Modelling Mixed Traffic Time Headway Distribution Induced by Bus Rapid Transit (BRT) in Cape Town, South Africa

2.1. Time Headway Distribution on Roadways and Freeways

Time headway distributions on roadways and freeways have been previously studied and documented in the literature. In 1993, Mei and Bullen investigated the number of headways measured on two southbound lanes of a four-lane motorway during the morning rush hour through probability distribution models. The lognormal distribution was the optimum model for the time headways in heavy traffic flow at both specified lanes, which was found to be with a 0.3 or 0.4 second shifting. In a study by [10] on interstate highways in Illinois, the United States, headways were recorded at traffic rates ranging from 140 to 1704 vehicles per hour per lane. The study recommended generating the time headway distribution for any traffic volume for both the right and left lanes using the lognormal model with a shift of 0.36 seconds. Thamizh [11] looked at the time headway data gathered from a divided urban arterial in Chennai City, India, with four lanes. It was discovered that the negative exponential statistical distribution could accurately simulate headways for all traffic flows and lane configurations. Bham and Ancha [12] also studied the temporal car-following headways for several sorts of highway sections in a steady state. A standard highway segment, lane shift, ramp merge, along with ramp weave part are all present at the data sites. In comparison to the shifted gamma distribution, the lognormal distribution with shifts was found to offer a superior fit for each of the locations cited. In Riyadh, Saudi Arabia, Al-Ghamdi [12] examined time headways observed on urban highways. Three categories of traffic flow were used to categorize the observed vehicle flow range: low (400 vehicles per hr), medium (400–1200 vehicles per hr), and high (>1200 vehicles per hr). His analysis found Erlang, shifted-exponential, and negative exponential distributions to be the best models.

He also noted that low-traffic situations had been the focus of the majority of the headway distribution studies that have been conducted. At large levels of traffic flow, the headway distribution modeling is nevertheless still hazy. Vehicle headways gathered from Finnish rural roadways were thoroughly analyzed by [13]. He concluded that low to moderate traffic levels with a low possibility of having short headways can be handled by the gamma distribution. Additionally, he asserted that the lognormal distribution could serve as a model for the follower headway distribution even though it is neither straightforward nor sufficiently realistic [13]. Zwahlen et al. [8], in 2007, studied the portability of the Ohio traffic load and lane traffic cumulative headway distributions. The study's findings demonstrated that for similar hourly traffic numbers, and for the most part, the headway distributions are the same for each lane. To examine the influence of specific factors on the time headway distribution, [15] used a synthetic Erlang distribution model to analyze headway data from multiple Japan-based sites. Besides, they looked for a law that relates to the model parameters. Results showed that each lane had a distinct background that could influence the distribution of time headways. Vehicle time headway distributions at varying flow rates on two- and four-lane roadways in India were examined by the researchers mentioned in the reference [6] using lognormal and log-Pearson statistical modelling methods. They also accessed the changes in the time headways of vehicles for several two-lane and four-lane routes throughout the morning and evening hours with mixed traffic. Their results not only captured the differences in headway distribution but also identified the headways of some types of vehicles, including the way vehicles maintain headways when headlights are turned on. In another study conducted in India by [14], the distribution of speed and time headway of vehicles in mixed vehicular traffic on four two-lane bidirectional roadways was examined using leader-follower vehicle pairs. Results showed that the speed and time headway distributions varied significantly and were found to have useful applications in capacity estimation, Level of Service (LOS) analysis, and the development of micro-simulation models. Time headway distributions were also modelled using three different probabilistic models (single, combined and mixed) by [15] in France. Results showed that the mixed models provided the best fit among several time headway samples. Using a traffic detector incorporated with laser sensors for sorting and analysis, the theoretical traffic flow models to analyse time headway distributions were developed, as mentioned in the reference [16]. The results gave a better understanding of headway in signalized arterials. In another study, [19] examined the impact of lane position on time headway in Isfahan, Iran, to evaluate driving behaviour at various highway lanes using headway distribution analysis. The results showed that the appropriate model for passing lanes differs from the one for middle lanes due to the different behavioural operations of drivers. In a study conducted by [17], time headway considering lateral distance was studied on a non-lane-based traffic flow using a novel approach, where time headways were divided into 5 intervals in form of a measuring criterion to evaluate time headway values and implications as “Unsafe (0-0.7 sec), non-lane-based car-following (0.9 sec), lane-based car-following (1.0 sec), overtaking time headway (1.3 sec), and free driving (larger than 2.5 sec)” [17]. Results indicated that “to differentiate between following and free-wheeling driving behavior, a reasonable criterion to use is the time headway of the overtaking operation”. Another closely related study conducted in China by [18] investigated traffic congestion and lane-changing patterns involving interactions between BRT and general traffic flow at a typical bottleneck along a BRT corridor. Results revealed abnormal lane violations resulting in a 16% reduction in the saturation rate of general traffic and 17% in bus travel time. Most of the studies discussed so far which have examined the time headway distributions of low, medium, and heavy mixed traffic flows were performed based on headway data of roadways or corridors without BRT dedicated lanes. The purpose of this paper is to report on the findings from a study that looked at how the presence of a BRT dedicated lane affected the time headway distribution of the high volume of mixed traffic that resulted on the adjoining lanes. The main thrust is to fit the headway data to probability distribution models and determine the most appropriate headway distribution model that best fits the data and the condition.

2.2. Hypothesis Testing and Estimation of Model Parameters

The technique known as “goodness of fit” is used to confirm and determine whether a probability distribution is suitable for simulating a specific phenomenon. The Akaike Information Criterion (AIC), the Schwartz or Bayesian Information Criterion (SIC or BIC), and the Hannan Quinn Information Criterion (HQIC) were the methodologies used in this investigation. Although the Chi-Squared (C-S), Kolmogorov-Smirnoff (K-S), and Anderson-Darling (A-D) goodness-of-fit statistics are still widely used today, they are not theoretically the best ways to compare distributions fit data. They cannot include censored, truncated, or binned data and are also restricted to accurate observations. The AIC, SIC, or BIC and HQIC, however, are statistical measures of fit generally known as information criteria. They are defined as follows:

A. AIC (Akaike Information Criterion): The Akaike Information Criterion is defined by the following model equation:

(1)

B. SIC (Schwarz Information Criterion, aka Bayesian Information Criterion BIC): The Schwarz Information Criterion, aka Bayesian Information Criterion, is defined as follows:

(2)

C. HQIC (Hannan-Quinn Information Criterion): The Hannan-Quinn Information Criterion is defined as follows:

(3)

The objective is to identify the model with the lowest information criterion value. Each formula has the -2ln [Lmax] term, which is an estimation of the model fit's deviation. Each formula's first component contains coefficients for k that indicate the severity of the penalty for the number of model parameters. Regarding punishing the loss of a degree of freedom, SIC [19] and HQIC [20] are stricter than AIC [21]. These three criteria are used to score each fitted model, whether it fits a copula, a time series model, or a distribution. Therefore, based on Maximum Likelihood Estimation (MLE), the model criterion with the lowest value of AIC, SIC, or BIC and HQIC provides the best fit. Amongst the three information criteria, the AIC ranking determines the best-fitted distribution.

2.3. Data Survey

The data were collected in Cape Town, South Africa. Traffic Characteristics data of volume and speed were surveyed at four designated sites 01 – 04 along the BRT route R27 in Cape Town, South Africa, for a three-month period. The Automatic Traffic Counter (ATC), installed at the four sites, was used to collect the data 24 hours a day for the entire period. Each location utilized two ATC loggers, one of which recorded BRT traffic flow data on the BRT dedicated lane, while the other recorded data on the adjoining lanes. Fig. (1) shows the general layout of the impact study location. The multilane provincial trunk route BRT corridor, R27 in Cape Town, South Africa, was selected for the field data survey. The sites, which are located along the trunk route, were selected after satisfying the following location criteria: (1) the geometric feature of roadway under investigation had the presence of a BRT dedicated lane with median design configuration and two adjoining lanes lying in parallel to it; (2) the roadway segments had relatively flat topographical terrain void of steep vertical slopes that could affect the free flow speed data, as well as pavement surfaces free from defects, such as rutting and potholes etc.; (3) the roadway segments were free from the influence of road intersections, on-street parking, broken down vehicles, traffic police check points, roundabouts and fuel filling stations along the route, which could also affect the free flow of traffic; (4) the segment lengths were long enough to allow for the setting up of the survey equipment at spots where there was sufficient free flow speed and the segment length was greater than the stopping sight distance (SSD) to reduce or eliminate the effect of intersections. Each pneumatic tube was placed and connected to the ATC loggers in accordance with the specifications and configurations and nailed in both lanes 1m apart at the four sites, as shown in Fig. (2). The loggers identified three vehicle categories or classes viz: passenger car (PC), medium vehicles (MV), and heavy vehicle (HV). A total of about 4560 headways were extracted for all the sites from the individual vehicle characteristics data and sorted on a Microsoft Excel worksheet. Table 1 presents the descriptive statistical characteristics of the headways collected from each of the four sites surveyed. Between sites 01 and 03, the mean headway decreased, while site 04 experienced a significant increase. This is due to the low traffic volumes observed at site 04. The explanation for this is that the road segment mainly carries traffic volumes from this work. The standard deviation exhibited similar behaviour. The trend in the coefficient of variation indicates that there is a cluster of headways on the roadway segments, particularly at sections near intersections where long queues are formed at peak traffic quickly because the introduction of BRT dedicated lanes has eliminated one of the three previously available lanes, causing long queues to build up quickly. The time headway variable's positive skewness and kurtosis all indicate that most of the distribution's headways are to the left of the mean value, which suggests that the road segment may have smaller headways or isolated clusters.

3.1. Lognormal Distribution

The well-known distribution model known as Lognormal is commonly employed in numerous research about headways. Additionally, modelling headways in instances where an automobile is followed is suggested by (Greenberg 1966). The mathematical expression for the lognormal distribution is:

(4)

where: τ denotes the shift's value in seconds; µ and σ are the “location” and “scale” variables of the lognormal distribution, respectively. The following can be deduced from the observed data::

(5)

(6)

(7)

3.2. Log Logistic Distribution

The log-logistic distribution is non-negative random variable probability distribution that is continuous and whose logarithm has a logistic distribution. Although it has heavier tails, it has a comparable shape to the log-normal distribution. Its cumulative distribution function can be expressed in closed form, unlike the log-normal distribution. The probability density function of a log-logistic distribution can be expressed as:

Probability Density Function:

(8)

and the Cumulative Distribution Function:

(9)

Where x > 0, α > 0, β > 0

3.3. Inverse Gaussian Distribution

The inverse Gaussian distribution is a family of continuous probability distributions with two parameters that has support on (0, ∞). It shares numerous characteristics with a Gaussian distribution. While the Gaussian represents a Brownian motion's level at a defined time, the inverse Gaussian defines the distribution of the time it takes for a Brownian motion with positive drift to achieve a fixed positive level. The probability density function of the Inverse Gaussian Distribution is given by:

(10)

3.4. Generalized Extreme Value Distribution (GEV)

A family of continuous probability distributions known as the Generalized Extreme Value (GEV) was created from the extreme value theory. It serves as a limit distribution of correctly normalized maxima of a series of independent random variables with similar distributions. As a result, it is utilized as an approximation to describe the maxima of lengthy (finite) sequences of random variables. The distribution has a continuous scale parameter (k > 0) and a continuous shape parameter (k > 0). The PDF and CDF for this distribution are described as follows:

(11)

Cumulative Distribution Function:

(12)

3.5. Burr Distribution

The Burr Distribution, also known as the generalized log-logistic distribution, is a continuous probability distribution for a non-negative random variable. To simulate household income, it is frequently utilized. Continuous shape parameters (k > 0; α > 0), continuous scale parameters (β > 0), and continuous location parameters (γ > 0) are present, and the PDF and CDF are given by:

Probability Density Function (PDF):

(13)

Cumulative Distribution Function (CDF):

(14)

To decide which of the distributions is the most appropriate model for headway data on each road segment, the five probability distribution models were subjected to goodness-of-fit tests earlier defined, also referred to as hypothesis testing. The following procedure was employed to select the best headway distribution model:

3.5.3. Step 3

The generated model parameters viz shape, scale, and location were applied to determine the P-value. The higher the P-value and the more it exceeded 0.05 in all three tests at a 5 percent level of significance and 95% level of confidence, the more compatible the model.

4.1. Parameters of the Probability Distribution

The distributions fitted to the time headways revealed varied estimated parameters, as shown in Table 3. Amongst these distributions, only the Lognormal, Inverse Gaussian, and Loglogistic have two parameters (shape and scale).

The Burr and GEV distributions have three which are the shape, scale, and location parameters. Each of the parameters of the distribution varied in different proportions across the four sites. Their magnitudes and differentials were significant in all sites investigated.