Edited by: Sakdirat Kaewunruen, University of Birmingham, United Kingdom
Reviewed by: Anna Granà, University of Palermo, Italy; Ampol Karoonsoontawong, King Mongkut's University of Technology Thonburi, Thailand
This article was submitted to Transportation and Transit Systems, a section of the journal Frontiers in Built Environment
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The percent timespent following (PTSF) is a key measure for estimating the levelofservice for twolane highways. However, PTSF is currently estimated indirectly using various measured field data. This study proposes two alternative direct models for estimating PTSF: (1) a general linear regression model using typicallyused input variables and (2) a nonlinear form using only the follower density. Both models are based on data generated from the corridor microsimulation (CORSIM) software. Comparison of the proposed models with the new directional PTSF analytical model of the Highway Capacity Manual shows that the proposed models are capable of predicting simulationgenerated PTSF quite closely with absolute mean error <3% using two different sets of validation datasets. The results of this research would be useful for a convenient and direct estimation of directional PTSF of twolane highways.
Traffic flow on twolane highways is characterized by some complex interactions among many contributing operational and geometric variables. The operational variables include directional distribution of traffic flow levels, vehicular traffic mix with slowmoving heavy vehicles or recreational vehicles, freeflow speed of the highway segment, and driver's aggressiveness factors. The geometric variables include available passing opportunities in terms of percent passing zones, dedicated traffic lanes, passing sight distance, and horizontal and vertical alignments. All these interactions ultimately lead to the formation of platoons of fastmoving vehicles behind slowmoving vehicles, subject to a time headway threshold.
In turn, the percent timespent following (PTSF) emerged as the most important and widely used traffic measure of performance among traffic practitioners for determining the level of service (LOS) of twolane highways (AlKaisy et al.,
In HCM, the LOS of twolane highways is determined based on the combination of PTSF and average travel speed (ATS) for Class I highways only, where ATS is estimated based on freeflow speed. For Class II highways, ATS is not used for determining LOS. Apparently, the assumption is that PTSF is not significantly and directly affected by the level of freeflow speed for both highway classes. The HCM2010 method also generalizes the estimation of PTSF without considering specified freeflow speed levels for both highway classes. For estimating PTSF, both HCM versions (Transportation Research Board,
The motivation for the present study is based on the following hypotheses: (a) the percent passing zones significantly affects PTSF under varying traffic, geometric, and driver behavior conditions, but the impact of nopassing zones is relatively low under different flow levels and directional splits, as reported in HCM (2010), (b) adjustment of PTSF for freeflow speed conditions and hourly trafficflow levels may be needed, and (c) PTSF can be estimated from an easily obtainable spotbased traffic measure, such as follower density. Based on these hypotheses, the purpose of this paper is to develop alternative direct models for estimating PTSF. Specifically, the two models are: (a) a generalized relationship that combines all input variables (traffic flow levels, percent passing zones, percent heavy vehicles, freeflow speeds, and driver sensitivity indicator) into a single relationship and (b) a model that involves a mobility indicator (follower density) that can be easily measured in the field.
The next sections present a review of existing methods, including the HCM procedures and alternative approaches. The following sections present the proposed methodology, including experimental setup, developed direct models, and discussion, followed by the conclusions.
The directional percent timespent following is defined by HCM (2010) as:
where
The HCM procedures for estimating PTSF have drawn a significant number of disagreements among researchers. Several researchers noted that the procedures produce inconsistent results with the surrogate 3s criterion and overestimate PTSF and in some cases, the range of the average difference was 20–40% (Luttinen,
Subsequently, other approaches for estimating PTSF have emerged. Durbin (
Bessa et al. (
Estimating PTSF using the HCM2010 method is also computationally timeconsuming as it involves using both (Equations 1 and 2) along with interpolated data for the adjustment factor. Thus, a direct relationship of PTSF based on all significant input variables would be convenient for use by traffic engineers and practitioners. A few mathematical forms of such a direct relationship are available, but most models are primarily based on hourly traffic flow only.
Alternative measures to PTSF have emerged for determining the LOS of twolane highways, including average travel speed of passenger cars (Brilon and Weiser,
Simulation methods are typically used for estimating PTSF in the absence of direct PTSF measurements in the field, although such methods employ numerous simplifying assumptions about the complex driver behavior in performing overtaking decisions (Polus and Cohen,
The traffic software integrated system (TSIS) module of the CORSIM microsimulation software, which is designed for twolane highways, was used in this study. The module was used to generate both calibration and validation datasets. The datasets were extracted using relevant input variables based on specified low and highrange values of the variables.
A 10 km segment of twolane highway with no restrictions on available sight distance and with a flat terrain was simulated in CORSIM as the test bed. The percent passing zones for both directions were placed alternatively after equal distance intervals on the highway segment (
Schematic diagrams of examples of percentage passing zone (Dashed lines mean passing is allowed).
Design of experiment setup for the test.
A: Percent passing zone  20  50 
B: Hourly flow (vph)  1,000  2,000 
C: Directional split of hourly flow (major direction)  50  70 
D: % Heavy vehicles (both directions)  5  10 
E: Freeflow speed (km/h)  80  100 
F: Driver's sensitivity indicator  50  100 
The simulation outputs of PTSF and follower density were extracted from the CORSIM simulation runs by varying six input variables (
An analysis of variance (ANOVA) was first conducted to determine the effect and significance of each of the input variables on PTSF. Then, a regression analysis was conducted to establish a generalized relationship between PTSF and the input variables. Finally, the calibrated models were tested against two validation datasets.
A total of 64 simulation points were tested for calibration, and another 64 simulation points with different percent of passing zones were tested for validation. Each simulation point was run with 10 iterations and with different random seeds for 1 h, and the average output of these iterations was considered since the variability of the standard deviations of all iterations was not that significant. Default values were adopted for passing in the oncoming lane and a nopassing lane was assumed.
Sample input and output of microsimulation models.
1  20  1,000  50  5  100  100  62.7  1.8 
2  20  1,000  50  5  100  50  57.3  1.6 
3  20  1,000  50  5  80  100  57.3  2.2 
4  20  1,000  50  5  80  50  57.5  2.2 
5  20  1,000  50  10  100  100  69.9  2.2 
60  50  2,000  70  5  80  50  83.4  12.7 
61  50  2,000  70  10  100  100  91.3  13.9 
62  50  2,000  70  10  100  50  89  14 
63  50  2,000  70  10  80  100  86.9  13.4 
64  50  2,000  70  10  80  50  85.4  13.9 
The vast majority of studies have used either a logarithmic or a negative exponential relationship of PTSF with directional traffic flow, but this study strives to develop two direct models: (a) a general linear model using all input variables and (b) a best fit nonlinear model using only the follower density.
An analysis of variance using Ftest was conducted to determine the main and interaction effects of all input variables on the directional PTSF. The variations in the mean PTSF provided a very quick indication of the overall central tendency of the PTSF because of individual and coupled interaction factors.
▪ The twoway hourly flow impacts the mean PTSF with high sensitivity. Then, directional split has also a significant impact on the mean PTSF.
▪ The freeflow speed has a significant impact on the mean PTSF. On the other hand, the percent passing zone, percent heavy vehicle, and driver's sensitivity indicator all have relatively milder influences on the mean PTSF.
Main effects on percent timespent following.
Thus, all factors showed somehow significant influence on the mean PTSF, which indicated a possibility for a strong relationship with PTSF.
▪ The percent passing zone's interaction with the hourly flow has the greatest impact on the mean PTSF as noted from the steeper slopes from the low to the high range inputs. Also, the variation in the percent passing zone itself does not influence the mean PTSF as the mean values lie within a range of 10 (between 70 and 80), indicating a very mild influence of the percent passing zone when it interacts as a couple with other input variables.
▪ For a specific hourly flow rate, the mean PTSF does not vary significantly since the values vary within a range of 10 when interacting with all input variables, except percent passing zone. However, the mean PTSF varies significantly because of the variation in the hourly flow rate at a specific set of other input variables.
▪ For all the remaining twoway interactions, the mean PTSF does not vary significantly as it varies mostly with a range of 5.
Interaction for percent timespent following in the major direction (Model 1) with
The preceding observations are in line with all PTSF models in the literature that are primarily based on hourly flow levels only. An adjustment of the base PTSF models follows to account for the effects of the percent passing zone, percent heavy vehicles, and other relevant factors. Thus, the abovementioned behavior for the central tendency of the PTSF data shows a possibility of direct linear relationship with all input variables either individually or combined in pairs, which is explored next.
After exploring different models for PTSF for different levels of interactions among all input variables, a refined PTSF model with simplified input variables was developed, as follows
In comparison with the HCM2010 models, the directional base PTSF model of Equation (3), called herein the base PTSF model, was applied to predict PTSF under the following sample cases: (a) nopassing zone = 60% and directional split = 50:50, (b) nopassing zone = 0% and directional split = 80:20, and (c) nopassing zone = 100%, directional split = 60:40. In all cases, freeflow speed = 80 km/h, heavy vehicles (both directions) = 10%, and driver sensitivity indicator = 100.
Since the HCM2010 PTSF models are not specified for any particular level of freeflow speed, it was assumed that (Equations 1 and 2) of the HCM are valid for a freeflow speed of 80 km/h.
Performance of the proposed PTSF model (Model 1).
Although the base PTSF model of Equation (3) was calibrated for only two different flow levels in the experimental setup (
Performance of proposed model 1 and HCM models using first validation dataset.
Proposed model 1  −0.68  3.93 
HCM 2010 BPTSF  −10.29  7.26 
HCM 2010 PTSF  9.38  4.31 
Proposed model 1  −3.43  4.44 
HCM 2010 BPTSF  −9.59  3.86 
HCM 2010 PTSF  −5.93  3.28 
Proposed model 1  −0.24  3.73 
HCM 2010 BPTSF  −13.21  7.38 
HCM 2010 PTSF  10.11  3.29 
The apparent large difference in the prediction of PTSF using the HCM2010 model (
In the absence of all the previously mentioned input variables, it is possible that PTSF can be estimated from the follower density (vpkpl) only. This measure can be easily obtained using some pointbased data collection system rather than collecting data from a spatial extent over a segment of highway that was the basis for PTSF. The follower density can be obtained using traffic density and percent of followers. Hence, in reality, the traffic density can be measured either using the surrogate occupancy measure or the average distance or time headway of observed vehicles. The percent followers can also be measured using the follower headway threshold of 3 s between vehicles under the observation time interval.
Hence, a bestfit curve (
where
Relationship between PTSF and follower density (Model 2).
Two different validation datasets were used to test the effectiveness of the proposed PTSF models. The first dataset (with 64 simulation points) is similar to the experimental setups of
Input variables of the second validation dataset.
1  20  1,200  60  5  60  70 
2  20  1,300  60  6  90  80 
3  50  1,500  50  12  80  75 
4  40  1,600  70  10  90  80 
5  30  1,630  68  9  70  85 
6  40  1,800  80  7  70  80 
7  50  1,900  80  8  90  95 
8  30  2,400  70  11  90  90 
Performance of the proposed and HCM models using the first and second validation datasets.
Proposed model 1  0.36  2.44 
Proposed model 2  0.19  4.92 
HCM 2010 BPTSF  −9.20  7.00 
HCM 2010 PTSF  11.46  7.81 
Proposed model 1  −2.42  6.30 
Proposed model 2  2.76  4.50 
HCM 2010 BPTSF  −6.20  5.27 
HCM 2010 PTSF  9.72  5.04 
Performance of the models for the second validation dataset.
This paper has presented two regression models for the direct estimation of the directional PTSF. In the first proposed model, all significant input variables that affect PTSF are used in a single relationship that can be used conveniently. This form eliminates the necessary additional computational efforts for adjusting the base PTSF in HCM 2010. However, the model includes several input variables similar to the HCM methodology. It was found that the PTSF value is also affected by the freeflow speed, where the HCM2010 PTSF estimation considers freeflow speed as another measure for Class I highways only to determine the overall LOS of the segment. The second proposed model of PTSF is a function of only the follower density which can be easily measured from the field. Both PTSF models showed better performance in estimating the simulated PTSF compared with the HCM methods. Although the proposed models were based on microsimulation, the models can be calibrated and validated with local conditions. It was observed that even though the correction term used in HCM2010 for the adjustment part along with the BPTSF, both analytical equations still produce high mean errors. Based on this research, the following comments are offered:
The analysis shows that factors such as freeflow speed and drivers' sensitivity indicator do also affect PTSF in addition to the typical input variables of hourly flow, directional split, percent of heavy vehicles, and percent passing zones.
Both proposed models for PTSF estimation are promising to be used generally after calibration of the coefficients using realsite specific data. Also, calibration of the internal carfollowing and passing decision parameters using field data, if available, would make the proposed models more useful instead of using the default internal values used in the CORSIM software.
Overestimation and underestimation of PTSF by HCM2010 do exists, but with a lower margin of mean errors than those reported in the literature.
Further research could test the proposed formulas by incorporating the variations in horizontal and vertical alignments of the simulated highway segment and passing opportunities for both sides at the same location if a significant impact on PTSF is expected.
The datasets generated for this study are available on request to the corresponding author.
FA developed the simulation models, data analysis, and model development. SE had developed the primary idea, reviewed the manuscript, suggested methodology revision, and reviewed the final draft also.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
coefficients based on opposing demand flow rate 

A  percent passing zone 
ATS  average travel speed (km/h) 
ATS_{pc}  average travel speed of passenger cars (km/h) 
B  hourly flow (vph) 
base directional PTSF (%)  
C  directional split of major flow 
D  percent heavy vehicles 
D_{f}  follower density (vpk) 
E  freeflow speed (km/h) 
adjustment factor based on percent nopassing zones and directional split  
F  driver's sensitivity indicator 
LOS  level of service 
PTSF  percent time spent following 
actual percent timespent following (%)  
PTSF1_{p}  percent time spent following for Model 1 
PTSF2_{p}  percent time spent following for Model 2 
v_{o}  opposing demand flow rate (vph) 
v_{d}  demand flow rate in the analysis direction (vph) 