Link parameters.
Abstract
This chapter is concerned with firstbest marginal cost pricing (MCP) in a stochastic network with both supply and travel demand uncertainty and perception errors within the travelers’ route choice decision processes. To account for the travelers’ perception error, moment analysis is adopted in this chapter to derive the mean and variance of total perceived travel time of the network. We then developed a Perceived RiskBased Stochastic Network Marginal Cost Pricing (PRSNMCP) model. Furthermore, in order to illustrate the effect of incorporating both stochastic supply and demand into the PRSNMCP model, the calculation of the PRSNMCP model is divided up into four scenarios under different simplifications of network uncertainties. Numerical examples are also provided to demonstrate the importance and properties of the proposed model. The main finding is that ignoring the effect of stochastic travel demand, capacity degradation, and travelers’ perception error may significantly reduce the performance of the firstbest MCP tolls, especially under high traveler’s confidence and network congestion levels.
Keywords
 marginal cost pricing
 moment analysis
 demand uncertainty
 supply uncertainty
 perception error
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 decisionmaking process [1, 2, 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 meanexcess 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. Supplyside sources refer to the capacity variations that can occur, due to several exogenous sources of uncertainty on the road sections or atgrade 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 nonrecurrent congestion [8, 9, 10]. Demandside 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 reliabilitybased 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 lognormal 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 firstbest 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 leastcost routes for their trips. As indicated earlier, due to various sources of uncertainty coming from both supplyside and demandside 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 firstbest 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 firstbest 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 85thpercetile 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.
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 reliabilitybased optimal toll design bilevel model. On the upper level, network performance is optimized from a road authority point of view including travel time reliability, while a dynamic userequilibrium is achieved from the viewpoint of travelers on the lower level. Boyles et al. [19] proposed a firstbest 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 NetworkMarginal Cost Pricing (SNMCP) for a riskneutral case and riskbased SNMCP (RSNMCP) for a riskbased case under the assumptions of lognormal demand and constant VMR across all OD pairs. Gardner et al. [21] consider the uncertainty in longterm travel demand and in daytoday network capacity, and discuss the benefit of responsive pricing and travel information.
In the abovementioned 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 logitbased 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 firstbest 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 derive the analytical function of SNMCP for a riskneutral case and RSNMCP for a riskbased case in a stochastic network with both supply and demand uncertainty, respectively. In Section 5, the analysis for the PRSNMCP is then described under different simplifications of network uncertainties. In Section 6, numerical examples with respect to a smallscale network and a mediumscale 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.
2. Framework of stochastic network model
2.1 Notations and assumptions
Consider a strongly connected network

travel demand between OD pair


mean travel demand between OD pair


variance of travel demand between OD pair


variancetomean ratio ( 

route flow on path


mean traffic flow on path


variance of traffic flow on path


column vector of mean route flow, where


traffic flow on link


mean traffic flow on link


variance of traffic flow on link


column vector of mean link flow, where


linkpath incidence parameter; 1 if link


total travel time of the system, where


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

travel time on path


mean travel time on path


variance of travel time on path


travel time on link


mean travel time on link


variance of travel time on link


perceived travel time on path


mean perceived travel time on path


variance of perceived travel time on path


perceived travel time on link


mean perceived travel time on link


variance of perceived travel time on link


total perceived travel time of the system, where


freeflow travel time on link


capacity of link

design capacity (upper bound) of link


degree of worstdegraded capacity for link



parameter, where


travelers’ perception error on link


perception error distribution of traveler, in this chapter

Before proceeding with the analysis, some assumptions are made to allow for the closedform formulation/calculation of the PRSNMCP model.
A1. The travel demand
A2. The route flow
A3. The
A4. The capacity degradation random variable
2.2 VI formulation for different stochastic network models
2.2.1 Stochastic networksystem optimal (SNSO) formulation
According to the Assumption A1 and A2, the OD travel demand, route flow
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
Let
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 SNSO model. In this section, we consider all the travelers to be riskneutral. 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 (SNSO) aims to minimize the expected total travel time. The VI formulation for the SNSO model can be obtained by finding
where
2.2.2 Riskbased SNSO (RSNSO) formulation
Up to this point, we have presented the riskneutral case. However, several empirical studies reveal that travel time reliability plays an important role in the traveler’s route choice decision process [1, 2, 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 decisionmaking process. Researchers have used the Travel Time Budget (TTB) to represent travelers’ riskbased travel behavior. Mathematically, the TTB associated with route
where
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:
From Eqs. (10) ∼ (12), the TTB of route and link satisfy the following conservation conditions:
Let
where
2.2.3 Perceived RSNSO 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
where
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:
From Eqs. (15) ∼ (17), the perceived TTB of the route and link satisfy the following conservation conditions
Let
where
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,
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 worstdegraded capacity as its lower bound (the lower bound to be a fraction of the design capacity), they derived the mean and variance of
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
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
where
where
where
Let
Using the BPR function of link travel time, we can derive the mean and variance of the link travel time as follows:
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:
3. Marginal cost pricing in a stochastic network (SNMCP) with both supply and demand uncertainty
3.1 Analysis of SNMCP
In this section, we discuss the SNMCP in the riskneutral 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 NetworkUser Equilibrium (SNUE) and Stochastic NetworkSystem Optimal (SNSO) models and established the firstmarginal 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
Let, then, the real gap between the marginal social and marginal private costs in a stochastic network be represented by
3.2 Calculation of SNMCP
In this study, we attempt to compute the value of SNMCP in the case of both link capacity and demand variation. To achieve this goal, we need to calculate
Differentiating Eq. (33) with respect to the mean link flow
By substituting Eqs. (30) and (34) into Eq. (32), the value of SNMCP in case of Stochastic Supply and Stochastic demand (SSSD) can be determined as follows:
Note that if we neglect the degradation of link capacity, Eq. (35) degenerates into the classical SNMCP model proposed by [18], which considers only the stochastic travel demand. Furthermore, they pointed out that the SNMCP toll is guaranteed to be positive when
4. Riskbased MCP (RSNMCP) in a stochastic network
4.1 Analysis of riskbased SNMCP
In the previous section, we know that the Stochastic NetworkUser Equilibrium (SNUE) flow pattern can be driven toward a SNSO flow pattern by charging a toll equal to the SNMCP. Meanwhile, the expected total travel time can be minimized. In this section, we consider the riskbased (averse or prone) case. The objective function of the RSNMCP 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 travel time. Therefore, the RSNMCP toll can be determined as
4.2 Calculation of RSNMCP
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 decisionmaking process. From Eqs. (32) and (36), we can see that the difference between SNMCP and RSNMCP is the term in the second parentheses of Eq. (36). This second term reflects the congestion toll on travel time reliability due to travelers’ riskbased behavior. Let us now turn our attention to
Differentiating Eq. (37) with respect to the mean link flow yields
By substituting Eqs. (31), (35), and (38) into Eq. (36), the value of RSNMCP in case of SSSD can be determined. In the same way, by neglecting the degradation of link capacity, the RSNMCP in case of SSSD degenerates into the classical RSNMCP model proposed by [18], which considers only the stochastic travel demand.
5. Formulation of perceived RSNMCP (PRSNMCP)
5.1 Model incorporating the travelers’ perception error
Up to this point, we have studied the SNMCP model and RSNMCP model based on the assumption that all the travelers have perfect knowledge about the network condition. However, in real life, due to the limitations of their own condition, travelers’ perception errors have to be incorporated into their route choice decision process. In view of this, it is necessary to investigate the RSNMCP model with travelers’ perception errors. In order to develop such a model, we need to make some additional assumptions on the perception error as follows:
A5. The perception error distribution of an individual traveler for a segment of road with unit travel time equals
A6. Traveler’s perception errors are independent for nonoverlapping route segments.
A7. Traveler’s perception errors are mutually independent over the population of travelers.
In order to compute the value of PSNMCP of each link in the stochastic network, we need to derive the perceived link travel time, based on moment analysis. According to Assumption A5, the perception error for unit travel time, denoted by
with conditional moment generating function (MGF)
where
where
From the first derivative of the equation above and evaluating at
where
The variance of the perceived travel time can be expressed as follows:
Using these equations, we can analyze the RSNMCP model with travelers’ perception errors. When taking travelers’ perception error into consideration, the objective function of the PRSNMCP model is to minimize the weighted sum of the mean and the variance of the total perceived travel time. Thus, the PRSNMCP toll can be given by
where
According to Eq. (46), it is clear that the value of PRSNMCP can be determined as long as
5.2 Calculation of PRSNMCP
In order to illustrate the importance of incorporating both stochastic supply and demand into the proposed PRSNMCP model, the calculation of PRSNMCP 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.
5.2.1 Case a: stochastic supply, stochastic demand (SSSD)
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 decisionmaking process. As of now, we have already obtained the values of
Differentiating Eq. (48) with respect to the mean link flow
Substituting Eqs. (30), (31), (34), (38), and (49) into Eq. (47), we can obtain the value of PRSNMCP in case of SSSD.
5.2.2 Case B: stochastic supply, deterministic demand (SSDD)
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
The expected total travel time can be simplified to
Differentiating Eq. (50) with respect to the mean link flow
The variance of the total travel time is described by
Differentiating Eq. (52) with respect to the mean link flow yields
With Eq. (26) we have
Differentiating Eq. (54) with respect to the mean link flow
By substituting Eqs. (20), (21), (51), (53), and (55) into Eq. (47), the value of PRSNMCP in case of SSDD can be determined.
5.2.3 Case C: deterministic supply, stochastic demand (DSSD)
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,
The expected total travel time is given by
Differentiating Eq. (56) with respect to the mean link flow
The variance of the total travel time is expressed as
Differentiating Eq. (58) with respect to the mean link flow yields
With Eq. (26) we have
Differentiating Eq. (60) with respect to the mean link flow
Thus the value of PRSNMCP in case of DSSD can be determined by substituting Eqs. (28), (29), (57), (59), and (61) into Eq. (47).
5.2.4 Case D: Deterministic supply, deterministic demand (DSDD)
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
The expected total travel time can be simplified to
Then we have
From Eq. (26) we can obtain
Consequently, we have, upon simplifying
By substituting Eqs. (63) and (65) into Eq. (47), the value of PRSNMCP in case of DSDD can be expressed as follows:
6. Numerical examples
The purpose of the numerical examples is to illustrate: (1) the effect of the
6.1 Effect of the VMR on the performance of SNMCP toll scheme
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) function with the following parameters:
Link  Freeflow travel time  Design capacity  Degradation parameter

Link  Freeflow travel time  Design capacity  Degradation parameter


1  3  2000  0.95  12  3  1000  0.95 
2  3  2000  0.95  13  3  2600  0.95 
3  3  2000  0.95  14  3  2000  0.95 
4  4.5  1800  0.95  15  3  1400  0.95 
5  7.5  1200  0.95  16  3  2000  0.95 
6  3  1000  0.95  17  3  800  0.95 
7  3  2000  0.95  18  3  2000  0.95 
8  3  1800  0.95  19  3  2000  0.95 
9  3  1800  0.95  20  3  4000  0.95 
10  4.5  1800  0.95  21  3  4000  0.95 
11  3  2000  0.95 
For the first example, we examine the effect of
6.2 Importance of incorporating supply and demand uncertainty
6.2.1 Effect of congestion on the performance of different PRSNMCP toll schemes.
We also use the traffic network shown in Figure 2 in the following test, in which both supply and travel demand uncertainty and travelers’ perception errors will be simulated. To demonstrate the effects of neglecting certain aspects of the stochasticity of the network, we compare the expected total perceived travel time under the four PRSNMCP scenarios discussed in Section 5.2. These four scenarios are analyzed under different congestion levels (the OD demand multiplier
In this example, we study the effect of congestion levels on the performance of different toll schemes with fixed
Demand multiplier ( 



Toll free  SSSD  SSDD  DSDD  DSSD  
0.8  132,261  129,158  129,171  129,184  129,300 
0.85  142,651  139,878  139,908  139,928  140,084 
0.9  153,870  151,438  151,474  151,502  151,695 
0.95  166,113  163,979  164,033  164,072  164,283 
1  179,550  177,688  177,757  177,801  177,996 
Figure 4 demonstrates the percentage improvements in the expected total perceived travel time related to Table 2. The “Improvement” in Figure 4 is, in this case, the percentage of improvement in the expected total perceived travel time from the toll free case compared to the SSSD tolls case, that is,
From the figure above, the improvement in the expected total perceived travel time obtained by the SSDD, DSDD, and DSSD tolls is lower than that obtained by the SSSD tolls. Besides, the gap between the expected total perceived travel time under the SSSD tolls and other toll schemes increases as
6.2.2 Effect of the VoR on the performance of different PRSNMCP toll schemes
By assuming the levels of congestion and




Toll free  SSSD  SSDD  DSDD  DSSD  
0.0068  179,233  177,335  177,384  177,413  177,510 
0.0085  179,291  177,394  177,445  177,491  177,620 
0.0104  179,344  177,468  177,525  177,568  177,716 
0.0129  179,432  177,560  177,623  177,668  177,844 
0.0165  179,550  177,688  177,757  177,801  177,996 
Based on Eq. (67) and Table 3, we can obtain the percentage improvements in the expected total perceived travel time, as shown in Figure 5. It can be seen that the discrepancies between the performance of the SSSD toll and that of other toll schemes become conspicuously larger as the
From the above analysis, it can be concluded that the discrepancies of these simplifications depend on both the congestion and
6.3 Analysis of the essentiality of incorporating the travelers’ perception error
The traffic network shown in Figure 2 is again adopted in examining the PRSNMCP model. By comparing the difference of the expected total perceived travel time achieved by the RSNMCP tolls (expressed by
6.4 Application to the Sioux Falls network in the PRSNMCP (SSSD) case
The final example illustrates the calculation of the PRSNMCP (SSSD) toll in a larger network. This example network is the wellknown mediumscale Sioux Falls network (see Figure 7), which consists of 24 nodes and 76 links. The link design capacity and link freeflow travel time can be found in [25]. Degradation parameter
O/D  4  5  6  10  14  19  22 

4  800  800  800  800  800  800  
5  800  800  800  800  800  800  
6  800  800  800  800  800  800  
10  800  800  800  800  800  800  
14  800  800  800  800  800  800  
19  800  800  800  800  800  800  
22  800  800  800  800  800  800 
In this example, we compare two scenarios. One is the toll free case, and the other one is the PRSNMCP toll scheme. Table 5 presents the link toll under the PRSNMCP scenario. By levying these tolls on each link, the network becomes smooth and efficient. At the equilibrium state, the expected total perceived travel time achieved by the toll free case and PRSNMCP toll scheme is 345,749 and 324,636, respectively. Therefore, the proposed PRSNMCP model is an efficient method in reducing traffic congestion.
Link  Link toll  Link  Link toll  Link  Link toll  Link  Link toll 

1  19.37  20  23.16  39  33.54  58  45.96 
2  12.81  21  17.97  40  68.29  59  20.13 
3  19.37  22  37.02  41  38.43  60  35.53 
4  16.08  23  95.53  42  22.83  61  20.13 
5  12.81  24  17.97  43  59.94  62  23.86 
6  34.19  25  37.12  44  38.43  63  32.55 
7  78.43  26  37.12  45  22.83  64  23.86 
8  34.19  27  33.15  46  29.97  65  7.68 
9  39.22  28  59.94  47  37.02  66  15.86 
10  125.82  29  17.53  48  17.53  67  29.97 
11  39.22  30  30.77  49  14.00  68  32.55 
12  46.12  31  125.82  50  3.07  69  7.68 
13  95.53  32  33.15  51  30.77  70  18.04 
14  16.08  33  18.94  52  14.00  71  22.83 
15  46.12  34  68.29  53  45.96  72  18.04 
16  73.25  35  78.43  54  15.33  73  5.24 
17  23.16  36  18.94  55  3.07  74  33.54 
18  15.33  37  24.96  56  35.53  75  15.86 
19  73.25  38  24.96  57  22.83  76  5.24 
7. Conclusions
To make pricing more efficient and effective, this chapter developed a reliabilitybased marginal cost pricing model. The new model explicitly accounts for both stochastic link degradation and stochastic demand of road network and perception errors within the travelers’ route choice decision process. We consider that the stochastic demand follows a lognormal distribution and the capacity degradation follows a uniform distribution, and
This chapter investigated possible defects associated with ignoring certain aspects of the stochastic behaviors of the network. Through numerical examples, we find that both link capacity degradation and stochastic demand play essential roles in the PRSNMCP model, especially under high travelers’ confidence level and network congestion. We further examined the effect of incorporating the travelers’ perception error into the RSNMCP tolls. The numerical example illustrates that travelers’ perception errors have a significant impact on the performance of the PRSNMCP tolls and, therefore, should not be neglected.
Acknowledgments
This research has been supported by the National Natural Science Foundation of China (Project No. 71701030 and 71971038), the Humanities and Social Sciences Youth Foundation of the Ministry of Education of China (Project No. 17YJCZH265), and the Fundamental Research Funds for the Central Universities of China (Project No. DUT20GJ210).
The MGF of
The firstorder moment is, from the first derivative evaluated at
Similarly, the secondorder moment of
Then we can obtain the variance of
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