. Furthermore, travelers are assumed to know exactly the time on each available route and can always choose the least-cost routes for their trips. As indicated earlier, due to various sources of uncertainty coming from both supply-side and demand-side of road network, it is unreasonable to assume that travel times are deterministic and known perfectly by all the travelers. Though several traffic equilibrium models have been developed for environments characterized by uncertainty in the past decades, such models have not been adopted in the analysis of first-best MCP. Intuitively, the variability and unreliability of travel times caused by network uncertainties directly influence the traveler's route choice behavior, thereby negatively affecting the performance of MCP. However, there is little theoretical basis for this intuition. At least, it is not yet clear to what extent the stochastic demand and supply and the travelers' perception error affect the performance of MCP. In this context, the study of first-best MCP under an uncertain environment is a necessary and urgent theoretical task. In addition, this investigation is also practically relevant. As indicated by [18], the recent change in the Electronic Road Pricing (ERP) toll adjustment scheme in Singapore involves the consideration of the 85th-percetile traffic condition (speed) to reflect the variability of traffic conditions. This involves determining optimal tolls in a stochastic environment, where both demand and capacity are subject to

in a stochastic traffic network.

uncertainty.

**50**

Although considerable research exists on congestion pricing and travel time reliability, relatively little research combines the two, especially regarding travelers' risk attitudes and/or the valuation of reliable travel [19]. Some examples of research that do combine congestion pricing and travel time reliability are included here. Li et al. [20] proposed a reliability-based optimal toll design bi-level model. On the upper level, network performance is optimized from a road authority point of view including travel time reliability, while a dynamic user-equilibrium is achieved from the viewpoint of travelers on the lower level. Boyles et al. [19] proposed a first-best congestion pricing model considering network capacity uncertainty and user valuation of travel time reliability, while [18] investigated marginal cost pricing in a stochastic traffic network in which demand uncertainty is explicitly considered. By assuming that all travelers have complete information about the road traffic condition, [18] derived an analytical function of Stochastic Network-Marginal Cost Pricing (SN-MCP) for a risk-neutral case and risk-based SN-MCP (RSN-MCP) for a risk-based case under the assumptions of lognormal demand and constant VMR across all OD pairs. Gardner et al. [21] consider the uncertainty in long-term travel demand and in day-to-day network capacity, and discuss the benefit of responsive pricing and travel information.

In the above-mentioned studies, MCP is analysis in a stochastic network, which considers either link supply uncertainty (e.g., see [19]) or stochastic travel demand (e.g., see [18]). In addition, to account for the travelers' perception error, researchers usually assume the commonly adopted Gumbel variate as the random error term and use the conventional logit-based Stochastic User Equilibrium (SUE) model. However, this approach may not reflect the travelers' perception of the random travel time exactly. Due to the variation of travel time, it is more rational to assume that the travelers' perception error is also dependent on the random perceived travel time [22]. Therefore, in order to explicitly consider both supply and demand aspects of a stochastic network and to reflect the travelers' perception error of the random travel time, this investigation extends [18] by (1) considering both the stochastic travel demand and link capacity degradation, and (2) incorporating travelers' perception error into the first-best MCP analysis.

The remainder of the chapter is organized as follows. The next section introduces the assumptions used in the analysis and presents the variational inequality (VI) formulation for different stochastic models. It also discusses the stochastic travel times under different sources of uncertainty. Then, Section 3 and Section 4

**Figure 1.** *Flow chart for the research process.*

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**.

~*ta* mean perceived travel time on link *a*

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

*<sup>a</sup>* free-flow travel time on link *a*∈ *A*

*ya* parameter, where *ya* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*εa Ta* j travelers' perception error on link *a*∈ *A*

*Ca* capacity of link *a*∈ *A*

A2. The route flow *F <sup>w</sup>*

ships among them

**53**

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

*<sup>T</sup>*~*T*<sup>~</sup> total perceived travel time of the system, where *<sup>T</sup>*~*T*<sup>~</sup> <sup>¼</sup> <sup>P</sup>

design capacity (upper bound) of link *a* ∈ *A* degree of worst-degraded capacity for link *a* ∈ *A*

p

closed-form formulation/calculation of the PRSN-MCP model.

opposed to the more commonly used normal distribution [18, 23].

dent random variable with a mean of *q<sup>w</sup>*and variance of *ε<sup>w</sup>*

1 þ *VMR=va*

*<sup>N</sup> <sup>χ</sup>*, *<sup>ϖ</sup>*<sup>2</sup> ð Þ perception error distribution of traveler, in this chapter *<sup>N</sup> <sup>χ</sup>*, *<sup>ϖ</sup>*<sup>2</sup> ð Þ follows a normal distribution with predefined and deterministic mean *χ* and variance *ϖ*<sup>2</sup>

Before proceeding with the analysis, some assumptions are made to allow for the

A1. The travel demand *Q<sup>w</sup>* between each OD pair is assumed to be an indepen-

*<sup>r</sup>* , and link flow *Va* are also assumed to be independent

variance-to-mean ratio (*VMR*) of the random travel demand in which *VMRw* ¼

*<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

random variables that follow the same statistical distribution as OD demand. The

A3. The *VMRs* of travel demand are assumed to be the same for all OD pairs in

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

According to the Assumption A1 and A2, the OD travel demand, route flow *F <sup>w</sup>*

and link flow *Va* are random variables, which consequently induce the random route/link travel times. As such, we have the following flow conservation relation-

> *r*∈*Rw F <sup>w</sup>*

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

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*,

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

*w* ∈*W*

*F <sup>w</sup>*

*VMRs* of route flows are equal to those of the corresponding OD demand.

order to derive the closed-form formulation of the PRSN-MCP model.

lower bound would be a fraction *θ<sup>a</sup>* of the design capacity).

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

*Va* <sup>¼</sup> <sup>X</sup>

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

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaT*<sup>~</sup> *<sup>a</sup>*

*<sup>q</sup>* , while *VMRw* is the

*<sup>r</sup>* , *w* ∈*W* (1)

*<sup>r</sup>* ≥0, *w* ∈*W*,*r*∈*Rw* (3)

*<sup>r</sup>* , ∀*a*∈ *A* (2)

*r* ,

~*ε a*

*t* 0

*εw*
