draft final report NGSIM ArterialLane Selection Mode Federal Highway Administration Cambridge Systematics, Inc.

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1 NGSIM ArterialLae Selectio Mode draft fial report prepared for Federal Highway Admiistratio prepared by Cambridge Systematics, Ic. with Itelliget Trasportatio Systems Laboratory Massachusetts Istitute of Techology March 25, 2007
2 NGSIM ARTERIAL LANE SELECTION MODEL DRAFT FINAL REPORT Charisma Choudhury Varu Ramaujam Vaibhav Rathi Tomer Toledo Moshe BeAkiva Itelliget Trasportatio Systems Laboratory Massachusetts Istitute of Techology DRAFT March 25, 2007
3 NGSIM  ArterialLae Selectio Mode Table of Cotets Executive Summary Itroductio Data Descriptio Study Area Descriptio Estimatio Dataset Statistics Data Samplig Sampled Dataset Characteristics Model Estimatio Model 1 Lae Choice at Itersectio Modelig Framework Model Structure Likelihood Fuctio Estimatio Result Model Compariso Model 2 Laechagig Withi Sectio Modelig Framework Model Formulatio The Gap Acceptace Model The LaeChage Executio Decisio Model Likelihood Fuctio Estimatio Result Model Compariso Implemetatio Verificatio Algorithm Specificatio Simulatio System Implemetatio Uit Testig Method Results Cambridge Systematics, Ic i
4 Table of Cotets, cotiued 5.0 Operatioal Validatio Data Descriptio GoodessofFit Measures System Calibratio Calibratio Process Calibratio Results System Validatio Measures of Effectiveess Validatio Results Summary ad Future Research A. MITSIMLab Implemetatio of the Models... 1 A.1 Withi Sectio Model... 1 A.2 Itersectio Model... 7 B. Base Model Toledo B.1 Model Specificatio... 1 B.1.1 Lae Shift Model... 1 B.1.2 Gap Acceptace Model... 2 B.2 Model Estimatio... 5 C. OrigiDestiatio Data... 1 D. Sample Results from the Verificatio Tests... 1 ii Cambridge Systematics, Ic
5 NGSIM  ArterialLae Selectio Mode List of Tables Table 2.1 Vehicle Distributio Table for Period 8:28 a.m. to 8:45 a.m Table 2.2 Vehicle Distributio Table for Period 8:45 a.m. to 9:00 a.m Table 2.3 Aggregate LaeSpecific Statistics for All Laes alog the Arterial Table 2.4 Distributio of Locatios of LaeChage Poitsfor Turig Vehicles, by the Number of Sectios before the Exit Poit Table 2.5 Distributio of Locatios of LaeChage Poits for Through Vehicles, by the Sectio Number ad Travel Directio Table 2.6 Summary Statistics o Number of Vehicle Observatios without Lead/Lag Vehicle i Adjacet Lae Table 2.7 Statistics Describig the Lead ad Lag Vehicles Table 3.1 Itersectio LaeChoice Estimatio Results Table 3.2 Itersectio LaeChoice Variable Defiitios Table 3.3 Model Compariso Table 3.4 LaeChagig Withi Sectio Estimatio Results Table 3.5 LaeChagig Withi Sectio Variable Defiitios Table 3.6 Model Compariso Table 5.1 Calibratio Parameters of the Combied Model Table 5.2 Improvemet Resultig from Calibratio Table 5.3 Compariso of LaeSpecific Couts Table 5.4 Compariso of LaeSpecific Speeds Table B.1 Estimatio Results of the Reestimated Lae Chagig Model... 5 Table C.1 Vehicle Distributio by OD Pair Period 8:28 A.M. to 8:45 A.M Table C.2 Vehicle Distributio by OD Pair Period 8:45 A.M. to 9:00 A.M Table C.3 Sample of U.S. 101 OrigiDestiatio File... 3 Table D.1 Target Lae Selectio Model Itersectio... 1 Table D.2 Target Lae Selectio Model WithiSectio... 2 Table D.3 Gap Acceptace Model Lead Gap WithiSectio... 3 Table D.4 Gap Acceptace Model Lag Gap WithiSectio... 4 Cambridge Systematics, Ic. iii
6 NGSIM  ArterialLae Selectio Mode List of Figures Figure 2.1 Lakershim Boulevard Arterial Sectio Figure 2.2 Schematic Represetatio of the Arterial Stretch Figure 2.3 Defiitios of the Lead ad Lag Vehicles ad the Gaps they Defie Figure 3.1 Itersectio Lae Selectio Figure 3.2 Structure of the Itersectio LaeSelectio Model Figure 3.3 Perspective of Myopic Drivers Figure 3.5 Schematic of PathPla Figure 3.6 Effect of PathPla Figure 3.7 Effect of Aticipated Delay Figure 3.8 Heterogeeity i ImmediateLae Choice Figure 3.9 Simple Model Structure Figure 3.10 Structure of the LaeChagig Model Figure 3.11 Hypothetical Sceario i a FourLae Roadway with Subject Vehicle i Lae Figure 3.12 Defiitios of the Frot, Lead ad Lag Vehicles ad Their Relatioship with the Subject Vehicle Figure 3.13 Tradeoff betwee CurretLae Iertia ad PathPla Effect Figure 3.14 Tradeoff betwee CurretLae Iertia ad PathPla Effect Figure 3.15 Variatio of Lead Critical Gap with Relative Lead Speed ad Alpha Driver Aggressiveess Figure 3.16 Variatio of Lag Critical Gap with Relative Lag Speed ad Alpha Driver Aggressiveess Figure 3.17 LaeShift Model Toledo Figure 4.1 Pseudocode of MITSIMLab Implemetatio of the ArterialLae Selectio Model Figure 5.1 Locatios of Sythetic Sesors Figure 5.2 Locatios of Sesors Figure 5.3 Compariso of Lae Distributios Sectio Figure 5.4 Compariso of Lae Distributios Sectio Cambridge Systematics, Ic. v
7 List of Figures, cotiued Figure 5.5 Compariso of Lae Distributios Sectio Figure 5.6 Distributio of Lae Chages for Through Vehicles Figure 5.7 Distributio of Lae Chages by Vehicle Turig Ito Arterial Figure 5.8 Distributio of Lae Chages by Vehicles Turig Off the Arterial512 Figure B.1 Structure of the LaeChagig Model... 1 Figure B.2 Defiitios of Gaps... 3 Figure C.1 Node Locatios ad Numberig for Lakershim Boulevard... 1 vi Cambridge Systematics, Ic.
8 NGSIM  ArterialLae Selectio Mode Ackowledgmets The authors would like to thak Vassili Alexiadis, Vijay Kovvali, ad Li Zhag from Cambridge Systematics, who provided the data for this research as well as umerous useful commets ad suggestios. We would also like to thak Meeakshy Vasudeva of Mitretek ad Joh Halkias ad James Colyar from FHWA for their commets ad support. Jo Bottom, Ke Courage, Jay Jayakrisha, Michael Huter, Tom Rioux, ad other members of the Expert Pael have eriched this research with their valuable feedback. Members of the NGSIM stakeholder groups also have provided useful commets that helped to improve the quality of the report. This report is based upo work supported by the Federal Highway Admiistratio uder cotract umber DTFH6102C Ay opiios, fidigs, ad coclusios or recommedatios expressed i this publicatio are those of the authors, ad do ot ecessarily reflect the views of the Federal Highway Admiistratio. Cambridge Systematics, Ic.
9 Executive Summary
10 NGSIM  ArterialLae Selectio Mode Executive Summary The objective of this research is to examie ad implemet models for lae selectio o cogested arterial corridors. The models iclude: a) Itersectio laechoice model ad b) Withisectio laechagig model. The itersectio laechoice model ivolves the laeselectio model for vehicles eterig the arterial from a side street. The process of lae choice maily ivolves prepositioig based o path pla cosideratios of the driver. This process may be sigificatly affected by the plaig capability ad aggressiveess of the driver, as well as his familiarity with the etwork. Maeuver to the target lae may ot be possible due to obstructios caused by eighborig vehicles, ad the immediate lae choice of the driver may ot be the same as the chose target lae. A latet class modelig methodology has bee used for this model to explicitly take driver heterogeeity ito accout. The laechagig model withi sectio ivolves targetlae choice, acceptig the gaps to make a lae chage towards the target lae, ad executio of the lae chage to the accepted gap. The choice of target lae is iflueced by eighborhood vehicle speeds ad positios as well as laespecific attributes like average speed, queue legth, ad desity, as well as the path pla of the driver ad driver characteristics (iertia ad aggressiveess). Both models are estimated with detailed trajectory data from Lakershim Boulevard, i Los Ageles, Califoria collected by the NGSIM team. The effect of path pla ad aggressiveess of the driver is foud to sigificatly affect both of the models. For each of the models, the improvemets i the goodessoffit are compared agaist simpler models estimated with the same data. The statistical comparisos show that the ewlydeveloped models had a sigificat improvemet i the goodessoffit. The model has bee implemeted i microscopic traffic simulator MITSIMLab ad validated agaist a base model reestimated with the arterial data. Measures used for validatio iclude laespecific traffic flow, laespecific speeds, ad umber of lae chages with distace. All measures show improvemet i the simulatio capabilities of the ew models. Cambridge Systematics, Ic. ES1
11 NGSIM  ArterialLae Selectio Mode 1.0 Itroductio Travelers o arterial etworks face special challeges regardig lae positioig strategies. Cogested arterial corridors cotai a set of varied drivig activities that differ by lae ad locatio. These activities ecompass trip destiatio activities (parkig, eterig trasit stops, right turs, left turs), trip origiatio activities (exitig a parkig spot, exitig trasit stops), ad complex routig behaviors (permissive left turs, pedestriaimpeded right turs). Drivers familiar with the etwork may be aware of these activities ad how they vary by lae ad locatio. These drivers ofte make appropriate tactical laepositioig decisios to miimize their travel times o these complex facilities. The familiarity of the driver with the etwork ad his plaig abilities therefore greatly ifluece his drivig decisios. These lae positioig decisios geerally maifest themselves iside laechagig models i existig simulatio systems. Laechagig models are ofte geeralized betwee freeway ad arterial facilities. O cogested arterial etworks where laespecific obstructios take place, existig models rely o stadard throughflow gap acceptace/laechagig logic to determie vehicle positioig behavior. Ofte, this approach does ot correctly model the complexity of the actual decisios made by travelers, which leads to urealistic spillbacks ad ueve queue distributio across laes. The objective of this research is to examie ad implemet models for lae selectio o cogested arterial corridors focusig o the tactical behaviors of drivers regardig laespecific, temporary, ad movig obstructios. The specific models discussed here are the itersectio laechoice model ad withisectio laechagig model. Parameters of both of the models are estimated with detailed vehicle trajectory data. I the itersectio laechoice model, driverspecific latet classes are used to model the effect of uobserved plaig capabilities ad of varyig levels of etwork familiarity o the part of the driver. I both models, driverspecific radom terms are icluded i differet model compoets to capture aggressiveess ad other driver/vehicle characteristics. The orgaizatio of the report is as follows: First, a short descriptio of the data is preseted. The detailed structures, model formulatio ad estimatio results of each of the two models are provided ext. We the preset the implemetatio details of the model, followed by model validatio results withi the simulator. Cambridge Systematics, Ic. 11
12 NGSIM  ArterialLae Selectio Mode 2.0 Data Descriptio 2.1 STUDY AREA DESCRIPTION The data used for estimatio of the two arterial models uder discussio represets urba traffic o Lakershim Boulevard i Los Ageles, Califoria. Detailed vehicle trajectory data was collected as a part of the Federal Highway Admiistratio s (FHWA) Next Geeratio Simulatio (NGSIM) project o a segmet of the arterial located ear the itersectio with U.S. Highway 101 (Hollywood Freeway) o Jue 16, Five video cameras were used to collect the trajectory data. These cameras were mouted o the top of a 36story buildig, 10 Uiversity Plaza, located adjacet to the U.S. 101 ad Lakershim Boulevard iterchage. Figure 2.1 provides a aerial image of the locatio withi which data was obtaied from the camera coverage. Figure 2.2 gives a schematic illustratio of the arterial segmet costitutig the study area. It also provides details regardig the referece idices used for demarcatig the origi ad destiatio poits (also termed as odes) ad the laes withi every sectio. Lae umberig is assiged startig from the leftmost lae. The study site is approximately 1,600 feet i legth. It cosists of four sigalized itersectios, ad three to four through laes i each directio through each sectio. Almost every sectio has exclusive turig bays i the approach leadig to a itersectio. Figure 2.1 Lakershim Boulevard Arterial Sectio Source: NGSIM Data Aalysis Report Cambridge Systematics, Ic. Cambridge Systematics, Ic. 21
13 NGSIM  ArterialLae Selectio Mode Figure 2.2 Schematic Represetatio of the Arterial Stretch Source: NGSIM Data Aalysis Report Cambridge Systematics, Ic. 2.2 ESTIMATION DATASET STATISTICS Owig to the use of two separate lae behavior models, the trajectory data was split ito two parts: oe part cotaiig all the vehicle observatios betwee the itersectios, ad the other part cotaiig all the vehicle observatios at itersectios. The first part of the dataset was used for estimatig the arterial 22 Cambridge Systematics, Ic.
14 NGSIM  ArterialLae Selectio Mode laechagig behavior model withisectio, while the secod part was used for estimatig the laeselectio behavior at the itersectio. The dataset cosists of a total of 2,442 vehicles observed over a 32miute period cosistig of two successive itervals, the first from 8:28 a.m. to 8:45 a.m., ad the ext from 8:45 a.m. to 9:00 a.m. A sigificat proportio of these vehicles (97 percet) were automobiles, as ca be see from the vehicle distributio table (Table 2.1) preseted below. Table 2.1 Vehicle Distributio Table for Period 8:28 a.m. to 8:45 a.m. Motorcycle Automobile Truck ad Buses All Time Period Vehicles % Vehicles % Vehicles % Vehicles % 8:28 a.m. to 8:30 a.m % % 8 6.0% % 8:30 a.m. to 8:35 a.m % % % % 8:35 a.m. to 8:40 a.m % % % % 8:40 a.m. to 8:45 a.m % % % % All 3 0.2% 1, % % 1, % Source: NGSIM Data Aalysis Report Cambridge Systematics, Ic. Table 2.2 Vehicle Distributio Table for Period 8:45 a.m. to 9:00 a.m. Motorcycle Automobile Truck ad Buses All Time Period Vehicles % Vehicles % Vehicles % Vehicles % 8:45 a.m. to 8:50 a.m % % % % 8:50 a.m. to 8:55 a.m % % 9 2.0% % 8:55 a.m. to 9:00 a.m % % 7 1.8% % All 1 0.1% 1, % % 1, % Source: NGSIM Data Aalysis Report Cambridge Systematics, Ic. As show i the schematic preseted above (Figure 2.2), there are a total of 11 origis ad 10 destiatios withi the study area. The distributio of vehicles over the differet origidestiatio pairs is preseted i the appedix (Table C.3). The available data elemets for each observatio (lae ID, sectio ID, speed, positio, time of observatio, etc.) were used to geerate all of the variables used i the model specificatio. More detailed iformatio o the Lakershim Boulevard dataset is available i the NGSIM data aalysis report (Cambridge Systematics, Ic. 2006) Data Samplig For estimatio purposes, the origial dataset was sampled radomly at the rate of oe per every five vehicles, with the objective of establishig a represetative dataset for arterial laechagig behavior. A secod stage of samplig was Cambridge Systematics, Ic. 23
15 NGSIM  ArterialLae Selectio Mode required to weed out faulty ad ocoformat observatios (through vehicles positioed o the turig bays, vehicles makig turs from the wrog laes etc.). The models developed i this study do ot have the capability to model these maeuvers. As previously metioed, all observatios withi itersectios were elimiated from the dataset used for the sectio model estimatio. All erroeous observatios (with improper lae or sectio idetities) were discarded. Sice the primary focus of the models is vehicle laechagig behavior o arterials, observatios o the side streets/o or off ramps also are excluded for estimatio purposes. A small portio of vehicles whose destiatios etailed exitig at poits alog the arterial segmet other tha ay of the four itersectios were removed from the sampled dataset. Vehicles whose behavior was coceivably icosistet with that expected i arterials also were discarded. Samplig also was executed i the time dimesio, at the rate of 1 per every 10 time istace observatios of every sampled vehicle. As the origial dataset was at oeteth of a secod time resolutio, this samplig step coverted the iformatio to a oesecod resolutio Sampled Dataset Characteristics The sampled dataset cosisted of 400 vehicles, of which 160 were orthboud, ad 240 were southboud. The average vehicle observatio duratio was 51.3 secods, with the maximum duratio of observatio beig 170 secods. Out of the 400 vehicles i the sampled dataset, 150 vehicles (aroud 37.5 percet) exited the arterial withi the stretch costitutig the study area, i.e., they had as their destiatio a side street at oe of the four itersectios withi the study area. Aggregate LaeSpecific Statistics The sectios i the arterial stretch are mostly three ad fourlae roadways, with exclusive turig bays wideig the sectio at the approach to every dowstream itersectio. For data aalysis ad estimatio purposes, laes have bee categorized o the basis of permitted vehicular movemets. Lae type 1 sigifies a shared through flow, right ad lefttur lae; lae type 2 deotes a shared through ad righttur lae; lae type 3 deotes a shared through ad lefttur lae; lae type 4 deotes a exclusive right tur bay; lae type 5 deotes a exclusive lefttur bay; ad lae type 6 deotes a extra tur bay that is adjacet to aother. Statistics o the relevat aggregate laespecific variables are preseted i Table 2.3. The presece of turig vehicles, ad the coflicts arisig due to their movemets i cojuctio with throughflowig vehicles, provides a reasoable explaatio for the low average speeds observed i both the through ad the 24 Cambridge Systematics, Ic.
16 NGSIM  ArterialLae Selectio Mode turig laes. The maximum queue legth values are observed durig red itervals at traffic sigals. The presece of the exclusive tur bays ad their relevace i lae usage of turig vehicles is a key issue that has to be cosidered while geeratig from trajectory data the explaatory variables for the laechagig model. Table 2.3 Aggregate LaeSpecific Statistics for All Laes alog the Arterial Average Speed (m/s), amog all vehicles Average Queue Legth (m), amog all vehicles Max Queue Legth (m), amog all vehicles Lae Type Leged: Lae Type 1: A shared through flow, right ad lefttur lae. Lae Type 2: A shared through ad righttur lae. Lae Type 3: A shared through ad lefttur lae. Lae Type 4: A exclusive right tur bay. Lae Type 5: A exclusive left tur bay. Lae Type 6: A extra tur bay that is adjacet to aother. LaeChagig Statistics I the sampled dataset, there were a total of 249 lae chages observed. Of these, 104 (41.8 percet) were made by turig vehicles, i.e. those who were exitig the arterial withi the observed stretch. The table below (Table 2.4) shows the distributio of the lae chages made by the turig vehicles over the differet sectios i their respective paths alog the arterial stretch. As oted previously, the arterial stretch i the study area covered a total of three sectios of the five covered by the video cameras (see Figure 2.1). Table 2.4 Distributio of Locatios of LaeChage Poits for Turig Vehicles, by the Number of Sectios before the Exit Poit Number of Sectios from Exit Poit Number of Lae Chages (by Turig Vehicles) Last Sectio Oe Sectio from Exit Two Sectios from Exit 0 0 Percetage of the Total Total Cambridge Systematics, Ic. 25
17 NGSIM  ArterialLae Selectio Mode The iformatio i Table 2.3 is idicative of the domiace of path plarelated cosideratios i the laechagig decisios of turig vehicles. For vehicles goig through the etire arterial stretch (also termed as the through vehicles), the distributio of lae chages over the three sectios covered fully withi the stretch, by their directio of travel, is give i the followig table. Table 2.5 Distributio of Locatios of LaeChage Poits for Through Vehicles, by the Sectio Number ad Travel Directio Sectio 2 (betwee itersectios 1 ad 2) a Sectio3 (betwee itersectios 2 ad 3) a Sectio 4 (betwee itersectios 3 ad 4) a Northboud Southboud Total Total a Refer to Figure 2.2. Gap Acceptace Statistics The most sigificat aspect of the arterial dataset was the relatively low percetage of observatios i which a lead or lag vehicle was preset i adjacet laes for a chose subject vehicle. I the model developed for this dataset, lead or lag vehicles are defied as the closest vehicles i the correspodig adjacet lae withi the same sectio as that which the subject vehicle curretly occupies (Figure 2.3). Figure 2.3 Defiitios of the Lead ad Lag Vehicles ad the Gaps they Defie Traffic directio Lag vehicle Lag gap Lead gap Lead vehicle Subject vehicle Due to factors icludig the effect of sigal operatios at the itersectios, several vehicle observatios (icludig laechagig observatios) did ot record the presece of either a lead or a lag vehicle, or both, i ay of the 26 Cambridge Systematics, Ic.
18 NGSIM  ArterialLae Selectio Mode adjacet laes. This aspect of the dataset is summarized through the umbers preseted i Table 2.6. Table 2.6 Summary Statistics o Number of Vehicle Observatios without Lead/Lag Vehicle i Adjacet Lae All Observatios LaeChagig Observatios Number % (Total = 16696) Number % (Total = 249) Lead Vehicle Abset Lag Vehicle Abset Table 2.7 presets the descriptive statistics for the lead ad lag vehicle i relatio to the subject vehicle. Relative speeds are defied as the speed of the lead vehicle or lag vehicle less the speed of the subject vehicle. The table also summarizes statistics for the accepted lead ad lag gaps. Accepted lead gaps vary from 0.22 meter to meters, with a mea of meters. Accepted lag gaps vary from 0.75 meter to meters. Table 2.7 Statistics Describig the Lead ad Lag Vehicles Variable Mea Stadard Deviatio Maximum Miimum Lead Relative Speed (m/sec) (0.35) 3.87 (3.55) 3.50 (15.92) (15.73) Lead Gap (m) (11.6) (18.74) (155.8) 0.22 ( ) Lag Relative Speed (m/sec) (0.35) 3.90 (3.65) 7.3 (15.62) (15.73) Lag Gap (m) 9.18 (3.51) (20.24) (152.28) 0.75 ( ) Note: Statistics are for the accepted gaps occurrig with a lead/lag vehicle preset (the statistics for all gaps, either accepted or rejected, havig a lead/lag vehicle are i paretheses). Cambridge Systematics, Ic. 27
19 NGSIM  ArterialLae Selectio Mode 3.0 Model Estimatio As oted previously, arterial lae selectio behavior is simulated by two submodels, the itersectio laechoice model, ad the withisectio laechagig model. The itersectio laechoice model ivolves the lae selectio of drivers eterig the arterial from a side street (Figure 3.1). The model also is applicable for lae selectio of drivers eterig the side street from arterials. It should be oted that sice vehicles travelig from a arterial lik to aother arterial lik are ot allowed to make lae chages withi the itersectio, the itersectio model is ot applicable there. These lae chages are captured i the withisectio model. Figure 3.1 Itersectio Lae Selectio The itersectio lae selectio is modeled as a twolevel decisio: the targetlae choice ad the immediatelae choice based o the target lae. The laechagig model withi sectio is modeled as a threestep process: targetlae choice, gap acceptace, ad executio of the lae chage i the accepted gap i the directio of the chose target lae. Parameters of both models are estimated with detailed vehicle trajectory data. The effect of uobserved driver/vehicle characteristics o the laechagig process is captured by driverspecific radom terms icluded i differet model compoets. I the data all side streets had traffic sigals. Cosequetly, the effect of stopcotrolled side street movemets could ot be captured. The framework, model structure, likelihood fuctio, ad estimatio results for the two models are preseted i the followig sectios. Cambridge Systematics, Ic. 31
20 NGSIM  ArterialLae Selectio Mode 3.1 MODEL 1 LANE CHOICE AT INTERSECTION Modelig Framework The itersectio laeselectio model cosists of two steps: choice of target lae ad choice of immediate lae. The structure of the model is show i Figure 3.2. The first step i the decisio process is latet sice the targetlae choice is uobservable ad oly the driver s actual chose laes are observed. Latet choices are show as ovals; observed oes are show as rectagles. Figure 3.2 Structure of the Itersectio LaeSelectio Model P ( l v, τ ) P (, ) i l v The target lae is the lae the driver perceives as best to be i cosiderig the attributes of the lae ad the path pla cosideratios. However, the driver may ot be able to maeuver to his target lae immediately after crossig the itersectio ad the observed lae of the driver may be differet from his target lae. The choice of target lae is a tactical decisio of the driver whereas the choice of immediate lae is govered by maeuverability cosideratios. It should be oted that oce the driver eters the arterial, the the withi sectio laechagig model will immediately take over Model Structure TargetLae Choice The targetlae choice of the driver ca be modeled as a multiomial logit (MNL) model. The targetlae choice is sigificatly affected by the plaig capability 32 Cambridge Systematics, Ic.
21 NGSIM  ArterialLae Selectio Mode of the driver ad his familiarity with the etwork. The choice set of the driver ca thus deped o driver characteristics. The drivers ca belog to either of the two classes: Class 1 Myopic Drivers Those who cosider the immediate sectio oly (Figure 3.3); ad Class 2 Drivers Who Pla Ahead Those who cosider more tha oe sectio ahead (Figure 3.4). Variables associated with the target lae of the driver also may vary depedig o the driver class. Figure 3.3 Perspective of Myopic Drivers???? Figure 3.4 Perspective of Drivers who Pla Ahead??? The probability that driver selects lae l as the target lae, coditioal o idividualspecific characteristics, ca be expressed as follows: Cambridge Systematics, Ic. 33
22 NGSIM  ArterialLae Selectio Mode P ( l υ, τ ) = T l l exp( β X ( τ ) + α υ ) t T j j t j C exp( β X ( τ ) + α υ ) (3.1.1) where, X l t = attributes of lae l for vehicle at time t, ca be fuctio of τ τ = idividual specific lookahead distace β = cofficiets υ = idividual specific radom effect, υ ~N(0,1) l α = coefficiet of idividual specific radom effect for lae l Variables likely to ifluece the targetlae choice of the driver iclude: Path Pla Variables Distace to the poit whe the driver eeds to be i a specific lae to follow his path, ad the umber of lae chages required to be i the correct lae; Lae Attributes Queue legths, average speeds, ad queue discharge rates; ad Drivig Style ad Capabilities Idividual driver/vehicle characteristics, such as the lookahead distace of the driver ad aggressiveess of the driver. These ifluecig variables ca differ amog drivers i the same itersectio with the same pathpla, depedig o their etwork kowledge ad experiece. ImmediateLae Choice The immediatelae choice of the driver also ca be modeled as a MNL model. The immediatelae choice is affected by the drivig effort eeded to reach a particular lae ad maeuverability cosideratios, ad is coditioal o the choice of target lae. The probability that driver selects lae i as the immediate lae, coditioal o target lae l ad idividualspecific characteristics, ca be described as follows: P ( i l, υ) = T i i exp( β X + α υ ) t T k k t k C exp( β X + α υ ) (3.1.2) where, X i t = attributes of lae i for vehicle at time t i α = coefficiet of idividual specific radom effect for lae i 34 Cambridge Systematics, Ic.
23 NGSIM  ArterialLae Selectio Mode Variables likely to ifluece the immediatelae choice of the driver iclude: Curret positio of the driver: Proximity of a give lae to the receivig lae closest to the driver; Neighborhood variables: Presece of other vehicles ad their actios, relative positio ad speed of the subject vehicle with respect to vehicles surroudig it, geometric elemets of the roadway, sigals ad sigs, ad available capacity of the lae; ad Drivig style ad capabilities: Idividual driver/vehicle characteristics, such as the aggressiveess of the driver ad performace capabilities of the vehicle (e.g., required turig radius) Likelihood Fuctio Probability that driver selects lae i is the joit probability of selectig lae i give target lae l ad the probability of choosig target lae l, which ca be expressed as: P ( i υ, τ ) = P ( i l, v) P ( l υ, τ ) (3.1.3) l C Ucoditioal probability of driver selectig lae i at a give time ca be expressed as: P ( i) = P ( i υ, τ ) p( τ ) f ( v) dv p ( τ ) v τ π 1 = 1 π1 immediate sectio more tha oe sectio ahead (3.1.4) where the probability that the driver belogs to a particular class, π1 or (1 π1), is estimated from the data. The parameters of the model are estimated by maximizig this fuctio. This loglikelihood fuctio is maximized to estimate the differet model parameters i the two levels of the model. I this study, the BroydeFletcher GoldfarbShao (BFGS) optimizatio algorithm implemeted i the statistical estimatio software GAUSS (Aptech Systems 1994) has bee used. BFGS is a quasinewto method, which maitais ad updates a approximatio of the Hessia matrix based o firstorder derivative iformatio (see, for example, Bertsekas 1999). GAUSS implemets a variat of BFGS attributed to Gill ad Murray (1972), which updates the Cholesky decompositio of the Hessia (Aptech Systems 1995). The itegratio of each idividual likelihood fuctio L over the idividualspecific characteristic υ is performed usig the umerical itegratio techique Legedre quadrature method i GAUSS. The likelihood fuctio is ot globally cocave. For example, if the sigs of all the coefficiets of the idividualspecific error term are reversed, the solutio is Cambridge Systematics, Ic. 35
24 NGSIM  ArterialLae Selectio Mode uchaged due to its symmetric distributio fuctio. To avoid obtaiig a local solutio, differet startig poits have bee used i the optimizatio procedure Estimatio Result The model parameters are estimated with detailed trajectory data usig the software Gauss7.0. The results are preseted i Table 3.1. The fial loglikelihood is the value of the loglikelihood at its maximum (after it reaches covergece). Loglikelihood represets the fit of the model to the empirical data. A higher value of fial loglikelihood (closer to zero) idicates a better fit. Table 3.1 Itersectio LaeChoice Estimatio Results Iitial Loglikelihood Fial Loglikelihood Number of Parameters 20 Number of Observatios 703 Variable Parameter Value TStat Target Lae Lae 2 costat Lae 3 costat Lae 4 costat Aticipated delay (secod) Laes away from turig lae (myopic) coefficietmyopic drivers Costat myopic drivers Heterogeity coefficiet myopic drivers Laes away from turig lae (with plaahead) coefficiet drivers who pla ahead Costat drivers who pla ahead Heterogeity coefficiet drivers who plaahead Expected maximum utility from immediate lae Immediate Lae Laes away from coectig lae: coefficiet Costat Heterogeity coefficiet Target lae dummy Laes away from target lae: coefficiet Costat Heterogeity coefficiet Coflict dummy Driver Class Driver populatio with >1 sectio plaahead (%) Cambridge Systematics, Ic.
25 NGSIM  ArterialLae Selectio Mode Explaatio of Variables Table 3.2 Itersectio LaeChoice Variable Defiitios Variable Variables Affectig TargetLae Choice Aticipated delay (secod) Laes away from turig lae Expected maximum utility from immediate lae Variables Affectig ImmediateLae Choice Laes away from coectig lae Targetlae dummy Laes away from target lae Coflict dummy Descriptio Delay associated with a lae. Fuctio of queue legth ad queue discharge rate of that lae (Equatio 3.1.6). Queue discharge rate is the iverse of the time required for queue dissipatio i a particular lae whe the sigal turs gree. Average queue discharge rates calculated over the trajectory data collectio period were used. Number of lae chages the driver has to make to take his desired tur. Differs depedig o the driver class. Myopic drivers are ot affected by turs dowstream of the immediate sectio. Maximum utility that ca be derived by choosig the lae as target lae. This is equivalet to logsum i ested logit models (BeAkiva ad Lerma). Laes away from the aturally coectig lae of the driver. Naturally coectig lae defied by geometry. Oe if the lae i cosideratio for immediate lae also is the target lae. Laes away from the lae that has bee selected as target lae. Oe if there is a vehicle obstructig maeuver to the lae i cosideratio. TargetLae Choice Pathpla of the driver has a importat role i the targetlae selectio. As described i the earlier sectio, we follow a latet class formulatio for the model. The probability of the driver beig a myopic driver (Class 1) or a driver who plas ahead (Class 2) is calculated alog with the other model parameters. The estimated probability that the driver is of Class 2 was foud to be 18.4 percet. The two classes of drivers were foud to have differet sesitivities to pathpla cosideratios, which i this case was modeled as a iteractio betwee the umber of laes away from the correct lae ad the aggressiveess of the driver. The fuctioal form best fittig the data was foud to be: δ v l 2 l ( e )(1 ) ( e ) v δ (3.1.5) Cambridge Systematics, Ic. 37
26 NGSIM  ArterialLae Selectio Mode where δ = 1 if the driver plasahead beyod immediate sectio e 1l e 2l = laes away from turig lae for myopic drivers = laes away from turig lae for drivers who plaahead (cosider pathpla beyod curret sectio) The effect of pathpla for each driver class is explaied i Figure 3.5. I this example, lae 4 is a righttur oly lae. Therefore, drivers who are goig straight have to chage later if they choose lae 4 i the previous sectio. Drivers who pla ahead are ot likely to choose this lae whereas drivers who do ot pla ahead may choose lae 4. Figure 3.5 Schematic of PathPla Figure 3.6 Effect of PathPla Probability Lae Plaahead = 1 sectio Plaahead > 1 sectio Both classes of driver were foud to have disutility for laes that are farther away from the laes that they eed to take i order to follow their path. This disutility is, however, less for aggressive drivers, sice they are more proe to 38 Cambridge Systematics, Ic.
27 NGSIM  ArterialLae Selectio Mode make aggressive lae chages later if eeded. The disutility was foud to be larger ad more sigificat for drivers who pla ahead (Class 2). A aticipated delay value was calculated for each class of driver based o what segmets they are cosiderig. The fuctioal form ca be expressed as follows: q% kl t 1 = k = 1, 2 kl 1+ exp( q ) q = d / r, q = q + d / r t 1l 1l 1l 2l 1l 2l 2l t t t t t (3.1.6) where q kl t d r ki t ki t = aticipated delay i lae l cosiderig k sectios ahead = queue legth i lae i i sectio k at time t (vehicles) = average queue discharge rate of lae i i sectio k (vehicles/sec) The effect of aticipated delay was ot foud to be sigificatly differet for the two classes of drivers. The sesitivity to aticipated delay ca be illustrated as follows: Figure 3.7 Effect of Aticipated Delay Effect of aticipated de lay o the utility of lae 1.2 Utility of lae Aticipated delay (sec) The utility of the target lae ca be expressed as follows: l l 1l 2l % % l U = β ( q )(1 δ ) ( q )( δ ) + ( e )(1 δ ) (3.1.7) v v 2l l ( e ) δ 0.915( EMU ) Cambridge Systematics, Ic. 39
28 NGSIM  ArterialLae Selectio Mode where l β = costat for lae l q% q% 1l 2l δ = 1 if the driver plasahead beyod immediate sectio e e 1l 2l = aticipated delay fuctio i lae l for myopic drivers = aticipated delay fuctio i lae l for drivers who plaahead = laes away from turig lae for myopic drivers EMU = laes away from turig lae for drivers who plaahead (cosider pathpla beyod curret sectio) l = expected maximum utility derived from selectig lae l as immediate lae The expected maximum utility term captures the maximum utility that ca be derived from selectig a particular lae as immediate lae. Mathematically, this refers to the followig: EMU l where, EMU U i l = It should be oted that the model structure is flexible eough to be applied to other scearios with a differet umber of available laes by recalibratig the lae costats based o the particular geometry. ImmediateLae Choice 1 2 i I E( max( U l,u l,...,u l,...,ul)) l = Expected maximum utility derived from lae l = utility of immediate lae i for idividual give target lae is l C 1,2,...,i,...,I i Immediate lae choices were foud to be iflueced by maeuverability cosideratios ad iertia. Maeuver to a give lae may ot be possible due to coflicts with eighborig vehicles. I the case of such obstructios or coflicts, the driver ca choose a immediately available lae as the target lae, or ca wait util the eighborig vehicle moves ad there are o obstructios to move to the iteded target lae. As a result, if there are coflictig vehicles i the directio of a lae, the driver was foud to have a lower preferece for that lae. Iertia effects are captured by variables like curret lae iertia ad umber of laes away from the coectig lae. The iertia effect was greater for aggressive drivers. Aggressive drivers ted to stay i their curret lae as log as possible ad the make aggressive chages if a lae chage is warrated by the path pla. Drivers also were foud to prefer laes that are closer to their target laes. The combied effect of iertia ad preferece for movig to laes earer to target laes for aggressive ad timid drivers are illustrated i Figure Cambridge Systematics, Ic.
29 NGSIM  ArterialLae Selectio Mode Figure 3.8 Heterogeeity i ImmediateLae Choice Heterogeeity i Immediate Lae Choice (CL=2,TL=1) Curret Lae =2 Target Lae =1 Probability Immediate Lae Normal Driver Aggressive Driver The utility of immediate lae i is summarized i the followig equatio. i 1.01 i i U = ( c ) ( l = 0) v where c i i v i i ( l ) 1.76γ i γ = 1 if maeuver to lae i is obstructed by adjacet vehicle = laes away from coectig lae l = laes away from target lae l, l C (3.1.8) Model Compariso The improvemet i the goodessoffit of the ew model was compared with a siglelevel laechoice model (Figure 3.8) estimated with same data. The statistical tests for comparig oested models imply that the ew model has a statistically sigificat improvemet i goodessoffit. The test results are preseted i Table 3.3. Cambridge Systematics, Ic. 311
30 NGSIM  ArterialLae Selectio Mode Figure 3.9 Simple Model Structure Table 3.3 Model Compariso Statistic Base Model New Model Log likelihood value Number of parameters (K) Akaike iformatio criteria (AIC) a Bayesia iformatio criteria (BIC) b a b AIC = LLK where, LL=Log likelihood, K=umber of parameters. BIC = LLK/2*l (N) where, LL=Log likelihood, K=umber of parameters, N=umber of observatios. 3.2 MODEL 2 LANECHANGING WITHIN SECTION The withisectio laechagig model estimated i this study is a extesio of the itegrated laechagig model (Toledo et al. 2003) previously used i the NGSIM Freeway Lae Selectio Model, which allows joit evaluatio of madatory ad discretioary cosideratios. The laechagig process used i this study models driver behavior as a threestep decisio process: choice of target laes, gap acceptace decisios, ad laechage executio decisios. A multiomial logit model is used to model the choice of target laes. Gap acceptace behavior is modeled by comparig the available gap legths i the adjacet lae leadig to the target lae to the critical gap legths. Where the available gap is acceptable, the executio decisio is modeled probabilistically as a fuctio of the driver s aggressiveess ad speed, together with a costat capturig other uobserved factors that ifluece his istataeous choice. The model requires that both the lead ad lag gaps are acceptable for the available gap to be cosidered acceptable. The effect of uobserved driver/vehicle characteristics o the laechagig process is captured by a driverspecific radom term icluded at all three decisio levels. Parameters of the model are estimated with detailed ad sampled disaggregate vehicle trajectory data Cambridge Systematics, Ic.
31 NGSIM  ArterialLae Selectio Mode Modelig Framework The laechagig process cosists of three steps: choice of target lae (ad thereby the immediate adjacet target lae); gap acceptace decisios; ad the laechage executio decisio. The structure of the model is show i Figure 3.9. It illustrates the decisio tree for a driver i a hypothetical sceario o a fourlae road with the driver i lae 3 at the currettime step, as depicted i Figure The lae umberig follows a icremetal order from the left to the right of the driver s travel directio. The first two steps i the decisio process are latet. The targetlae choice of a driver is uobservable. Also, at ay give istat, the fact that the available adjacet gap i the immediate target lae is acceptable to the driver is ot discerible from the observatios. Oly the driver s laechagig actios, costitutig a laechage executio to the left or right, are observed. Latet choices are show as ovals, ad observed oes are show as rectagles. Figure 3.10 Structure of the LaeChagig Model Note: Illustrated for the hypothetical sceario of Figure Figure 3.11 Hypothetical Sceario i a FourLae Roadway with Subject Vehicle i Lae 3 Cambridge Systematics, Ic. 313
32 NGSIM  ArterialLae Selectio Mode The target lae is the lae the driver perceives as best to be i, depedig upo the prevalet drivig coditios ad his immediate destiatio. The choice set for the targetlae selectio icludes all the laes i the curret sectio alog the directio of travel to which the vehicle ca legitimately move. 1 Each of the ovals show at the first level represets a lae available for the driver to select as a target lae at the currettime step i the give sceario. The choice of the target lae implicitly fixes the immediate adjacet lae the driver targets movig to, which is located o the left or the right of the curret lae depedig upo the choice of the target lae. Give that i the sceario cosidered the driver is i lae 3 i the currettime step, a choice of lae 3 as the target lae meas that the driver decides ot to pursue a laechage ad to cotiue i his curret lae. If the driver perceives that movig to aother lae would improve his coditio, he will choose that lae as his target lae. I the above example, the immediate target lae for the driver would be to his left if he chooses either lae 1 or 2 as his target lae, while it would be to his right if he chooses lae 4 as his target lae. The driver would evaluate the gaps i his eighborhood (the appropriate adjacet lae, i.e., left or right) if he chooses a lae other tha his curret lae as his target lae i the currettime step. The available lead ad lag gaps i the immediate target lae are compared with respective latet critical gaps, based o which the driver decides to accept or reject the gap. For the gap to be acceptable, both the lead ad the lag gaps have to be acceptable, i.e., greater tha the respective critical gap values. If the driver perceives that the gap is acceptable, he cosiders the fial laechage executio step. He ca either decide to execute the lae chage i the give istat (fial observable decisio captured as CHANGE RIGHT or CHANGE LEFT respectively), or ot to execute the laechage (fial observable decisio captured as NO CHANGE). I the istaces where the driver chooses his curret lae as the target lae or does ot fid the available gaps i the immediate target lae acceptable, he does ot cosider the executio step, ad the fial observable decisio reflects NO CHANGE. This decisio process is repeated at every time step. Explaatory variables for laechagig behavior ca be classified ito the followig types of cosideratios: Neighborig Vehicles Ifluece Presece of other vehicles ad their actios, relative positio ad speed of the subject vehicle with respect to vehicles surroudig it, geometric elemets of the roadway, sigals ad sigs, etc.; Aggregate Drivig Coditios Average desity, speeds, ad queue legths i each lae; 1 The choice set for a through vehicle's target lae does ot iclude tur bays that are exclusive oly for turig vehicles, while the choice set for a turig vehicle icludes all laes i its sectio other tha the tur bay which oly allows for vehicles turig i a directio opposite to that of the subject vehicle Cambridge Systematics, Ic.
33 NGSIM  ArterialLae Selectio Mode Path Pla Variables Distace to the poit where the driver eeds to be i a specific lae to follow his path, ad the umber of lae chages required to be i the correct lae; ad Drivig Style, Capabilities, ad Curret Drivig State Idividual driver/vehicle characteristics, such as the aggressiveess of the driver ad his plaig capabilities, as well as variables defiig curret drivig state like speed, acceleratio, etc. I urba arterial coditios, owig to the relatively short duratio a driver speds i the mai traffic stream (which is the study area of this project), the path pla variables are expected to domiate over other comfortrelated cosideratios i laechagig decisios. Also, the presece of sigalized itersectios, ad of queuedup laes at these itersectios, is expected to play a key role i laechagig behavior. (This is i essece captured by the itersectio model.) The relative magitude of the ifluece these cosideratios exert o driver behavior will actually deped upo the driver s plaig capabilities ad other characteristics. The ifluece of these idividual driver characteristics will be resposible for the correlatio amog successive decisios made by each driver i the give pael dataset. Therefore, modelig driver heterogeeity to capture these correlatios is vital Model Formulatio The TargetLae Model The targetlae (TL) choice set cosists of all the laes that are available for the driver to move i give his curret positio o the roadway ad directio of travel. The utilities of these laes are give by: U = β * X + α * υ + ε lae i I (3.2.1) where, i i i i t t t t I t describes the set of all laes available to the driver at time step t; i U t is the utility of lae i to driver at time t; i X t β is a vector of explaatory variables cotaiig laespecific attributes for lae i at time t; is the correspodig vector of parameters; i i ε t is the radom term associated with the lae utility U t ; υ i α is a driver specific radom term that represets uobservable characteristics of the driver (i particular, characterizig driver s aggressiveess), thus capturig correlatios betwee observatios of the same driver over time. It is assumed to be ormally distributed i the drivers populatio; ad is the parameter ofυ specific to lae i. Cambridge Systematics, Ic. 315
34 NGSIM  ArterialLae Selectio Mode i Assumig that the radom terms ε t are idepedetly ad idetically Gumbel distributed over all the laes i the targetlae choice set, the choice probabilities of lae i Є I t for driver at time t, coditioal o the idividualspecific error term ( υ ) are give by: i t i ( t υ ) j ( t υ ) exp V P ( lae i υ ) = lae i I exp V t t j I V υ are the coditioal systematic utilities of the alteratives, give by: i i i t t t (3.2.2) V = β * X + α * υ lae i I (3.2.3) The explaatory variables idetified to be most sigificat ad ifluetial i the estimated model iclude: Geeral aggregate lae attributes, such as queue legth ad average speed of traffic i the lae before the upcomig sigalized itersectio, ad other laespecific attributes depedig upo the curret positio of the driver (e.g., umber of lae chages required of the driver to reach the target lae from curret lae, geeral preferece to cotiue i the curret lae of travel at each istat, etc.) Path pla variables (e.g., the distace to a poit where the driver must be i a specific lae(s) to cotiue alog his path, iteractig with the umber of laechages eeded i order to be i these laes) Driverspecific characteristics like aggressiveess, ifluecig the choice for a particular lae durig travel through the maistream traffic. These latet characteristics are captured by the idividualspecific error termυ. I this model, the preferece for each driver towards his curret lae of travel also is modeled as depedig upo his aggressiveess, i additio to a geeral preferece exhibited by all drivers irrespective of their aggressiveess (captured i the first class of explaatory variables) The Gap Acceptace Model I the targetlae model the driver chooses the target lae. The immediate laechagig choice is determied as a cosequece of the targetlae selectio. Next, the driver decides whether or ot the chose lae chage ca be udertake by evaluatig the gaps i the correspodig adjacet lae. Coditioal o the targetlae choice, the gap acceptace model idicates whether a decisio for laechage executio usig the existig gaps i the currettime step would be cosidered or ot. The adjacet gap i the target lae is defied by the lead ad lag vehicles i that lae, as show i Figure The lead gap is the clear spacig betwee the rear of the lead vehicle ad the frot of the subject vehicle. Similarly, the lag gap is 316 Cambridge Systematics, Ic.
35 NGSIM  ArterialLae Selectio Mode the clear spacig betwee the rear of the subject vehicle ad the frot of the lag vehicle. Note that oe or both of these gaps may be egative if the vehicles overlap. Figure 3.12 Defiitios of the Frot, Lead ad Lag Vehicles ad Their Relatioship with the Subject Vehicle Adjacet gap Traffic directio Lag vehicle Lag gap Lead gap Lead vehicle Subject vehicle Frot spacig Frot vehicle The driver compares the available lead ad lag gaps to the correspodig critical gaps, which are the miimum acceptable gap legths. A available gap is acceptable if it is greater tha the critical gap. Critical gaps are modeled as radom variables. Their meas are fuctios of explaatory variables. The idividualspecific term i this mea fuctio captures correlatios betwee the critical gaps of the same driver over time. Critical gaps are assumed to follow logormal distributios to esure that they are always oegative: T gd, cr g gd g gd ( Gt ) β X t α υ εt g { lead lag d { right left l = + +,,, (3.2.4) where, G is the critical gap g i the directio of chage d, measured i meters; gd, cr t gd X t is a vector of explaatory variables; g β is the correspodig vector of parameters; gd gd 2 ε t is a radom term: ε t N ( 0, σ g ) ; ad g α is the parameter of the driverspecific radom term υ. The gap acceptace model assumes that the driver must accept both the lead gap ad the lag gap to cosider laechage executio. The probability of acceptig the adjacet gap, coditioal o the idividual specific term υ ad the choice of directio of chage d t, is therefore give by: Cambridge Systematics, Ic. 317
36 NGSIM  ArterialLae Selectio Mode ( t t, υ ) ( t t t, υ ) ( t, υ ) P ( accept lag gap dt, υ ) = lead dt lead dt (, cr lag dt lag dt, ) (, cr t > t t υ t > t t, υ ) P acceptig adjacet gap i directio d d = P AG = d d = P accept lead gap d P G G d P G G d dt { Right, Curret, Left (3.2.5) is the chose directio of chage for driver at time t, which is implied by the targetlae choice. lead ad lag gaps i this directio, respectively. G ad lead d t lag d Gt are the available 1 if the available adjacet gaps to the left of the driver are acceptable, give the left lae is the immediate target lae AG t = 1 if the available adjacet gaps to the right of the driver are acceptable, give the right lae is the immediate target lae 0 otherwise Assumig that critical gaps follow logormal distributios, the coditioal g lead, lag is acceptable is give by: probabilities that gap {, ( l l, υ ) T gd g gd g ( Gt ) ( β X t + α υ ) gd gd, cr gd gd, cr ( υ ) ( ) ( ) P G > G d = P G > G d = l Φ t t t t t t σ g where Φ[ ] deotes the cumulative stadard ormal distributio. (3.2.6) The gap acceptace decisio is primarily affected by eighborhood variables such as the subject speed relative to that of the lead ad lag vehicles. Also, driver s characteristics, such as aggressiveess, play a key role i decidig the average lead or lag gap that gets accepted The LaeChage Executio Decisio Model The driver cosiders the laechage executio decisio step if he chooses a target lae that is differet from his curret lae, ad fids the adjacet gaps i the immediate target lae i the directio of the chose target lae acceptable i the curret timestep. Give the above latet decisios, the decisio to execute the lae chage i the currettime step is modeled as a probabilistic choice. The choice set i this decisio step cotais two alteratives, to execute the lae chage (ito the immediate target lae) or ot. The radom utility (ad the correspodig systematic utility) goverig this choice is give by: 318 Cambridge Systematics, Ic.
37 NGSIM  ArterialLae Selectio Mode U = { β * X + α * υ + ε if l = 1 or 1 lt t t t t 0 if l = 0 ad V = { β * X + α * υ if l = 1 or 1 lt t t t 0 t if l = 0 t (3.2.7) The probability for executig the lae chage i the curret time istat t is give by P( l AG, υ ) = { 1/(1 + exp( V l t )) if AG = {1,1 where, t t t t 0 if AG = 0 t (3.2.8) l t is the laechagig actio. 1 a lae chage is executed at time t to the left l t = 1 a lae chage is executed at time t to the right 0 otherwise i X t is a vector of explaatory variables capturig the factors ifluecig the driver s executio decisio i the currettime step. They may iclude variables like curret istataeous speed amog others that defie his istataeous state; β is the correspodig vector of parameters; i ε t is the radom term associated with this utility fuctio; υ is a driverspecific radom term that represets uobservable characteristics of the driver (i particular, characterizig driver s aggressiveess), that would ifluece his laechagig executio decisios; ad i α is the parameter ofυ. The fial observed lae chage is a istataeous process, ad the essece of the executio decisio step is to model this istataeous decisiomakig process of the driver. It is more relevat i this particular study, where there exist a umber of successive observatios where the driver faces large ad uchagig adjacet gaps (icludig those istaces where there are o lead or lag vehicles). It is observed that the driver chages lae i oe of these istaces. The executio step i a way models the way the driver differetially evaluates these seemigly similar scearios. Aother key aspect that affects this istataeous decisio is the legth of the decisio itself. I datasets where the time steps of observatio vary across the drivers, it is possible to estimate the ifluece of this attribute i the driver s fial decisios. Cambridge Systematics, Ic. 319
38 NGSIM  ArterialLae Selectio Mode Likelihood Fuctio I this sectio, the likelihood fuctio of laechagig actios observed i the data is preseted. If l t deote the laechagig actio observed of driver at time t ( l t Є{1,1,0), the probability of the observed laechagig actio at each time step coditioed o the latet idividual characteristics υ is give by: j ( t υ ) = ( t, t, t υ ) P l P TL AG l (3.2.9) j TL j where TL t deotes the evet of driver selectig lae j as the target lae durig time istat t, with TL deotig the choice set for target laes. Here, j ( ) j P TL, l, AG υ = P( l AG, υ )* P(AG d, υ )* P( TL υ ) (3.2.10) where, t t t t t t t t d t is the directio of lae chage for driver at time t as chose based o the choice of target lae j; j P( TL υ ) is give by equatio (3.2.2); t P(AGt dt, υ ) is give by equatio (3.2.5); ad P( l t AGt, υ ) is give by equatio (3.2.8). If a driver is observed over a iterval of T time istats, ad assumig that these observatios, coditioed o the driver characteristics, are idepedet of each other, the probability for the etire sequece of observatios l for driver is give by: T j ( υ ) = ( t, t, t υ ) P l P TL l AG (3.2.11) t= 1 j TL As stated earlier i the model structure, the idividualspecific characteristic is assumed to follow a ormal distributio over the etire populatio of drivers. The ucoditioal likelihood L for the sequece of observatios l characterizig the drivig trajectory of idividual is give by: ( υ ) ( υ ) L P( l ) P l f (3.2.12) = = υ Assumig that the observatios over differet drivers are idepedet of each other, the likelihood of observig the complete set of observatios i the give dataset, for all N idividuals (or vehicles), is give as the product of every driver sequece of observatios as i equatio (3.2.10). The loglikelihood fuctio for the etire dataset of observatios is hece give by: υ 320 Cambridge Systematics, Ic.
39 NGSIM  ArterialLae Selectio Mode N L = l( L ) (3.2.13) = 1 This loglikelihood fuctio is maximized to estimate the differet model parameters i the two levels of the model. I this study, the BroydeFletcher GoldfarbShao (BFGS) optimizatio algorithm implemeted i the statistical estimatio software GAUSS (Aptech Systems 1994) has bee used. As oted above, BFGS is a quasinewto method, which maitais ad updates a approximatio of the Hessia matrix based o firstorder derivative iformatio (see, for example, Bertsekas 1999). GAUSS implemets a variat of BFGS attributed to Gill ad Murray (1972), which updates the Cholesky decompositio of the Hessia (Aptech Systems 1995). The itegratio of each idividual likelihood fuctio L over the idividualspecific characteristic υ is performed usig the umerical itegratio techique Legedre quadrature method i GAUSS. The likelihood fuctio is ot globally cocave. For example, if the sigs of all of the coefficiets of the idividualspecific error term are reversed, the solutio is uchaged due to its symmetric distributio fuctio. To avoid obtaiig a local solutio, differet startig poits have bee used i the optimizatio procedure Estimatio Result Table 3.4 LaeChagig Withi Sectio Estimatio Results Fial LogLikelihood Number of Observatios 16,696 Number of Vehicles 400 Number of Parameters 22 Parameter Variable Name Estimate TStat TargetLae Selectio Model CurretLae Dummy Path Pla impact: No. of lae chages to exit lae Path pla impact: No. of lae chages to exit iteracted with distace from exit Expoet of dist. to exit i o. of laes to exit dist. to exit iteractio No. of lae chages from curret lae = No. of lae chages from curret lae >= Queue legth ahead i lae (umber of vehicles) Frot vehicle rel. speed egative, iteracted with frot veh. Gap (m/s per m) Extra Bay Dummy α CL Cambridge Systematics, Ic. 321
40 NGSIM  ArterialLae Selectio Mode Gap Acceptace Model Lead Critical Gap Lead gap costat lead,tl V t (m/s) σ lead α lead Lag Critical Gap Lag gap costat lag,tl V t (m/s) σ lag α lag Executio Decisio Level Itercept α lead V t Table 3.5 LaeChagig Withi Sectio Variable Defiitios Variable Name CurretLae Dummy Path Pla impact: No. of lae chages to exit lae Path pla impact: No. of lae chages to exit iteracted with distace from exit Expoet of dist. to exit i o. of laes to exit dist. to exit iteractio Number of lae chages from curret lae = 1 Number of lae chages from curret lae >= 2 Queue legth ahead i lae Frot vehicle rel. speed egative, iteracted with frot veh. Gap (m/s per m) Extra Bay Dummy α CL Defiitio 1 if the lae is the curret lae of the driver, 0 otherwise Number of lae chages the driver has to make from the target lae i order to follow his path (to take the tur/exit) Number of lae chages the driver has to make from the target lae i order to follow his path (to take the tur/exit)* (remaiig logitudial distace to the tur/exit ) raised to a estimated expoet The expoet of the remaiig logitudial distace i the precedig variable Dummy variable. Oe if Number of lae chages required to reach the target lae from the curret lae of the driver, zero otherwise Number of lae chages required to reach the target lae from the curret lae of the driver if the umber of lae chages required to reach the target lae>=2, 0 otherwise Number of vehicles ahead i target lae Iteractio of relative speed of frot vehicle i the curret lae with the available frot spacig (Referece Equatio ) 1 if the target lae is a extra bay, 0 otherwise Heterogeeity term for iertia 322 Cambridge Systematics, Ic.
41 NGSIM  ArterialLae Selectio Mode Lead gap costat lead,tl V t (m/s) σ lead α lead Lag gap costat lag,tl V t (m/s) σ lag α lag Itercept α lead V t Costat i lead gap fuctio Relative speed differece with lead vehicle Stadard deviatio of lead gap fuctio Heterogeeity term for lead gap Costat i lag gap fuctio Relative speed differece with lag vehicle Stadard deviatio of lag gap fuctio Heterogeeity term for lag gap Itercept of executio level Heterogeeity term for executio level Speed of subject vehicle TargetLae Model The targetlae utility ca be expressed as follows: TargetLae Utility Specificatios: V = * δ * e * e * (d ) i ic i i e x it e x p ( ) t t t t t * ( k < = 3 ) * k * ( k > = 4 ) * k * υ * δ ic t i i i i ic t t t t t i i i * ( q ) * ( q < = 3 ) * ( q > = 4 ) + t t t c c ( δ ) * [ * m i ( 0,f r ) / (1 + e x p ( f s ) )] t t (3.2.14) where, δ = curret lae dummy, 1 if lae i is curret lae, 0 ow q ic t i t v i t i k = umber of lae chages required from curret lae to lae i ( i j ) t i t exit t c t c t = queue ahead i lae i = average speed ahead i lae i where j t = curret lae of idividual at time t e = umber of lae chages reqd. to take the desired exit/tur from lae i d = remaiig distace to exit/tur fr = frot vehicle rel. speed fs = frot vehicle spacig t Cambridge Systematics, Ic. 323
42 NGSIM  ArterialLae Selectio Mode As ca be see from the estimatio results, the two most sigificat variables i the targetlae selectio model are the path plarelated variables (i.e., umber of lae chages to exit lae, ad umber of lae chages to exit lae iteracted with the remaiig distace to madatory laechagig poit), ad the curretlae iertia variables (e.g., curret lae dummy ad distace from curretlae variable). The tradeoff betwee the effects caused by these two variables i the utility of the target laes is illustrated i the followig figures (Figures 3.12 ad 3.13), for a stadard case of four laes (which represets the typical case i the curret dataset), with lae 4 beig the exit lae. Figure 3.13 Tradeoff betwee CurretLae Iertia ad PathPla Effect Lae Prob. values Lae 1 Lae 2 Lae 3 Lae CL=1 CL=2 CL=3 CL=4 Note: Distace to exit = 410m, Turig/exit lae = Lae Cambridge Systematics, Ic.
43 NGSIM  ArterialLae Selectio Mode Figure 3.14 Tradeoff betwee CurretLae Iertia ad PathPla Effect Lae Prob. values Lae 1 Lae 2 Lae 3 Lae CL=1 CL=2 CL=3 CL=4 G ε Note: Distace to exit = 75m, Turig/exit lae = Lae 4. Whe the driver is far from his exit, curretlae iertia domiates his targetlae selectio, while as the driver approaches his tur, the pathpla effects take over. Figure 3.12 illustrates the driver s prefereces for the laes i the immediate viciity of his curret lae, all other thigs beig equal, whe he is still reasoably far from his exit poit (a distace of 410 m implies over 2 sectios distace i the study dataset). The order of preferece shifts to laes i the viciity of the exit lae (Lae 4) as the driver comes closer to his exit ad the path pla cosideratios take over. This pheomeo is exhibited i Figure With referece to the study dataset, at 75m from his exit, the driver is more or less eterig his fial sectio. I this situatio, there is a high probability for choosig Lae 4 irrespective of the curret lae of the driver, as depicted by the figure. Gap Acceptace Model The critical gap of the driver ca be expressed by the followig equatio: Critical Gap Specificatio: = e x p ( V υ + ε ), le a d T L, c r le a d, T L le a d t t t le a d t 2 ~ N ( 0, ) G ε = e x p ( V υ + ε ) la g T L, c r la g, T L la g t t t la g t 2 ~ N ( 0, ) (3.2.15) P ( a c c e p t i g g a p ) = P ( a c c e p t i g le a d g a p ) P ( a c c e p ti g la g g a p ) Cambridge Systematics, Ic. 325
44 NGSIM  ArterialLae Selectio Mode Where, V = V V V V t gi t gi gi t t t = speed of subject vehicle at time t = speed of vehicle associated with gap g of subject at time t i directio of target lae i The ifluece of the icluded explaatory variables o the critical gap legths are summarized i the figures below. Figure 3.15 Variatio of Lead Critical Gap with Relative Lead Speed ad Alpha Driver Aggressiveess Critical Lead Gap Variatio Critical Lead Gap (l(m)) iu = 1 iu = 0 iu = relv = 10 relv = 0 relv = 10 Rel. Lead Speed (m/s) 326 Cambridge Systematics, Ic.
45 NGSIM  ArterialLae Selectio Mode Figure 3.16 Variatio of Lag Critical Gap with Relative Lag Speed ad Alpha Driver Aggressiveess Critical Lag Gap Variatio Lag Critical Gap (l (m)) iu = 1 iu = 0 iu = relv = 10 relv = 0 relv = 10 Rel. Lag Speed Based o the respective defiitios of relative lead ad lag speed, the above figures illustrate their impact o gap acceptace decisios. The impact of driver heterogeeity also is summarized i these charts. Cosiderig a uvaryig lead gap, the driver is more likely to cosider it safe to move ito the selected lae if the lead vehicle s speed relative to his ow speed is high. This is corroborated by Figure 3.14, where the critical gap distace decreases with icreasig relative lead speed, implyig a icrease i gap acceptace probability. The driverspecific term iu represets aggressiveess, ad as expected, we see a icrease i the critical gap distace as iu value decreases, implyig that timid drivers prefer larger gaps. Similar coclusios are supported by Figure 3.15 for the lag critical gap. The chart shows that the lag critical gap icreases with a icrease i the lag vehicle s speed relative to the subject, as there is a icrease i perceived risk while acceptig the gap whe the lag vehicle is closig i. Also, the impact of heterogeeity o lag critical gap is similar to that o the lead critical gap, with the critical gap distace decreasig as the driver aggressiveess ( iu value) icreases Model Compariso The improvemet i the goodessoffit of the ew model was compared with a simpler laechagig model, which is illustrated i Figure The referece model has the structure of the laeshift model proposed by Toledo (2003). I this model, the driver evaluates the curret ad the adjacet laes ad decided Cambridge Systematics, Ic. 327
46 NGSIM  ArterialLae Selectio Mode whether to make a lae chage or ot. The model was reestimated with the same arterial data. The statistical tests for comparig oested models imply that the ew model has a statistically sigificat improvemet i goodessoffit. The test results are preseted i Table 3.6. Figure 3.17 LaeShift Model Toledo 2003 Lae shift LEFT CURRENT RIGHT Gap acceptace NO CHANGE CHANGE LEFT NO CHANGE CHANGE RIGHT NO CHANGE Table 3.6 Model Compariso Statistic Base Model New Model Log likelihood value Number of parameters (K) Akaike iformatio criteria (AIC) a Bayesia iformatio criteria (BIC) b a b AIC = LLK where, LL=Log likelihood, K=umber of parameters. BIC = LLK/2*l(N) where, LL=Log likelihood, K=umber of parameters, N=umber of observatios Cambridge Systematics, Ic.
47 NGSIM  ArterialLae Selectio Mode 4.0 Implemetatio Verificatio This sectio describes the implemetatio verificatio of the proposed arterial models withi the microscopic traffic simulator MITSIMLab. This sectio first provides a overview of the MITSIMLab system. The implemetatio of the combied mergig model withi MITSIMLab is explaied ad fially, the verificatio method ad the results are preseted. The detailed implemetatio code is provided i Appedix A. 4.1 ALGORITHM SPECIFICATION Simulatio System MITSIMLab is a microscopic traffic simulatio laboratory developed to evaluate Advaced Traffic Maagemet Systems (ATMS) ad Advaced Traveler Iformatio Systems (ATIS) at the operatioal level. MITSIMLab ca represet a wide rage of traffic maagemet systems ad it ca model the respose of drivers to realtime traffic iformatio ad cotrol. This eables MITSIMLab to simulate the dyamic iteractios betwee traffic maagemet systems ad drivers. MITSIMLab cosists of three mai modules: Microscopic Traffic Simulator (MITSIM); Traffic Maagemet Simulator (TMS); ad Graphical User Iterface (GUI). MITSIM represets traffic ad etwork elemets. It represets the movemets of idividual vehicles i detail. The road etwork is represeted by odes, liks, segmets (liks are divided ito segmets with uiform geometric characteristics), ad laes. Traffic cotrols ad surveillace devices are represeted at the microscopic level. Travel demad is iput i the form of timedepedet OD flows from which idividual vehicles wishig to eter the etwork are geerated. Alteratively, idividual vehicles ca be loaded i the etwork at their exact startig times. Behavior parameters (e.g., desired speed, aggressiveess, ad aticipatio time) ad vehicle characteristics are assiged to each vehicle/driver. MITSIM moves vehicles accordig to acceleratio ad lae chagig models. The acceleratio model captures the drivers respose to coditios ahead as a fuctio of relative speed, headway, ad other traffic measures. The laechagig model distiguishes betwee madatory ad discretioary laechages. Mergig is classified as a madatory laechage. Drivers resposes to traffic sigals, speed limits, icidets, ad tollbooths are also captured. The drivig behavior models implemeted i MITSIMLab were estimated ad validated by Ahmed (1999), Toledo (2003), ad Choudhury (2005). Cambridge Systematics, Ic. 41
48 NGSIM  ArterialLae Selectio Mode TMS mimics the traffic cotrol system i the etwork uder cosideratio. A wide rage of traffic cotrol ad route guidace systems ca be simulated. These iclude itersectio cotrols, ramp cotrol, freeway mailie cotrol, laecotrol sigs, variable speed limit sigs, portal sigals, variable message sigs, ad ivehicle route guidace. TMS ca represet differet desigs of such systems with logic at varyig levels of sophisticatio (pretimed, actuated, or adaptive). A extesive GUI is used for both debuggig purposes ad demostratio of traffic impacts through vehicle aimatio. A detailed descriptio of MITSIMLab appears i Yag ad Koutsopoulos (1996), ad Yag et al. (2000) Implemetatio Withi Sectio Model The withi sectio model is implemeted as a special case i the laechagig model. The fuctio itself is computatioally simple, but is called may times i the simulatio: oce for every simulated vehicle at every time step. This makes it importat to miimize the commuicatio betwee the laechagig fuctio ad the rest of the system whe implemetig the model. Therefore, the laechagig part of the laechagig model has bee implemeted directly withi MITSIMLab, rather tha as a separate program that iterfaces with it. The pseudocode of the MITSIMLab implemetatio of the combied laechagig model is show i Figure 4.1. The fuctio CalcVehicleStats, which is called at every time step of the simulatio, iterates all vehicles o the etwork ad calculates their movemets. Amog other thigs, it calls the fuctio makelaechagigdecisio that checks whether there are exteral reasos forcig or forbiddig the vehicle from makig lae chages (such as messages from lae use sigs to stay i the curret lae or icidets blockig it). If there are o such special evets, the fuctio checkforlookaheadlc is called. This fuctio calls the fuctio TLProbabiltyLookAhead that calculates the utilities of all laes i that segmet ad calculates ad returs the probability of selectig each lae of the segmet usig these utilities. Depedig o the curret lae of the driver, these probabilities are allocated to the probability of choosig to chage to the left lae or the right lae ad the probability of choosig the curret lae. Based o a radom umber draw it the makes a choice betwee them (shift to the left, shift to the right, stay i the curret lae) ad returs this choice. This value also is retured by makelaechagigdecisio to the CalcVehicleStats fuctio. If the value of the variable retured by makelaechagigdecisio idicates that the driver is lookig to chage laes, the fuctio ExecuteLaeChage is called. This fuctio evaluates the lag ad lead critical gaps by callig the fuctio lccriticalgap ad comparig them to the correspodig available gaps. If the gaps are accepted, the vehicle is physically moved to the ew lae. 42 Cambridge Systematics, Ic.
49 NGSIM  ArterialLae Selectio Mode Figure 4.1 Pseudocode of MITSIMLab Implemetatio of the ArterialLae Selectio Model 4.2 UNIT TESTING Method This verificatio step is desiged to esure that the MITSIMLab code is a correct implemetatio of the laechagig model. We verified the code by comparig results from the MITSIMLab implemetatio with those obtaied from a Cambridge Systematics, Ic. 43
50 NGSIM  ArterialLae Selectio Mode idepedet spreadsheet implemetatio of the same model. The uit tests icluded the followig steps: 1. Creatio of a idepedet spreadsheet implemetatio of the model i Excel. This implemetatio was used as a bechmark to compare the results of the MITSIMLab implemetatio ad, thus, to idetify ay mistakes i the implemetatio. This is ecessary sice the computatios i the model are beyod what ca reasoably be doe by had. However, the two implemetatios were made by two differet researchers to reduce the risk that mistakes that may have bee made i the MITSIMLab implemetatio might be repeated i the spreadsheet implemetatio, ad so would ot be idetified i the verificatio tests. 2. Implemetatio of a log file i MITSIMLab that records the iputs; itermediate variable values (e.g., probabilities of lae choice, critical gaps, ad gap acceptace probabilities); ad the output (lae chages) from the laechagig fuctio. 3. The recorded iputs from the laechagig logs were used i the spreadsheet implemetatio to geerate the correspodig outputs. 4. The outputs from the two implemetatios were compared. Ay differeces betwee the two would idicate that some errors i either of the implemetatios exist. These errors were ivestigated ad corrected. Steps 1 through 4 were repeated util the results from the two implemetatios are idetical. I order to record the required iformatio, a dedicated MITSIMLab output file was created. The format of this file is detailed i Appedix D Results The uit tests were performed for the four elemets i the laechagig model implemetatio: 1. The itersectio lae choice model usig the probabilities of choice of the various alteratives; 2. The lead critical gap; 3. The lag critical gap; ad 4. The executio level. 44 Cambridge Systematics, Ic.
51 NGSIM  ArterialLae Selectio Mode 5.0 Operatioal Validatio This sectio describes the operatioal validatio process ad results. The dataset used i this study, the details of the calibratio, the validatio process that was applied, ad the results obtaied are preseted. The calibratio process ivolves adjustig the values of the parameters of the behavioral models ad estimatig travel demad, i the form of OD flows, o the etwork beig studied, i order to obtai a better fit of the model output with the actual traffic flow. I this study, the trajectory data collected from the same site was used for calibratio ad validatio i absece of other suitable data. 5.1 DATA DESCRIPTION Trajectory data from Lakershim Boulevard, Los Ageles, Califoria collected by the NGSIM Team has bee used for calibratio ad validatio of the model. Sythetic sesor couts ad speeds were geerated from the trajectory data for this purpose. Three sets of sesor couts ad speeds were calculated for each sectio: begiig, ed, ad midpoits. The locatios of the sythetic sesors are show i Figure 5.1. Figure 5.1 Locatios of Sythetic Sesors There was o route choice ivolved i the etwork. Exact vehicle OD flows were calculated from the trajectory data. The calibratio process therefore oly ivolved calibratio of the drivig behavior parameters. Details of the OD data are provided i Appedix C. The geerated OD flows ca be dowloaded from The total dataset was available for a 32miute period (8:28 a.m. to 9:00 a.m.). The first 22 miutes of data was used for calibratio ad the remaiig 10 miutes was used for validatio. Cambridge Systematics, Ic. 51
52 NGSIM  ArterialLae Selectio Mode 5.2 GOODNESSOFFIT MEASURES A umber of goodessoffit measures are used to evaluate the overall performace of a simulatio model. Popular amog them are the root mea square error (RMSE) ad the root mea square percet error (RMSPE). These statistics quatify the overall error of the simulator. Percet error measures directly provide iformatio o the magitude of the errors relative to the average measuremet. The two measures are give by: 1 N sim obs N = 1 ( ) 2 RMSE = Y Y (5.1.1) 2 N sim obs 1 Y Y RMSPE = obs N = 1 Y (5.1.2) obs Y ad sim Y are the averages of observed ad simulated measuremets at spacetime poit, respectively calculated from all available data (i.e., several days of observatios ad/or multiple simulatio replicatios). RMSE ad RMSPE, however, pealize large errors at a higher rate relative to small errors. Other measures iclude: Mea error ( ME ); ad Mea percet error ( MPE ). ME ad MPE idicate the existece of systematic uder or overpredictio i the simulated measuremets. These measures are give by: 1 N sim obs N = 1 ( ) ME = Y Y (5.1.3) N sim obs 1 Y Y MPE = N (5.1.4) Y obs = 1 obs sim where Y ad Y are the averages of observed ad simulated measuremets at spacetime poit, respectively calculated from all available data. 52 Cambridge Systematics, Ic.
53 NGSIM  ArterialLae Selectio Mode 5.3 SYSTEM CALIBRATION Calibratio Process Aggregate calibratio ca be formulated as a optimizatio problem, which seeks to miimize a fuctio of the deviatio of the simulated traffic measuremets from the observed measuremets. The formulatio preseted here assumes that the drivig behavior parameters are stable over the period of observatio. The formulatio is show below. The first ad secod terms i the objective fuctio are a measure of deviatio betwee observed ad simulated measuremets. The first costrait shows the depedece of simulated measuremets o the drivig behavior parameters ad the etwork coditios. mi β,od i = 1 N sim obs T 1 sim obs ( M M i ) W ( M M i ) sim s.t. M = S where, ( β ) β = Drivig behavior parameters; N = Number of days for which sesor data is available; sim M = Simulated measuremets; obs M i = Observed measuremets for day i; (5.1.6) S = The simulatio model fuctio, which geerates simulated traffic measuremets; ad W = Variacecovariace matrix of the sesor measuremets. I this study, the sesor measuremets that will be used for the calibratio are the sythetic sesor couts ad speeds geerated usig the first 22 miutes of available trajectory data (8:28 a.m. to 8:50 a.m.). The umber of behavioral parameters i the simulatio model is very large. It is ot feasible to calibrate all of them. Therefore, a few parameters that have the most sigificat effect o the simulatio results have bee selected. The focus was o calibratig these parameters, while fixig the other parameters to their previously estimated (default) values. Previous experiece has show that the simulatio results are most sesitive to the followig parameters: Sesitivity parameters of the acceleratio ad deceleratio fuctio; Parameters of the desired speed distributio; Itercepts ad variaces (costats ad sigmas) i the critical gap fuctios; ad Path pla variables. Cambridge Systematics, Ic. 53
54 NGSIM  ArterialLae Selectio Mode Calibratio Results Based o previous experiece ad sesitivity test results, the followig parameters of the combied model were selected for calibratio: Acceleratio ad deceleratio costats; Desired speed mea ad sigma; Itercepts (costats) ad variace (sigmas) of critical gap; Costat i the executio level; ad Itercept (costat) of lae 3 i the itersectio lae choice model. The calibratio parameters are listed i Table 5.1. Table 5.1 Model/Variable CarFollowig a Desired Speed a Gap Acceptace ItersectioLae Choice Withi Sectio Calibratio Parameters of the Combied Model Parameter Value Calibrated Parameter Explaatio of Calibrated Parameter Iitial Calibrated Acceleratio costat Deceleratio costat Mea Variace Lead gap costat Costat term for acceleratio i MITSIMLab carfollowig model (Ahmed 1999) Costat term for deceleratio i MITSIMLab carfollowig model (Ahmed 1999) Mea of the ormally distributed desired speed of the driver Variace of the ormally distributed desired speed of the driver Costat term for critical ormal lead gap Lead gap sigma Variace for critical ormal lead gap (log ormally distributed) Lag gap costat Lag gap sigma Costat term for critical ormal lag gap Variace for critical ormal lag gap (log ormally distributed) Lae 3 costat Costat term for lae Targetlae dummy Away from exit lae Curretlae dummy Executio costat Oe, if immediate lae ad target lae are the same Number of lae chages eeded for gettig to the lae that cotiues to the path of the driver Oe, if the curret lae is the target lae Costat for target lae a Geeral parameters of MITSIMLab. These variables are described i Ahmed (1999). 54 Cambridge Systematics, Ic.
55 NGSIM  ArterialLae Selectio Mode Amog these parameters, the desired speed parameter, the curretlae dummy, ad the executio costat made the most sigificat cotributio i improvig the performace of the model. Whe these parameters were ucostraied, the model performed better (objective fuctio for calibratio improved sigificatly) compared to the case whe these parameters were fixed to the origially estimated values. The driverspecific variables (desired speed, plaahead distace ad the coefficiets of aggressiveess) are expected to vary sice the model was validated at a differet site from the estimatio data collectio site. The improvemets after the calibratio are preseted i Table 5.2. Table 5.2 Improvemet Resultig from Calibratio LaeSpecific Couts Before Calibratio After Percet Improvemet RMSE (vehicles per 20 miutes) % RMSPE % LaeSpecific Speeds RMSE (mph) % RMSPE % The origial MITSIMLab laechagig model reestimated with arterial data (referred to as base model) parameters also were calibrated i a similar maer. The calibrated parameters of the base model iclude the followig: Acceleratio ad deceleratio costats; Desired speed mea ad sigma; Curretlae dummy; Laes away from exit lae; ad Itercepts (costats) ad variace (sigmas) of mal critical gap: Lead gap costat; Lead gap sigma; Lag gap costat; ad Lag gap sigma. Cambridge Systematics, Ic. 55
56 NGSIM  ArterialLae Selectio Mode 5.4 SYSTEM VALIDATION The purpose of system validatio is to determie the extet to which the simulatio model replicates the real system. At this step, the behavior parameters obtaied i the system calibratio step are fixed, ad the model predictios are compared agaist the secod set of traffic measuremets, which were ot used for calibratio. The validatio process is comparative. I this study, the goodessoffit statistics of the mergig model are compared with those of the base MITSIMLab model that icludes a laeshift model (Toledo 2003). The details of the model are attached i Appedix C. I this study, the sesor measuremets that will be used for the validatio are the sythetic sesor couts ad speeds geerated usig the last 10 miutes of available trajectory data (8:50 a.m. to 9:00 a.m.) i the orth boud directio Measures of Effectiveess The validity of the calibrated model was tested usig the several measures of effectiveess (MOE) that were obtaied from the sythetic sesor data ad from the summaries of the trajectory data. These icluded measures related to the mailie traffic coditios as well as measures related to the mergig lae (auxiliary lae) traffic coditios: Laespecific flows; Laespecific speeds; Lae distributios by locatio; Number of lae chages per vehicle; Locatio of lae chages; ad Number of icomplete trips Validatio Results LaeSpecific Flows Laespecific flow (vehicle/uit time) was compared amog the observed data, ew arterial models ad default MITSIMLab models. As see i Table 5.2, the ew model performs better tha the base model i terms of all measures. 2 Accurate sigal iputs (the logic cotrollig the actuated sigals) was ot available for the curret study. This could have sigificatly affected validatio results. The effect of the sigal time variatio was less i the orth boud directio. For this reaso calibratio ad validatio was performed oly i this directio. 56 Cambridge Systematics, Ic.
57 NGSIM  ArterialLae Selectio Mode Table 5.3 Compariso of LaeSpecific Couts Base New Percet Improvemet RMSE (vehicles per five miutes) % RMSPE % ME (vehicles per five miutes) % MPE % LaeSpecific Speeds Speed distributio i laes was compared amog the observed data, ew arterial models ad default MITSIMLab models. As see i Table 5.3, the ew model performs better i terms of RMSE ad RMSPE. The sigificat improvemet, however, is i ME ad MPE. RMSE ad RMSPE ted to pealize large errors. The results therefore imply that there are large discrepacies i some of the observatios of the ew model that cotribute to the large RMSE ad RMSPE values. The average errors still are ot sigificatly high. As discussed before, the limitatios associated with absece of accurate sigal iputs could have resulted such errors. Table 5.4 Compariso of LaeSpecific Speeds Base New Percet Improvemet RMSE (mph) % RMSPE % ME (mph) % MPE % Lae Distributios by Locatio Distributio of vehicles i laes was compared amog the observed data, base model (default MITSIMLab models), ad the ew models. The results for the orthboud sectios are preseted i Figure 5.3 to Figure 5.5. I each sectio distributios are calculated i three locatios (referred to as statios ): Statio 1 I the begiig of the sectio; Statio 2 I the middle of the sectio; ad Statio 3 At the ed of the sectio. The locatios of the sesors are preseted i Figure 5.2. As observed i the figures, the ew models have a better replicatio of the observed data. Particularly, i may cases, the base model ted to overpredict lae 4 occupacy ad uderpredict the through lae occupacies. Cambridge Systematics, Ic. 57
58 NGSIM  ArterialLae Selectio Mode Figure 5.2 Locatios of Sesors Figure 5.3 Statio 1 Statio 2 Statio 3 Statio 1 Statio 2 Statio 3 Statio 1 Statio 2 Statio 3 Compariso of Lae Distributios Sectio 1 Sectio 1  Statio 1 Lae Distributio Observed Base Model New Model Lae Sectio 1  Statio 2 Lae Distributio Observed Base Model New Model Lae Sectio 1  Statio 3 Lae Distributio Observed Base Model New Model Lae 58 Cambridge Systematics, Ic.
59 NGSIM  ArterialLae Selectio Mode Figure 5.4 Compariso of Lae Distributios Sectio 2 Sectio 2  Statio 1 Lae Distributio Observed Base Model New Model Lae Sectio 2  Statio 2 Lae Distributio Observed Base Model New Model Lae Sectio 2  Statio 3 Lae Distributio Observed Base Model New Model Lae Cambridge Systematics, Ic. 59
60 NGSIM  ArterialLae Selectio Mode Figure 5.5 Compariso of Lae Distributios Sectio 3 Sectio 3  Statio 1 Lae Distributio Observed Base Model New Model Lae Sectio 3  Statio 2 Lae Distributio Observed Base Model New Model Lae Sectio 3  Statio 3 Lae Distributio Observed Base Model New Model Lae Number of Lae Chages per Vehicle The distributio of the umber of lae chages made by vehicles, classified o the basis of their trajectory for the validatio time iterval, is preseted as aother measure of validity of the ew model i compariso to existig MITSIMLab models. The simulated results are compared with those extracted from the observed trajectory data set. Vehicles are classified o the basis of their trajectory ito three groups: 510 Cambridge Systematics, Ic.
61 NGSIM  ArterialLae Selectio Mode 1. Through Vehicles Vehicles travelig through the etire arterial sectio, eterig the study area from its southermost poit ad exitig through its orthermost poit; 2. Turig I Vehicles Vehicles eterig the study area from a side street at a itersectio, ad exitig through its orthermost poit; ad 3. TurigOut Vehicles Vehicles exitig the study area through a side street at a itersectio. Figures 5.6, 5.7, ad 5.8 depict the compariso of this measure betwee the simulatio results of the ew model, the base, ad the observatio dataset. Figure 5.6 Distributio of Lae Chages for Through Vehicles No. of Vehicles 30 Simulated  New Model Simulated  Base Observed No. of LC Cambridge Systematics, Ic. 511
62 NGSIM  ArterialLae Selectio Mode Figure 5.7 Distributio of Lae Chages by Vehicle Turig Ito Arterial No. of Vehicles Simulated  New Model Simulated  Base Observed No. of LC's Figure 5.8 Distributio of Lae Chages by Vehicles Turig Off the Arterial No. of Vehicles Simulated  New Model Simulated Base Observed No. of LC's The above figures illustrate that the ew model better replicates the observed behavior i compariso to the base model curretly existig i MITSIMLab. It ca be observed that the base model cosistetly forecasts a higher proportio of vehicles with more tha oe lae chage, irrespective of the vehicle trajectory. This overpredictio ca be attributed to the structure of the base model, which allows for oly the immediate adjacet laes i the targetlae choice set of a driver, thereby rederig him myopic. As a result, discretioary laechagig motivatios are overemphasized i compariso to madatory cosideratios withi the base model, ad a disproportioate umber of lae chages are simulated. This short comig is redressed to a extet by the ew model, which accords a lesser degree of sigificace to discretioary motivatios ad forecasts a more represetative spread of lae chages across differet vehicle trajectories. I particular, the results for the ew model validate the hypothesis, as also 512 Cambridge Systematics, Ic.
63 NGSIM  ArterialLae Selectio Mode idicated by the observatios, that lae chages i a arterial settig occur predomiatly with a madatory motivatio, resultig i a greater proportio of lae chages amog turig vehicles as compared to through vehicles. Number of Icomplete Trips I most cases there were o icomplete trips (vehicles missig their turs) recorded. A mior umber of icomplete trips (1 to 2 out of 2,240) were recorded i some cases. This is ot uusual give the stochastic ature of the simulator. Cambridge Systematics, Ic. 513
64 NGSIM  ArterialLae Selectio Mode 6.0 Summary ad Future Research A itersectio lae choice ad a withisectio laechagig model are preseted i this report. Both models are estimated with detailed trajectory data collected by the NGSIM team from Lakershim Boulevard, i Los Ageles, Califoria. The itersectio laechoice model ivolves the lae choice of vehicles eterig the arterial from a side street. The choice is modeled as a twostep process: targetlae choice, followed by immediatelae selectio based o the targetlae selectio. The choice of target lae was, however, uobserved ad oly the fial maeuvers of the driver were observed. The choice of target lae is iflueced more by pathpla variables ad laespecific attributes, whereas immediate lae choices are govered by maeuverability cosideratios. The heterogeeity of the driver populatio was explicitly take ito accout i the model formulatio. I particular, the plaig capability ad aggressiveess of the driver were allowed to vary amog drivers. The laechagig model withi sectio ivolves targetlae choice, acceptig the gaps to make a lae chage towards the directio of the target lae, ad executio of the lae chage to the accepted gap. The choice of target lae is, however, uobserved ad oly the fial lae actios are observed. The choice of target lae was foud to be iflueced by eighborhood vehicle speeds ad positios; laespecific attributes like average speed, queue legth, ad desity; ad factors such as path pla of the driver ad driver characteristics (iertia ad aggressiveess). Gap acceptace was iflueced by relative speeds of lead ad lag vehicles. Factors such as path pla ad aggressiveess of the driver were foud to sigificatly affect the models. The goodessoffit of the ew models were compared agaist simpler models estimated with the same data i each case. Statistical tests of estimatio results showed sigificat improvemet i the goodessoffit of the fial ew models. The model was implemeted i microscopic traffic simulator MITSIMLab. The accuracy of the implemetatio of the model was checked through Uit Tests. These ivolved compariso of MITSIMLab probability calculatios agaist Excel implemetatio of the same models. The ew models were validated agaist base models. These models cosisted of a rulebased itersectio laechoice model ad a simpler withisectio lae choice model, reestimated with the arterial data. The measures of validatio icluded compariso of the laespecific flows ad speeds, lae distributios i differet locatios, umber of lae chages per vehicle, ad umber of icomplete trips. The validatio results show improvemet i the simulatio capabilities of the ew models. Cambridge Systematics, Ic. 61
65 NGSIM  ArterialLae Selectio Mode The followig directios ca be explored to ehace the model: The curret study area did ot have ay bus stops or ostreet parkig. I additio, pedestria data was ot extractable. Cosideratio of the effects of these variables ca further erich the model. Acceleratio behavior i arterials may be differet from freeways. Previous research has show improvemets i the simulatio capability after combiig acceleratio models with laechagig models. The curret model does ot iclude acceleratio ad this ca be explored further i future research. Oe issue of iterest withi the domai of driver behavior models is the ability to capture the effect of the time resolutio of data o the estimatio results. The itroductio of a third level (executio level) i the laechagig model structure to model the laechage executio decisio, facilitates the explicit cosideratio of the impact of time resolutio o drivig decisios. This is a aspect of research related to this work that ca be explored i the future. Oe critical feature that has ot yet bee icorporated withi the model structure for laechagig ad acceleratio decisios of a mailie driver has bee that of statedepedecy amog the successive decisios of a idividual i traffic. The explicit modelig of state depedecy withi driver behavior models would form a very importat extesio to the curret work. 62 Cambridge Systematics, Ic.
66 Appedix A MITSIMLab Implemetatio of the Models
67 NGSIM  ArterialLae Selectio Mode Appedix A. MITSIMLab Implemetatio of the Models A.1 WITHIN SECTION MODEL //The fuctio is called every simulatio if the driver is eligible to make a lae chage vector <double> TS_Vehicle::TLProbabilityLookAhead(TS_Lae*plae, vector <it> & chage, vector <double> & LCdistace, vector <it> & coected) { it = plae>localidex(); it ex_id; // exit lae local idex : VR 01/06/07 RN_Segmet* psegmet = plae>segmet(); float *c = theparameter>targetlaeparams(); // cfc jue 3 it rules = lae_>crules(); double uc=0, ul=0, ur=0; //calculatio of CL utility TS_Vehicle* av = fidfrotbumperleader(plae); TS_Vehicle* bv = fidfrotbumperfollower(plae); double vld, spacig, relv, fspacig; float heavy_eighbor = 0.0; if (av) { else { relv = Mi(0,av>curretSpeed_  this>curretspeed_); vld = Mi(desiredSpeed_, av>curretspeed_) ; heavy_eighbor = (av>istype(vehicle_small))? 0.0 : 0;//c[18]; spacig = this>gapdistace(av); fspacig = spacig; relv = 0; vld = desiredspeed_ ; fspacig = 0; spacig = distace_; if (extlae_) spacig += extlae_>legth(); if (bv){ heavy_eighbor = (bv>istype(vehicle_small))? heavy_eighbor : 0; // c[18]; double rel_spacig; Cambridge Systematics, Ic. A1
68 NGSIM  ArterialLae Selectio Mode Appedix if (spacig > 0) { rel_spacig = relv/(1+exp(spacig)); else { rel_spacig = 0; float tailgate_dummy = 0; TS_Vehicle* behid = this>vehiclebehid() ; if (behid) { double gap_behid = behid>gapdistace(this); float des = tssegmet()>desity(); tailgate_dummy = (gap_behid <= c[9] && des <= c[10])? c[8] : 0; uc = c[6]+ c[7] * rel_spacig + c[15] * spacig + c[20] * aggresiveess(); double vld_right=0; double vld_left=0; //calculatio of RL utility oly if RL is i choice set if( (plae = lae_>right()) && (!flag(flag_vms_lane_use_right) plae>localidex() < VmsLaeUsePivotToIdex(flag())) && >= 0) { (rules & LANE_CHANGE_RIGHT!attr(ATTR_GLC_RULE_COMPLY)) && plae>istherebadevetahead(this) float heavy_eighbor_right = 0.0; TS_Vehicle* av = fidfrotbumperleader(plae); TS_Vehicle* bv = fidfrotbumperfollower(plae); if (av) { vld_right = Mi(desiredSpeed_, av>curretspeed_) ; heavy_eighbor_right = (av>istype(vehicle_small))? 0.0 : 0 ;// c[18]; else { vld_right = desiredspeed_ ; if (bv){ heavy_eighbor_right = (bv>istype(vehicle_small))? heavy_eighbor : 0 ;// c[18]; ur = c[16]; //calculatio of LL utility : oly if LL is i choice set A2 Cambridge Systematics, Ic.
69 NGSIM  ArterialLae Selectio Mode Appedix if( (plae = lae_>left()) && (!flag(flag_vms_lane_use_left) plae>localidex() < VmsLaeUsePivotToIdex(flag())) && >= 0) { //double vld_left; (rules & LANE_CHANGE_LEFT!attr(ATTR_GLC_RULE_COMPLY)) && plae>istherebadevetahead(this) float heavy_eighbor_left = 0.0; TS_Vehicle* av = fidfrotbumperleader(plae); TS_Vehicle* bv = fidfrotbumperfollower(plae); if (av) { vld_left = Mi(desiredSpeed_, av>curretspeed_) ; heavy_eighbor_left = (av>istype(vehicle_small))? 0.0 : 0 ;// c[18]; else { vld_left = desiredspeed_ ; if (bv){ heavy_eighbor_left = (bv>istype(vehicle_small))? heavy_eighbor : 0 ;// c[18]; ul = c[16]; //cfc: calculatig lae specific variables it laenum= psegmet>laes(); vector <double> ulae(laenum); vector<double> laedesity(laenum); vector<double> laespeed(laenum); vector<double> laequeue(laenum); // VR: 01/06/07 vector<it> la_av(laenum); // VR: 01/06/07 vector <it> dummycl(laenum); vector<it> dummyrl(laenum); vector<it> dummyll(laenum); vector<it> hovdummy(laenum); for (it j = 0; j <laenum; j ++) { //loop through all laes to fill i the probability vector float vel, que; it lae_av; it likid; likid = this>lik()>code(); TS_Lae* qlae = (TS_Lae*) psegmet>lae(j); laedesity[j]=qlae>desity(); laespeed[j]=qlae>calcspeed(); laequeue[j] = qlae>queueahead(this>positio()); Cambridge Systematics, Ic. A3
70 NGSIM  ArterialLae Selectio Mode Appedix vel = laespeed[j]; que = laequeue[j]; ulae[j]=0; //iitialize if ((qlae>ishovlae())&&(!(qlae>doesnotallow(this)))) {hovdummy[0]=1; ulae[0]= c[19];// cfc jue 24 if ( (j < coected[0]) (j> coected[coected.size()1]) ) { TS_Node * dode = (TS_Node *)lik()>dnode(); short it dliks = dode>dliks(); for(it ii=0;ii<dliks;ii++) { TS_Lik * target = (TS_Lik *)dode>dlik(ii); if ( (target>dnode()>code()  dode>code() == 1) (target>dnode()>code()  dode>code() == 1) ) { if (!qlae>isdowlaesinextlik(target)) { la_av[j] = 0; else {la_av[j] = 1; else { la_av[j] = 1; lae_av = la_av[j]; // if ( ((coected[0]  j)*(coected[0]  )) < 0 ) // { // la_av[j]=0; // //else // { // la_av[j]=1; // lae_av = coected[0]; A4 Cambridge Systematics, Ic.
71 NGSIM  ArterialLae Selectio Mode Appedix if ( likid == 3) { if (lae_av > 0) { lae_av = lae_av  1; it lv = la_av[j]; lae_av = la_av[j]; it dest, orig, likid; dest = this>desnode()>code(); orig = this>orinode()>code(); likid = this>lik()>code(); if (extlik_){ if ((this>extlik()>code() < 10) && (laenum > 3) && (la_av[laenum1] == 1) && (this>lik()>dnode()>code()!= 0) && (this>lik()>dnode()>code()!= 5)) { it lv; lv = la_av[laenum1]; if (lv > 1) { if (this>curretspeed() == 0) { it speedcheck =1; // ulae[0]=c[19];//c[19] is HOV lae dummy //allocate ucl, url,ul; switch (){ case 0://leftmost lae { ulae[0]=ulae[0]+uc; ulae[1]=ur; dummycl[0]=1; dummyrl[1]=1; Cambridge Systematics, Ic. A5
72 NGSIM  ArterialLae Selectio Mode Appedix break; default://aylae { ulae[]=uc; ulae[1]=ul; dummycl[]=1; dummyll[1]=1; if (!=(laenum1)) {//ot rightmost lae ulae[+1]=ur; dummyrl[+1]=1; // else //rightmost lae // {ulae[]=0; //default //switch double ulae0=ulae[0];//debug:cfc double ulae1=ulae[1];//debug:cfc double ulae2=ulae[2];//debug:cfc double ulae3=ulae[3];//debug:cfc // modified by VR : 01/06/07 for (it k=0;k<laenum;k++){ //allocatig costats it umchage=chage[k]; // distace from exit lae ulae[k]=ulae[k]+c[14]+c[11]*(umchage)*pow(lcdistace[k]/1000,c[17])+c[1]*(umchage); if ( (laenum == 6) && (k == 0) ) { ulae[k] = ulae[k]+c[22]; // VR : 01/06/07 float uq; // utility compoet for lae queue legth if (laequeue[k] <= 3) { uq = laequeue[k]*c[21]; else { uq = c[21]*3; A6 Cambridge Systematics, Ic.
73 NGSIM  ArterialLae Selectio Mode Appedix // VR : 01/06/07 float uclc; // utility compoet for o. of LC from curret lae if (abs(k) <= 1) { uclc = c[5]*abs(k); else { uclc = c[10]*abs(k); ulae[k]=exp(ulae[k]+c[3]*laedesity[k]+c[4]*laespeed[k]+ uclc + uq); //c[5]=coeff of each additioal lc reqd // VR : 01/06/07 ulae0=ulae[0];//debug:cfc //for double sum=0; for (it i=0;i<laenum;i++){ ulae[i] = la_av[i]*ulae[i]; // VR 01/06/07 sum=sum+ulae[i]; vector <double> problae(laenum); for (it i=0;i<laenum;i++){ problae[i]=ulae[i]/sum; double prob=problae[i]; retur problae; # edif A.2 INTERSECTION MODEL /* This fuctio is called whe turig vehicles are at the ed of a segmet before they eter the arterial */ Void TS_Vehicle::assigNextItLae() { it i, = lae_>dlaes(); extlae_ = NULL; if ( == 0) retur; Cambridge Systematics, Ic. A7
74 NGSIM  ArterialLae Selectio Mode Appedix // First, radomly choose a startig lae it k = (it) theradomizer>uradom(0, ); // Fid a lae that is i correct directio ad has the permissio to use if (segmet()>dowstream()!= NULL) { //this is the withi sectio case whe there is a dowstream segmet i same lik, cfc for (i = 0; i < ; i ++) { extlae_ = (TS_Lae *)lae_>dlae((i + k) % ); if (!(extlae_>iswroglae(this) retur; else if (extlik_!= NULL) { extlae_>doesnotallow(this))) { // this is the last segmet i this lik, there is a ew lik dowstream if (<2){ for (i = 0; i < ; i ++) { extlae_ = (TS_Lae *)lae_>dlae((i + k) % ); //<2 if (!extlae_>doesnotallow(this) && retur; (extlik_ == extlae_>lik())) { else if ((lae_>lik()>code())>9){//>=2,oly turig vehicles 9 is hardcoded lik umber for Lakershim NW vector <it> _cotiuiglaes; lookahead // list of local idexes to coected laes i the ext segmet. ca be differet based o vector <it> reqdchage();// is the umber of laes i dowstream segmet.reqdchage ca be differet based o lookahead // calculate the path pla variable, assumig all vehicles have paths float lookahead_= LookAheadDistace() ;// lookahead distace of the driver it myopic=0;// 1 if driver does ot lookahead beyod immediate ext sectio, 0 otherwise if (lookahead_<extlik_ >legth()){ myopic =1; for (it j=0;j< ;j++){ TS_Lae* qlae = (TS_Lae*) lae_>dlae(j);// qlae is j th lae the ext segmet i path it chage=0; it check=path()>lik(2)>code(); it check2=(qlae>lik()>code()); if ((check!=check2)&&(extlik_!=qlae>lik())) {//cfc 14 Ja chage=100; A8 Cambridge Systematics, Ic.
75 NGSIM  ArterialLae Selectio Mode Appedix reqdchage[j]=100; //qlae ot i path else { for (it i = 0; i < ; i ++) { _cotiuiglaes.push_back(i); if (checkiflookaheadevets()) { chage = checkmadatoryevetlc(); else { chage = checkmadatorylookahead_discrete(qlae, _cotiuiglaes); reqdchage[j]=chage; //check //for j vector <double >probl=iterlaechoice(lae_,reqdchage, myopic);//this returs (prob immediate lae choices) //fial check for (it p = 0; p < ; p ++) { TS_Lae* templae = (TS_Lae *)lae_>dlae(p); if (extlik_!= templae>lik()) { probl[p]=0; double cdf=probl[1]; it kk = (it) theradomizer>uradom(0, 1); it m ; for (m =  1; m > 0 && kk > cdf; m ) { cdf += probl[m]; extlae_ = (TS_Lae *)lae_>dlae(m); retur; //if >1 ad turig vehicle Cambridge Systematics, Ic. A9
76 NGSIM  ArterialLae Selectio Mode Appedix /* this fuctio returs probability of itersectio lae choice, called by assignextitlae */ vector <double> TS_Vehicle::iterLaeChoice(TS_Lae*plae, vector <it> & chage, it myopic) { float *d = theparameter>arterialparams(); // cfc dec 06 it curret = plae>localidex(); //idex of aturally coectig lae it laenum= chage.size();// total umber of laes i lik // defie the vectors vector <double> ulae(laenum);// utility of target laes vector <double> problae(laenum);//probabilities of target laes vector <double> ulae_1(laenum);// utility of immediate laes give target lae 1 vector <double> ulae_2(laenum);// utility of immediate laes give target lae 2 vector <double> ulae_3(laenum);// utility of immediate laes give target lae 3 vector <double> ulae_4(laenum);// utility of immediate laes give target lae 4 vector <double> problae_i(laenum);//probability vector of immediate laes vector <double> logsum(laenum);// utility of immediate laes give target lae 4 vector <it> check(laenum);// ext lik id vector <it> check2(laenum);//qlae id //immediate lae probabilities are calculated first for (it j = 0; j <laenum; j ++) { //loop through all dowstream laes i ext segmet i path to fill i the probability vector TS_Lae* qlae = (TS_Lae*) lae_>dlae(j); RN_Segmet *qsegmet = qlae >segmet(); check[j]=path()>lik(2)>code(); check2[j]=(qlae>lik()>code()); if ((check[j]==check2[j])&&(extlik_==qlae>lik())) {//qlae is i path double commo_ulae=(d[11]/(d[12]+d[13]*aggresiveess()))*abs(curretj);//+ d[18]*gap; ulae_1[j]=commo_ulae+(j==1)*d[14]+abs(1curret)*(d[15]/(d[16]+d[17]*aggresiveess())); double debug1=(j==1)*d[14]; double debug2=abs(1curret)*(d[15]/(d[16]+d[17]*aggresiveess())); ulae_2[j]=commo_ulae+(j==2)*d[14]+abs(2curret)*(d[15]/(d[16]+d[17]*aggresiveess())); if (laenum>2) ulae_3[j]=commo_ulae+(j==3)*d[14]+abs(3curret)*(d[15]/(d[16]+d[17]*aggresiveess())); else {ulae_3[j]= ; if (laenum>3) ulae_4[j]=commo_ulae+(j==4)*d[14]+abs(4curret)*(d[15]/(d[16]+d[17]*aggresiveess())); else {ulae_4[j]= ; double debug33=ulae_1[0]; double debug43 =ulae_2[1]; double debug53= ulae_4[3]; A10 Cambridge Systematics, Ic.
77 NGSIM  ArterialLae Selectio Mode Appedix ulae_1[j]=exp(ulae_1[j]); ulae_2[j]=exp(ulae_2[j]); ulae_3[j]=exp(ulae_3[j]); ulae_4[j]=exp(ulae_4[j]); // calculatio of logsum logsum[j]=log(mi(.00001,(ulae_1[j]+ulae_2[j]+ulae_3[j]+ulae_4[j]))); //check else{//ot i path, check==check2 ulae_1[j]= ; ulae_2[j]= ; ulae_3[j]= ; ulae_4[j]= ; logsum[j]=0; //for double q=ulae_1[0]; double q2=ulae_1[3]; //target lae probabilities are calculated ext for (it hh = 0; hh <laenum; hh ++) { //loop through all dowstream target laes TS_Lae* qlae_tl = (TS_Lae*) lae_>dlae(hh); if ((check[hh]==check2[hh])&&(extlik_==(qlae_tl>lik()))) {//qlae_i is i path //assigig costats for target lae switch (hh){ case 0: { ulae[hh]=0; break; case 1: { ulae[hh]=d[0]; break; //left most lae lae0 is the base case 2: { ulae[hh]=d[1]; break; default: { ulae[hh]=d[2]; break; Cambridge Systematics, Ic. A11
78 NGSIM  ArterialLae Selectio Mode Appedix //addig other variables it lq=qlae_tl> Vehicles(); double dischargeq= qlae_tl>dischargerate(qlae_tl);//this is discharge rate (historic)of that lae. Should be a iput. For time beig hardcoded. double delay; if (lq>1) {delay=1/(1+exp(lq/dischargeq)); else { delay=2; it Chage=chage[hh]; ulae[hh]=ulae[hh]+d[3]*delay+ myopic *(d[4]/(d[5]+d[6]*aggresiveess()))*chage[hh]+ (1 myopic)*(d[7]/(d[8]+d[9]*aggresiveess())) * chage[hh]+d[10] * logsum[hh]; ulae[hh]=exp(ulae[hh]); //if check==check2 else{ulae[hh]=exp( );// qlae_i ot i path double debug3=ulae[0]; double debug4 =ulae[1]; double debug5= ulae[3]; //all ds laes double sum=0; // calculate sum for (it i=0;i<laenum;i++){ sum=sum+(check[i]==check2[i])*ulae[i]; //calculate probabilty for (it i=0;i<laenum;i++){ if (extlik_!=(lae_>dlae(i)>lik())) {problae[i]=0; else{ problae[i]=ulae[i]/sum; // select target lae double cdf2=problae[laenum1];// sice local idex starts from 0 it r = (it) theradomizer>uradom(0, 1); it TL ; for (TL = laenum  1; TL > 0 && r > cdf2; TL ) { cdf2 += problae[tl]; A12 Cambridge Systematics, Ic.
79 NGSIM  ArterialLae Selectio Mode Appedix // ow calculate immediate lae probability double sum_i=0; // calculate sum_i switch (TL){ case 0: {//chose target lae is lae 1 for (it h=0;h<laenum;h++){ sum_i=sum_i+(check[h]==check2[h])*ulae_1[h]; for (it h=0;h<laenum;h++){ TS_Lae* qqqlae= (TS_Lae*) lae_>dlae(h); if (extlik_==(qqqlae>lik())){ problae_i[h]=(check[h]==check2[h])*ulae_1[h]/sum_i; else { problae_i[h]=0; break; case 1: {//chose lae is lae 2 for (it h=0;h<laenum;h++){ sum_i=sum_i+(check[h]==check2[h])*ulae_2[h]; for (it h=0;h<laenum;h++){ TS_Lae* qqqlae= (TS_Lae*) lae_>dlae(h); if (extlik_==(qqqlae>lik())){ problae_i[h]=(check[h]==check2[h])*ulae_2[h]/sum_i; else { problae_i[h]=0; break; case 2: {//chose lae is lae 3 for (it h=0;h<laenum;h++){ sum_i=sum_i+(check[h]==check2[h])*ulae_3[h]; for (it h=0;h<laenum;h++){ TS_Lae* qqqlae= (TS_Lae*) lae_>dlae(h); if (extlik_==(qqqlae>lik())){ Cambridge Systematics, Ic. A13
80 NGSIM  ArterialLae Selectio Mode Appedix else { problae_i[h]=0; break; problae_i[h]=(check[h]==check2[h])*ulae_3[h]/sum_i; case 3: {//chose lae is lae 4 for (it h=0;h<laenum;h++){ sum_i=sum_i+(check[h]==check2[h])*ulae_4[h]; for (it h=0;h<laenum;h++){ TS_Lae* qqqlae= (TS_Lae*) lae_>dlae(h); if (extlik_==(qqqlae>lik())){ problae_i[h]=(check[h]==check2[h])*ulae_4[h]/sum_i; else { problae_i[h]=0; break; double debug63=problae_i[0]; double debug64 =problae_i[1]; double debug65= problae_i[2]; double debug66= problae_i[3]; retur problae_i; //ed of fuctio A14 Cambridge Systematics, Ic.
81 Appedix B Base Model Toledo 2003
82 B. Base Model Toledo 2003 NGSIM  ArterialLae Selectio Mode Appedix B.1 MODEL SPECIFICATION This sectio describes the structure ad the detailed specificatio of the laechagig model. The model is a itegrated lae chagig model, i which the driver joitly evaluates madatory ad discretioary cosideratios. The lae chagig process cosists of two steps: 1) choice of a lae lae shift; ad 2) gap acceptace decisios. This decisio process is latet, sice the lae shift choice is uobservable; oly the driver s lae chagig actios are observed. The structure of the model is show i Figure B.1. Latet choices variables are show as ovals, ad observed oes are show as rectagles. The lae lae shift is the directio of chage (or decisio ot to chage) that the driver perceives as best to udertake. The CURRENT brach correspods to a situatio i which the driver decides ot to pursue a lae chage. I the RIGHT ad LEFT braches, the driver perceives that movig i these directios, respectively, would improve his/her coditio. I these cases, the driver evaluates the adjacet gap i the lae i the chose directio ad decides whether the laechage ca be executed or ot. Oly if the driver perceives that the gap is acceptable the laechage is executed (CHANGE RIGHT or CHANGE LEFT); otherwise, the driver does ot execute the lae chage (NO CHANGE). This decisio process is repeated at every time step. Figure B.1 Structure of the LaeChagig Model Lae shift LEFT CURRENT RIGHT Gap acceptace NO CHANGE CHANGE LEFT NO CHANGE CHANGE RIGHT NO CHANGE B.1.1 Lae Shift Model The laeshift (LS) choice set icludes up to three alteratives: the driver may chose to stay i the curret lae (CL); target shiftig either to the right lae (RL); or to the left lae (LL). The utilities of these alteratives are give by: Cambridge Systematics, Ic. B1
83 NGSIM  ArterialLae Selectio Mode Appedix where, U ( t) = X ( t) β + α υ + ε ( t) i = CL, RL, LL i i i i i i U ( t) is the utility of lae i to driver at time t; (B.1.1) X ( t ) is a vector of explaatory variables; i i β is the correspodig vector of parameters; i ε ( t ) is the radom term associated with the lae shift utility; υ is a driver specific radom term that represets uobservable characteristics of the driver/vehicle, thus capturig correlatios betwee observatios of the same driver over time; υ is assumed to be ormally distributed i the drivers populatio; ad i α are the parameters of υ. CL RL LL Assumig that the radom terms ε ( t), ε ( t) ad ε ( t) are idepedetly ad idetically Gumbel distributed, the choice probabilities of lae lae shifts, coditioal o the idividual specific error term ( υ ) are give by: i ( V t υ ) j ( V t υ ) exp ( ) P ( i( t) υ ) = i I = CL, RL, LL exp ( ) j I { i V ( t) υ are the coditioal systematic utilities of the alteratives, give by: i i i i V ( t) υ = X ( t) β + α υ i = CL, RL, LL (B.1.2) (B.1.3) Laeshift utility fuctios may deped o explaatory variables from the four categories discussed above. Variables should reflect the coditios i the immediate eighborhood i each lae (e.g., relative leader speed i each lae, presece of heavy vehicles, ad tailgatig); path pla cosideratios (e.g., the distace to a poit where the driver must be i specific laes ad the umber of laechages eeded i order to be i these laes); ad kowledge of the system (e.g., avoidig the left lae before permissive left turs or avoidig oramp mergig laes). I most cases, iformatio about the driver s style ad characteristics is ot available. Thus, these characteristics are captured by the idividual specific error termυ. B.1.2 Gap Acceptace Model The gap acceptace model captures drivers decisios to execute the lae chage. The driver evaluates the adjacet gap i the target lae, which is defied by the lead ad lag vehicles i that lae (Figure B.2). The lead gap is the clear spacig betwee the rear of the lead vehicle ad the frot of the B2 Cambridge Systematics, Ic.
84 NGSIM  ArterialLae Selectio Mode Appedix subject vehicle. Similarly, the lag gap is the clear spacig betwee the rear of the subject vehicle ad the frot of the lag vehicle. Note that both these gaps may be egative if the vehicles overlap. Figure B.2 Defiitios of Gaps Adjacet gap Traffic directio Lag vehicle Lag gap Lead gap Lead vehicle Subject vehicle Frot spacig Frot vehicle The driver compares the available space lead ad lag gaps to the correspodig critical gaps, which are the miimum acceptable space gaps. A available gap is acceptable if it is greater tha the critical gap. Critical gaps are modeled as radom variables. Their meas are fuctios of explaatory variables. The idividual specific error term captures correlatios betwee the critical gaps of the same driver over time. Critical gaps are assumed to follow logormal distributios to esure that they are always oegative. The lead ad lag critical gaps are, respectively:, ( ) l G lead i cr ( t) = X lead i ( t) β lead + α lead υ + ε lead ( t), ( ) l G lag i cr ( t) = X lag i ( t) β lag + α lag υ + ε lag ( t) where, lead i, cr lag i, cr G ( t ) ad G ( t ) are, respectively, the lead ad lag critical gaps i the target lae measured i meters; lead i lag i X ( t ) ad X ( t ) are vector of explaatory variables affectig the lead ad lag critical gaps, respectively; lead β ad lag β are the correspodig vectors of parameters; lead lag ε ( t) ad ( t) ε lag ( t) N 0, 2 ( σ lag ) 2 ε are radom terms: ε lead ( t) N ( 0, σ lead ) ; ad lead lag α ad α are the parameters of the driverspecific radom term ad lag critical gaps, respectively. ad υ i the lead The gap acceptace model assumes that the driver must accept both the lead gap ad the lag gap to chage laes. The probability of executig a lae Cambridge Systematics, Ic. B3
85 NGSIM  ArterialLae Selectio Mode Appedix chage, coditioal o the idividual specific term ad the lae shift is, therefore, give by: i ( chage i shift directio ( ), υ ) ( ( ) 1 ( ) ) P i t = P l t = i t,υ = ( accept lead gap ( ), υ ) ( accept lag gap ( ), υ ) P i t P i t lead i lead i ( ( ), cr lag i lag i ( ) ( ), ) ( ( ), cr > υ > ( ) ( ), υ ) P G t G t i t P G t G t i t where, { RL, LL i is the lae shift (that requires a laechage); = (B.1.6) lag i G ( t ) ad G ( t ) are the available lead ad lag gaps i the lae i the lae lead i shift, respectively; ad i l ( t ) is a idicator to the laechagig actio: i 1 a lae chage i the directio i is executed at time t l( t) = 0 otherwise Assumig that critical gaps follow logormal distributios, the coditioal probability that the lead gap is acceptable is give by: lead i lead i, cr ( ( ) > ( ) ( ), υ ) = lead i lead i, cr ( l ( ( )) l ( ( )) ( ), υ ) lead i lead i lead lead l ( G ( t) ) ( X ( t) β + α υ ) P G t G t i t P G t > G t i t = Φ σ lead Φ[ ] deotes the cumulative stadard ormal distributio. where (B.1.7) Similarly, the coditioal probability that the lag gap is acceptable is give by: lag i lag i, cr ( ( ) > ( ) ( ), υ ) = lag i lag i, cr l ( ( )) l ( ( )) ( ), υ lag i lag i lag lag l ( G ( t) ) ( X ( t) β + α υ ) P G t G t i t ( ) P G t > G t i t = Φ σ lag (B.1.8) The gap acceptace decisio is primarily affected by eighborhood variables, such as the subject relative speeds with respect to the lead ad lag vehicles. Path pla variables, capturig the ecessity of the laechage, may also affect critical gaps. B4 Cambridge Systematics, Ic.
86 B.2 MODEL ESTIMATION NGSIM  ArterialLae Selectio Mode Appedix Estimatio results of the proposed laechagig model are preseted i Table B.1. Table B.1 Estimatio Results of the Reestimated Lae Chagig Model Fial LogLikelihood Number of Observatios Number of Vehicles 160 Number of Parameters 19 Variable Name Parameter Estimate Tstat Target Lae Selectio Model Curret Lae Dummy Path Pla impact : No. of lae chages to exit lae Path pla impact : No. of lae chages to exit iteracted with distace from exit Expoet of dist. to exit i o. of laes to exitdist. to exit iteractio Queue legth ahead i lae Frot vehicle rel. speed (give whe frot leadig vehicle is preset) αcl Gap Acceptace Model Lead Critical Gap Lead gap costat lead,tl V t (m/s) σlead αlead Lag Critical Gap Lag gap costat lag,tl V t (m/s) σlag αlag Cambridge Systematics, Ic. B5
87 Appedix C OrigiDestiatio Data
88 C. OrigiDestiatio Data NGSIM  ArterialLae Selectio Mode Appedix For validatio, exact vehicle OD flows were calculated usig the Lakershim trajectory data. The OD files ca be dowloaded from: papers/od_lakershim.zip The ode locatios ad umberig associated with the OD files are preseted i Figure C.1. Summary of OD is give i Table C.1. Sample of the OD file is preseted i Table C.2. Figure C.1 Node Locatios ad Numberig for Lakershim Boulevard Cambridge Systematics, Ic. C1
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