- Original Paper
- Open Access

# Headway distribution models of two-lane roads under mixed traffic conditions: a case study from India

- Rupali Roy
^{1}Email authorView ORCID ID profile and - Pritam Saha
^{1}

**10**:3

https://doi.org/10.1007/s12544-017-0276-2

© The Author(s) 2017

**Received:**26 April 2017**Accepted:**28 November 2017**Published:**12 December 2017

## Abstract

### Introduction

The time headway of vehicles is of fundamental importance in traffic engineering applications like capacity, level-of-service and safety studies. Further*,* the performance of traffic simulation depends on inputs into the simulation process and ‘accurate vehicle generation’ is critical in this context. Thus, it is important to define headway distribution pattern for the purpose of analyzing traffic and subsequently, taking infrastructure related decisions. In so far, majority of the researches on this subject are based on homogeneous traffic and effects of mixed traffic especially on two-lane roads are yet to be culminated. The present study, thus, aimed at investigating headway distributions on such roads under mixed traffic situation.

### Methods

Field study was conducted on two-lane highways in India that exhibits heterogeneity in its traffic composition. Contestant headway distribution models were evaluated and four distribution functions namely, log-logistic, lognormal, Pearson 5 and Pearson 6 were considered while modeling the headway data. The appropriate models were selected using a methodology based on K-S test and subsequent field validation.

### Results

Log-logistic distribution was found appropriate at moderate flow whereas, at congested state of flow it was Pearson 5. However, at unstable flow nearing capacity, both, following and non-following components of headways were observed to follow different distributional characteristics. Nomographs are developed for calculating the headway probabilities at different flow levels considering the appropriate distribution models. ‘Probability of headway less than‘t’s’ increases with the flow rate and rate of such increase is considerably high for headways 7.5 s or more. This attributes to the fact that at heavy flow more vehicles are entrapped inside platoons and they move in following with shorter headways. Further, a comparison was made between the headway probabilities obtained in the current study and road segments that exhibit more or less homogeneous traffic. It was found that at moderate flow level proportion of shorter headways are considerably high under mixed traffic.

### Conclusions

The present paper demonstrates the effect of mixed traffic on distribution of time headways of two-lane roads. Presence of slower vehicles in such traffic leads to frequent formation of platoons, thereby, increases the risk taking behavior of drivers’ while overtaking. As a result, proportion of shorter headways increases resulting in highly skewed observations. Thus, the study proposed Log-logistic distribution under medium flow since it can model events which have ‘increased initial rate’ and Pearson 5 under heavy flow to model ‘highly skewed data’. The model outputs were accordingly compared with other studies and were found to explain the mixed traffic characteristics satisfactorily. The present study, thus, creates a starting point of further initiatives aimed at establishing a robust method of modeling headways on two-lane roads with mixed traffic.

## Keywords

- Two-lane roads
- Mixed traffic
- Time headway
- Distributions
- Goodness-of-fit

## 1 Introduction

Time headway is an important microscopic traffic flow parameter and defined as the time interval, usually measured in seconds, between successive vehicles in the traffic stream. Study on this parameter is important particularly in context to capacity analysis, safety studies, car following and lane changing behavior modelling and level of service evaluation [1]. Capacity of a roadway and saturation flow rate of an intersection is the reciprocal of minimum time headway; in a way that rear end collision does not occur even in the event of sudden stop of leading vehicles. Further, assessment of LOS on highways is predominantly based on the percentage of vehicles that move in following with shorter headways [2]. Defining headway distribution function is, therefore, considered as fundamental in traffic flow study and its’ simulation issues. A key component of determining the performance of a simulation model is the generation of inter-arrival times as an input into the simulation process. However, most of the previous studies on this subject are based on homogeneous traffic and considered low and medium flow of traffic. In context to mixed traffic, there have been a very few research that focused on modeling headways under congested or heavy traffic flow in so far. Accordingly, there is a need to develop an appropriate headway model especially under mixed traffic situation while taking decisions on transportation infrastructure development [3]. At the same time, traffic congestion in many western countries has been a critical issue and transportation researchers have been trying to address it effectively. They are, therefore, in search of having a unified method of describing headways for the purpose of traffic characterization and modeling at congested state of flow; this is due to the fact that the conventional approach does not exhibit compatibility under such flow even if the traffic is homogeneous in character.

The negative exponential distribution is conventionally applied in describing the headway data. However, there have been a number of researchers who reported the use of several other models in order to explain the headway distribution pattern more explicitly if the prevailing traffic is heterogeneous in character and car-following interaction is frequent at increased flow level. This is more acute on two-lane roads where such interaction is reasonably frequent due to the presence of opposing flow and aggravates further especially under mixed traffic composed of a wide variety of vehicles in terms of static and dynamic characteristics. Statistically, the headway data can be described as exponential if the co-efficient of variation is unity resulting in an exact 45^{0} plot of mean and standard deviation [4, 5]. However, research on heterogeneity effect into traffic flows indicates that co-efficient of variation is likely to deviate under such traffic mostly because of complicated traffic flow dynamics, thereby, making exponential model inappropriate in describing headways.

Over the past few decades, significant researches have been conducted while suggesting alternative distribution functions compatible to mixed traffic. The log-logistic and lognormal distributions were observed to describe headway data well at flow levels that corresponds to peak hour’s traffic than off-peak hours, indicating a better approximation of congested traffic [6, 7]. By the same token, Pearson 5 and Pearson 6 distributions also exhibit their aptness at heavy flow and they provide a fairly decent fit under such flow while describing the headways [8]. After almost a century, this search is not yet over. In fact, transportation researchers are still working on reliable and practical descriptions of headways particularly if the prevalent traffic is extremely heterogeneous in character and composed of a wide variety of vehicles including the non-motorized ones.

There have been a number of studies that investigated headway distributions at low and medium traffic flow rates. However, studies that have been devoted to headway modelling at high flow, in which all vehicles are in the car-following state, are still insignificant [4]. Thus, the premise on which the present study is situated considers the effect of heterogeneity on headway distribution models and accordingly encompasses traffic flow ranging from moderate to congested, wherein such effect usually aggravates owing to frequent interaction between vehicles. Accordingly, an attempt was made to find the theoretical distribution models which fit well to headway data: log-logistic, lognormal, Pearson 5 and Pearson 6 distributions were considered in the present study. Appropriate models of headway distributions were selected using a methodology based on goodness-of-fit test (K-S test) considering 5% level-of- significance.

## 2 Research motivations

Over the past few decades, there have been a number of researchers who suggested several theoretical models for describing headways. Couple of experiences in urban settings with Indian traffic indicates that hyperlang distribution is best to describe the headway characteristics under mixed traffic conditions [9] whereas negative exponential distribution exhibits its compatibility over a wide range of traffic flow levels if the traffic consists of substantial percentage of smaller vehicles such as motorized two-wheelers [5]. Khasnabis and Heimbach [10] studied headway distribution models for two-lane rural highway under varying traffic flow levels in North Carolina. They applied six headway distribution models, namely, Erlang, negative exponential, Pearson Type III, Schuhl models, and their combinations and found Schuhl model appropriate for the headway distributions. Panichpapiboon [11] investigated and characterized the time-headway distributions of vehicles travelling on an urban expressway in Bangkok, Thailand and concluded that GEV distribution is most effective in modelling time headways. On the other hand, the exponential distribution was found to be the least effective distribution. Al-Ghamdi [4] studied time headways to establish boundaries in terms of traffic flow rate for each flow level. The negative exponential distribution was found to reasonably fit at the low flow state (less than 400 vph) whereas shifted exponential and gamma distributions were found to fit for medium flow (400–1200 vph). The Erlang distribution gave a decent fit for high flow state (more than 1200 vph). Based on a study in Finland, Luttinen [12] concluded that the gamma distribution is effective under low-to-moderate traffic volumes where probability for shorter headways is low and suggested that lognormal distribution could be considered as a model for the follower headway distribution. Riccardo and Massimiliano [8] analyzed headway distribution on rural two-lane two-way roads and suggested that the inverse Weibull distribution fits the headway data well. Abtahi et al. [13] studied different headway distribution models in the passing and middle lanes in urban highways under heavy traffic condition and concluded that lognormal and gamma models with 0.24 s and 0.69 s shifts exhibit their aptness in passing and middle lane respectively. Ren and Qu [14] studied headway distribution type analysis selecting three distributions namely, exponential, inverse gaussian and lognormal distribution. They found that the majority of headway samples follow an inverse Gaussian distribution.

The mixed models are more flexible to represent headways considering them into following and free-following components. Griffiths and Hunt [15] opined that double displayed negative exponential distribution is appropriate to model the vehicular headways. Zhang et al. [16] made an attempt to calibrate and examine the performance of various headway mixed models. The goodness-of-fit was checked by K-S test and also, visualized by Q-Q plot. The test result showed that the double displaced negative exponential distribution model provided the best fit to the urban freeway headway data and shifted lognormal distribution fits the general purpose lane headways very well. Besides, there are several other mixed models that have been developed and tested over the years in predicting headway distributions; they are respectively combined normal distribution and shifted negative exponential distribution, combined negative exponential distribution and shifted negative exponential distribution [1], generalized queuing model (GQM) [16] and semi Poisson distribution etc. Some mixed distributions were developed based on the assumption that headway consists of two components; following and free. Quite a few important models have been derived in accordance with this concept; they are Cowan M1–M4 [17], the generalized queuing model [18], and the semi-poisson model. Cowan’s M3 model is, however, extensively investigated and applied because of its simplicity and easy approximation of describing longer headways [19].

Further, several studies investigated car-following interactions to identify free-moving vehicles in the traffic stream. An experience on Swedish roads reveals that vehicles could be considered free beyond 6 s headway when interaction of vehicles is usually observed to be nonexistent [20]. Similar results were also obtained by couple of studies conducted on two-lane roads [21, 22]. Dey and Chandra [23] proposed two continuous statistical distribution models, gamma and lognormal for desired time gap and time headway of drivers in a steady car-following state on two-lane roads under mixed traffic conditions. Mei and Bullen [24] reported that in a car-following situation time headways are log normally distributed and converge to the shifted lognormal distribution with a shift of 0.3–0.4 s at higher traffic volumes. Kumar and Rao [25] proposed that negative exponential distribution adequately describes the headways at low to moderate flow levels. A relationship was developed between the proportions of vehicles in platoons and mean headways with an assumption that the vehicles travelling at headways less than 2.0 s are in platoons. A study on mixed traffic, however, indicates that at car-following state headway between two vehicles depends on the length of the lead vehicle [26].

Above studies demonstrate various statistical distributions that could be applied in modelling the time headways. However, most of them are based on homogeneous traffic, thereby, making it intrinsic to investigate the effect of heterogeneity in the distribution models. Further, such effect aggravates in the event of heavy flow when interaction between vehicles is considerably high. This has been the motives of the present study wherein an attempt was made to develop accurate models considering a systematic analysis of headway data under mixed traffic.

## 3 Field data collection

Video photographic survey technique was adopted while collecting field data. A reference line was marked on the pavement and two observation points were chosen for installing the video cameras; one in each direction, for recording the time when front and rear end of a vehicle cross the reference line. Heavy flow situation was investigated next to a bottleneck created by closing one lane of the two-lane carriageway and also, by stopping traffic movements for about two minutes to examine the discharged flow on the study section. Several such trials were made in order to ensure adequate sample size for the purpose of analysis. Traffic police help was taken for conducting the study.

Sample data collected on the selected highway section

Type of vehicles | Time in (Front axle) | Time out (Rear axle) | Lapsed time (Sec) | Speed, kmph | Headway (sec) |
---|---|---|---|---|---|

Car | 00:15:49.000 | 00:15:49.850 | 0.850 | 21.18 | |

Two wheeler | 00:15:50.000 | 00:15:50.270 | 0.270 | 26.67 | 00:00:01 |

Car | 00:15:51.000 | 00:15:50.875 | 0.875 | 20.57 | 00:00:01 |

Two wheeler | 00:15:55.000 | 00:15:51.310 | 0.310 | 23.23 | 00:00:04 |

Car | 00:15:57.000 | 00:15:55.825 | 0.825 | 21.82 | 00:00:02 |

Three wheeler | 00:16:01.000 | 00:16:01.450 | 0.450 | 24.00 | 00:00:04 |

Two wheeler | 00:16:03.000 | 00:16:03.290 | 0.290 | 24.83 | 00:00:02 |

Two wheeler | 00:16:04.000 | 00:16:04.320 | 0.320 | 22.50 | 00:00:01 |

Two wheeler | 00:16:06.000 | 00:16:06.290 | 0.290 | 24.83 | 00:00:02 |

- | - | - | - | - | - |

- | - | - | - | - | - |

## 4 Headway distribution modelling

In an attempt to find an appropriate distribution function for describing headway data, the compatible statistical models have been critically reviewed and compared. Adams, as early as 1936, established that Poisson distribution could be applied to vehicle arrivals [29] and since then it has been extensively employed in many theories of traffic flow. Statistically, inter-arrival time i.e. headway between successive arrivals would follow negative exponential distribution in the event of Poisson arrivals. Therefore, negative exponential distribution is commonly used for headway modelling. However, the assumption of Poisson arrival could be well applied if traffic volume is considerably low and also, there are two factors that limit the application of negative exponential distribution to the headways.

Firstly, negative exponential distribution spreads over the entire range of headways from zero to infinity and the probability density is maximum at h(t) = 0. This seems to be impractical since a vehicle has a finite length and finite speed resulting in an existence of a minimum finite headway.

Secondly, at increased traffic flow levels formation of platoons is frequent resulting in an increase of shorter headways. Majority of them cluster within a certain short range thereby, hypothesizing of negative exponential distribution over the entire range becomes inaccurate.

Besides, studies indicate that distribution of headways varies with the rate of traffic flow [2, 4].Such variation could be classified as random, intermediate and constant state respectively for low, medium and heavy flow of traffic: at random state there is no interaction between successive vehicles and their arrival is independent to each other, whereas, a constant state reflects following state of flow with constant headways near to capacity [31]. Since past few decades, significant efforts have been made in developing suitable distribution models for headways at low and medium traffic flow levels. However, attempts in describing them at heavy flow, in which all vehicles are in the car-following state, is still insignificant [4]. This has been the motives of taking initiatives for proposing distribution functions that exhibit aptness in modelling headways at moderate to heavy flow.

Several researchers have reported that the log-logistic distribution model is better for fitting headway data of peak hours than off-peak hours i.e. at congestion state [6]. In fact, the logistic distribution affords a good approximation to the normal distribution and the log-logistic distribution in the same way approximates well the lognormal distribution which, apparently seems to be appropriate in car following state. Pearson 3 distribution, which is basically a very general case of negative exponential distribution and normal distribution as well, could be used for headway modelling at moderate flow levels [32, 33]; this is perhaps the most general function that traffic engineers use under such traffic. Similarly, Pearson 5 and Pearson 6 distributions reveal their appropriateness in time headway modelling under heavy flow of traffic [8].

Thus, the present study considers four distributions namely, log-logistic, lognormal, Pearson 5 and Pearson 6 for headway modelling at moderate and heavy flow of traffic. Eqs*.* 1–4 demonstrates the probability density function of the log-logistic, lognormal, Pearson 5 and Pearson 6 distributions and investigations were accordingly made with these functions.

Where, h = time headway; α = Shape parameter >0; β = Scale parameter >0; μ = Mean; *σ*= Standard deviation; Γ = Gamma function; *B* = Beta function

## 5 Empirical investigations

### 5.1 Descriptive statistics of the headway data

Descriptive statistics of the headways

v/c ratio | Mean (sec) | Median (sec) | Standard deviation (sec) | Co-efficient of variation | Skewness | Kurtosis | Sample Size (Nos.) | |
---|---|---|---|---|---|---|---|---|

West bound traffic | 0.6 | 5.70 | 4.50 | 6.11 | 1.07 | 2.05 | 4.63 | 744 |

0.7 | 4.86 | 1.50 | 5.54 | 1.14 | 2.23 | 5.48 | 1302 | |

0.8 | 4.49 | 1.50 | 4.92 | 1.09 | 2.16 | 4.89 | 1110 | |

0.9 | 3.97 | 1.50 | 4.21 | 1.06 | 2.41 | 6.92 | 990 | |

1.0 | 3.66 | 1.50 | 4.10 | 1.12 | 2.84 | 9.97 | 782 | |

East bound traffic | 0.6 | 6.00 | 1.50 | 7.05 | 1.17 | 2.14 | 4.77 | 812 |

0.7 | 4.69 | 1.50 | 5.05 | 1.08 | 2.12 | 5.04 | 738 | |

0.8 | 4.50 | 1.50 | 5.17 | 1.15 | 2.30 | 6.12 | 852 | |

0.9 | 3.60 | 1.50 | 5.12 | 1.43 | 3.29 | 11.71 | 314 | |

1.0 | 3.55 | 1.50 | 4.78 | 1.34 | 2.89 | 9.82 | 356 |

Further, in the event of negative exponential distribution, the mean and standard deviation should have equal value resulting in a 45^{0} plot. Thus, statistically the headway data can be described as exponential if the co-efficient of variation is unity. However, Table 3 indicates that it exceeds unity at all the flow scopes pertaining to moderate and heavy flow. A study on urban roads of Riyadh reported that usually such co-efficient of variation falls in the range of 0.5 to less than 1.5 corresponding to the flows that range between 500 and 2000 vph [4]. However, the present investigation elucidates a higher range, from about 1.05 to 1.35, even when the observed flow is in the range of 900–1000 vph. This is evidently attributable to the effect of wide variety of vehicles in the traffic composition.

^{0}plot signifying inappropriateness of the negative exponential distribution in describing headways. The present study accordingly explicates the applicability of four distribution functions that seem to be compatible to such traffic while modelling headways; they are respectively log-logistic, lognormal, Pearson 5 and Pearson 6.

### 5.2 Statistical modelling of time headways

The statistical models should be used to fit the observed headway data in an attempt to find an appropriate model of headway distribution. Negative exponential distribution is normally used to describe the distribution pattern of highways. However, there have been a number of researchers who reported the use of several other models in order to explain the headway distribution pattern more explicitly. The extent of fit of the selected distributions to the data points is examined based on goodness-of-fit tests. In traffic engineering two such tests are commonly used: the chi-square test and the Kolmogorov–Smirnov (K-S) test.

In the present study, K-S test is chosen to measure goodness-of-fit of the selected headway models to the observed headways. The reason behind this decision was due to certain advantages that the K-S test offers over the chi-square test; K-S test can use data with a continuous distribution and there is no minimum frequency per test interval [1]. The K-S test statistic is calculated by determining the difference between the cumulative percentage of the measured frequency and the cumulative percentage of the expected frequency. The K-S test statistic, “D” was computed at the desired significance level for the selected distributions and the distribution which is expected to give the smallest “D” value was considered as the best fitted model.

The null hypotheses for each test were as follows: ‘the compatibility hypothesis of headway distribution with fitted model was rejected (P-value < α) or not rejected (P-value > α)’. The ‘*p*-*value’* is defined as the probability of obtaining a result equal to or “more extreme” than what was actually observed, when the null hypothesis is true [13, 34]. A critical value is the point on the scale of the test statistic beyond which the null hypothesis is rejected, and is derived from the level of significance (α) of the test. If the test statistic exceeds the critical value for the predefined significance level then the null hypothesis is rejected. The null hypothesis cannot be rejected when the hypothesized distribution passes the K-S test [31]. However, if the hypothesized distribution is rejected by the K-S test, it signifies that the distribution is not good enough to model the empirical distribution.

Goodness-of-fit test details and the estimated parameters of the fitted distribution model

v/c ratio | K-S test statistic (D-value) | Estimated Parameters (α | P –Value | Significance level | Critical value | Hypothesis test | ||||
---|---|---|---|---|---|---|---|---|---|---|

Log logistic (2P) | Pearson 5 (2P) | Pearson 6 (3P) | Lognormal | |||||||

West bound traffic | 0.6 | 0.26467 | 0.33164 | 0.33281 | 0.3420 | 1.7277; 2.5374 | 0.22670 | 0.05 | 0.37543 | Accept |

0.7 | 0.33262 | 0.36204 | 0.36038 | 0.3643 | 1.6063; 1.9653 | 0.08819 | 0.05 | 0.39122 | Accept | |

0.8 | 0.41250 | 0.36275 | 0.39565 | 0.3652 | 1.9385; 4.2904 | 0.09016 | 0.05 | 0.39256 | Accept | |

0.9 | 0.65284 | 0.35135 | 0.49904 | 0.3805 | 2.2151; 4.7249 | 0.07057 | 0.05 | 0.38925 | Accept | |

1.0 | 0.69634 | 0.44933 | 0.44968 | 0.5168 | 2.4433; 4.9689 | 0.04634 | 0.05 | 0.40925 | Reject | |

East bound traffic | 0.6 | 0.28394 | 0.33164 | 0.33831 | 0.3492 | 1.7301; 2.7077 | 0.23508 | 0.05 | 0.34890 | Accept |

0.7 | 0.33563 | 0.36242 | 0.35428 | 0.3686 | 1.8991; 2.1644 | 0.13807 | 0.05 | 0.39122 | Accept | |

0.8 | 0.41250 | 0.36484 | 0.39565 | 0.3829 | 1.9199; 4.1315 | 0.09857 | 0.05 | 0.38431 | Accept | |

0.9 | 0.62579 | 0.37341 | 0.49904 | 0.4596 | 2.6198; 4.8490 | 0.09186 | 0.05 | 0.39122 | Accept | |

1.0 | 0.69634 | 0.43762 | 0.46675 | 0.4964 | 3.4817; 6.5551 | 0.03644 | 0.05 | 0.41926 | Reject |

Ideally, the modelling paradigm will have some bearing on the system behaviour if expected probabilities are approximately equal to empirical probabilities. However, since ‘traffic composition’ and also, ‘drivers’ behaviour’ in the pilot study segment are different to some extent, the data points do not lie on the same line and deviate from the anticipated line of agreement. The ‘deviation’ is shown by the ‘blue zone’ which explicates that the model may be valid in the sense that it represents the system behaviour with reasonable amount of accuracy.

## 6 Discussion

Developing an appropriate headway distribution model is an important step in traffic modelling and simulation. Thus, it is imperative to characterise the vehicle time headways statistically using a distribution function. Conventionally, negative exponential distribution function is used; however, several studies on mixed traffic have shown that it does not capture the characteristics of mixed traffic especially on two-lane roads where interaction takes place in both the directions. Besides, arrival pattern of vehicles changes considerably with the flow which may consequence different distributions to work better at different flow conditions. The lookout of the current study was, therefore, to investigate the appropriate headway distribution model on such roads under mixed traffic.

Comparison of headway probabilities [probability of headways less than‘t’ sec] at different lane volumes: current study and case studies with homogeneous traffic

Headway (sec) | Probability at lane volume of 600 veh/h | Probability at lane volume of 700 veh/h | ||||
---|---|---|---|---|---|---|

Current study | Zwahlen et al. 2007 [31] | Khasnabis and Heimbach 1980 [10] | Current study | Zwahlen et al. 2007 [31] | Khasnabis and Heimbach 1980 [10] | |

<1.5 | 0.0638 | 0.0000 | 0.1714 | 0.0952 | 0.0000 | 0.2046 |

<4.5 | 0.2853 | 0.0812 | 0.6348 | 0.3574 | 0.0968 | 0.7222 |

<7.5 | 0.4215 | 0.1853 | 0.7827 | 0.4808 | 0.1926 | 0.8772 |

<10.5 | 0.5718 | 0.2134 | 0.8464 | 0.6423 | 0.2190 | 0.9358 |

<13.5 | 0.6256 | 0.2871 | 0.8896 | 0.7271 | 0.2881 | 0.9634 |

<16.5 | 0.7065 | 0.3256 | 0.9187 | 0.7499 | 0.3384 | 0.9784 |

<19.5 | 0.7565 | 0.4012 | 0.9383 | 0.7995 | 0.4285 | 0.9870 |

<22.5 | 0.7893 | 0.4115 | - | 0.8426 | 0.4369 | - |

Impedance to faster vehicles and availability of passing opportunities, effect of which is manifested in the amount of platooning, have significant impact on headway distribution pattern and thus, affects traffic modelling and simulation. Usually, platoon formation is infrequent under homogeneous traffic and, thus, small proportion of shorter headways is observed in the traffic stream. However, once the proportion of slower vehicles starts increasing, formation of platoons is quite evident resulting in an increase of shorter headways. A look into the table (see Table 5) indicates that at moderate flow levels it is about 10% when the traffic is more or less homogeneous, which however, increases upto 30–35 in case of mixed traffic. This fact was further compared with a study wherein road segment exhibits more or less homogeneous traffic but carries truck traffic upto 20%. Since they usually travel at a speed that is much lower than the posted speed, a large proportion of faster vehicles are compelled to move with platoons resulting in a simultaneous increase of shorter headways (see Table 5). However, the car-following behaviours are quite different under mixed traffic; a large number of drivers adopt headways which are less than safe headway and a few drivers even take considerable amount of risk to overtake.

## 7 Conclusions

Over the past few years, road traffic has tremendously increased across the globe and as a result of it traffic congestion has been a critical issue in many countries worldwide. At times, traffic analysts face difficulties in characterizing and modeling such congested flow of traffic, thereby, addressing this issue effectively. Since a key component of determining the performance of a simulation model is the generation of inter-arrival times or headways as an input into the simulation process, it is imperative to have reliable and practical descriptions of headways. At congested state of flow, the conventional approach is, however, found ineffective due to large proportion of shorter headways. Further, modelling of time headways under mixed traffic is still vague even after decades because of inherent complexities of analysing such traffic and no significant efforts have been made so far in regard to this. This was the motives for taking up a systematic investigation of some intriguing characteristics of time headway distributions under such traffic particularly at moderate and heavy flow.

Empirical investigations with the headway data reveal that at all flow scopes’ pertaining to moderate and heavy flow, median is less than mean, signifying concentration of shorter headways. This is attributable to high risk-ability of driver population which results in safety reduction. Further, it was observed that effect of heterogeneity results in a deviation from the conventional model and accordingly, makes exponential model inappropriate in describing headways. Assessment of contestant headway distribution models elucidates that at heavy flow under such traffic four distribution functions namely, log-logistic, lognormal, Pearson 5 and Pearson 6 exhibit their aptness is describing headways. Subsequently, the appropriate models were selected using a methodology based on K-S test and also, compatibility hypothesis of empirical distribution; log-logistic distribution fits well to the observed data at moderate flow whereas, Pearson 5 distribution provides a decent fit particularly at congested state of flow.

While Pearson 5 distribution indicates acceptable statistical validity in describing headways at heavy flow, its acceptability was rejected on the basis of hypothesis test at unstable flow nearing capacity. Field experiences with the mixed traffic on two-lane roads indicate that at such flow level existence of both following and non-following headways are significant, thereby, displaying different trends in distributional characteristics. This was further investigated by plotting the nomographs wherein probability of headway less than 7.5 s or more was observed to increase almost sharply with the flow; this attributes to frequent formation of platoons and higher amount of car following interaction.

In order to further validate the results, a pilot study was conducted and it was observed that the difference between the calculated probabilities of field data and the theoretical probabilities obtained from nomograph was very marginal. The present study, thus, creates a starting point of further initiatives aimed at establishing a robust method of modeling headways on two-lane rural highways with mixed traffic.

## Declarations

### Acknowledgements

The part of the analysis presented in the paper has used the data collected in the CSIR-CRRI, New Delhi sponsored project “Development of Indian Highway Capacity Manual (INDO-HCM)”. The authors sincerely acknowledge CSIR-CRRI. The authors are grateful to the anonymous referees for their suggestions to improve the paper.

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