Role of Day-to-Day Learning in Achieving Equilibrium after ... · 3 incorporates a Bayesian-Stochastic learning Automata model previously developed by Yanmaz-4 Tuzel and Ozbay to

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Role of Day-to-Day Learning in Achieving Equilibrium after a Major �etwork Change: 1

Application to �ew Jersey Turnpike Toll Road 2

3 Ozlem Yanmaz-Tuzel, PhD (Corresponding Author) 4 Senior Quantitative Analyst 5 Property & Portfolio Research, Inc. - CoStar 6 33 Arch Street, 33rd Floor 7 Boston, MA 8 Phone: (617) 443-3143 9 Fax: (877) 770-3950 10 e-mail: [email protected] 11 12 13 Kaan Ozbay, Ph.D. 14 Professor, 15 Department of Civil and Environmental Engineering, 16 Rutgers, The State University of New Jersey, 17 623 Bowser Rd. Piscataway, NJ 08854 USA, 18 Tel: (732) 445-2792 19 Fax: (732) 445-0577 20 e-mail: [email protected] 21

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23 Abstract: 239 24 Word count: Text (5238) + Figures (9) = 7488 25 Submission Date: November 15, 2010 26

27

28

Paper accepted for Presentation in the 29

Transportation Research Board’s 90th

Annual Meeting, Washington, D.C., 2011 30

TRB 2011 Annual Meeting Paper revised from original submittal.

2

Abstract 1

This paper presents an experimental dynamic traffic assignment framework that 2 incorporates a Bayesian-Stochastic learning Automata model previously developed by Yanmaz-3 Tuzel and Ozbay to study day-to-day updating mechanism of travelers’ learning and adaptation 4 to major changes in transportation networks. The main objective of this paper is to examine the 5 day-to-day evolution of travel patterns in a traffic network when major disturbances are 6 introduced into the transportation system. The dynamic traffic flow evolution and network-level 7 interactions of driver departure time and route choice decisions are captured within the traffic 8 flow simulator. The proposed integrated learning and dynamic traffic assignment framework is 9 tested using the NJTPK network. The case study investigates the impacts of the addition of a 10 new interchange (15X) on the Eastern Spur on day-to-day departure-time and route choice 11 behavior of NJTPK travelers, and the impacts of toll structure change on day-to-day departure-12 time behavior of the travelers. The calibration and validation results have shown that the 13 proposed framework that integrates day-to-day learning with DTA rk can successfully capture 14 day-to-day update of traffic flow after the imposed disruptions. The sensitivity analysis 15 confirmed that proposed day-to-day learning framework is a crucial component of the DTA, 16 particularly while investigating the traveler behavior during a transient period after a major 17 disruption. Ignoring the impacts of travel experiences, and travelers’ learning behavior on the 18 evolution of traffic conditions result in lower prediction capabilities, and failure to capture the 19 day-to-day evolution of travel trends. 20

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I�TRODUCTIO� 1

Modeling of traffic flows and travel times on congested road networks is crucial for 2 predicting, controlling or managing congestion, and analyzing the need for infrastructure 3 provision or improvement. Dynamic traffic assignment (DTA) refers to the assignment process 4 that incorporates the traffic flow dynamics varying over time. DTA approach is useful to analyze 5 how congestion forms and dissipates under time-varying conditions. Currently, many analytical 6 models and simulation-based models are under development in attempt to understand the 7 evolution of traffic congestion over a given time period. 8

This paper has a unique objective of investigating the travelers' behavior during a 9 transient period after a major disruption to the network equilibrium. This disruption may be the 10 addition of a new road, interchange or a congestion pricing scheme. To achieve this goal an 11 experimental DTA framework that incorporates a Bayesian-Stochastic Learning Automata 12 (Bayesian-SLA) model previously developed by Yanmaz-Tuzel and Ozbay (1) is employed to 13 study day-to-day updating mechanism of travelers’ learning and adaptation under major system 14 disruptions. The availability of vehicle-by-vehicle trip data for before and after the disruptions 15 has enabled us to build a unique experimental set up that can be used to evaluate the modeling 16 capability of our proposed integrated framework for capturing time dependent learning behavior 17 of travelers in the presence of a major event that disturbs network equilibrium. 18

The most common approach employed to capture the interaction between travel choice 19 and network performance has been to solve an equilibrium DTA problem where a time-20 dependent yet pre-determined trip matrix for the design period to be evaluated is assigned onto a 21 network. Under equilibrium conditions the travel choice is assumed to be governed by the 22 Wardrop principle which states that all used routes have equal and minimum travel time costs. At 23 this equilibrium point no user can decrease his/her travel time cost by switching to another 24 choice since travel costs on all the used routes are equal. 25

Equilibrium DTA approaches assume that network conditions from day-to-day and 26 within different periods of a day are in steady-state, predicate rigid behavioral tendencies a 27 priori, and try to attain either User Equilibrium (UE) or System Optimum (SO) (3). Moreover, 28 travelers are assumed to be rational, exploring each alternative’s relevant attributes and trading 29 off the utilities derived from them. The decision strategy serves to generate a choice from a 30 choice set for the alternative that provides the individual with the maximum utility. The question 31 of whether equilibrium actually takes place or is a mathematical construct is a very old question 32 (4). Horowitz (5) showed that day-to-day flow dynamics may oscillate around Stochastic UE 33 (SUE), or may even converge to some non-equilibrium point. Chang and Mahmassani (6) and 34 Friesz et al. (7) performed experiments for modeling transition of disequilibria from one state to 35 another. The results lead to the conclusion that a day-to-day adjustment process can lead to an 36 equilibrium state under fixed supply and demand conditions. However, the fact that there are 37 always changes in supply, demand, and traffic propagation, in combination with the stochasticity 38 of all the involved parameters and drivers’ day-to-day disequilibrium route choice behavior, 39 makes the concept of equilibrium highly questionable (4). Moreover, these models exclude 40 driver learning (based on past experiences and personal characteristics) which can significantly 41 affect traveler choice behavior on a specific day. In reality, the natural mechanism of traveler 42 choice is based on traveler’s behavioral tendencies, past experiences, and the traffic conditions 43 encountered (1). This issue implies the need for day-to-day modeling of users' learning 44 mechanism through a day-to-day traffic assignment model. 45


4

DTA models integrated with day-to-day learning framework aim to model traveler’s day-1 to-day learning and adaptation behavior and provide insight on how the traffic flow pattern 2 evolves over time. In day-to-day modeling, behavioral approaches are integrated into the 3 equilibrium paradigm, where the sequences of states that occur as the system reaches to 4 equilibrium are linked through a learning model based on travelers’ past experiences. These 5 intermediate stages are important for evaluation of the transportation system, because the 6 transportation system is often in disequilibrium due to travelers’ gradual response to 7 continuously changing conditions. These models predict travelers’ choices for any given day 8 based on his/her experienced choices in the previous days. Day-to-day models thus reflect the 9 travelers’ learning and forecasting mechanisms. 10

Several studies have focused on modeling drivers’ day-to-day route choice behavior 11 adjustment (e.g. 7,8,9,10,11,12,13,14,15,16,17,18,19). An extensive review of these models can 12 be found in Yang and Zhang (20). Moreover, several simulation tools have been proposed to 13 model day-to-day learning behavior of travelers including DYNASMART which is based on a 14 mesoscopic simulator that treats traffic individually but moves them according to macroscopic 15 flow principles (11,12,21) and DRACULA (22,23) which is based on a microscopic simulator 16 both on demand and supply levels. 17

This paper presents a new DTA framework to examine the day-to-day evolution of travel 18 patterns in a transportation network when major disturbances are introduced into the 19 transportation system. Proposed DTA framework integrated with day-to-day learning combines 20 three main sub-models. The first sub-model namely, demand model represents the day-to-day 21 variability in total demand. It takes the observed total demand values within the analysis period 22 and simulates for each traveler the preferred departure time choice and route to be taken. This 23 information is then passed to the traffic simulator, the second sub-model, which calculates 24 macroscopic flow conditions for each simulation interval. At the end of each day, a new learning 25 model proposed by Yanmaz-Tuzel and Ozbay (1), the third sub-model, stores the experienced 26 travel history, and updates each individual’s probability profile based on this experience. 27

The developed integrated day-to-day learning and DTA framework is tested using the NJ 28 Turnpike (NJTPK) network. The case study investigates the impacts of the addition of a new 29 interchange (15X) on the Eastern Spur (just south of EXIT 16E) on day-to-day departure-time 30 and route choice behavior of NJTPK travelers, and the impacts of toll structure change on day-31 to-day departure-time behavior of the travelers. Next sections present the details of the 32 application network, followed by the proposed DTA framework, and the results of the 33 application. 34

Data Sources 35

The proposed integrated day-to-day learning and DTA framework is tested and verified 36 on NJTPK network. NJTPK is a 148 mile-toll road extending from the Delaware Memorial 37 Bridge in the South of New Jersey to George Washington Bridge in New York City. Currently, 38 the road has 28 interchanges, commonly referred to as exits, with an average daily traffic that 39 exceeds 700,000 vehicles. To minimize queuing delays, NJTPK has minimal number of toll 40 plazas located at the exits. The interchanges connect to NJ’s major highways and vast 41 transportation network, institutions, and economic hubs. 42

The database includes vehicle-by-vehicle traffic and travel time information for all the 43 trips observed on NJTPK between December 2005 and December 2006. The data were collected 44


5

from E-ZPass recordings of each vehicle at each time of the day (2). This time period includes 1 the observed trips after two major disruptions imposed to NJTPK network, i.e. installation of 2 15X interchange on December 2005, and change in congestion pricing structure on January 3 2006. The calibration and verification of the proposed day-to-day learning model covers the 4 period between December 2005 and December 2006. 5

The first major disruption is the installation of 15X Interchange. After nearly three years 6 of construction, NJTPK Authority opened the $250 million Interchange 15X on the Eastern Spur 7 (just south of EXIT 16E) on December 1, 2005. The new interchange serves the Secaucus 8 Junction rail transfer station. Before opening of Interchange 15X, the only alternative traveling to 9 Lincoln Tunnel area was to travel through Interchange 16E. However, since December 2005, 10 Interchange 15X became a viable alternative for these travelers. 11

The second major disruption was imposed one month later. In January 2006, NJTPK 12 Authority eliminated the E-ZPass peak period discounts and E-ZPass peak users started to pay 13 the same amount of toll as the cash users. In this new toll structure, toll amounts were increased 14 by around 18% and 5% for E-ZPass peak and off-peak users, respectively; while cash tolls were 15 kept the same. 16 17

METHODOLOGY 18

The proposed integrated day-to-day learning and DTA framework makes use of the 19 concept of demand and supply submodels which interact with each other. Demand side is 20 modeled via microscopic simulation to describe individual decisions, and supply side is modeled 21 via macroscopic simulation to depict movement of vehicles and both sub models evolve over 22 time from one day to another. The travel costs experienced by the travelers are then re-input to 23 the demand model for the next day. Thus, there is no pre-determined requirement for the process 24 to converge to a stable “equilibrium” state. In fact, the transportation system never reaches to a 25 single state but continuously changes and evolves from one day to the next as the travelers learn 26 about the system and react to changes within the system. The proposed traffic simulator performs 27 for the time period between 6 am and 10 am. The demand entering to the transportation network 28 during this time period is obtained from observed O-D trip matrices. The general form of the 29 simulator is as follows: 30

1. Initialization: For each traveler in the network set individual characteristics, and an 31 initial departure time choice probability profile. Set day counter n=0 32

2. OD demand generation: Increment day counter n=n+1. Select the total am peak 33 demand for each origin-destination pair from observed O-D trip matrices. 34

3. Travel Choice: For each individual traveling on day n select a departure time period 35 (pre-peak, peak or post-peak) and/or route (interchange 15X or interchange 16E), based 36 on their choices, travel costs, and arrival time to their destinations on previous days. 37

4. Traffic Loading: Load the O-D matrix on day n depending on departure time and/or 38 route choice of each individual. 39

5. Simulation Results: For each time interval calculate travel time, traffic flow, speed and 40 density variables. 41

6. Individual Travel Experience: For each traveler calculate experienced travel cost and 42 determine whether the selected action is favorable or unfavorable. 43


6

7. Learning: Update the probability profile of each traveler using the learning parameters. 1 In particular, increase the probability of selecting favorable actions, and decrease the 2 probability of selecting unfavorable actions for the next day. Return to step 2. 3 4 The overall structure of the integrated DTA framework is presented in FIGURE 1. Our 5

objective is to determine the probability distribution of individual day-to-day states as proposed 6 and discussed in (e.g. 9,14,18,24,25,26). 7

8 FIGURE 1 Flow chart of the day-to-day dynamic traffic assignment framework 9

10

Travel Choice 11

Consider a transportation network [M;L;S] with nodes m, (m=1,2,…M), links l, 12 (l=1,2,….L) and OD pairs s = (i, j) (s=1,2,…..S). The demand functions for each OD pair are 13

time-dependent, given by ��, � ∈ �, and give rise to path flows ℎ�� The experienced travel 14

time for a path p which carries a flow generated at time t is ��, ℎ�. On any given day, each 15

driver traveling from origin i to destination j maintains a probability profile for the available 16 alternatives and updates his/her probability profile based on previous travel choices, exhibiting a 17 tendency to search for satisfying choice rather than the best behavior. 18

Annual and daily traffic trends observed at NJTPK indicate that traffic demand continues 19 to increase and shows similar behavior before and after the implemetation of time-of-day pricing 20


7

(27). Based on the results of the same study by (27), it can be safely assumed that majority of the 1 NJTPK users make departure time choices (between peak and off-peak periods), or route choice 2 between interchanges 15X and 16E. This is in fact an expected outcome since NJTPK is 3 practically the only alternative for a large number of trips to various important employment 4 centers in and outside the state including New York City. NJ has an excellent rail system, but it 5 does not provide a viable alternative to most of the NJTPK users who reside far from train and 6 who want to travel at peak periods can shift to peak hours. 7

Thus in this study two different choice mechanisms are considered. Travelers exiting 8 from interchanges 16E or 15X make both route (which interchange to take) and departure time 9 (which period to travel) choice, while all travelers using the remaining interchanges only make 10 departure time choice. Next sections provide in depth description of both choice models. 11

For the departure time choice model, an input set X composed of the explanatory 12 variables is considered. Output set D = {d1,d2,d3} includes actions composed of three choices (1: 13 pre-peak from 6:00 am to 7:00 am, 2: peak from 7:00 am to 9:00 am, and 3: post-peak from 9:00 14 am to 10:00 am). 15

The viable options for travelers making simultaneous route and departure time choice 16 consist of 6 choices. D = {d1,d2,d3,d4,d5,d6} includes actions composed of six choices (1: pre-17 peak Interchange 16E, 2: peak Interchange 16E, 3: post-peak Interchange 16E, 4: pre-peak 18 Interchange 15X, 5: peak Interchange 15X, 6: post-peak Interchange 15X). 19

Each traveler has a probability profile for each travel choice. On day n=1 it is assumed 20 that the probability profile represents a random variation around the observed frequency on day 21 n=0: 22 23

�,��,�� = ��,�� ∗ �,�� ∀�, � � ! ∀" (1) 24

25 where; 26 i: Origin index (i=1,2, ….27) 27 j: Destination index (j=1,2, ….27) 28 k: Individual traveler index (k=1,2,….*n) 29 n: Day index 30 r: Travel choice index 31 �,��,��: Probability of selecting choice r at day n=1 for individual k traveling from origin i to 32

destination j 33 ��,�� : Random variation for individual k traveling from origin i to destination j 34 �,��: Observed frequency of choice r from origin i to destination j 35

36 Depending on the individual probability profiles Monte Carlo simulation is used to 37

generate discrete choices, and travel demand at each choice is generated. 38

Traffic Loading 39

Generated travel demand based on individual probability profiles at each departure time 40 period is loaded to the transportation network. The traffic loading model is a mesoscopic 41 simulation of vehicle movement on the network. The model uses a small time step so that 42 congestion details can be accurately modeled. During each small time interval, vehicles keep 43


8

their variables constant (position, and speed). At time each interval, the variables are computed 1 for all vehicles and the simulation proceeds to the next time step. The simulation is finished 2 when all vehicles entering between 6 am and 10 am finish their journeys. The major steps of the 3 traffic simulation on day n are as follows: 4

5 1. Initialization: Set simulation clock t=0, generate vehicle macro-particles by distributing 6

travel demand at each departure time period equally within each time interval 7 2. Vehicle movement: Load the macro-particles to the transportation network 8 3. Simulation Results: Store aggregate measures of traffic volume, travel time, speed, 9

density for each link and O-D pair 10 4. Termination: If all drivers have finished their journey, terminate the day; otherwise 11

increment the simulation clock and return to step 2. 12

13 The traffic simulation algorithm satisfies the following constraints: 14

1. Flow conservation: The flow conservation require that for any given time instant, the 15 flow entering to any node, together with the demand generated at that node, must all exit 16 from this node to the next link unless the node is a destination. Mathematically, this 17 constraint can be expressed as: 18 19 ∑ $%��%∈&�� = �� + ∑ (%��%∈)�� (2) 20

21

In the above equation $%�� denotes the inflow rate to link l to destination j at time 22

instant t, �� denotes the travel demand from node i to destination j at time t, and (%�� is the 23

exit flow rate from link l to destination j at time t. Moreover, *�� is the set of links whose 24

starting nodes are i, and +�� is the set of links whose ending nodes are i. 25

Then, total flow on link l, ℎ%��, due to demand from origin i to destination j is the sum 26

of vehicles entering to link l from previous link and demand generated at link l minus the number 27 of vehicles exiting link l: 28

29

ℎ%�� = $%�� − (%�� + �%�� ∀- ∈ ., ∀��, �� ∈ � (3) 30

31

Then total flow on link l at time t, ℎ%��: 32 33 ∑ ℎ%��,�� = ℎ%�� (4) 34

35 2. FIFO constraint: FIFO condition states that a traffic stream generated at time t” cannot 36

overtake another stream that has started earlier at time t �� < �"�. In order to satisfy FIFO 37 condition when path costs are additive, certain conditions must be satisfied by the link 38 cost functions. The FIFO condition may be stated as: 39 40

�" > � ⇒ �" + �%��"� > � + �%�� (5) 41


9

1

At each time interval, link travel time function, �%�, is linked to instantaneous inflow $%�, 2

link volume ℎ%�, density "%� and speed 3%�, and other link parameters 456%�such as length and 3 capacity: 4 5

�%� = 7�$%�, ℎ%�, "%�, 3%�; 456%�� (6) 6

7 This mesoscopic description can also be seen as a cellular automata description where 8

each link and node corresponds to a unique cell. Congestion occurs on link cells, and route 9 decisions occur in node cells. Inside any given link, there is no underlying representation of the 10 vehicles, except the fact that they follow FIFO rule. The vehicle interaction is described only by 11

travel time function�%�. This approach implies vertical queuing since there is no restriction 12

imposed on $%� or ℎ%�. 13 Travel time at each time interval and link is calculated based on well-known speed-14

concentration relationship. The speed-flow relationships are (28): 15 16

"%�9� = "%� + : �;< Δ>?@ �$%�9� − (%�9� + �%�9�� (7) 17

3%� = �3A − 3B� :1 − �<D�E@F + 3B (8) 18

��%� = GH<I<D (9) 19

where 20 "%�9�= Concentration on link l at time interval t+1 21 J%= Number of lanes 22 Δ>?= Length of link l 23 3%�= Mean speed at link l during t-th time step 24 3A= Free flow speed 25 3B= Minimum speed on the facility 26 "B= Maximum concentration 27 ��%�= Travel time at link l during ��K time step 28 L= a parameter to be calibrated 29 �%�9�= Demand generated at link l at time t+1. 30 31 32 The observed travel times for NJTPK include the link travel times, as estimated in eqn-33

10, and service and waiting times at the exit toll plazas. To model the toll plaza delays at each 34 interchange, we have used a macroscopic toll plaza delay model developed by Lin (27). Even 35 though, this model is a relatively simple formulation, it has been validated by Ozmen-Ertekin et 36 al. (30) that macroscopic model results are comparable (within average error of 2.6% to 6.4%) 37 with the PARAMICS microscopic model for the NJTPK toll plazas. 38

According to this macroscopic model total delay experienced at toll plazas by each 39 vehicle can be expressed as follows: 40

! = !M + !� + ! + !N + !O (10) 41


10

where; 1 !M: deceleration delay (s/veh) 2 !�: incremental delay (s/veh) 3 !: service time (s/veh) 4 !N: acceleration delay (s/veh) 5 !O: initial queue delay (s/veh) 6

7 Deceleration delay is the extra travel time incurred while drivers decelerate before 8 reaching a toll booth (27): 9 10

!M = �IPIQ�RSMI (11) 11

where, 12 V: Speed of the approaching vehicle (m/s) 13 d: Deceleration rate (m/s2) 14 Vb: Speed at toll booth (m/s) 15 16 Acceleration delay depends on the free flow speed and the acceleration characteristics of 17 vehicles (27): 18 19

!N = �IPIQ�RSNI (12) 20

21 where, 22 a: Acceleration rate (m/s2) 23

24 Incremental delay experienced by each vehicle refers to the random variations in toll 25

processing times and vehicle arrivals (27): 26 27

!� = 900T W�X − 1� + Y�X − 1�S + Z[\.^._` (13) 28

29 where, 30 T: Analysis period (h) 31 X: Volume-to-capacity ratio 32 C: Capacity (veh/lane/hour) 33 *: number of toll lanes 34 35 Similarly, initial queue delay can be formulated as (27): 36 37

!O = �a��bQ��9c��de (14) 38

where, 39


11

fg: Total number of vehicles present at toll lanes at beginning of T (veh) 1 C: Toll lane group capacity (vph) 2 t: Duration of oversaturation within T (h) 3 u: Delay parameter 4

Travel Experience 5

After determining the performance measures for each traveler, depending on the 6 experienced utility and deviation from the desired arrival time, whether the action is favorable or 7 not is determined. Using the explanatory variables obtained from traveler survey (31), a user-8 specific utility function is derived for each departure time choice based on the proposed 9 Bayesian-SLA framework by Yanmaz-Tuzel and Ozbay (1). Eqn-15 shows parameters for each 10

utility function. Each parameter, h.., follows a normal distribution, with mean, ij.., and standard 11

deviation values, kj.., provided in (1). Thus, parameters of the utility function for each traveler 12

are different. This departure time choice model considers travelers’ trip characteristics (travel 13 time (tt), toll), desired arrival time characteristics (departure time (dt), early (ea)/ late (la) arrival 14 amount) and socio economic characteristics (income, education (ed), age, employment (emp), 15 gender). 16

17 l�,�m = h��,�m + hM�!��,�m + hnNo��,�m + h%N-��,�m + �h�B%%�p--��,�m + �h�� qpro��,�m +18 �h�� qpro ∗ ��,�m + �h��B%%� qpro ∗ �p--��,�m + �h��B%%�� ∗ �p--��,�m + �hnMo!��,� +19 �hsn�to !o��,� + �hnuor ��,� + �h to !o��,� + v�,� (15) 20

21

The travel utility function for route choice, l�,�m , includes travel time, ��,�m , departure 22

time, !��,�m , early arrival amount, o��,�m , late arrival amount, -��,�m , inertia effect on choice r for 23

driver k, w�m (takes value of 1 if choice r is the current choice ; 0 otherwise); where h.. are the 24 coefficient for each corresponding variable. 25

26 l�,�m = h��,�m + hM�!��,�m + hnNo��,�m + h%N-��,�m + �h�B%%�p--��,�m + hxw�m + v�,� (16) 27

28 After calculating the experienced travel utility value and early/late arrival amount 29

associated with each individual travel choice; whether the selected choice is favorable or not is 30 determined. In particular, it is assumed that travelers exhibit a tendency to search for satisfying 31 choices rather than the best behavior; thus they do not have the cognitive ability to process all the 32 information simultaneously and are happy with a good solution. To incorporate this kind of 33 behavior, bounded rationality (BR) approach (first introduced by (32) and used in many studies 34 in transportation field including (33)) is included in the behavior updating mechanism. In 35 particular, the travelers will switch routes and/or departure times if the difference between 36 experienced costs on the selected choice and the travel cost on the best choice that day. If the 37 difference is acceptable the traveler increases the probability to select the current choice for the 38 next day, otherwise decrease the probability to select the current choice, and increases the 39 probability to select the best choice of that day. The mechanism for the indifference threshold 40 can be formulated as follows: 41

42


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y�,� = z1 �7 l�,�m − lgn{�,� ≥ ∆�l�,�m , ~ℎo�o 0 ≤ ∆�≤ 1 0 p�ℎo�~��o � (17) 1

2

where y�,� is a binary variable that takes value of 1 if the difference between the utility of the 3

current choice r, l�,�m , and the utility of the best choice, lgn{�,� is acceptable for individual k on 4

day n, and 0 otherwise; ∆� is the acceptability threshold. Behaviorally, small ∆� value indicates 5

less tolerance of small cost differences compared with large ∆� value. If ∆� takes zero value, 6 traveler is intolerant of any difference in travel cost, and would switch for even the smallest cost 7

difference. The acceptability threshold ∆� reflects individual attitudes and preferences, and thus 8 should vary across the population (33). In this study, a normally distributed acceptability 9

threshold with mean i∆� and standard deviation k∆� is assumed (33). 10

Learning Model 11

In this paper, commuters’ day-to-day learning behavior on the basis of experienced travel 12 choices and user-specific characteristics is modeled via Bayesian-SLA theory proposed by the 13 authors (1). Specifically, each user updates his/her choice based on the rewards/punishments 14 received due to selected actions in the previous days. At the end of each day, favorable actions 15 are rewarded, while unfavorable actions are punished. After determining favorable and 16 unfavorable actions, a linear reward-penalty reinforcement scheme is considered to update day-17 to-day learning behavior of NJTPK users, and to investigate commuters’ response to new route 18 inclusion while selecting their departure times and routes. The parameters of the learning model 19 are calibrated and validated using the same network and same system disruptions based on the 20 real data collected as part of the NJTPK Value pricing study 21

As discussed in (1), the learning parameters for departure time choice follow Beta 22 distribution. Mean values for the parameters (a,b) are (0.062,0.0067), and standard deviations are 23 (0.0046,0.0021), respectively. 24

25

�� = �)�S.S�,�.�� NP�.��.R��.��PN��.��.��.�R (18a) 26

�� = �)�Z�.��,Z.�� g9�.�Z��.R��.��Pg��.��.��S��.� (18b) 27

28 where; 29 B(.): Beta distribution. 30 a: Reward parameter 31 b: Punishment parameter 32

33 Similarly, the learning parameters for departure time and route choice follow Beta 34

distribution (34). Mean values for the parameters (a, b) are (0.029,0.0029), and standard 35 deviations are (0.011,0.00093), respectively. 36

37

�� = �)��.�Z�,�.��Z� �N9�.��.��.��PN��.��.��.�� (19a) 38

�� = �)�Z�.��,��.�a� �g9�.��.��.��Pg��.��.��.�� (19b) 39

40


13

CALIBRATIO� A�D VALIDATIO� 1

Calibration and validation are important processes in the development and application of 2 DTA models. These processes ensure that the models accurately replicate the observed traffic 3 condition and driver behavior. 4

Model calibration is a process whereby the values of model parameters are adjusted so as 5 to match the simulated model outputs with observations from the study site. It is usually 6 formulated as an optimization problem to determine the best set of model parameter values in 7 order to minimize the discrepancies between the observed and simulated values (35). The 8 calibration process then modifies the values of the model parameters {β}, to find the best set of 9 values which minimizes function F. The proposed objective function F minimizes the difference 10 between observed and simulated link volumes and travel times: 11

12

min� � = ∑ ∑ �~� �OD,<��POD,<EQ�OD,<EQ� �S + ~S ��D,<��P��DEQ�,%��D,<EQ� �S��% (20) 13

where; 14 h: Set of parameters to be calibrated 15 t: Time interval 16 l: link index 17 ��{�u: Simulated link flows in t 18 ��Bg{: Observed link flows in t 19 ��{�u: Simulated link travel time in t 20 ��Bg{: Observed link travel time in t 21 w.: weight parameters 22

23 FIGURE 2 illustrates the solution algorithm for the calibration process. It is an iterative 24

procedure to try to match the simulated results with those observed from the study site. 25


14

1 FIGURE 2 Flowchart for the calibration process 2

3 After determining the optimal set of parameters from the calibration process, a validation 4

process is performed in order to determine whether the simulation model successfully replicates 5 the real system. Two different measures were considered in the validation process: 6

1. Root mean square error (RMSE) 7 2. Mean percentage error (MPE) 8

The formulations of these measures are as follows: 9

�� = ¡ �¢∗e ∑ ∑ �OD,<��POD,<EQ�OD,<EQ� �S¢%��e�� (21) 10

11

�4 = �¢∗e ∑ ∑ |OD,<��POD,<EQ�|OD,<EQ�¢%��e�� (22) 12

13

where ��,%{�u and ��,%Bg{ are the simulated and observed measurements for link l during aggregated 14

time period t, �¤ is the sample mean, k is the sample standard deviation, * is the total number of 15


15

links and T is the total number of time periods. RMSE measure penalizes large errors at a higher 1 rate than small errors, while MPE indicates the existence of systematic under or over-prediction 2 in the simulated variables (35). 3 4

MODEL APPLICATIO� 5

This section analyzes the effectiveness of the proposed day-to-day DTA framework in 6 evaluating the impacts of major disruption on day-to-day traffic flows of real transportation 7 networks. FIGURE 3 depicts the NJTPK network and the location of each interchange. The 8 detailed results of the calibration and validation of the integrated day-to-day learning and DTA 9 framework applied to NJTPK are presented in the next paragraph. 10

11 FIGURE 3 NJTPK network (36) 12

13 FIGURE 4 summarizes the MPE and RMSE plots for traffic volume calculations. The 14

proposed integrated framework has a good performance with MPE values ranging around 0.107, 15 and RMSE values ranging around 0.235 between December 2005 and December 2006. 16 Similarly, FIGURE 5 summarizes the MPE and RMSE plots for travel time calculations. These 17


16

results are fairly consistent regardless of network congestion levels, and the simulation model 1 and the real system show fairly good agreement. The relative magnitude of errors is similar to 2 well accepted DTA simulation softwares, such as DynaMIT, developed for FHWA (37, 38) and 3 AIMSUN (39). Both softwares obtained RMSE values between 0.2 and 0.3 when the DTA on 4 real transportation networks was calibrated with observed day-to-day traffic and travel time data. 5

6 FIGURE 4 Mean percentage error and Root mean square error for traffic volume 7

8 FIGURE 5 Mean percentage error and Root mean square error for travel time 9


17

Convergence Properties 1

One major issue while investigating the day-to-day impacts of policy implications is to 2 understand how long it takes for the transportation system to converge to a new steady state. This 3 information is crucial in terms of understanding travelers’ behavioral responses to the major 4 system changes that would help both researchers and policy makers in identifying expected 5 impacts of future transportation management strategies. 6

To this extent, this section focuses on the overall system changes in terms of departure 7 time and route choice after January 2006 toll structure change and December 2005 Interchange 8 15X installation, respectively. For each major disturbance, changes in the average traffic volume 9 during peak and peak shoulder (pre-peak and post-peak) periods were analyzed. To evaluate the 10 performance of integrated day-to-day learning and DTA framework in terms of predicting 11 traveler departure time and route choice behavior, simulated and observed traffic volumes were 12 compared. In order to reduce the impacts of seasonal changes each monthly average volume is 13 normalized. 14

FIGURE 6 summarizes the behavioral changes (simulated and observed) after January 15 2006 toll structure change for peak and peak shoulder periods. Analysis results reveal that the 16 DTA framework can successfully capture the trends in peak and peak shoulder periods after the 17 disturbance. Furthermore, the observed and simulated traffic volume results show that February 18 2006 is the transient period where the travelers learn the prevailing conditions of the disturbed 19 transportation system (fast-moving system). In this period, a rapid decrease for peak period and 20 rapid increase for peak shoulder periods is observed. After February 2006, experienced and 21 simulated traffic conditions exhibit a more steady state until November 2006. Between 22 November 2006 and December 2006 the change in the traffic conditions exhibits an increased 23 rate. This trend may be explained due to winter conditions and increased leisure trips due to 24 holiday season. Since travelers are familiar with the transportation system this short transient 25 period and rapid changes in this period is expected, and supported by larger learning parameters. 26 Travelers being aware of the structure of the whole transportation system can adapt themselves 27 to the new conditions and exhibit faster learning behavior. 28


18

1

2 FIGURE 6 Simulated vs. observed departure time choice, peak shoulder 3

4 A similar analysis is conducted to investigate the impacts of Interchange 15X installation. 5

To determine the duration of the transient period and understand the convergence behavior to the 6 new equilibrium, traveler choice behavior is analyzed by observing the changes in average traffic 7 volume exiting interchanges 15X and 16E during am period in the northbound direction. 8

FIGURE 7 summarizes the behavioral changes (simulated and observed) at Interchange 9 15X and Interchange 16E. Analysis results reveal that proposed framework can successfully 10 capture the trends in demand at interchanges 15X and 16E after March 2006. Furthermore, the 11 observed and simulated traffic volume results show that the transient period after this major 12 disturbance is much longer compared with January 2006 toll structure change. In fact, by 13 September 2006, we observe that the demand for Interchange 15X is still increasing at a positive 14 rate. Within this transit period, instead of a rapid change we observe slower responses from the 15 travelers. As pointed out in (34), NJTPK users exhibit resistance to change their behavior 16


19

resulting in slower learning and adapting rates to the new conditions. This type of behavioral 1 pattern causes a longer transient period where no rapid changes in the demand is observed. After 2 September 2006, the transient period diminishes and transportation system starts to reach to a 3 new steady state. In particular, the demand for Interchange 15X still continuous to increase, 4 however rate of increase diminishes resulting in reduced fluctuations in traffic flow conditions. 5

6

7 FIGURE 7 Simulated vs. observed route choice 8

9 The overall convergence properties of observed and simulated traffic conditions reveal 10

that, when the travelers are familiar with the form of the disturbance imposed to the system, such 11 as changes in an existing pricing application, the transient period is rather short and we observe 12 rapid changes and fast learning rates during this period. On the other hand, when the disturbance 13


20

is more significant, such as an infrastructural change in the transportation system, the transient 1 period becomes longer, and we observe lower learning rates where travelers are hesitant to make 2 drastic behavioral changes. 3

Sensitivity Analysis 4

In this section sensitivity analysis is conducted to investigate the impacts of changes in 5 the model specifications on the validity of the proposed framework. In particular, in order to see 6 the impacts of learning component on the performance of the proposed model, learning 7 component is removed from the proposed framework. 8

FIGURE 8 and FIGURE 9 show the trend in MPE and RMSE values for traffic volume 9 and travel time, respectively, when learning component is excluded from the simulation model. 10

When learning component is removed from the integrated DTA framework, it is observed 11 that both MPE and RMSE values increase compared with the error values calculated when day-12 to-day learning behavior of individuals is included into the model. Since the initial probability 13 profile was assumed to be the observed frequency values, the estimated error values are smaller 14 during first 10 days. However, as the transportation system evolves from day-to-day and drivers 15 learn the new system conditions the error values start to increase. Moreover, neither MPE nor 16 RMSE values show any tractable trend. 17

This sensitivity analysis confirms that proposed day-to-day learning framework is a 18 crucial component of the DTA framework, particularly while investigating the traveler behavior 19 during the transient period after a major disruption. Ignoring the impacts of travel experiences 20 and travelers’ learning behavior on the evolution of traffic conditions results in lower prediction 21 capabilities, and failure to capture the day-to-day evolution of travel trends. 22

23

24 FIGURE 8 MPE and RMSE for traffic volume, no learning component 25


21

1 FIGURE 9 MPE and RMSE for travel time, no learning component 2

Conclusions and Discussions 3

This paper presented a new experimental DTA framework to examine the day-to-day 4 evolution of travel patterns in a traffic network when major disturbances are introduced into the 5 transportation system. The dynamic traffic flow evolution and network-level interactions of 6 driver departure time and route choice decisions are captured via a traffic flow simulator. The 7 approach uses microscopic simulation to model the behavior of drivers on the demand side, and 8 uses macroscopic simulation to obtain system variables such as link travel time, volume and 9 density. Bayesian-SLA framework developed by the authors (1) is used to model day-to-day 10 update mechanism of the transportation network. 11

The proposed model has been tested and verified on NJTPK. In particular, two major 12 disruptions were considered. The first major disruption is the installation of 15X Interchange on 13 December 2005. The second major disruption was imposed one month later. In January 2006, 14 NJTPK Authority eliminated the E-ZPass peak period discounts and E-ZPass peak users started 15 to pay the same amount of toll as the cash users. 16

The calibration and validation results have shown that the proposed integrated day-to-day 17 learning and DTA framework can successfully capture day-to-day update of traffic flow after the 18 imposed disruptions. The proposed framework performed reasonably well with MPE values 19 ranging around 0.107, and RMSE values ranging around 0.235 between December 2005 and 20 December 2006 for traffic. Similarly, the MPE values range around 0.118 and RMSE values 21 range around 0.257 for the same time period. 22

In order to investigate the impacts of traveler learning behavior on capturing the day-to-23 day evolution of the travel trends, day-to-day learning component is removed from the DTA 24 framework. The sensitivity analysis confirmed that proposed day-to-day learning framework is a 25 crucial component of the DTA framework, particularly while investigating the traveler behavior 26


22

during the transient period after a major disruption. Ignoring the impacts of travel experiences 1 and travelers’ learning behavior on the evolution of traffic conditions resulted in lower prediction 2 capabilities, and failure to capture the day-to-day evolution of travel trends. 3

The overall convergence properties of observed and simulated traffic conditions reveal 4 that, when the travelers are familiar with the form of the disturbance imposed to the system, such 5 as changes in an existing pricing application, the transient period is rather short and we observe 6 rapid changes and fast learning rates during this period. On the other hand, when the disturbance 7 is more significant, such as an infrastructural change in the transportation system, the transient 8 period becomes longer, and we observe lower learning rates where travelers are hesitant to make 9 drastic behavioral changes. 10

11

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