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WORKING PAPER SERIES: NO. 2015-01

Revenue Maximizing Dynamic Tolls for Managed Lanes:

A Simulation Study

Caner Gocmen

Nomis Solutions

Robert Phillips

Columbia University, Nomis Solutions

Garrett van Ryzin,

Columbia University

2015

http://www.cprm.columbia.edu

Revenue Maximizing Dynamic Tolls for Managed Lanes: A

Simulation Study

Caner G¨o¸cmen∗

Robert Phillips†

Garrett van Ryzin‡

October 28, 2015

Abstract

In recent years, public-private partnership schemes for highway construction have become in-

creasingly popular. In a typical private-public partnership, a private company builds additional

lanes on existing highways in return for the right to charge a toll on the additional lane for a

speciﬁed period of time and to keep all or part of the resulting revenue. We address the question

of how an operator should set and update tolls in order to maximize expected revenue when

drivers have access to a free alternative. We address this problem through stochastic simulation

of a freeway with both toll lanes and free lanes. We assume that drivers choose whether to

travel on the toll (managed) lane or the free (unmanaged) lane based on the current congestion

in each lane and on the current toll. We use a mesoscopic traﬃc model to represent the traﬃc

dynamics in each lane and calibrate the model using data from the SR 91 highway in Orange

County, California. Our baseline is a myopic policy in which the operator sets tolls to maximize

expected revenue from each vehicle. We compare this policy with time-of-use policies that can

anticipate the likely pattern of future demand and consider both non-adaptive policies which

cannot update the toll based on current conditions and adaptive policies which can. We ﬁnd

that the best-performing policies raise tolls prior to anticipated peaks in order to divert traﬃc

to the unmanaged lanes and thereby increase congestion on those lanes and decrease congestion

on the managed lanes – an approach we call jam-and-harvest. When a peak is present, the

myopic policy compares poorly to non-adaptive policies that anticipate expected demand but

do not adapt to current conditions. We conﬁrm and extend these observations using simpliﬁed

stylized models.

∗Nomis Solutions

†Columbia Business School and Nomis Solutions

‡Columbia Business School

1

In an era of increasing traﬃc congestion and limited budgets for infrastructure improvement, the

idea of allowing a private company to foot the bill for a highway project in return for a share of

future toll revenues is tempting to many public transportation agencies. In the past ﬁve years a

handful of such projects have been built in the United States. Examples include the LBJ Freeway

(near Dallas Texas) and the 495 Express (in Northern Virginia near Washington DC). A number of

similar projects have been proposed or are under construction. As of 2012, 32 states and Puerto Rico

had passed legislation enabling such public-private partnerships for highway construction (Perez

et al. (2012)). The vast majority of these projects consist of a managed lane scheme in which

drivers have a choice between a number of managed lanes for which a usage toll is charged and a

number of unmanaged lanes that are always free to use. (A managed lane scheme is sometimes

called a high occupancy and toll (HOT) scheme. Under this nomenclature, the toll lanes are called

the HOT lanes and the free lanes are called the general purpose or GP lanes.) In a managed lane

scheme, the driver always has the choice of choosing the parallel unmanaged lanes if she does not

want to pay the toll – this distinguishes managed lane schemes from traditional toll roads in which

the only alternative available to the driver is to take a diﬀerent route. The motivation of an arriving

driver to choose the managed lanes is the possibility of less congestion and faster travel time than

if she chooses the unmanaged lanes. In this paper we consider the problem of setting and updating

the tolls for a managed lane in order to maximize expected revenue.

A driver approaching the managed lanes is informed of the current toll by large digital signs. These

are placed suﬃciently far from the entrance that the driver has the time to decide whether to

chose the managed or unmanaged lanes. The frequency of toll changes is governed by regulation –

for current projects the minimum interval between toll changes ranges from three to ﬁve minutes.

Managed lane schemes use a gantry-based toll system in which a transponder is used to identify

vehicles at the point of entry. If an entering vehicle does not carry a transponder, the license plate

is photographed and optical character recognition technology is used to identify the license plate

number of the vehicle. There are no toll booths and there is no need for vehicles to slow down as

they enter the managed lanes. Vehicle owners are typically billed monthly for the tolls accumulated

in the prior month.

In this paper, we consider the problem faced by an operator who is seeking to set the tolls of a set

of managed lanes over time in order to maximize revenue. Because the incremental cost incurred by

an additional vehicle using the managed lanes is essentially zero, maximizing revenue is equivalent

to maximizing short-run proﬁt. In managed lane schemes, the current traﬃc conditions on both

the managed and unmanaged lanes are continually monitored by sensors and the operator can use

this information in setting the toll. In practice, operators will also have additional information

available on factors such as weather or lane closures that could inﬂuence the revenue-maximizing

2

0 2 4 6 8 10 12 14 16 18 20 22

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Hour

Volume (Cars/Hour)

Monday

Tuesday

Wednesday

Thursday

Friday

Figure 1: Average hourly volumes for SR 91 Eastbound between January 2009 - July 2011.

toll, however for simplicity assume that the only information available to the operator the travel

time diﬀerence between the managed and unmanaged lanes.

An important characteristic of managed lane projects is that demand – particularly weekday de-

mand – follows predictable patterns. Managed lane projects are usually built to alleviate congestion

in areas that experience either a morning rush hour or an afternoon rush hour or both. The Cal-

ifornia State Route 91 (SR 91) in Orange County, California was one of the ﬁrst managed lane

projects in the United States. Average Eastbound traﬃc for summer weekdays is shown in Figure

1. On each weekday, Eastbound traﬃc has a small morning peak between about 7:00 and 9:30

AM followed by a much higher afternoon peak between 2:00 and 6:00 PM. Westbound traﬃc has

a higher morning peak and a much lower afternoon peak. The patterns vary somewhat by day of

the week but are generally stable.

We consider two broad categories of pricing policies. Under a non-adaptive policy, tolls are published

at least one day prior to the day of operation. Tolls may vary by time-of-day and/or day-of-week

but they cannot be changed based on current conditions. Non-adaptive pricing has historically been

the norm for managed lane schemes such as the SR 91. The average hourly tolls for the Westbound

managed lanes for the SR 91 for weekdays in June, 2011 are shown in Figure 2. Tolls can vary

dramatically over the course of a day – in this case from a low of $1.30 to a high of $9.75. The fact

that the strong traﬃc peaks evident in Figure 1 persist in spite of such dramatic diﬀerences in tolls

across the day implies that there are a large number of drivers who are either unwilling or unable

to change departure times away from periods characterized by both high congestion and high tolls.

3

Under adaptive pricing tolls can be changed periodically in response to current conditions. We

consider two types of adaptive policies. In the myopic policy, the toll is set that maximizes the

expected revenue from vehicles arriving in the next time interval. In a linear adjustment policy, a

time-of-use policy is used as a baseline for tolls. If the travel-time diﬀerential between the managed

and unmanaged lanes is greater than expected the toll is increased above the baseline: if the

diﬀerential is less than expected, tolls are decreased.

1 3 5 7 9 11 13 15 17 19 21 23

0

1

2

3

4

5

6

7

8

9

10

Hour

Toll ($)

Monday

Tuesday

Wednesday

Thursday

Friday

Figure 2: Average Westbound weekday hourly tolls for the SR 91 managed lanes in June, 2011.

Source:www.912expresslanes.com/schedules.asp

One of our key ﬁndings is that the majority of the beneﬁts from actively managing tolls comes from

carefully setting and updating tolls around peaks. In particular, a revenue-maximizing operator

should increase tolls very high in anticipation of a peak. This encourages drivers to take the

unmanaged lanes which has the dual eﬀects of decreasing congestion in the managed lanes and

increasing congestion in the unmanaged lanes. These two eﬀects combine to make the managed

lanes more attractive to future drivers, enabling the controller to charge higher tolls than would

have been otherwise possible. In this situation, we ﬁnd that non-adaptive time-of-use policies can

signiﬁcantly outperform the adaptive myopic policy which considers only the current states of the

managed and unmanaged lanes in setting the toll. More generally, we ﬁnd that successful policies

tend to charge higher tolls in all situations than the myopic policy. However, in cases in which

total traﬃc demand is low or declining, the uplift of such policies relative to the myopic policy is

small. On the other hand, when traﬃc demand is relatively high and increasing – as it would be

entering a peak – both intelligent time-of-use and linear adjustment policies can yield substantially

more revenue than the myopic policy.

4

Up to the last few years, the majority of managed lane schemes were operated by public entities who

were not trying to maximize proﬁt. The criteria used for setting such tolls was often to keep traﬃc

moving freely in the managed lanes subject to some constraints such as maximum tolls. Chung and

Recker (2011) provide an overview of the approaches taken by public managed lane operators in

the United States. However revenue-maximizing toll policies are very diﬀerent from those that seek

to maintain free-ﬂow in the managed lanes. The problem of setting revenue-maximizing tolls for a

managed lane scheme has received very little attention in the literature. To our knowledge, the only

exception is Yang (2012). However that paper not consider that congestion on the unmanaged lanes

is inﬂuenced by the tolls on the managed lanes – which we ﬁnd to be a very important consideration

in setting the revenue-maximizing tolls. G¨o¸cmen (2013) in his thesis derives some structural results

for the case in which traﬃc in each lane is modeled as a queue and demand is stationary. While

this provides insight into the problem, it does not apply directly to the real-world cases of interest

in which traﬃc dynamics are more complex and demand is not stationary.

In the remainder of this paper, we describe our simulation model and how the underlying modules

were calibrated to the SR 91 data. We then compare the performance of diﬀerent toll-setting policies

using data from both the Eastbound and the Westbound SR 91 traﬃc. A key ﬁnding is that policies

that anticipate future traﬃc and set tolls accordingly outperform the myopic policy that sets tolls

based only on current conditions. The advantage of policies that incorporate forecasts of future

traﬃc demand is particularly important immediately before peak periods. To better understand

the performance of diﬀerent policies, we used a set of stylized models in which we could vary the size

and duration of peak demand to see how these features inﬂuenced performance. In the last section,

we discuss our results and their implications for toll operators as well as potential extensions.

1 Simulation Model Description

Our simulation represents a highway that is 10 miles long and consists of ﬁve unmanaged lanes

and two managed lanes with a single entry and exit point. This is approximately the length of the

managed lanes for the SR 91. We assume that traﬃc arrives at a random rate. The arrival rate

is exogenous, that is, it is not inﬂuenced by the current toll or traﬃc. Arriving vehicles choose

either the managed lanes or the unmanaged lanes based on the travel time diﬀerence between the

managed and unmanaged lanes and the toll. A consumer choice model estimates the fraction of

arriving vehicles that choose the managed lanes based on the current toll.

A schematic diagram of the discrete time simulation model shown in Figure 3. In each time

increment, the demand generation module determines how much new traﬃc arrives to the system.

5

Demand

Generation

Consumer

Choice

Traffic

Simulation

Total

Traffic

Traffic

by Lane

Travel time Difference

Toll

Schedule

Figure 3: Modules in the Simulation Model.

Based on the current toll and time diﬀerential between the managed and unmanaged lanes, the

consumer choice module determines the proportions of the arriving traﬃc that choose the managed

and unmanaged lanes. The traﬃc module uses that information to update the traﬃc on the

managed and unmanaged lanes. This information is then passed to the consumer choice module

which determines the fraction of the incoming traﬃc that chooses the managed and the unmanaged

lanes. This process repeats itself until the stopping time for the simulation is reached. In the

numerical study, we start with an empty highway and simulate the system for a whole day.

We calibrated the simulation model using publicly data on the SR 91 available from the California

Freeway Performance Measurement System (PeMS) 1. We used data from the 10 mile managed

lane section of the SR 91 that runs from the SR 55 interchange to the Riverside County line. These

managed lanes are known as the “SR 91 Express Lanes” and are operated by the Orange County

Transportation Authority who uses a time-of-use based tolling schedule for the managed lanes

which it updates every few months. In setting the tolls, the primary stated goal of the authority is

to maintain free ﬂow speed on the managed lanes (The Orange County Transportation Authority,

2013).

1http://pems.dot.ca.gov/

6

1.1 Demand Generation Module

The demand generation module computes sample paths of total traﬃc demand that are based

on both the mean traﬃc arrival rate patterns shown in Figure 1 and the serial correlation of

traﬃc. Total traﬃc arriving to the system is initially generated as hourly demands which are

then distributed to demands at ﬁve-minute intervals. Starting from midnight, hourly demand is

generated by an autoregressive model of order three:

Yt=βt+α1,tYt−1+α2,tYt−2+α3,tYt−3+εt,(1)

where Ytis the traﬃc volume for hour t;βt, α1

t, α2

tand α3

tare coeﬃcients and εtis a normally

distributed error term. We estimated the parameters of (1) using Ordinary Least Squares (OLS)

applied to historic SR 91 total traﬃc demands. The resulting parameter estimates by hour are

shown in Tables 7 and 8 for Eastbound and Westbound traﬃc respectively.

In the next step we distribute the hourly traﬃc into ﬁve-minute intervals. For this purpose we

used ﬁve-minute traﬃc volume data for July 2011. We omitted the ﬁrst week of July due to the

Independence Day holiday. For each hour, we calculated the fraction of hourly demand occuring in

each 5-minute interval. By averaging those fractions across all days in our dataset, we calculated

the average proportion of hourly demand each 5-minute interval for each hour. More details of the

regression and the results are given in the Appendix.

1.2 Consumer Choice Module

The Consumer Choice Module takes in total highway demand at ﬁve minute intervals produced

by the demand generation module and allocates the demand between the managed lanes and the

unmanaged lanes based on the current travel-time diﬀerential and toll. Our approach is similar to

the models described in (Xu, 2009; Yin and Lou, 2009). We use historical data on lane choice for

the SR 91 to estimate the parameters of a consumer choice model as in Liu et al. (2004) and Liu

et al. (2007).

Denote the expected travel time savings from choosing the managed lanes at time tby ∆T(t) and the

toll by p(t). Let U(k)

in (t) = gik(t) + εin denote the utility that driver nreceives at time tby choosing

alternative i=u, m, where uand mdenote the unmanaged and managed lanes, respectively. The

index kdenotes the structure used for the deterministic part of the utility function. The term εin

accounts for the unobserved component of driver n’s utility from choosing alternative i. For the

unmanaged lanes. Without loss of generality we set guk(t) to zero. We considered three diﬀerent

7

formulae for driver utility:

gm1(t) = βT(t)∆T(t) + βp(t)p(t),

gm2(t) = βT(t) log(∆T(t)) + βp(t)p(t),

gm3(t) = βT(t)(∆T(t))2+βp(t)p(t).

The ﬁrst formula corresponds to the case in which a driver’s utility increases linearly with the

expected time savings. The second corresponds to the case in which drivers get less sensitive to the

expected travel time savings as it increases, and the third formula corresponds to the case where

they become more sensitive. In all cases we allow βTand βpto vary by period.

For both lanes, we assume that the random term in the utility function is independently and

identically distributed across drivers according to a Type I Extreme Value distribution. After

evaluating the utility of both alternatives, each driver chooses the alternative that provides the

highest utility. The probability that a driver chooses alternative iat time tis given by the logit

function (Ben-Akiva and Lerman, 1985)

P(k)

i(t) = egik(t)

1 + egmk(t).(2)

We estimated βTand βpusing maximum likelihood estimation (MLE). We use the same VDS

sensor data that was used to calibrate the demand generation module. We analyzed the lane choice

decisions of Eastbound commuters on SR 91 from Monday through Friday during the last two

weeks of July 2011. Traﬃc in this direction has an afternoon peak as shown in Figure 1.

Figure 4 shows the minimum, maximum and average hourly time savings observed in our dataset.

Not surprisingly, the expected travel time savings is highest during the afternoon peak. There

is also signiﬁcant variation between the time savings observed throughout this two week period.

Figure 5 shows the managed lane share of traﬃc for Eastbound traﬃc. The managed lane share

peaks in the afternoon when both congestion and tolls are at their highest. During the oﬀ-peak

hours, the managed lanes command a very low share of the traﬃc passing through this segment of

the highway.

Estimation of the model parameters, as well as the model parameters themselves and the ﬁt for

diﬀerent models are described in the appendix. The model that showed the best ﬁt with sensible

coeﬃcient signs was gm3(t) = βT(t)(∆T(t))2+βp(t)p(t), which is the function we use in the simula-

tion model. Note that this model implies that the utility of the managed lines rises at an increasing

rate with the time diﬀerence.

8

1 3 5 7 9 11 13 15 17 19 21 23

0

5

10

15

20

25

30

35

40

45

Hour

Average Time Savings (min)

Average

Minimum

Maximum

Figure 4: Average hourly time savings on the SR 90 Eastbound.

1.3 Traﬃc Module

The traﬃc module takes in demand for both the managed and unmanaged lanes at ﬁve minute

intervals as calculated by the consumer choice module and calculates new travel times for the

managed and unmanaged lanes. The resulting travel time diﬀerential is fed back to the consumer

choice model as an input to the allocation of vehicles between managed and unmanaged lanes in

the next time period. As vehicle density increases in a set of lanes, the speed of the vehicles in

those lanes will decrease. Figure 6 shows a scatter plot of the weekday speed and density for the

unmanaged lanes in the SR 91 during the ﬁrst four weeks of July 2011. At low density – less than

approximately 35 vehicles/mile/lane – vehicles can pass freely and the average speed is at or close

to the “free-ﬂow speed” for the highway. As traﬃc density increases above this level, average speed

begins to drop. At approximately 60-70 vehicles/mile/lane, average speed tends to stabilize at or

near the so-called “jam speed”.

Traﬃc simulation models can be categorized as macroscopic, microscopic, and mesocopic, Macro-

scopic simulation works at the level of ﬂows and does not represent the behavior of individual

vehicles. Macroscopic simulation is not appropriate for our purposes because it cannot track lane

choice by vehicles based on current tolls and congestion. Mesoscopic and microscopic simula-

tions keep track of individual vehicles. Microscopic simulation models the behavior of individual

drivers and how they change their behavior with changing road conditions. MITSIM, VISSIM and

PARAMICS are some of the most well known microscopic simulation models (Olstam and Tapani,

2004). However, the level of detail that microscopic simulations can capture comes at a heavy

9

0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hour

Market Share

Figure 5: Hourly market share of the managed lanes on the SR 90 Eastbound.

cost: there are many more parameters that need to be estimated such as a desired speed for each

driver, propensity to pass, desired following distance, etc. In addition, the time required to run

and the time required to run a microscopic simulaiton model is orders of magnitude greater. For

this reason, microscopic simulation is typically used to model the details of individual intersections

and is not well suited to our purposes.

Mesoscopic simulation is typically used to model traﬃc behavior over stretches of highway of the

length that we are considering. In mesoscopic simulations the road is divided into segments and

at each time step the vehicles are moved from one segment to another based on speed, ﬂow, and

density relationships. Mesoscopic models have proven eﬀective at realistically representing the

behavior of traﬃc over longer stretches while requiring relatively few parameters to be calibrated.

We based the logic in the traﬃc module on the mesoscopic simulations used in the DYNASMART

(Jayakrishnan et al., 1994) and the DynaMIT (Ben-Akiva et al., 2002) models.

We divide the highway into nine segments with equal length L. The segment length was chosen as

the distance that a vehicle moving at the free-ﬂow speed would traverse during one time increment.

The number of lanes in each segment is denoted by wand the average space that a vehicle occupies

(including its headway at jam density) is denoted by `. As a result, the physical capacity of each

segment is wL/`. We split each segment into two parts: moving and queuing. Vehicles that are

queued up to join the next segment will be in the queuing part and the remainder of the vehicles

in the segment will be in the moving part. The lengths of both parts are dynamic and depend on

the number of vehicles in the queuing part. Given that there are nqvehicles queued, the length of

10

the queuing part is nq`/w. Accordingly, the length of the moving part is L−nq`/w.

At each time step in our simulation, the vehicles in the moving part traverse the segment at a speed

v(k) calculated according to the speed-density relationship

v(k) =

66.8−0.14k, if k≤25,

15.0 + 69.33(1 −(k/100)2.22)7.69,if k > 25,

(3)

where kis the density of the moving part of the segment at the beginning of the time step, All

speeds are in miles-per-hour (mph) and densities are in vehicles/mile/lane unless otherwise noted.

After a vehicle has traversed a segment it has two options. If there is space in the next segment, it

passes on to that segment and travels on that segment for the remainder of the time step. Otherwise

it joins a queue and the end of the segment. The parameters of the model were estimated using

historic data from the SR 91 as described in the appendix.

At each time increment, the movement of vehicles is calculated in three steps. In the ﬁrst step,

vehicles in the moving segment move according to the speed-density relationships in (3), and the

vehicles that reach the end of the segment move into a queue to await transition into the next

segment. We update the position of each vehicle starting from the one that is closest to the

highway’s end and move towards the beginning of the highway. In the next step, vehicles move

from the queues at the end of each segment to the next segment. Vehicles are allowed to pass

into the next segment until it reaches jam density. The last step involves moving the vehicles that

just changed segments. If a vehicle waited in the queue for at least one iteration, then it is moved

according to the prevailing moving part speed of its previous segment. If the vehicle joined the

queue in that iteration, it completes its movement by traveling the amount it was not allowed to

complete before joining the queue.

Once the vehicles are moved, we calculate the expected travel times for each segment. The expected

travel time for each segment consists of the time it takes for a vehicle to traverse the moving part

of a segment (Tm), and the waiting time in the queue (Tw). Let qdenote the number of vehicles

waiting in the queuing part of a segment, then,

Tm=L−nq×`×w

v(k),

and

Tw=nq/d,

where dis the moving average of the discharge rates observed in the previous periods. Those times

are calculated for each segment and are in turn used to calculate the expected travel time for both

11

0 20 40 60 80 100

0

10

20

30

40

50

60

70

80

Density (vechiles/mile/lane)

Speed (mph)

Figure 6: Speed-density relationship for SR 91.

parts of the highway.

This approach to modeling traﬃc ﬂow is identical in structure to DYNASMART. It is also similar

to DynaMIT, although we use a slightly diﬀerent approach to calculate moving times and waiting

times. These models have been used in many studies and have been extensively validated (Han

et al., 2006; Roelofsen, 2012; Ben-Akiva et al., 2010).

We calibrated our model to the historic speed-density relationship for the unmanaged lanes in the

SR 91 as shown in Figure 6. The black curve in this ﬁgure is the speed-density relationship that

we ﬁt to the underlying. The parameters of this speed-density relationship are shown in Table ??

and details of the calibration can be found in the Appendix. Since the current SR 91 policy sets

tolls to encourage free-ﬂow conditions in the managed lanes, there is no data for more congested

conditions in the managed lanes. For this reason, we use the same speed-density relationship in

Equation 3 for both parts of the highway. Since the managed lanes run parallel to the unmanaged

lanes and are of similar quality, this is a reasonable assumption.

2 Numerical Studies

We used the simulation model described in the previous section to compare the expected revenue

generated by diﬀerent policies. We ﬁrst considered two cases based on the SR 91 traﬃc. In the

ﬁrst case, we calibrated the simulation model using data from the weekday Eastbound SR 91. The

12

Eastbound SR 91 has a very pronounced afternoon peak and a much weaker morning peak. In

the second case, we calibrated the simulation model using the weekday Westbound SR 91, which

displays a strong morning peak and a weaker afternoon peak. These two cases provided a reasonable

comparison of revenue generation among the policies considered but do not provide much insight

into the drivers of revenue. In order to understand how tolling and revenue is inﬂuenced by various

aspects of demand, we constructed a series of highly stylized demand models that allowed us to

vary the height and duration of peak demand among other parameters. In this section, we ﬁrst

discuss the diﬀerent tolling policies that we tested, then describe the results on the Eastbound and

Westbound SR 91 data followed by the results on the stylized demand models.

2.1 Policies

The tolling policies considered include adaptive policies in which the tolls can be adjusted in

response to current conditions as well as non-adaptive policies in which tolls are established in

advance and do not change based on current conditions. In each case, we consider diﬀerent intervals

over which the toll can change. We use a discrete time approach in which we divide the planning

horizon into Tintervals, and tdenotes the interval index. The number of vehicles that arrive in

an interval is random, and is denoted by the random variable D(t). We assume that the toll stays

constant over each interval. This is realistic; all current managed lane schemes enforce a minimum

interval between toll changes – for example, ﬁve minutes in the case of the LBJ Project. The

number of vehicles in the lanes and their locations are denoted by xi(t) for i=u, m.

The revenue maximization problem for a non-adaptive policy is

max

T

X

t=1

EhD(t)P(k)

m(t)ip(t)

s.t. xi(t+ 1) = fi(D(t), p(t), xm(t), xu(t)),∀t∈ {1, . . . , T −1}, i ={u, m},

p(t)≥0,∀t∈ {1, . . . , T },

where the mapping fi(.), for iin {u, m}, updates the list of vehicles and their locations in every

period. The solution to this problem is a Time-of-Use policy since it tells the toll manager how

much to charge at each point in time independent of the real-time state of the system.

13

The discrete-time counterpart of the adaptive policy is

max

T

X

t=1

EhD(t)P(k)

m(t)ip(t, xu(t), xm(t))

s.t. xi(t+ 1) = fi(D(t), p(t, xu(t), xm(t)), xm(t), xu(t)),∀t∈ {1, . . . , T −1}, i ={u, m},

p(xu(t), xm(t)) ≥0,∀t∈[0, T ].

In each case, determination of the optimal policy given the underlying parameters is intractable.

For this reason, we use various well-established approaches to approximate the parameters of the

optimal policies based on the data.

2.1.1 Myopic Pricing

The Myopic Policy is an adaptive policy that sets the toll that maximizes the expected revenue

rate for the next period based on the current travel time diﬀerence:

p(t) = argmaxp≥0P(k)

m(t)p.

We use Brent’s method (Brent, 2002) which is a robust derivative-free approach to estimate the

myopic toll for every period. Because the myopic policy is a simple and easy to understand policy,

we use it as the baseline for policy comparisons.

2.1.2 Time-of-Use Tolling

We use Time-of-Use Tolling to refer to a non-adaptive policy that speciﬁes a toll for each time

period based on the anticipated demand in each period, and the parameters of the consumer

choice and the traﬃc ﬂow modules. Determining the optimal Time-of-Use Policy is a stochastic

optimization problem. Such problems are typically solved by iterative algorithms that sequentially

update a trial solution using the stochastic gradient of the objective function:

xk+1 =xk+akgk(xk),

where kis the iteration index, xk∈Rndenotes the current solution, ak∈Rn

+is the updating step

size that decreases in k, and gk(.) is the stochastic gradient estimate.

Since no direct measurements of the gradient are available in our case, we employ the ﬁnite dif-

ferences stochastic approximation (FDSA) method that estimates the gradient by calculating the

14

diﬀerence quotient one-by-one for each decision variable using the Monte Carlo method. Kiefer and

Wolfowitz (1952) introduced this method for univariate optimization problems and Blum (1954)

extended it to the multivariate case. Let ωdenote the outcome of a random process, and y(x, ω)

be a function whose value depends on x∈Rnand the realization of the random outcome ω. Using

the FDSA method the gradient estimate of y(.) for each iteration is obtained by

(ˆgk(xk))i=

m

X

j=1

y(xk+eick, ωkj )−y(xk−eick, ωki )

2ck

,∀i= 1, . . . , n,

where cki is a small positive scalar that decreases in k, and ei∈Rnis the unit vector in direction

i. For the Eastbound Case, we used the sequences: ck= 0.5/k1/6,ak=a/(A+k) with a= 1

and A= 100 when nmax = 1000, and a= 5 and A= 500 when nmax = 5000. For the Westbound

Case, we used the sequences ck= 1/k1/6and ak= 0.5/(200 + k), and perform 1000 iterations in

the stochastic approximation procedure. The resulting parameters can be found in Table 16 of

Appendix A.4.

Typically, smoothness and the diﬀerentiability of the objective function are required to establish

convergence (Spall, 2003). In our problem the objective function is not tractable and we can assert

neither smoothness nor diﬀerentiability. Thus, convergence is not guaranteed. Furthermore, due to

the ill-behaved nature of the problem, the ﬁnal set of decision variables may depend on the initial

starting points. So, the Time-of-Use policies that are generated by the stochastic approximation

procedures are heuristics. We stop the algorithm after a predetermined number of iterations denoted

by nmax.

2.1.3 Linear Adjustment Policy (LAP)

The Linear Adjustment Policy is an adaptive policy that takes a set of base time-of-use tolls

{¯p(t)}T

t=1 and travel time savings {∆¯

T(t)}T

t=1 as inputs. Every time the toll is updated, it compares

the current travel time savings to the base values. If the system is more congested than expected,

the policy increases the toll relative to the base toll. If there is less congestion than expected, the

policy decreases the toll. The form of this policy is given by

p(t, ∆T(t)) = ¯p(t) + α+(t)(∆T(t)−∆¯

T(t))+−α−(t)(∆ ¯

T(t)−∆T(t))+,(4)

where α+(t) and α−(t) are positive scalars.

For the Linear Adjustment Policy, we start with a set of time-of-use tolls calculated as described

above. Then, we estimate the adjustment factors using the FDSA method. Details of the calculation

15

of the adjustment factor can be found in the Appendix in Section A.4.

2.2 Eastbound SR 91

In this example we analyze the Eastbound traﬃc scenario. The calibration of the demand generation

and consumer choice model components for this direction were described earlier in Sections 1.1 and

1.2. We generated 1,000 sample paths for the traﬃc demand and performed our analysis on the

same set of sample paths in every case.

Table 1 reports the average revenues and the 90% conﬁdence intervals for the Myopic Policy with

diﬀerent update intervals. From the results we can see that there is a slight but consistent decrease

in the expected revenues as the update interval increases. However, since all conﬁdence intervals

contain zero we cannot conclude that this decrease is statistically signiﬁcant. Thus, the tolling

frequency does not appear to have a signiﬁcant eﬀect on the performance of the Myopic Policy.

Figure 7 shows the average myopic toll (60 min. tolling interval) and the average hourly traﬃc

load. During the oﬀ-peak hours, the average toll is relatively stable. During the peak hours, the toll

increases as the congestion build-ups in the unmanaged lanes, and later decreases to its oﬀ-peak

value.

We now consider Time-of-Use policies. We explored the performance of a hourly time-of-use tolling

schedule to match the real-life implementations of such policies. Before starting the stochastic

approximation procedure, we obtained two diﬀerent starting points. For the ﬁrst one we assumed

that the demand is deterministic and equal to its certainty equivalent (CE) values. In the second

case, we optimized over hundred randomly drawn sample paths – an approach known assample

average approximation SAA. We used the Nelder-Mead nonlinear optimization heuristic, and we

tried one hundred diﬀerent random starting points in each case.

We set the upper bound on the toll to $100. Figure 8(a) depicts the tolls obtained through the

stochastic approximation procedure and the average hourly traﬃc load. The 2-tuple in the legend

indicates the starting point and the number of iterations performed, respectively. The second part of

the ﬁgure reports the market shares of the managed lanes for the tolling schedules given in the ﬁrst

Tolling Interval 1 min. 5 min. 10 min. 15 min. 20 min. 30 min. 60 min.

Avg. Rev. $125,157 $125,095 $124,859 $124,647 $124,547 $124,510 $124,511

C.I. Lower Bound -$3,500.61 -$3,254.63 -$2,219.23 -$1,184.97 -$626.42 -$473.75 -

C.I. Upper Bound $2,207.17 $2,086.27 $1,522.01 $912.19 $554.71 $475.85 -

Table 1: Average revenues and conﬁdence intervals for the Myopic Policy.

16

● ● ● ● ●●●●●●●●●●●

●

●

●

●

●

● ● ● ●

Hour

Toll ($)

0 2 4 6 8 10 12 14 16 18 20 22

0 2 4 6 8 10 12 14

0 2000 4000 6000 8000 10000

Cars/Hour

●Avg. Myopic Toll (60 min.)

Avg. Demand

Figure 7: Myopic tolls and mean hourly demand.

part of the ﬁgure. The structure of all three policies are very similar. When the traﬃc load is low,

the tolls are also quite low and stable in the region of $3. A few hours before the peak arrival traﬃc

is observed, the tolls go up to very high levels and eﬀectively divert all arrivals into the unmanaged

lanes. By diverting almost all arriving vehicles into the unmanaged lanes, the toll operator achieves

two goals: he reserves capacity in the managed lanes for the peak hours and increases congestions

in the unmanaged lanes. These two eﬀects combine to increase the attractiveness of the managed

lanes during the peak hours – which enables the operator to extract more revenue from arriving

traﬃc just when the volume of arrivals is highest. We term this a jam and harvest approach. From

Table 2 and Figure 9 we can see that this approach translates into substantial revenue improvements

over the Myopic Policy. When the Time-of-Use Policy sets its tolls high, a minuscule amount of

revenue is earned since almost all drivers choose the unmanaged lanes. By forgoing the revenue in

this period of time, the operator earns substantially more revenues when the jamming period ends

and the harvest period begins.

We note that, in several cases the policies recommend a toll at the upper bound of $100. This would

suggest that, during these periods, the optimal policy is to forgo almost all revenue from incoming

Static Policy (CE, 5k) (SAA, 1k) (SAA, 5k)

Avg. Rev. $153,086.51 $152,029.13 $151,532.68

% Imp. over Myopic 22.38% 21.53% 21.13%

(1 min. tolling update)

Table 2: Performance of the static Time-of-Use Policies for the Eastbound Case.

17

●●●●●●●●●●●●●

●●

●●●

●

●

●●●●

Hour

Toll ($)

0 2 4 6 8 10 12 14 16 18 20 22

0 20 40 60 80 100

0 2000 4000 6000 8000 10000

Cars/Hour

●(CE, 5k)

(SAA, 1k)

(SAA, 5k)

Avg. Demand

(a) Time-of-use tolls and average hourly

traﬃc load.

Hour

Man. Lanes Market Share

0 2 4 6 8 10 12 14 16 18 20 22

0.0 0.1 0.2 0.3 0.4

(CE, 5k)

(SAA, 1k)

(SAA, 5k)

(b) Market share of managed lanes. (1 min. granular-

ity)

Figure 8: Time-of-use tolls and the corresponding market share of managed lanes for the Eastbound

case.

traﬃc in favor of diverting that traﬃc to the unmanaged lanes in order to increase congestion in

those lanes.

Almost 70% of the daily revenues come between the hours of 4-8 pm Thus, we calibrate the Linear

Adjustment Policy to those hours. We use the (CE, 5k) Time-of-Use Policy to form the base tolls

and time savings since this policy resulted in the highest expected revenue among the non-adaptive

policies considered. We allow α−and α+to vary hourly. From the results in Table 15, we can

see that α−is always much higher than α+. This is a result of the non-linearity of the speed-

density relationship shown in Figure 6; travel times get increasingly more sensitive to density as

traﬃc increases up to the jam density. The operator realizes substantially more revenue when the

time diﬀerential is high. Because of the serial correlation of traﬃc demand, higher demand now is a

strong indicator that future demand is likely to be higher so raising current tolls is likely to increase

the future time-diﬀerential enabling even higher revenue later. On the other hand, if traﬃc is lower

than expected, it is likely that future traﬃc is also lower and there is a motivation to reduce current

tolls towards the Myopic Policy tolls in order to generate higher revenue from current traﬃc.

Table 3 shows the average revenues for diﬀerent tolling intervals for the Linear Adjustment Policy

along with the 90% conﬁdence intervals for the revenue diﬀerences compared to the Linear Ad-

justment Policy with a 60 minute tolling interval. In this case, there appears to be no advantage

to increasing the frequency of updates. Furthermore, from Figure 9 we can see that the Linear

Adjustment Policy results in tolls that are similar in structure to the Time-of-Use Policy.

18

●●●●●●●●●●●●●●●

●

●●

●

●

●●●●

Hour

Hourly Rev. ($)

0 2 4 6 8 10 12 14 16 18 20 22

0 10000 20000 30000 40000

●Myopic (5 min.)

(CE, 5k)

(SAA, 1k)

(SAA, 5k)

LinTD (1 min.)

Figure 9: Average hourly revenues from diﬀerent policies for the Eastbound Case.

Tolling Interval 1 min. 5 min. 10 min. 15 min. 20 min. 30 min. 60 min.

Avg. Rev. $167,338 $167,328 $167,456 $167,595 $168,058 $167,119 $167,709

% Imp. over

Policy (1 min.) 33.7% 33.7% 33.8% 33.9% 34.3% 33.35% 34.0%

C.I. Lower Bound -$14,473.18 -$8,684.09 -$10,190.90 -$7,006.63 -$8,920.62 -$9,970.29 –

C.I. Upper Bound $15,213.38 $9,444.61 $10,695.74 $7,233.73 $8,221.04 $11,149.70 –

Table 3: Average revenues, percent improvement over Myopic Policy with 1 minute period and

conﬁdence intervals for the Linear Adjustment Policy for the Eastbound case.

19

Myopic Time-of-Use LAP Computational

1 min. CE, 5k 20 min. Bound

Avg. Rev. $125,157 $153,087 $168,058 $179,444

Rel. Gap 30.25% 14.69% 6.34% –

Table 4: Summary of policies and comparison to the computational upper bound for the Eastbound

case.

So far we analyzed each policy separately. Now, we compare their performance to each other and

also to a computational upper bound where the operator is assumed to know the whole traﬃc pat-

tern for each day and can set tolls that maximize revenue given that knowledge. The computational

upper bound is obtained by computing the revenue-maximizing tolls for each sample path and then

averaging them. Consistent with the SR 91 policy, we assumed that tolls could be changed only

hourly.

Table 4 shows the expected revenue from three of the policies compared to the computational upper

bound for the Eastbound Case. The Myopic Policy generates the least revenue of the three. We

attribute this to the inability of the Myopic Policy to set tolls that anticipate future traﬃc demand.

The Time-of-Use Policy outperforms the Myopic Policy by more than 20%. This uplift comes almost

entirely from additional peak revenue resulting from the jam and harvest policy. Making the policy

adaptive using linear adjustments adds an additional 10% of revenue. Furthermore, the Linear

Adaptive Policy achieves 93.7% of the computational upper bound, which is impressive given that

the computational upper bound assumes full knowledge of demand for each sample path.

2.3 Westbound SR 91 Results

As shown in Figure 1, Westbound traﬃc experiences a higher morning peak and a lower afternoon

peak than Eastbound traﬃc. Furthermore, the larger morning peak is less pronounced than the

afternoon peak for the Eastbound traﬃc. Parameters for the Westbound demand model were

generated in the same fashion as the Eastbound case and can be found in the appendix. The same

methods were used to estimate the Time-of-Use tolls and the adjustment parameters as in the

Eastbound case.

Figure 10 shows the average hourly demand, the time-of-use tolls and the corresponding market

share of managed lanes for the Westbound case. The Time-of-Use Policy again seeks to jam the

unmanaged lanes by setting the toll very high between 6-7 AM in anticipation of the morning peak.

As a result, the managed lanes’ market share dips and most of the incoming vehicles choose the

unmanaged lanes, which in turn creates increased congestion in the unmanaged lanes.

20

●●●●●

●

●

●

●

●

●●●●●●●●●●●●●●

Hour

Toll ($)

0 2 4 6 8 10 12 14 16 18 20 22

0 2 4 6 8 10 12 14 16 18

0 2000 4000 6000 8000

Cars/Hour

●Toll

Avg. Demand

(a) Time-of-use tolls and average hourly

traﬃc load.

Hour

Man. Lanes Market Share

0 2 4 6 8 10 12 14 16 18 20 22

0.00 0.05 0.10 0.15 0.20 0.25 0.30

(b) Market share of managed lanes. (1 min. granular-

ity)

Figure 10: Time-of-use tolls and the market share of managed lanes for the Westbound case.

The revenues obtained from diﬀerent policies in the Westbound Case are given in Table 5. Consis-

tent with the Eastbound Case, we can see that increasing the tolling frequency for adaptive policies

does not result in signiﬁcant additional revenue. The Time-of-Use and Linear Adjustment policies

both outperform the Myopic Policy and the Linear Adjustment Policy outperforms the Time-of-Use

Policy. However, these policies generate less additional revenue relative to the Myopic Policy than

in the Eastbound Case. The gap between the Linear Adjustment Policy, and the computational

upper bound is also higher. A potential explanation for these two observations stems from the traf-

ﬁc load being spread out more evenly compared to the Eastbound case. As a result, the operator

does not have the same scope for increasing unmanaged lane congestion by diverting traﬃc into the

managed lanes. Furthermore, the mean traﬃc demand in the Eastbound case appears to be closer

to a threshold on the magnitude of peak demand at which the value of Time-of-Use pricing changes

dramatically. This means that more of the sample paths fall on either side of this threshold and

the value of full knowledge of demand is greater than when the peak was more pronounced in the

Eastbound Case. We will make this discussion more concrete in the next section.

2.4 Stylized Models

Our analysis of the SR 91 Westbound and Eastbound traﬃc showed that both adaptive and non-

adaptive time-of-use policies generated signiﬁcantly more revenue than myopic pricing. In all

Time-of-Use policies, the recommended tolls had a jam and harvest character – that is, they raise

21

Policy Myopic Time-of-Use LinTD Comp. Bound

Tolling Interval 1 min. 60 min. 60 min. 1 min. 60 min. 60 min.

Revenue $167,325 $167,031 $171,831 $181,454 $181,680 $210,698

Rel. Gap 20.59% 20.73% 18.45% 13.88% 13.77% –

(vs. Comp. Bound)

% Imp. over Myopic – – 2.69% 8.44% 8.58% 25.92%

(1 min. tolling update)

Table 5: Revenues from diﬀerent policies for the Westbound case.

tolls in advance of an anticipated peak in order to divert more traﬃc into the unmanaged lanes.

This has the dual eﬀect of increasing congestion in the unmanaged lanes and reducing congestion in

the managed lanes, thereby making the managed lanes relatively more attractive during the peak

which enables the controller to set higher tolls during the peak and the resulting increased revenue

more than makes up for the reduced oﬀ-peak revenue. This suggests that a critical characteristic

of a successful policy is the ability to anticipate peaks and set tolls accordingly.

While the character of the Time-of-Use policies was similar for both the Eastbound and Westbound

cases, the gain over the Myopic Policy was signiﬁcantly diﬀerent. In particular, the magnitude of

the gain was considerably greater on the Eastbound traﬃc pattern, which had a very high afternoon

peak, than on the Westbound traﬃc pattern in which the peak was less pronounced. This raises

the question of the extent to which the results depend upon the magnitude and duration of peak

demand relative to oﬀ-peak demand. To address this question, we created some simple stylized

models of traﬃc demand.

2.5 Stylized Models

Each of the stylized models considers a 24-hour day and has the general form shown in Figure 11.

Traﬃc at the beginning of the day is set at the oﬀ-peak level. Traﬃc then rises linearly through a

transition period to a peak. Traﬃc demand remains at the peak level for a number of hours, then

decreases linearly through another transition period back to oﬀ-peak levels. We assume that both

transition periods are of equal length. To create diﬀerent scenarios, we vary the peak demand, the

oﬀ-peak demand, the length of the peak, and the length of the transition periods.

In each scenario, we assumed deterministic arrival rates and used the same traﬃc model and choice

models described in Sections 1.3 and 1.2. We calculated the tolls and revenues generated by both

the non-adaptive Time-of-Use Policy and the Myopic Policy. Since traﬃc demand is deterministic,

we did not test any adaptive models. We used the Nelder-Mead heuristic with 20 diﬀerent randomly

chosen starting points to compute the Time-of-Use tolls.

22

Figure 11: Demand structure of the stylized deterministic model.

Table 6 reports the gap between Time-of-Use and Myopic policies for diﬀerent combinations of the

settings. The Time-of-Use Policy outperforms the Myopic Policy in all cases. This supports the

idea that anticipating peaks and pricing accordingly is a characteristic of any successful policy.

The relative beneﬁts of time-of-use pricing generally (but not uniformly) increase as a function

of peak hourly demand, length-of-peak, and transition length. The most striking characteristic of

the relative performance is the existence of some sharp transitions. When peak hourly demand is

less than 8,000 cars/hour, the additional beneﬁts from the Time-of-Use Policy are always less than

3.0%, regardless of the settings of the other parameters. When demand rises above 9,000 vehicles

per hour and the length of the peak is two or three periods, the beneﬁts from the Time-of-Use

Policy increase substantially.

The results in Table 6 show a number of these sharp transitions, notably from 8,000 to 9,000 vehicles

per hour when the peak length is three and from 9,000 to 10,000 cars/hour when the peak length is

two. These transitions result from the non-linearity of the speed-volume relationship combined with

the nature of the jam-and-harvest policy. When total peak demand – measured as a combination of

peak vehicles/hour and length of peak – is relatively low, the controller is unable to divert enough

vehicles into the unmanaged lanes to signiﬁcantly inﬂuence the travel time diﬀerential. In this

case, the additional beneﬁt from time-of-use pricing is low. However, when peak hourly/demand

is high, the controller can signiﬁcantly inﬂuence travel time diﬀerential by using high tolls early to

divert vehicles into the unmanaged lanes. This increases congestion in the unmanaged lanes and

reduces congestion in the managed lanes relative to the Myopic Policy, which enables the controller

to generate higher revenue during the peak. The beneﬁts of a jam and harvest policy in this cases

are substantial – over 50% when peak demand is high.

The existence of these thresholds is a likely explanation of the diﬀerence in the revenues from the

Time-of-Use and the Linear Adjustment Policies relative to the computational upper bound on

revenue as noted in the previous section. If the mean peak demand is signiﬁcantly high, than the

23

Table 6: Revenue gap between Time-of-Use and Myopic polices for diﬀerent traﬃc patterns in the

stylized deterministic model in Figure 11.

Transition Length 0 1 2

Length of Peak 1 2 3 1 2 3 1 2 3

Peak Hourly Dem.

7000 1.37% 1.24% 1.09% 1.41% 1.51% 1.61% 0.67% 1.23% 0.95%

8000 0.95% 1.22% 1.08% 1.09% 1.11% 0.98% 1.23% 1.01% 1.36%

9000 1.28% 4.70% 21.63% 1.64% 7.70% 17.53% 2.24% 9.14% 23.79%

10000 2.35% 22.28% 37.70% 2.93% 23.40% 38.39% 4.41% 34.74% 53.25%

(a) Oﬀ-peak demand is 4000 vehicles/hour.

Transition Length 0 1 2

Length of Peak 1 2 3 1 2 3 1 2 3

Peak Hourly Dem.

7000 2.53% 2.29% 2.69% 2.28% 2.21% 2.63% 2.45% 2.33% 2.45%

8000 2.75% 2.49% 2.65% 2.42% 1.81% 1.84% 2.29% 2.20% 1.97%

9000 2.80% 6.33% 12.94% 3.28% 9.09% 21.39% 3.75% 10.67% 31.85%

10000 3.84% 22.79% 40.63% 8.44% 25.96% 40.17% 13.13% 38.66% 54.97%

(b) Oﬀ-peak demand is 5000 vehicles/hour.

Transition Length 0 1 2

Length of Peak 1 2 3 1 2 3 1 2 3

Peak Hourly Dem.

7000 1.09% 1.16% 1.27% 0.85% 1.03% 0.97% 1.23% 1.20% 1.38%

8000 1.19% 0.99% 0.58% 1.04% 1.25% 0.96% 0.67% 0.48% 0.80%

9000 3.63% 5.40% 14.13% 3.92% 7.90% 17.83% 5.31% 11.60% 41.17%

10000 4.00% 17.50% 36.90% 4.67% 34.65% 54.69% 24.66% 50.81% 66.29%

(c) Oﬀ-peak demand is 6000 vehicles/hour.

24

vast majority of sample paths will fall in the region in which Time-of-Use policies are eﬀective –

corresponding to the parameters in Table 6 for which the beneﬁts of the Time-of-Use Policy are

high, then using the same jam and harvest policy every day will be eﬀective at capturing much of

the available revenue for each sample path. If, on the other hand, the mean demand is close to the

threshold, then jam and harvest will be very eﬀective on days when demand is above the threshold

but counter-productive on days when demand falls below the threshold. In this case, knowledge of

the full pattern of demand would be much more valuable than if demand is consistently above or

below the threshold.

3 Discussion

A key insight from this work is that the most important aspect of maximizing revenue from managed

lanes is managing around peaks. If traﬃc is always low, no policy can do appreciably better than

the Myopic Policy of individually maximizing the expected revenue from each entering vehicle.

However, in the presence of peaks, policies that anticipate future traﬃc demand and set tolls

accordingly outperform the Myopic Policy in every case studied. In particular, the best-performing

policies raise tolls well above the Myopic Policy in the periods before the peak in order to divert

traﬃc from the managed to the unmanaged lanes. This increases congestion in the unmanaged

lanes while reducing congestion in the managed lanes and increases the relative attractiveness of

the managed lanes to arriving traﬃc. This allows the toll operator to charge higher rates in the

peak than would have proﬁtable under the Myopic Policy, more than making up for any revenue

lost in the oﬀ-peak period. These jam and harvest policies are very robust and outperform the

Myopic Policy not only using both realistic Eastbound and Westbound traﬃc demand from the SR

91 but also in highly stylized models whenever there is a substantial peak.

For each of the cases studied with uncertain demand, the non-adaptive Time-of-Use Policy out-

performed the Myopic Policy and the adaptive Linear Adjustment Policy outperformed both. The

revenue from each policy is shown along with the computed upper bound for both Eastbound and

Westbound traﬃc in Figure 12. For the Eastbound traﬃc, the greatest gain was from the Time-of-

Use policy with a smaller relative beneﬁt from making this policy adaptive. The opposite pattern

is observed for Westbound traﬃc. We conjecture that the higher gain for the Time-of-Use for the

Eastbound traﬃc is due to the more pronounced peak than in the Westbound traﬃc.

We showed using deterministic stylized models that the beneﬁt from a Time-of-Use Policy over a

Myopic Policy is highly sensitive to the length and magnitude of the peak. When the peak demand

is small and/or of short duration relative to the oﬀ-peak demand, Time-of-Use policies gain little

25

$-

$50,000

$100,000

$150,000

$200,000

$250,000

Eastbound Westbound

Myopic ToU LinTD Upper Bnd

Figure 12: Revenue generated by diﬀerent policies for Eastbound and Westbound traﬃc shown

with the computational upper bound for revenue in each direction.

over a Myopic Policy. However, when peak demand is high and/or of long duration, the gain from

a Time-of-Use Policy can be 50% or more. Furthermore, the data shows sharp transitions between

a regime in which Time-of-Use provides relatively small additional beneﬁts (5% or less) and a

regime in which the beneﬁts are 20% or higher. We attribute the existence of these transitions

to the nature of the “jam and harvest” policy combined with the non-linearity of the underlying

speed-density relationship.

Policies that sought to maximize revenue consistently set tolls that were higher than the myopic

policy. The diﬀerence can be attributed in part to the existence of two negative externalities

associated with a vehicle that chooses the managed lanes. The ﬁrst is the additional congestion in

the managed lanes created by that vehicle. The second negative externality is that, by choosing

the managed lanes, the vehicle is not contributing to congestion in the unmanaged lanes. The

combination of these two eﬀects means that each vehicle that chooses the managed lanes rather than

the unmanaged lanes leads to a reduced time diﬀerential, making the managed lanes less attractive

to future arrivals. In this sense, the economics similar to Paris Metro Pricing as described by

Odlyzko (1999). In Paris Metro Pricing, two identical units of capacity – such as train cars –

are priced diﬀerently with the idea that some customers will be willing to pay more to use the

less congested alternative. To the extent that customers prefer less crowded capacity, PMP can

26

improve both revenue and customer satisfaction compared to pricing both units identically. While

the fundamental idea of tolling for managed lanes is similar, the calculation of tolls is far more

complicated due to the dynamic and non-linear relationship between traﬃc and the time-diﬀerential.

A somewhat counter-intuitive result from the simulations was that the time interval at which tolls

were updated did not have a strong inﬂuence on revenue in the range from one minute to one

hour. For example, for the SR 91 adjusting the the Myopic Policy every minute resulted in a

gain of only .5% relative to adjusting the toll every hour. Similarly, the Linear Adjustment Policy

showed no signiﬁcant gain from being adjusted at intervals of a minute versus intervals of an hour.

These results need further validation, however, they suggest that frequent toll changes may not be

necessary to capture the majority of the revenue available.

Our model was parameterized using cleansed data that excluded exceptional situations such as lane

closures, traﬃc accidents and unforeseen weather conditions. For this reason, our analysis almost

certainly underestimates the value of adaptive policies relative to non-adaptive policies. Speciﬁcally,

exceptional events are more likely to lead to higher congestion rather than lower congestion in the

unmanaged lanes. An adaptive policy would detect this diﬀerence and adjust the tolls accordingly,

extracting more revenue than a non-adaptive policy. However, in reality, exceptional conditions

will typically be managed directly by the operator who can use knowledge about the nature and

expected duration of the exceptional conditions to set and update the tolls.

Our analysis has assumed that traﬃc demands were generated exogenously – that is, that total

demand in each time period was not inﬂuenced by either the toll or the travel-time diﬀerential. In

actuality, some drivers may have the ﬂexibility to change their travel plans – either by choosing a

diﬀerent departure time or a diﬀerent route – in order to avoid high tolls and/or heavy congestion.

From the initial paper by Wardrop (1952), extensive research has been performed on traﬃc equilib-

rium models which assume that at least some drivers will choose their routes in order to minimize

travel time given congestion and/or tolls. Incorporation of such strategic behavior on the part of

drivers may change some of the results and is a topic for on-going research. However, we note that

the existence of predictable traﬃc jams at “rush hours”, often combined with high tolls on many

urban highways suggests that many drivers are unable or unwilling to change their routes and/or

departure times easily. We suspect that the incorporation of strategic behavior on the part of some

subset of drivers would not change the qualitative nature of the optimal tolling policies.

27

Acknowledgments

The authors would like to acknowledge the very detailed comments by Philipp Afeche that signiﬁ-

cantly improved the paper.

References

Ben-Akiva, M., M. Bierlaire, H. Koutsopoulos, R. Mishalani. 2002. Real Time Simulation of Traﬃc

Demand-Supply Interactions Within DynaMIT, chap. 2. Kluwer Academic Publishers, 19–36.

Ben-Akiva, Moshe, Haris N Koutsopoulos, Constantinos Antoniou, Ramachandran Balakrishna.

2010. Traﬃc Simulation with DynaMIT, chap. 10. Springer, 363–398.

Ben-Akiva, Moshe, Steven Lerman. 1985. Discrete choice analysis: theory and application to predict

travel demand. MIT press.

Blum, Julius R. 1954. Multidimensional stochastic approximation methods. The Annals of Math-

ematical Statistics 25(4) 737–744.

Brent, Richard P. 2002. Algorithms for minimization without derivatives. Dover Publications.

Chung, C. L., W. Recker. 2011. State-of-the-art assessment of toll rates for high-occupancy and

toll lanes. Proceedings of the Transportation Research Board 90th Annual Meeting.

G¨o¸cmen, Caner. 2013. Pricing managed lanes. Columbia University.

Han, Lee D, Fang Yuan, Shih-Miao Chin, Holing Hwang. 2006. Global optimization of emergency

evacuation assignments. Interfaces 36(6) 502–513.

Jayakrishnan, R., H. S. Mahmassani, T. Hu. 1994. An evaluation tool for advanced traﬃc infor-

mation and management systems in urban networks. Transporation Research C 2129–147.

Kiefer, Jack, Jacob Wolfowitz. 1952. Stochastic estimation of the maximum of a regression function.

The Annals of Mathematical Statistics 23(3) 462–466.

Lam, Terence, Kenneth Small. 2001. The value of time and reliability: measurement from a value

pricing experiment. Transportation Research Part E: Logistics and Transportation Review 37(2)

231–251.

Liu, Henry X, Xiaozheng He, Will Recker. 2007. Estimation of the time-dependency of values

of travel time and its reliability from loop detector data. Transportation Research Part B:

Methodological 41(4) 448–461.

28

Liu, Henry X, Will Recker, Anthony Chen. 2004. Uncovering the contribution of travel time

reliability to dynamic route choice using real-time loop data. Transportation Research Part A:

Policy and Practice 38(6) 435–453.

Odlyzko, Andrew. 1999. Paris metro pricing for the internet. Proceedings of the First ACM

Confrence on Electronic Commerce. 140–147.

Olstam, Johan Janson, Andreas Tapani. 2004. Comparison of Car-following models. Swedish

National Road and Transport Research Institute.

Perez, Benjamin G., Charles Fuhs, Colleen Gants, Reno Giordano, David H. Ungemah. 2012. Priced

managed lane guide. Tech. Rep. FHWA-HOP-13-007, US Department of Transportation, Federal

Highway Adminsitration.

Roelofsen, Mark. 2012. Dynamic modelling of traﬃc management scenarios using dynasmart. Tech.

rep., University of Twente.

Spall, James C. 2003. Introduction to stochastic search and optimization: estimation, simulation,

and control, vol. 64. Wiley-Interscience.

Sullivan, Edward. 2000. Continuation study to evaluate the impacts of the sr 91 value-priced express

lanes. Tech. rep., Cal Poly State University.

The Orange County Transportation Authority. 2013. The sr-91 express lanes.

http://www.91expresslanes.com/. Accessed on 6.2..2013.

Wardrop, J. G. 1952. Some theoretical aspects of road traﬃc research. Proceedings of the Institute

of Civil Engineers 11(1) 325–378.

Xu, Shunan. 2009. Development and test of dynamic congestion pricing model. Master’s thesis,

Massachusets Institute of Technology.

Yang, Li. 2012. Stochastic traﬃc ﬂow modeling and optimal congestion pricing. Ph.D. thesis,

University of Michigan.

Yin, Yafeng, Yingyan Lou. 2009. Dynamic tolling strategies for managed lanes. Journal of Trans-

portation Engineering 135(2) 45–52.

29

A Model Calibration

A.1 Demand Generation Module

To calibrate the hourly demand generator we used the hourly ﬂow data for the SR 91 from January

2009 to August 2011. We combined the volume information from the managed and unmanaged

lanes to calculate the total volume of traﬃc using the highway. Traﬃc data for managed lanes

comes from VDS 1208156 and for unmanaged lanes we use the data from VDS 1208147. Figure

1 depicts the average hourly traﬃc volumes for each day of the week for Monday through Friday

for both Eastbound and Westbound traﬃc on the SR 91. The demand pattern is similar for all

days so we used an average across days to create a mean demand for each hour. Based on this

data, we used Ordinary Least Squares to estimate the parameters of Equation 1 using Ordinary

Least Squares (OLS). The resulting parameter estimates by hour are shown in Tables 7 and 8 for

Eastbound and Westbound traﬃc respectively.

t βtα1

tα2

tα3

tStd. Dev. of Residual (εt)

0 116.94 0.67 -0.01 -0.03 180.51

1 152.92 0.73 -0.19 0.04 159.63

2 259.67 0.73 -0.09 0.00 83.25

3 303.68 0.71 -0.20 0.06 62.11

4 379.54 1.67 -0.38 -0.12 130.38

5 288.02 2.35 -0.20 -0.44 165.62

6 792.27 1.59 -0.11 -0.23 204.16

7 1091.10 0.77 0.35 -0.05 195.74

8 1540.70 0.47 0.37 -0.06 220.13

9 1818.52 0.56 0.04 0.01 354.20

10 197.21 1.37 -0.14 -0.22 270.91

11 810.85 1.01 -0.02 -0.05 287.93

12 452.39 0.95 -0.12 0.20 315.61

13 2122.24 1.10 -0.24 -0.10 388.58

14 4729.05 1.13 -0.28 -0.49 519.94

15 6479.56 1.14 -0.25 -0.89 734.43

16 2618.14 0.87 -0.11 -0.20 644.00

17 1599.20 0.78 0.00 -0.04 538.35

18 1371.06 0.69 -0.01 0.12 513.08

19 3602.23 0.73 -0.27 0.01 528.49

20 4118.83 0.78 -0.29 -0.15 505.42

21 1082.86 1.00 -0.12 -0.11 424.01

22 212.01 1.09 0.00 -0.26 565.98

23 -280.13 0.90 -0.16 0.01 442.78

Table 7: Hourly demand model parameters for the Eastbound direction.

30

t βtα1

tα2

tα3

tStd. Dev. of Residual (εt)

0 692.42 0.10 0.09 0.01 180.51

1 218.53 0.69 0.11 -0.12 159.63

2 310.82 0.89 -0.13 -0.02 83.25

3 881.90 1.12 -0.44 -0.11 62.11

4 1626.93 3.23 -1.71 -1.00 130.38

5 1565.48 1.85 0.33 -1.65 165.62

6 2309.88 0.96 0.21 -1.52 204.16

7 856.04 0.63 0.21 0.05 195.74

8 1339.35 0.66 -0.11 0.23 220.13

9 2555.56 0.43 0.06 0.08 354.20

10 2736.31 0.50 -0.02 0.04 270.91

11 1855.47 0.83 -0.10 -0.07 287.93

12 1030.70 0.85 0.10 -0.14 315.61

13 30.46 0.80 0.14 0.07 388.58

14 -350.95 0.98 -0.12 0.27 519.94

15 813.96 1.07 -0.04 -0.09 734.43

16 323.55 1.03 0.04 -0.12 644.00

17 73.59 0.65 0.36 -0.03 538.35

18 -277.89 0.60 -0.03 0.33 513.08

19 282.95 0.94 -0.37 0.14 528.49

20 417.60 1.03 -0.18 -0.05 505.42

21 295.30 0.94 -0.07 -0.03 424.01

22 -216.51 0.90 0.07 -0.07 565.98

23 390.16 0.97 -0.26 -0.08 442.78

Table 8: Hourly demand model parameters for the Westbound direction.

31

0 2 4 6 8 10 12 14 16 18 20 22

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Hour

Cars

Real Data

Simulated Data

(a) Comparison of means

0 2 4 6 8 10 12 14 16 18 20 22

0

200

400

600

800

1000

1200

1400

Hour

Cars

Real Data

Simulated Data

(b) Comparison of standard deviations

0 2 4 6 8 10 12 14 16 18 20 22

0.4

0.5

0.6

0.7

0.8

0.9

1

Hour

Cars

Real Data

Simulated Data

(c) Comparison of autocorrelations with a lag of one

Figure 13: Demand model validation

In order to test the validity of this approach we generated 1000 sample paths and compared their

statistics to the real-life traﬃc data. Figure 13 depicts the hourly means, standard deviations and

autocorrelation (with a lag of one) for both the data and the sample paths on the Eastbound

traﬃc. As can be seen from the ﬁgure, the statistics of the generated demand matched those of the

real-life data quite well. Similar ﬁts were obtained for the Westbound traﬃc (not shown). Tables

7 and 8 gives the parameters for the ﬁtted demand model for Eastbound and Westbound traﬃc

respectively.

To start generating hourly loads from midnight and onwards, we need starting values for the hours

21, 22 and 23. For simplicity, we sampled the demand for these three hours from normal distri-

butions whose means, standard deviations and pairwise correlations match their real-life values.

These statistics are shown in Tables 9 and 10.

32

Eastbound Westbound

Mean Std. Dev. Mean Std. Dev.

Hour 21 5887.20 862.36 2958.30 529.14

Hour 22 4940.18 1142.69 2339.71 506.18

Hour 23 3351.17 1103.87 1633.21 356.85

Table 9: Mean and standard deviation of traﬃc volume for the hours used to start the demand

generation module.

Eastbound Westbound

Hours 21 & 22 0.78 0.93

Hours 21 & 23 0.62 0.77

Table 10: Correlations between hours used to start the demand generation module.

A.2 Consumer Choice Module

The SR 91’s policy leads to high correlation between the tolls and time savings. To estimate βTand

βpwe use an approach similar to Lam and Small (2001) in which we exploit the variation in time

savings during the peak hours when the tolls do not change very much. Figure 14 plots the average

ratio of expected time savings to tolls in ﬁve minute intervals. There is signiﬁcant variation in that

ratio during both peak and early morning hours. For our analysis we choose the afternoon peak

hours, speciﬁcally between the hours of 2 PM and 8 PM, since the traﬃc volume is signiﬁcantly

higher compared to the early morning hours.

0 2 4 6 8 10 12 14 16 18 20 22 24

0.5

1

1.5

2

2.5

3

3.5

Hour

Time Savings to Tolls Ratio

Figure 14: The average ratio of time savings to tolls.

33

Now, we are in a position to use MLE to estimate βTand βp. The dataset we use contains the

number of vehicles that chose the unmanaged and managed lanes between 2 PM and 8 PM in ﬁve

minute granularity. It also contains the expected time savings the vehicles choosing the managed

lanes enjoyed as well as the tolls they paid. There are 390,310 vehicles that chose the unmanaged

and 140,931 that chose the managed lanes in the dataset.

In the estimation procedure we treat the aggregated ﬁve minute market share of the managed lanes

as the dependent variable. The weight of each observation is equal to the total number of vehicles

that pass through the unmanaged and managed lanes in that ﬁve minute time frame.

We evaluated the performance of the three diﬀerent structures introduced for guk . For each struc-

ture, we tested four models which correspond to cases where the coeﬃcients are allowed to vary

over time or kept ﬁxed. In the former case, diﬀerent values for coeﬃcients are estimated on a hourly

basis. The coeﬃcients for a given point in time are determined by taking the weighted average

of the estimates corresponding to the current and upcoming hour, where the weights are obtained

proportionally. For example, the coeﬃcients at 2:20 PM would be the weighted averages of the

coeﬃcients for 2:00 PM and 3:00 PM with the weights 2/3 and 1/3, respectively. As discussed

in the next, we considered three possible transformations for the travel-time savings (∆T), ln(∆),

∆T, and ∆T2. The resulting coeﬃcients and LLE’s are shown in Tables 11, 12 and 13. We chose

Model 3-2, which corresponds to the case of k= 3, as it has the best ﬁt among models with the

correct signs on the coeﬃcients. For the hours outside of the 2-8 PM range, we use the coeﬃcients

corresponding to 8 PM

The goodness-of-ﬁt plots for Model 3-2 are given in Figure 15. The color of each point in the scatter

plots depicts its weight. The darker a point is, the more weight it has. From the plots we can see

that the model ﬁts reasonably well to the data and there is no evident bias.

We did not account for high occupancy vehicles in our choice model. Under SR 91’s tolling policy,

vehicles that have at least three occupants (HOV3+) can use the managed lanes for free except

between 4 PM-6 PM when they have to pay 50% of the toll. According to traﬃc counts performed

on the eastbound direction of SR 91 between 3:30 PM5 and 5:30 PM, only 3.7% of the total vehicles

that entered into the unmanaged and managed lanes were in the HOV3+ category (Sullivan, 2000).

Because this percentage is so low, we do not feel that omitting the HOV3+ category signiﬁcantly

inﬂuenced our results.

34

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0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Predicted Market Share

Actual Market Share

(a) Actual vs. predicted market shares

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●

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−1.0 0.0 0.5 1.0 1.5 2.0 2.5

−1.0 0.0 0.5 1.0 1.5 2.0 2.5

Predicted Log Odds

Actual Log Odds

(b) Actual vs. predicted log of odds

Figure 15: Consumer choice model goodness-of-ﬁt plots.

A.3 Traﬃc Module

We calibrated the traﬃc model using the ﬁrst four weeks of July, 2011. This period was chosen

because there was no rain. The data was obtained through PeMS for VDS 1208147 with an

aggregation level of 5 minutes. We deleted approximately 17% of the observations as outliers

resulting from lane closures and accidents, resulting in 5,760 observations.

We set the jam density to 100/vehicles/mile/lane because there were only three observations with

densities greater than 100. We removed the observations from our data set. We set the minimum

speed vmin to 15 mph in accordance with the average speed observed when density reached 100

vehicles/mi/ln. We set the simulation time-step to a minute. The moving average of the last

ﬁve periods gave a fairly accurate representation of the discharge rate from each segment under

congestion. As shown in Equation 3, to ﬁt the historic speed and density observations, we divided

the data into two regions, one in which the density is below 25 vcl/mi/ln and one in which the

density is greater than 25 but less than 100 vcl/mi/ln. We ﬁt a line to the ﬁrst part and a geometric-

decay model with the form shown in Equation 3 to the second region. We breakpoint density of

25 vcl/mi/ln minimized the sum of squared residuals from both regions of the ﬁtted model. We

used OLS regression to estimate the parameters for the ﬁrst (linear) regime and NLS regression

to estimate the parameters in the second (nonlinear) regime. After the estimation procedure we

readjusted β2to ensure continuity at the breakpoint density. Table ?? shows the parameters that

35

were estimated using this procedure.

A.4 Numerical Study Data and Results

In the stochastic approximation procedure we used the same cksequence as we did in the calibration

of the Time-of-Use Policy. Table 14 shows the parameters used in the sequence ak. We performed

1000 iterations for each parameter. We used the same updating intervals as in the Myopic Policy

and calibrated a diﬀerent set of parameters for each one. The starting points were obtained through

the application of Brent’s method over 100 randomly chosen sample paths. Tables 15 and 16 show

the results for the Eastbound and Westbound traﬃc respectively.

36

Variable Model 1-1 Model 1-2 Model 1-3 Model 1-4

Toll −0.2460∗∗∗ −0.2253∗∗∗

(0.0011) (0.0012)

Toll - Hour 14 −0.4781∗∗∗ −0.3999∗∗∗

(0.004) (0.0067)

Toll - Hour 15 −0.2352∗∗∗ −0.3156∗∗∗

(0.0022) (0.0043)

Toll - Hour 16 −0.1595∗∗∗ −0.1611∗∗∗

(0.0018) (0.0028)

Toll - Hour 17 −0.1444∗∗∗ −0.1670∗∗∗

(0.0018) (0.0034)

Toll - Hour 18 −0.1770∗∗∗ −0.1495∗∗∗

(0.0022) (0.0040)

Toll - Hour 19 −0.3354∗∗∗ −0.2958∗∗∗

(0.003) (0.0048)

Toll - Hour 20 −0.4520∗∗∗ −0.3999∗∗∗

(0.0051) (0.0074)

Time Savings 0.0567∗∗∗ 0.0357∗∗∗

(5e-04) (5e-04)

Time Savings - Hour 14 −0.1969∗∗∗ −0.0453∗∗∗

(0.0041) (0.0065)

Time Savings - Hour 15 0.0534∗∗∗ 0.0922∗∗∗

(0.0014) (0.0025)

Time Savings - Hour 16 0.0588∗∗∗ 0.0348∗∗∗

(8e-04) (0.0013)

Time Savings - Hour 17 0.0690∗∗∗ 0.0462∗∗∗

(8e-04) (0.0014)

Time Savings - Hour 18 0.0503∗∗∗ 0.0241∗∗∗

(8e-04) (0.0016)

Time Savings - Hour 19 −0.0092∗∗∗ 0.0155∗∗∗

(0.0014) (0.0023)

Time Savings - Hour 20 −0.1217∗∗∗ −0.0201∗∗∗

(0.0038) (0.0051)

Log Likelihood -16294.6593 -7443.8744 -9295.4047 -6943.2609

***p < 0.01

Table 11: Parameter estimates for models with untransformed variables.

37

Variable Model 2-1 Model 2-2 Model 2-3 Model 2-4

Toll −0.1874∗∗∗ −0.2488∗∗∗

(0.0016) (0.0019)

Toll - Hour 14 −0.4792∗∗∗ −0.5423∗∗∗

(0.0042) (0.0081)

Toll - Hour 15 −0.2186∗∗∗ −0.1438∗∗∗

(0.0027) (0.0060)

Toll - Hour 16 −0.1237∗∗∗ −0.1528∗∗∗

(0.0021) (0.0057)

Toll - Hour 17 −0.1000∗∗∗ −0.2540∗∗∗

(0.0021) (0.0074)

Toll - Hour 18 −0.1340∗∗∗ −0.1128∗∗∗

(0.0027) (0.0055)

Toll - Hour 19 −0.3128∗∗∗ −0.3009∗∗∗

(0.0036) (0.0061)

Toll - Hour 20 −0.4491∗∗∗ −0.3859∗∗∗

(0.0052) (0.0096)

log(Time Savings) 0.1395∗∗∗ 0.0988∗∗∗

(0.0044) (0.0050)

log(Time Savings) - Hour 14 −0.5207∗∗∗ 0.3300∗∗∗

(0.0141) (0.0260)

log(Time Savings) - Hour 15 0.1559∗∗∗ −0.1408∗∗∗

(0.0077) (0.0176)

log(Time Savings) - Hour 16 0.4856∗∗∗ 0.2072∗∗∗

(0.0073) (0.0175)

log(Time Savings) - Hour 17 0.5443∗∗∗ 0.5396∗∗∗

(0.0067) (0.0207)

log(Time Savings) - Hour 18 0.3453∗∗∗ 0.0456∗∗∗

(0.0060) (0.0131)

log(Time Savings) - Hour 19 −0.0380∗∗∗ 0.0779∗∗∗

(0.0072) (0.0137)

log(Time Savings) - Hour 20 −0.4543∗∗∗ −0.1193∗∗∗

(0.0154) (0.0270)

Log Likelihood −22175.6920 −9437.6691 −10465.0480 −9022.4410

Num. obs. 720 720 720 720

***p < 0.01

Table 12: Parameter estimates for models with log. of time savings.

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Variable Model 3-1 Model 3-2 Model 3-3 Model 3-4

Toll −0.1954∗∗∗ −0.1882∗∗∗

(8e-04) (8e-04)

Toll - Hour 14 −0.4547∗∗∗ −0.4153∗∗∗

(0.004) (0.0045)

Toll - Hour 15 −0.1994∗∗∗ −0.2700∗∗∗

(0.0021) (0.0028)

Toll - Hour 16 −0.1360∗∗∗ −0.1307∗∗∗

(0.0016) (0.0019)

Toll - Hour 17 −0.1136∗∗∗ −0.1083∗∗∗

(0.0015) (0.0020)

Toll - Hour 18 −0.1340∗∗∗ −0.1345∗∗∗

(0.0019) (0.0027)

Toll - Hour 19 −0.2859∗∗∗ −0.2887∗∗∗

(0.0028) (0.0037)

Toll - Hour 20 −0.4290∗∗∗ −0.4110∗∗∗

(0.0050) (0.0056)

(Time Savings)20.0016∗∗∗ 0.0010∗∗∗

(1e-04) (1e-05)

(Time Savings)2- Hour 14 −0.0235∗∗∗ −0.0029∗∗∗

(6e-04) (6e-04)

(Time Savings)2- Hour 15 0.0033∗∗∗ 0.0045∗∗∗

(1e-04) (1e-04)

(Time Savings)2- Hour 16 0.0011∗∗∗ 6e−04∗∗∗

(2e-05) (0.0000)

(Time Savings)2- Hour 17 0.0018∗∗∗ 9e−04∗∗∗

(3e-05) (0.0000)

(Time Savings)2- Hour 18 0.0018∗∗∗ 9e−04∗∗∗

(4e-05) (0.0000)

(Time Savings)2- Hour 19 −9e−04∗∗∗ 0.0012∗∗∗

(1e-04) (1e-04)

(Time Savings)2- Hour 20 −0.0082∗∗∗ −0.0014∗∗∗

(3e-04) (3e-04)

Log Likelihood -15642.3036 -7174.4661 -12972.6198 -6309.6336

Num. obs. 720 720 720 720

***p < 0.01

Table 13: Parameters estimates for models with time savings squared.

39

α+α−

Hour 16 Hour 17 Hour 18 Hour 19 Hour 16 Hour 17 Hour 18 Hour 19

a 0.5 0.5 0.5 1 5 2 5 10

A 100 100 100 100 100 100 100 100

α1.5 1.8 1.5 1.5 1.8 1.5 1.5 1.5

Table 14: Parameters of akused in the calibration of LinTD for the Eastbound example.

Tolling Interval 1 min. 5 min. 10 min. 15 min. 20 min. 30 min. 60 min.

Hour 16 (0.44,1.36) (0.35,1.17) (0.36,1.26) (0.22,1.11) (0.34,1.13) (0.48,1.04) (0.44,1.25)

Hour 17 (0.62,1.60) (0.52,1.39) (0.53,1.37) (0.47,1.53) (0.52,1.94) (0.58,1.81) (0.51,1.91)

Hour 18 (0.54,1.59) (0.44,1.05) (0.45,1.67) (0.39,1.46) (0.47,2.61) (0.41,2.29) (0.37,2.58)

Hour 19 (0.24,1.57) (0.3,4.60) (0.01,1.43) (0.20,3.44) (0.14,3.73) (0.34,3.94) (0.10,1.41)

Table 15: Stochastic approximation procedure results for the (α+, α−) pairs for the Eastbound

example.

Tolling Interval 1 min. 60 min.

Hour 5 (0.15,3.5) (1.38,3.45)

Hour 6 (0.18,2.58) (1.35,1.89)

Hour 7 (0.25,2.50) (0.77,4.17)

Hour 8 (0.42,4.88) (0.57,3.76)

Hour 9 (0.28,2.49) (0.38,4.70)

Hour 10 (0.25,1.63) (0.28,1.95)

Hour 11 (0.23,1.53) (0.18,2.19)

Hour 12 (0.38,1.47) (0.39,2.71)

Hour 13 (0.28,1.67) (0.36,3.00)

Hour 14 (0.29,1.76) (0.23,3.44)

Hour 15 (0.35,1.37) (0.24,2.53)

Hour 16 (0.41,2.45) (0.57,2.56)

Hour 17 (0.28,1.57) (0.59,0.45)

Hour 18 (0.17,1.74) (0.23,1.42)

Table 16: Stochastic approximation procedure results for the (α+, α−) pairs for the Westbound

example.

40