time headway distribution of probe vehicles on single and multiple lane highways

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KSCE Journal of Civil Engineering (2013) 17(4):824-836 DOI 10.1007/s12205-013-0212-5 824 www.springer.com/12205 Transportation Engineering Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways Rattaphol Pueboobpaphan*, Dongjoo Park**, Youngchan Kim***, and Sangho Choo**** Received April 29, 2012/Accepted August 17, 2012 ··································································································································································································································· Abstract Time headway distribution modeling is fundamental to many aspects of traffic flow studies such as capacity estimation, safety analysis, and microscopic simulation. Existing time headway distribution models have focused on the behavior of general vehicles. We examine the distribution of sampled vehicle headway (e.g., probes) on both single and multiple lane highway traffic streams. This study is divided into three parts: an empirical study, a simulation analysis, and an analytical derivation. The empirical study uses probe data obtained from Houston, Texas that was collected as part of the Automatic Vehicle Identification system of the Houston Transtar system. In the empirical study, a shifted negative exponential distribution was found to give the closest fit for both single and multiple lane cases. We found that if the volume level of the probes is low, regardless of the volume level of general vehicles, the headway of the probes followed a shifted negative exponential distribution. In the simulation study, we found that the time headway of probes does not necessarily follow the time headway distribution of general vehicles. Rather, it depends on many variables such as the volume level of general vehicles, the market penetration of probe vehicles, and the number of lanes. However, when the volume level of general vehicles is low, the headway of probes tends to follow the shifted negative exponential distribution at all levels of market penetration, together with the general vehicles. We analytically proved that if the time headway of general vehicles follows the shifted negative exponential distribution, then the time headway of the probes is the same as that of the general vehicles. Keywords: probe vehicles, headway distribution, multiple lanes, highway, negative exponential distribution ··································································································································································································································· 1. Introduction Time Headway Distribution (THD) modeling is fundamental to many aspects of traffic flow studies such as capacity estimation, safety analysis, and microscopic simulation (Hossanin et al., 1999; Jin et al., 2009; Toledo, 2010). Most of the existing THD models focus on the behavior of general vehicles in a single lane (Sadeghhosseini and Benekohal, 2002; Abdennour and Al- Ghamdi, 2002) or multiple lanes (Akcelik and Chung, 1994; Hagring, 1998; Yin et al., 2009). With the advent of ITS(Intelligent Transportation Systems), a number of probe-based information systems have become popular and important. Such systems include Automatic Vehicle Identification (AVI), the Global Positioning System (GPS), and Beacon, where instrumented vehicles provide information about the system. An advantage of these systems is that they can automatically collect detailed location information on the instrumented vehicles active in the system. This location data can subsequently be converted into traffic information such as link travel times, route travel times, and origin-destination volumes. These data have been applied to the Advanced Traffic Information System (ATIS) to provide travel time information to road users (Park et al., 1999; Choi and Chung, 2001; Rilett and Park, 2001; Eisele and Rilett, 2002; Park et al., 2002; Yim and Cayford, 2002; Pattanamekar et al., 2003, Lam and Chan, 2004; Kim et al., 2007; Park et al., 2008; Kim and Chang, 2008; Kim, 2009; Kim et al., 2009). Another application of probe data is incident detection and management (Sermons and Koppelman, 1996; Mussa, et al., 1998; Hellinga and Knapp, 2000; Cheu et al., 2002). A number of studies have used the time headway information of probes (Parkany and Bernstein, 1995; Rilett and Park, 1999). However, existing time headway models do not consider market penetration and traffic volume. In this context, the goal of this study is to analyze probe headway distribution in single and multiple lane traffic streams. The relationship between the THD of general vehicles and probes under different traffic conditions is also examined. We expect that knowledge of the THD of probe vehicles can be used to explain the distribution type of general vehicle headway. Moreover, it can be applied directly to the *Assistant Professor, Dept. of Transportation Engineering, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand (E-mail: ratta- [email protected]) **Professor, Dept. of Transportation Engineering, The University of Seoul, Seoul 130-743, Korea (Corresponding Author, E-mail: [email protected]) ***Member, Professor, Dept. of Transportation Engineering, The University of Seoul, Seoul 130-743 Korea (E-mail: [email protected]) ****Member, Assistant Professor, Dept. of Urban Design & Planning, Hongik University, Seoul 121-791, Korea (E-mail: [email protected])

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KSCE Journal of Civil Engineering (2013) 17(4):824-836DOI 10.1007/s12205-013-0212-5

− 824 −

www.springer.com/12205

Transportation Engineering

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Rattaphol Pueboobpaphan*, Dongjoo Park**, Youngchan Kim***, and Sangho Choo****

Received April 29, 2012/Accepted August 17, 2012

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Abstract

Time headway distribution modeling is fundamental to many aspects of traffic flow studies such as capacity estimation, safetyanalysis, and microscopic simulation. Existing time headway distribution models have focused on the behavior of general vehicles.We examine the distribution of sampled vehicle headway (e.g., probes) on both single and multiple lane highway traffic streams. Thisstudy is divided into three parts: an empirical study, a simulation analysis, and an analytical derivation. The empirical study usesprobe data obtained from Houston, Texas that was collected as part of the Automatic Vehicle Identification system of the HoustonTranstar system. In the empirical study, a shifted negative exponential distribution was found to give the closest fit for both single andmultiple lane cases. We found that if the volume level of the probes is low, regardless of the volume level of general vehicles, theheadway of the probes followed a shifted negative exponential distribution. In the simulation study, we found that the time headwayof probes does not necessarily follow the time headway distribution of general vehicles. Rather, it depends on many variables such asthe volume level of general vehicles, the market penetration of probe vehicles, and the number of lanes. However, when the volumelevel of general vehicles is low, the headway of probes tends to follow the shifted negative exponential distribution at all levels ofmarket penetration, together with the general vehicles. We analytically proved that if the time headway of general vehicles followsthe shifted negative exponential distribution, then the time headway of the probes is the same as that of the general vehicles. Keywords: probe vehicles, headway distribution, multiple lanes, highway, negative exponential distribution

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1. Introduction

Time Headway Distribution (THD) modeling is fundamentalto many aspects of traffic flow studies such as capacity estimation,safety analysis, and microscopic simulation (Hossanin et al.,1999; Jin et al., 2009; Toledo, 2010). Most of the existing THDmodels focus on the behavior of general vehicles in a single lane(Sadeghhosseini and Benekohal, 2002; Abdennour and Al-Ghamdi, 2002) or multiple lanes (Akcelik and Chung, 1994;Hagring, 1998; Yin et al., 2009).

With the advent of ITS(Intelligent Transportation Systems), anumber of probe-based information systems have becomepopular and important. Such systems include AutomaticVehicle Identification (AVI), the Global Positioning System(GPS), and Beacon, where instrumented vehicles provideinformation about the system. An advantage of these systems isthat they can automatically collect detailed location informationon the instrumented vehicles active in the system. This locationdata can subsequently be converted into traffic information suchas link travel times, route travel times, and origin-destination

volumes. These data have been applied to the Advanced TrafficInformation System (ATIS) to provide travel time information toroad users (Park et al., 1999; Choi and Chung, 2001; Rilett andPark, 2001; Eisele and Rilett, 2002; Park et al., 2002; Yim andCayford, 2002; Pattanamekar et al., 2003, Lam and Chan, 2004;Kim et al., 2007; Park et al., 2008; Kim and Chang, 2008; Kim,2009; Kim et al., 2009).

Another application of probe data is incident detection andmanagement (Sermons and Koppelman, 1996; Mussa, et al.,1998; Hellinga and Knapp, 2000; Cheu et al., 2002). A numberof studies have used the time headway information of probes(Parkany and Bernstein, 1995; Rilett and Park, 1999). However,existing time headway models do not consider market penetrationand traffic volume. In this context, the goal of this study is toanalyze probe headway distribution in single and multiple lanetraffic streams. The relationship between the THD of generalvehicles and probes under different traffic conditions is alsoexamined. We expect that knowledge of the THD of probevehicles can be used to explain the distribution type of generalvehicle headway. Moreover, it can be applied directly to the

*Assistant Professor, Dept. of Transportation Engineering, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand (E-mail: [email protected])

**Professor, Dept. of Transportation Engineering, The University of Seoul, Seoul 130-743, Korea (Corresponding Author, E-mail: [email protected])***Member, Professor, Dept. of Transportation Engineering, The University of Seoul, Seoul 130-743 Korea (E-mail: [email protected])

****Member, Assistant Professor, Dept. of Urban Design & Planning, Hongik University, Seoul 121-791, Korea (E-mail: [email protected])

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Vol. 17, No. 4 / May 2013 − 825 −

aforementioned applications instead of using general vehicleheadway data. In particular, this approach might be useful whenprobe data are much easier to obtain than general vehicle data, orwhen only probe data are available.

The remainder of the paper is organized as follows. In Section2, we introduce notation and provide a definition of probe timeheadway on single and multiple lane traffic streams from ananalytical perspective. In Section 3, we analyze real-world probedata collected as a part of the Automatic Vehicle Identification(AVI) system of the Houston Transtar system, and the beststatistical distribution models under various traffic conditions areidentified. In Section 4, a simulation technique is used toexamine the distribution type of probes. Lastly, a probe THDmodel is analytically derived from that of the general vehicles inSection 5. Our conclusions are given in Section 6.

2. Problem Definition

The time headway is the difference in arrival time betweensuccessive vehicles at an observation point. Similarly, the timeheadway of probe vehicles on a single lane can be defined as thedifference in arrival times between successive probe vehicles atan observation point. The time headway of a probe vehicle onmultiple lanes is defined as a minimum different value betweenthe current probe vehicle at an observation point and successiveprobes on all lanes. Figs. 1(a) and (b) illustrate the time headwayof probe vehicles on single and multiple lane traffic streams. Fig.1(c) shows how to find probe headway on a time-space diagram.Analytically, the time headway between probe vehicles on singleand multiple lane traffic streams are defined by Eqs. (1) and (2),respectively.

(1)

and (2)

3. Empirical Test

3.1 Data Collection and Experimental Study DesignThe test bed freeway is US-290, which is a radial urban

freeway in Houston, Texas. It is part of the Houston trafficmonitoring system that currently includes approximately 365 kmof freeway and 113 km of High Occupancy Vehicle (HOV) lanesinstalled with an AVI system. AVI station numbers 24, 27, and41 were used as the observation points for a three-lane, four-lane,and single-lane freeway section, respectively. Note that stationnumber 41 is located at the median lane, which is a e (HOV) laneand was used as a single lane traffic stream in this study. Figure 2illustrates the US-290 test bed freeway and the observationpoints.

Weekday headway data from ten days in April, 1997 (April14-18 and 21-25) were used in the analysis. The study timeperiod was defined as lasting from 6:00 AM to 6:00 PM. Onlythe Eastbound headway data were employed in this study. Due tothe nature of the AVI system, the information for non-probe

vehicles was not available. AVI vehicle volume ranged from 42-168, 20-160, and 25-160 veh/hr/lane (vphpl) for the single lane,

hp tpn tp

c–=

hp min tp l,c tp j,

c–( ); l∀ 1 L;,= = j∀ 1 L,=

Fig. 1. Time Headway between Probe Vehicles: (a) Single Lane, (b)Multiple Lane, (c) Probe Headway in Time-Space Diagram

Fig. 2. Test Bed Freeway US-290 and the Observation Points

Rattaphol Pueboobpaphan, Dongjoo Park, Youngchan Kim, and Sangho Choo

− 826 − KSCE Journal of Civil Engineering

three-lane, and four-lane freeways, respectively. The data were subsequently aggregated into 10 min increments

and the number of probe vehicles per 10 min period wascalculated. For each of the three test sites, the 720 10 min periodswere further aggregated based on the observed probe volumelevel. The number of observations, the probe vehicle volumerange, the mean headway, the headway standard deviation, andthe coefficient of variation for each data set are shown in Table 1.

There are 10, 14, and 18 data sets for the single, three, and fourlane freeways, respectively. As expected, both the mean andstandard deviation of headway decreased as the probe vehiclevolume increased. Note that the mean and standard deviation ofheadway decreased as the number of lanes increased.Interestingly, the coefficient of variation is approximately 1.0 forall observations, indicting a one-to-one relationship between themean and standard deviation. This implies that the probe

Table 1. Time Headway Data of AVI VehiclesNumber of

Lanes Data Set No. of Probe Observations

Probe Vehicle Volume Range (veh/10 min)

Mean Headway (sec)

Standard Deviation of Headway (sec)

COV (STD / Mean)

1*

1 155 7 - 8 99.1 99.5 1.02 130 9 - 10 69.3 56.6 0.83 170 13 - 14 49.8 49.9 1.04 159 15 - 16 41.9 43.2 1.05 232 17 - 18 35.5 37.7 1.16 149 19 - 20 32.7 31.7 1.07 208 21 - 22 28.7 26.4 0.98 247 23 - 24 26.8 24.9 0.99 173 25 - 26 24.4 20.0 0.8

10 106 27 - 28 22.9 24.3 1.1

3

1 251 10 - 15 46.8 42.3 0.92 1083 16 - 20 34.7 34.8 1.03 1853 21 - 25 27.1 27.6 1.04 2690 26 - 30 22.2 22.4 1.05 2841 31 - 35 18.9 20.3 1.16 2313 36 - 40 16.0 17.0 1.17 2250 41 - 45 14.5 15.5 1.18 2347 46 - 50 12.8 13.4 1.09 2016 51 - 55 11.6 11.6 1.0

10 1592 56 - 60 10.6 10.7 1.011 679 61 - 65 9.8 9.8 1.012 930 66 - 70 9.0 9.3 1.013 433 71 - 75 8.4 9.3 1.114 382 76 - 80 8.2 9.9 1.2

4

1 249 17 - 22 31.3 30.3 1.02 614 23 - 27 24.4 24.2 1.03 1135 28 - 32 20.5 20.6 1.04 1711 33 - 37 17.4 17.4 1.05 2372 38 - 42 15.2 15.4 1.06 2996 43 - 47 13.6 13.9 1.07 3175 48 - 52 12.3 12.6 1.08 2973 53 - 57 11.1 11.3 1.09 3958 58 - 62 10.1 10.2 1.0

10 4164 63 - 67 9.4 9.5 1.011 3989 68 - 72 8.7 8.6 1.012 3614 73 - 77 8.1 8.4 1.013 2998 78 - 82 7.6 7.7 1.014 1780 83 - 87 7.1 6.9 1.015 1241 88 - 92 6.8 6.7 1.016 650 93 - 97 6.5 6.6 1.017 789 98-102 6.2 7.3 1.218 205 103-107 5.9 4.9 0.8

*Data in probe volume ranges between 11-12 veh/10 min were not considered since the number of observations in this range was less than 100.

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Vol. 17, No. 4 / May 2013 − 827 −

headway variable comes from a Poisson process (i.e., randomness).It is not accidental that Negative Exponential (NE) or ShiftedNegative Exponential (SNE) is derived from Poisson distributionand also has a special characteristics of equal mean and standarddeviation (for SNE, the values of mean and standard deviationare not exactly equal, but depend on the shifted value).

The next step was to identify the distribution type of probes’headway. Based on previous studies, three distributions weretested: SNE, log-normal, and Pearson type III. SNE, log-normal,and Pearson type III were found to provide good fits withheadway under low, medium, and high traffic conditions,respectively (Buckley, 1968; Tolle, 1971; Mei and Bullen, 1993).The test was made based on a Chi-square test at a 0.05 level ofsignificance with the shifted values ranging from 0.0 to 3.0 s atan increment of 0.1 s. In this study, the shifted value that gave thehighest number of passing fit test for each volume range waschosen as the suitable shifted value. With the chosen best-shiftedvalue, the number of passing fit test for each volume range wascounted and used to represent how well the theoreticaldistributions fit with the empirical probe headway data.

3.2 Results and DiscussionThe results of the Chi-square fit tests are shown in Table 2. We

note that SNE is the best distribution for both single and multiplelane(s), while log-normal is the worst. Note that in Table 2, bothsingle and multiple values were found to be the best shiftedvalue(s). This may be attributed to the low hourly volumes ofAVI probes per lane. When volumes are low, the highestobserved number of AVI probes for the four-lane test bed wasabout 107 per 10 min, or 642 vehicles/hr. It has been shown thatNE and SNE generally provide a good fit for traffic data underlow volume conditions; i.e., random conditions (Buckley, 1968;

Gerlough and Huber, 1975). Accordingly, we hypothesize thatwhen the volume of probe vehicles is low, the time headway ofprobes follows the NE or SNE. This hypothesis will be examinedagain in the next section. In addition, from Table 2, we observethat the suitable shifted values are very small. This result seemsacceptable because the minimum allowable headway of probevehicles should be nearly zero on a multiple lane traffic stream.We also observe that the suitable shifted value does not have anysystematic relationship with the number of lanes and/or thedistribution types.

To test whether the levels of congestion affected the results, thetests were repeated for the peak period. The results are shown inTable 3. During the morning peak period (6:30 to 8:30 A.M.), theaverage speed of the test bed freeway was about 35 to 50 km/hr,implying that the level of traffic congestion was severe. FromTable 3, we note that the AVI vehicle’s headway of this periodtends to follow SNE because the number of passing fit test for allthree cases are more than half. In this sense, we hypothesize thateven if the level of traffic congestion is not light (i.e., the trafficvolume level of the general vehicles is not low), if the volumelevel of the probes is low, then probe headway follows the SNE.In other words, if the volume level of the probes is low, thenprobe headway follows the SNE regardless of the volume levelof the general vehicles.

4. Simulation Study

Our empirical study was limited such that only certain volumeconditions were measured; consequently, numerous hypothesescould not be addressed. To analyze the relationship betweengeneral vehicle headway and probe vehicle headway, a sensitivityanalysis using simulated data was constructed.

Table 2. Chi-square Test Result of AVI Time Headway DataNo. of Lanes

1 3 4

Suitable Shifted Value (sec)SNE 0.0 - 2.0 0.0, 0.1, 0.4 0.5Lognormal 0.0 - 0.9 0.0 - 1.0 0.0Pearson type III 0.0 - 0.3 0.0 - 1.3 0.1 - 1.7

Number Passing Fit Test SNE 9 (10)* 9 (14)* 9 (18)*

Lognormal 1 (10)* 1 (14)* 1 (18)*

Pearson type III 9 (10)* 4 (14)* 5 (18)*

*The number in parenthesis represents the number of volume ranges for each scenario.

Table 3. Chi-square Test Result of AVI Time Headway Data during Peak Period No. of Lane

1 3 4

Suitable Shifted Value (sec)SNE 0.0 - 2.0 0.0 - 0.7 0.2 - 0.4Lognormal - - 0.0 - 1.3Pearson type III 0.0 - 0.3 0.0 - 1.6 0.0 - 0.2, 0.4 - 0.8

Number of passing fit testSNE 8 (9)* 5 (8)* 6 (11)*

Lognormal 0 (9)* 0 (8)* 2 (11)*

Pearson type III 8 (9)* 3 (8)* 2 (11)*

*The number in parenthesis represents the number of volume ranges during the peak period of each number of lane.

Rattaphol Pueboobpaphan, Dongjoo Park, Youngchan Kim, and Sangho Choo

− 828 − KSCE Journal of Civil Engineering

4.1 Experimental Study DesignThere are two primary assumptions underlying the simulation

analysis. First, we assumed that there is no interaction betweengeneral vehicles (and also between probe vehicles, since they area subset of the general vehicles) on the same lane, and on adifferent lane or lanes. Second, the multiple lane freeway hasequal lane utilization characteristics. In other words, volumelevels of general vehicles on each lane were assumed to be equal,implying that the distributions of general vehicle headway oneach lane are the same (i.e., the distribution shape (function),mean, standard deviation, and shifted value are the same). Inorder to avoid any confusion when the variables/parameters ofheadway are referred to in different situations (such as for generalvehicles or probe vehicles, and for single or multiple lanes or theuniversal case--any number of lanes), the abbreviations in Table 5are used hereafter.

In order to simulate the headway of probes on a single lane, thedistribution of general vehicle headway on a lane was firstassumed. The key parameters included the type of distributionfunction, mean, standard deviation, and shifted value. A series ofgeneral vehicle headways was generated based on an assumeddistribution. The series of headways was converted into the

observed passing times of general vehicles at the observationpoint. A geometric distribution was then used to classify thegenerated vehicles as non-probe vehicles or probe vehicles.Probe headway was calculated by finding the difference betweenthe passage times of the current probe and the next probe. Thenext probe vehicle was again treated as the current probe vehicle.By repeating this process, a set of probe headways on a singlelane was obtained. The additional process for the multiple lanecase involved generating the observed passing time of probevehicles on all lanes by using the same concepts for the singlelane case. Probe headway on multiple lanes was obtained byselecting the minimum value of the differences of passage timesbetween the current probe and the next probes from all lanes, asshown in Eq. (2).

Ten volumes of general vehicles on single lanes (VOLGS)were assumed in this study; these were classified as low (lessthan 400 vphpl), medium (400 to 1,200 vphpl), and high (greaterthan 1,200 vphpl) traffic conditions following the method of Al-Ghamdi (2001). The distribution functions for the low, medium,and high traffic conditions were assumed to be SNE, Pearsontype III, and log-normal, respectively, based on previous research(Buckley, 1968; Tolle, 1971; Mei and Bullen, 1993). In addition,the mean of general vehicle headway in a single lane (MEANGS)µ, which was measured from the origin and is equal to the shiftedmean headway plus a shifted value , was assumed to be afunction of VOLGS as shown in Eq. (3).

(3)

By assuming the shifted value of THD of a general vehicle ona single lane (SHIFTGS), the shifted mean headways correspondingto each VOLGS were calculated by subtracting the SHIFTGSfrom the MEANGS. The standard deviation of the generalvehicles’ headway on a single lane (STDGS) was estimatedbased on the following Al-Ghamdi (2001) formula in Eq. (4).

(4)

A sensitivity analysis on the number of lanes (one, two, three,

µ′ ∆+

µ 3600V

------------=

σ 0.784– 0.777µ′+=

Table 4. Effect of Number of Data on Calculated Chi-square ValueSmall no. of data Large no. of data

Range Obs. Freq.

Theo. Freq.

Chi. Val.

Obs. Freq.

Theo. Freq.

Chi. Val.

0-2 10 20 10.0 20 40 20.02-4 30 40 3.3 60 80 6.74-6 50 50 0.0 100 100 0.06-8 50 40 2.0 100 80 4.08-10 30 30 0.0 60 60 0.010-12 20 30 5.0 40 60 10.012-14 20 10 5.0 40 20 10.014-16 20 10 5.0 40 20 10.016-18 10 10 0.0 20 20 0.018-20 10 10 0.0 20 20 0.0Sum 250 250 30.3 500 500 60.7

Table 5. Abbreviations of Terminologies*

No. Terms Universal Case Single Lane Multiple LanesGeneral Vehicles

1 Volume of general vehicles VOLG VOLGS VOLGM2 Time headway distribution of general vehicles THDG THDGS THDGM3 Mean of general vehicles' headway MEANG MEANGS MEANGM4 Standard deviation of general vehicles' headway STDG STDGS STDGM5 Shifted value of time headway distribution of general vehicles SHIFTG SHIFTGS SHIFTGM

Probe Vehicles6 Volume of probe vehicles VOLP VOLPS VOLPM7 Time headway distribution of probe vehicles THDP THDPS THDPM8 Mean of probe vehicles' headway MEANP MEANPS MEANPM9 Standard deviation of probe vehicles' headway STDP STDPS STDPM10 Shifted value of time headway distribution of probe vehicles SHIFTP SHIFTPS SHIFTPM

*These terms are used in both simulation and derivation part.

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Vol. 17, No. 4 / May 2013 − 829 −

and four lanes) was performed. Each type of freeway wasassumed to have ten VOLGS. At each VOLGS, there were twoSHIFTGS (0.5 and 1.0 s) and five market penetration rates (0.2,0.4, 0.6, 0.8, and 1.0). Therefore, the total number of cases in thisstudy was 400 (4 different numbers of lanes × 10 volume levels× 2 shifted values × 5 market penetrations). The details of thestudy design are summarized in Table 6.

For each case (two lanes of freeway with VOLGS, SHIFTGS,and market penetration of 400 vphpl, 0.5 s, and 0.2, respectively),ten different simulations were performed. At each simulation,40,000 probe headway data elements were generated. The firsthalf of the data set (the first 20,000 data elements) was used tocalculate the mean of the probe vehicle headway (MEANP) andthe standard deviation of the probe vehicle headway (STDP).The second half of the data set was used for a goodness of fit testwith three theoretical distributions (SNE, Pearson type III, andlog-normal) with the MEANP and STDP obtained from the firsthalf of the data set. The test was made based on a Chi-square testat a 0.05 level of significance with the shifted value of the THDof probe vehicles (SHIFTP) ranging from 0.0 to 3.0 s inincrements of 0.1 s.

Since each case consists of ten different simulations, this studyused a summary of results instead of showing all results. Thenumber of passing fit tests was divided into two categories: 0-5and 6-10. Similar to the real-world probe data analysis describedin Section 4, the shifted value(s) that provided the highestnumber of passings was selected.

4.2 Analysis of Results, and DiscussionThe analyses were divided into two parts: an analysis of the

distribution type that is the closest to probes’ headway, and ananalysis of the relationship between parameters of the headwayof the general vehicles and probes.

4.2.1 Probe Distribution TypeThe fit results are summarized in Tables 7 and 8 for the cases

of SHIFTGS of 0.5 and 1.0 s, respectively. For a low trafficcondition, we observed that SNE and Pearson type III provided agood fit to the probes’ headway for any of the number of lanes

and market penetrations studied. For the medium trafficcondition, SNE and Pearson type III provided a good fit onlywhen the market penetration was low (i.e., 0.2 and 0.4 for onelane, and only 0.2 for more than one lane). However, at thistraffic condition, SNE was found to provide a better fit resultcompared to the Pearson type III distribution, particularly whenthe VOLGS ranged from 1000 to 1200 vphpl. For the high trafficcondition, we observed that SNE and Pearson type III provided agood fit for the case of a one-lane freeway with 0.2 marketpenetration and 0.5 s of SHIFTGS. In addition, we note that thelog-normal distribution did not provide a good fit with theprobes’ headway for any of the scenarios studied (all combinationsof lanes, traffic conditions, and market penetrations).

From the aforementioned results, we hypothesize that VOLGS,the number of lanes on the freeway, and the market penetrationof probe vehicles, are the important factors for the distribution ofthe headway of probes. In addition, SNE and Pearson type III aresuitable for modeling probe headway distribution at both the lowand medium traffic volume levels. This is particularly true whenmarket penetration is low (i.e., 0.2). It should be noted that themarket penetration of probe vehicles in Houston, Texas was lessthan approximately 20%. This result is similar to the empiricalAVI data part in Section 4, wherein SNE was found to be the bestdistribution among the three theoretical distributions to describethe probes’ headway. We also observed that the log-normaldistribution did not fit well with the probes’ headway at all trafficvolume levels, which confirms the findings described in Section 4.

4.2.2 Relationship between Parameters of General Vehi-cles and Probe Headway

We expected that there would be some relationship betweenparameters of the general vehicles’ headway distribution and theprobes’ headway distribution. First, the MEANP obtained fromthe simulation result were thoroughly analyzed in order tounderstand how this value is related to the parameters of thegeneral vehicles’ headway. Based on this analysis, a number ofrelationships were examined between MEANP and theparameters of the general vehicles’ headway. These relationshipswere compared to the simulated values for which the measures

Table 6. Experimental Study Design of SimulationTraffic Volume

LevelHourly Volume

(vphpl)Original Distribution of General Vehicles

Mean(sec)

Shifted Value (sec)

Shifted Mean(sec)

Standard Deviation(sec)

Marketpenetration

Low200

SNE18.0

0.5, 1.0

17.5, 17.0 17.5, 17.0

0.2, 0.4, 0.6, 0.8, 1.0

300 12.0 11.5, 11.0 11.5, 11.0400 9.0 8.5, 8.0 8.5, 8.0

Medium

600

Pearson III

6.0 5.5, 5.0 3.5, 3.1800 4.5 4.0, 3.5 2.3, 1.9

1000 3.6 3.1, 2.6 1.6, 1.21200 3.0 2.5, 2.0 1.2, 0.8

High1400

Lognormal2.6 2.1, 1.6 0.8, 0.4

1600 2.3 1.8, 1.3 0.6, 0.21800 2.0 1.5, 1.0 0.4, -*

*This case is not applicable since the calculated STDGS is negative.

Rattaphol Pueboobpaphan, Dongjoo Park, Youngchan Kim, and Sangho Choo

− 830 − KSCE Journal of Civil Engineering

Table 7. Summary of Fit Distributions for SHIFTGS = 0.5 sec

Traffic VolumeLevel

VOLGS(vphpl) THDGS* Market

PenetrationDistribution with the Number of Passing Fit Test Greater than 5*

1 Lane 2 Lanes 3 Lanes 4 Lanes

Low

200 Sne

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.6 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.8 Sne, Pear Sne, Pear Sne, Pear Sne, Pear1.0 Sne, Pear Sne, Pear Sne, Pear Sne, Pear

300 Sne

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.6 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.8 Sne, Pear Sne, Pear Sne, Pear Sne, Pear1.0 Sne, Pear Sne, Pear Sne, Pear Pear

400 Sne

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.6 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.8 Sne, Pear Sne, Pear Sne, Pear Sne, Pear1.0 Sne, Pear Sne, Pear Sne, Pear Sne, Pear

Medium

600 Pear

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Pear Pear0.60.81.0 Pear

800 Pear

0.2 Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear0.60.81.0 Pear

1000 Pear

0.2 Sne, Pear Sne, Pear Sne Sne, Pear0.4 Sne, Pear0.60.81.0 Pear

1200 Pear

0.2 Sne, Pear Sne, Pear Sne Sne0.4 Sne, Pear0.60.81.0 Pear

High

1400 Log

0.2 Sne, Pear0.40.60.81.0 Log

1600 Log

0.2 Sne, Pear0.40.60.81.0 Log

1800 Log

0.2 Sne, Pear0.40.60.81.0 Log

*Sne, Pear, and Log are used to represent Shifted Negative Exponential, Pearson type III, and Lognormal, respectively.

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Vol. 17, No. 4 / May 2013 − 831 −

Table 8. Summary of Fit Distributions for SHIFTGS = 1.0 sec

Traffic VolumeLevel

VOLGS(vphpl) THDGS* Market

PenetrationDistribution with the Number of Passing Fit Test Greater than 5*

1 Lane 2 Lanes 3 Lanes 4 Lanes

Low

200 SNE

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.6 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.8 Sne, Pear Sne, Pear Sne, Pear Sne, Pear1.0 Sne, Pear Sne, Pear Sne, Pear Sne, Pear

300 SNE

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.6 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.8 Sne, Pear Sne, Pear Sne, Pear Sne1.0 Sne, Pear Sne, Pear Sne

400 SNE

0.2 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.4 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.6 Sne, Pear Sne, Pear Sne, Pear Sne, Pear0.8 Sne, Pear Sne, Pear Sne1.0 Sne, Pear Sne

Medium

600 Pear

0.2 Sne, Pear Sne Sne, Pear Sne0.4 Sne, Pear0.60.81.0 Pear

800 Pear

0.2 Sne, Pear Sne, Pear Sne0.4 Sne, Pear0.6 Sne, Pear0.81.0 Pear

1000 Pear

0.2 Sne, Pear Sne Sne0.4 Sne, Pear0.60.81.0 Pear

1200 Pear

0.2 Sne, Pear0.40.60.81.0 Pear

High

1400 Log

0.20.40.60.81.0 Log

1600 Log

0.20.40.60.81.0 Pear, Log

1800** Log

0.2 - - - -0.4 - - - -0.6 - - - -0.8 - - - -1.0 - - - -

*Sne, Pear, and Log are used to represent Shifted Negative Exponential, Pearson type III, and Lognormal, respectively.**This case is not applicable since the calculated STDGS is negative.

Rattaphol Pueboobpaphan, Dongjoo Park, Youngchan Kim, and Sangho Choo

− 832 − KSCE Journal of Civil Engineering

of effectiveness were the Average Percentage Error (APE) andthe Maximum Percentage Error (MPE). The APE was defined asthe average relative error between the actual and the estimatedvalue in terms of percent, and the MPE was defined as themaximum relative error between the actual value and theestimated value in terms of percent.

The relationship between MEANP and the parameters ofgeneral vehicle headway is shown in Eq. (5).

(5)

It can be seen that the mean headway of the probes is a

function of the mean headway of general vehicles, and aninverse function of market penetration and the number of lanes.The estimated MEANP obtained from Eq. (5) and the simulatedMEANP were compared, and the results are shown in Table 9.From Table 9, we note that both APE and MPE are very low forall numbers of lanes and all traffic conditions. Based on theseresults, we conclude that Eq. (5) is a good representation of therelationship between MEANP and the parameters of the generalvehicles’ headway.

In order to find the relationship between STDP and theparameters of the general vehicles’ headway, an approach similarto the aforementioned procedures was implemented. We foundthat any systematic relationship for all situations (allnumbers of lanes with all traffic conditions), except the casewhen the traffic condition was low, does not exist. TheSTDP when the traffic volume was low (i.e., volume lessthan 400 vph) can be estimated using Eq. (6). Comparisonsbetween the estimated values of STDP and the actual values aregiven in Table 10.

(6)

We note that the APE and MPE values increased with theincrement of the number of lanes. However, these values are stillquite low, even for a four-lane freeway, thereby implying that Eq.

µpµ′ ∆+

rL--------------=

σpσ ∆+

rL------------ ∆

L---–=

Table 9. Comparisons between the Estimated and Actual MeanProbe Headway

No. of Lane Traffic Condition APE (%) MPE (%)

1

Low 0.54 1.67Medium 0.40 2.00High 0.32 1.44Unified* 0.42 2.00

2

Low 0.48 1.53Medium 0.39 1.61High 0.38 1.46Unified* 0.42 1.61

3

Low 0.59 1.57Medium 0.39 1.84High 0.38 2.01Unified* 0.45 2.01

4

Low 0.53 1.53Medium 0.38 2.14High 0.36 1.61Unified* 0.42 2.14

Unified*: The comparisons were done by accounting the estimated andactual MEANP from all volume levels.

Table 10. Comparisons between Estimated and Actual StandardDeviation of Probe Headway for Low Traffic Condition

No. of Lane APE (%) MPE (%)1 0.78 2.832 0.77 2.923 0.87 3.164 0.89 4.21

Table 11. Relationships between Standard Deviation of Probe Headway and Mean Probe Headway, R-square, APE, and MPENo. of Lane Traffic Condition Relationship R-square APE (%) MPE (%)

1

Low 0.9997 1.41 3.80Medium 0.9970 8.11 32.37High 0.9956 22.95 259.41Unified* 0.9989 12.44 210.02

2

Low 0.9997 1.34 4.36Medium 0.9986 4.20 18.84High 0.9989 3.58 18.47Unified* 0.9993 5.57 39.92

3

Low 0.9997 1.20 3.90Medium 0.9990 3.23 15.28High 0.9988 3.17 14.56Unified* 0.9994 5.17 37.53

4

Low 0.9998 1.20 3.70Medium 0.9992 2.72 13.00High 0.9989 2.84 11.96Unified* 0.9995 4.65 32.97

Unified*: Regression analyses were done by accounting the MEANP and STDP from all volume levels.

σp 1.0022µp 0.7945–=σp 1.0063µp 2.0518–=σp 1.0476µp 1.6938–=σp 1.0188µp 1.7263–=σp 0.9988µp 0.3589–=σp 0.9805µp 0.7010–=σp 0.9856µp 0.5052–=σp 1.0075µp 0.6532–=σp 1.0027µp 0.2474–=σp 0.9668µp 0.3474–=σp 0.9589µp 0.2328–=σp 1.0064µp 0.3742–=σp 1.0004µp 0.1696–=σp 0.9622µp 0.2066–=σp 0.9519µp 0.1318–=σp 1.0021µp 0.2382–=

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Vol. 17, No. 4 / May 2013 − 833 −

(6) provides a very good estimation of STDP for this trafficcondition. Thus, we suggest that Eq. (6) represents therelationship between STDP and the parameters of the generalvehicles’ headway for a low traffic condition.

Another method for estimating STDP for all situations wasimplemented by performing regression analyses between theSTDP and MEANP. This approach was deemed acceptable sincethe MEANP for all situations can be estimated from Eq. (5) witha small error. The regression results and their corresponding R-square values, APE, and MPE are shown in Table 11. We notethat the R-square values for all cases are very high (>0.99).However, the MPE values are also quite high, particularly for thecases of medium and high traffic conditions, whereas the APEvalues are acceptable (low) except for only cases of a one-lanefreeway with medium and high traffic conditions. We consideredthe estimated equations of STDP for the low traffic condition tobe acceptable in terms of both APE and MPE. By comparing theunified and individual (low, medium, and high) regressionequations, we suggest that the individual regression equationsshould be used in practice because the overall errors aresubstantially lower compared to those from the unified equation.From these results, we conclude that MEANP and STDP areaffected by the number of lanes, market penetration of theprobes, and VOLGS (since µ and σ can be calculated fromVOLGS).

5. Analytical Derivation

As seen from the simulation part, the probe THD does notnecessarily follow the same distribution as the THD for generalvehicles. However, Tables 7 and 8 show that when the trafficcondition was low (i.e., the THD of general vehicles on a singlelane (THDGS) follows SNE), the probes’ headway alsofollowed SNE. This result implies that the THD of probevehicles (THDP) and THDGS are the same only for some cases(THDGS follows SNE). In this section, we analytically provethat if the THDGS is SNE, then the THDP (for both single andmultiple lanes) is also SNE.

5.1 Single Lane CaseThe general vehicles’ headway (h) at a given point is a random

variable for a specific observation point at any specific time ofday. If the current observed vehicle is a probe, the next arrivingvehicle may or may not be a probe. If the right next arrivingvehicle is not a probe, successive and the n-th next vehicle alsomay or may not be a probe. Thus, the relationship between theprobes’ headway and the general vehicles’ headway is describedby Eq. (7).

or or (7)

or

If it is assumed that the next probe arrives as the next n-thvehicle, then the probes’ headway is a function of the generalvehicles’ headway, and the order of the next arriving probe is asfollows:

(8)

Because the general vehicles’ headway on a single lane, h, isassumed to follow SNE with MEANGS, µ, and the SHIFTGS,∆, the THD of general vehicles can be written as:

; (9)

By considering the inverse function of Eq. (8) together withEq. (9), the THD of probe vehicles on a single lane (THDPS) canthen be obtained using the transformation technique shown inEq. (10).

;

(10)

From Eq. (10), we see that THDPS still follows SNE withthe new mean nµ (the mean of the probe vehicles’ headwayon a single lane: MEANPS), and the new shifted value n∆(the shifted value of the THD of probe vehicles on a singlelane: SHIFTPS). Therefore, if the THDGS is SNE, then theTHDPS is also SNE.

5.2 Multiple Lane CaseWe prove the multiple lane case by following the procedure

of Hagring (1998). In order to derive the THD of generalvehicles on multiple lanes (THDGM) and the capacity of theroadway, Hagring (1998) assumed that THDGS is Cowan’sM3, which could be considered the more complex form of NEand SNE. We used Eq. (10) as THDPS in order to derive thedistribution form of the THD of probe vehicles on multiplelanes (THDPM).

From Haight (1963), the starting density function can bewritten as:

(11)

From Eq. (11), the MEANPS (µp) was substituted by thenew mean obtained from Eq. (10), which is nµ. However, theterm was used in this study instead of the term nµ. Itis because this part assumes SNE so all parameters related tothe shifted value, such as the mean, should be subtracted bythe new shifted value (i.e., n∆). The probability distributionF(hp) was obtained by integrating Eq. (10) with respect to hp.Then, the starting density function of Eq. (11) can be writtenas:

hp h=h h 2h=+=h h h + + 3h== = =h … h+ += nh=

hp nh=

f h( ) 1µ ∆–-----------e

h ∆–µ ∆–-----------–

= h ∆≥

f hp( ) 1dhp

dh------------------ f h hp( )( )= 1

n--- 1µ ∆–-----------e

hp

n----- ∆–

µ ∆–-------------–

= 1nµ n∆–------------------e

hp n∆–nµ n∆–------------------–

=

hp n∆≥

f0 hp( ) 1µp----- 1 F hp( )–( )=

nµ n∆–

Rattaphol Pueboobpaphan, Dongjoo Park, Youngchan Kim, and Sangho Choo

− 834 − KSCE Journal of Civil Engineering

(12)

The starting distribution function can then be obtained asfollows:

(13)

By assuming independency between lanes, the relationbetween the starting distribution function for the whole roadand for the individual lanes is given by:

(14)

For an L-lane highway, Eq. (15) was obtained from Eqs.(13) and (14). Note that only the first part of Eq. (13) (thecase of ) was considered in this study. This is becausethe value of headway was restricted to be no less than theshifted value.

(15)

By taking the derivative of the preceding equation with respectto hp, g0(hp) can be obtained as follows:

(16)

Using the same concept given by Eq. (11), the starting densityfunction for a highway with L lanes can be written as:

(17)

Therefore,

(18)

The cumulative THDPM can then be obtained by substitutingEq. (16) into (18) as follows:

(19)

By taking derivative of the preceding equation with respect tohp, the THDP on an L-lane highway, g(hp), is given by

(20)

If we assume that each parameter in Eq. (20) is the same for alllanes, then Eq. (20) can be reformulated as follows:

(21)

Equation (21) looks similar to SNE with a mean of nµ/L (themean of the probe vehicles’ headway on a multiple lane:MEANPM). However, the exponential term the value subtractedin the denominator (n∆/L) is different from that in the numerator(n∆). If the exponential term is the same as n∆/L, Eq. (21) ischanged into Eq. (22), which illustrates that g(hp) is absolutelySNE with the new mean (MEANPM) nµ/L, and the new shiftedvalue n∆/L (the shifted value of the THD of probe vehicles onmultiple lanes: SHIFTPM).

(22)

Equations (21) and (22) were tested with the same set ofprobe headway data (at a low traffic condition). We found thatboth provided good results; however, Eq. (22) providedslightly better results than Eq. (21). Thus, it can be said that ifthe THD of a probe on a single lane is SNE, then the THD of

f0 hp( )

1nµ n∆–------------------ 1 1 e

hp n∆–nµ n∆–------------------–

–⎩ ⎭⎨ ⎬⎧ ⎫

–⎝ ⎠⎜ ⎟⎛ ⎞

, hp n∆≥

1nµ n∆–------------------ 1 0–( ) , 0 hp n∆≤ ≤⎩⎪⎪⎨⎪⎪⎧

=

1nµ n∆–------------------ e

hp n∆–nµ n∆–------------------–

⋅ , hp n∆≥

1nµ n∆–------------------ , 0 hp n∆≤ ≤

⎩⎪⎪⎨⎪⎪⎧

=

F0 hp( ) 1 f0 t( ) tdhp

∫–=

1 1nµ n∆–------------------ e

t n∆–nµ n∆–------------------–

⋅ tdhp

∫–=

1 e

hp n∆–nµ n∆–------------------–

– , hp n∆≥

1nµ n∆–------------------hp , 0 hp n∆<≤⎩⎪⎨⎪⎧

=

1 G0 hp( )– 1 F0l hp( )–( )l∏=

hp n∆≥

G0 hp( ) 1 1 1 ehp nl∆l–

nlµl nl∆l–-----------------------–

–⎩ ⎭⎨ ⎬⎧ ⎫

–⎝ ⎠⎜ ⎟⎛ ⎞

l∏–=

1 ehp nl∆l–

nlµl nl∆l–-----------------------–

l∏–=

1 e

hp nl∆l–nlµl nl∆l–-----------------------

l∑–

–=

g0 hp( )dG0 hp( )

dhp------------------- 1

dhp-------- 1 e

hp nl∆l

nlµl nl∆l–-----------------------

l∑–

–⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

= =

e

hp nl∆l–nlµl nl∆l–-----------------------

l∑– 1

nlµl nl∆l–----------------------

l∑–⎝ ⎠

⎛ ⎞⋅–=

e

hp nl∆l–nlµl nl∆l–-----------------------

l∑– 1

nlµl nl∆l–----------------------∑⋅=

g0 hp( ) 1nlµl nl∆l–----------------------

l∑⎝ ⎠⎛ ⎞ 1 G hp( )–( )=

G hp( ) 1 g0 hp( )1

nlµl nl∆l–----------------------

l∑----------------------------–=

G hp( ) 1 11

nlµl nl∆l–----------------------

l∑---------------------------- e

hp nl∆l–nlµl nl∆l–-----------------------

l∑– 1

nlµl nl∆l–----------------------

l∑⋅ ⋅–=

1 e

hp nl∆l–nlµl nl∆l–-----------------------

l∑–

–=

g hp( ) ddhp--------G hp( )=

e

hp nl∆l–nlµl nl∆l–-----------------------

l∑– 1

nlµl nl∆l–----------------------

l∑⋅=

g hp( ) 1nµL

------ n∆L

------–------------------ e

hp n∆–nµL

------ n∆L

------–------------------–

⋅=

g hp( ) 1nµL

------ n∆L

------–------------------ e

hpn∆L

------–

nµL

------ n∆L

------–------------------–

⋅=

Time Headway Distribution of Probe Vehicles on Single and Multiple Lane Highways

Vol. 17, No. 4 / May 2013 − 835 −

a probe on a multiple lane also tends to be SNE. Even thoughthe goal of this part of the analysis was not achieved asexpected, we can conclude that THDPM follows SNE aswell.

In summary, in the first part, the statement “If THDGS is SNEthen the THDPS is also SNE” was proved. In the second part, weobserved that “If THDPS is SNE then the THDPM tends tofollow SNE as well.” Therefore, by combining both, weconclude that “If THDGS is SNE then the THDPM tends tofollow SNE as well in a multiple lane highway.”

6. Conclusions

We examined the distribution and related parameters of probevehicles’ time headway on a single and multiple lane trafficstream. This study is divided into three parts: an empiricalstudy using probe data, simulation, and an analyticalderivation. The empirical data study was performed using dataobtained from Houston, Texas that was collected as part of theAutomatic Vehicle Identification system of the HoustonTranstar system. In the empirical study, SNE was found to givethe closest fit to the observed THDP for both single andmultiple lanes compared to other distribution types (log-normaland Pearson type III). Furthermore, we found that if the volumelevel of the probes was low, then regardless of the volume levelof general vehicles, the probes’ headway followed the SNE. Inthe simulation study, THDP was found not necessary to followthe same distribution of THDGS; however, it depends on manyvariables (i.e., VOLGS, the market penetration of probevehicles, and the number of lanes). These variables alsoaffected the MEANP and STDP. However, when the VOLGSis low, the probes’ headway tended to follow SNE (which is thesame as the THDGS) at all levels of market penetration. Thederivation analytically proved that THDP follows SNE if theTHDGS follows SNE.

In future work, we recommend that this study be extended by i)relaxing the simulation assumptions such as equal lane utilizationand independence between/among lanes; ii) collecting probe datathat includes information for general vehicles; and iii) collectingprobe data under various market penetrations of probes.

Notations

f(h) = Probability density function of general vehicles’headway on a single lane traffic stream

f(hp) = Probability density function of probes’ headway on asingle lane traffic stream

f0(hp) = Starting density function of probes’ headway on asingle lane traffic stream

F(hp) = Probability distribution function of probes’ headwayon a single lane traffic stream

F0(hp) = Starting distribution function of probes’ headway ona single lane traffic stream

F0l(hp) = Starting distribution function of probes’ headway onlane l

g(hp) = Probability density function of probes’ headway on amultiple lanes traffic stream

g0(hp) = Starting density function of probes’ headway on amultiple lanes traffic stream

G(hp) = Probability distribution function of probes’ headwayon a multiple lanes traffic stream

G0(hp) = Starting distribution function of probes’ headway ona multiple lanes traffic stream

h = Time headway of general vehicles on a single lanetraffic stream

hp = Time headway of probes on a traffic stream with anynumber of lane

j = Index variable for a lane on which current probe isobserved

l = Index variable for a laneL = Number of lanen = Index variable for a vehiclenl = Index variable for a vehicle on lane lr = Market penetration of probes

= Arrival time of the current probe at the observation point= Arrival time of the current probe on lane j= Arrival time of the next probe vehicle which is the n-

th general vehicle behind the current probe= Arrival time of the next probe which is the n-th general

vehicle on lane l counted from the observation pointV = Volume of general vehicles on a lane∆ = Shifted value of general vehicles’ headway distribution

on a lane∆l = Shifted value of general vehicles’ headway distribution

on lane lµ = Mean of general vehicles’ headway on a laneµl = Mean of general vehicles’ headway on lane l

= Shifted mean of general vehicles’ headway (i.e.,) on a lane

µp = Mean of probes’ headway on a traffic stream withany number of lane

σ = Standard deviation of general vehicles’ headwayon a lane

σp = Standard deviation of probes’ headway on a trafficstream with any number of lane

Acknowledgements

This work was supported by the 2011 Research Fund of theUniversity of Seoul.

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