Seasonal Influences on Driver Behaviour: A Review of Car-Following Dynamics in Hot and Cold Climates

Understanding how seasonal variations influence driver behaviour is essential for crafting efficient traffic management systems and enhancing global road safety. Different seasons change weather conditions, such as temperature, precipitation, and visibility, which can significantly impact driving patterns [1]. Despite the importance of this topic, there is a notable gap in the literature, particularly in comparative studies of driving behaviour across different seasons and climatic conditions. This review synthesises existing research on the influence of seasonal changes on car-following dynamics and driver performance.

Seasonal changes can profoundly affect various aspects of driving, including speed, headway, and driver aggressiveness. For example, high temperatures in the summer can lead to increased driver stress and fatigue, potentially resulting in more aggressive driving behaviours. In contrast, winter conditions, characterised by lower temperatures, snow, and ice, can lead to cautious driving patterns and increased headways [2]. Kehagia et al. [3] highlight the critical role of Road Safety Audits (RSA) in identifying infrastructure deficiencies and safety risks on Greek roads, with proposed measures grounded in behavioural studies. Similarly, Alruwaili and Xie [4] demonstrate how connected vehicle (CV) technologies can mitigate driver unawareness and traffic conflicts, particularly on horizontal curves, while accounting for the influence of weather conditions on safety outcomes. Likewise, Adeliyi et al. [5] examine road traffic accident severity using machine learning models, identifying key contributing factors such as casualties, weather, lighting conditions, and the number of vehicles involved, with the J48 pruned tree model outperforming other predictive approaches. Furthermore, Bal and Vleugel [6] address the environmental challenges posed by freight transport, exploring how substituting maritime transport with road and rail alternatives can reduce emissions without compromising logistics. Understanding these seasonal effects, drivers' behaviour, and road safety modelling techniques are essential for tailoring traffic regulations, enhancing road safety measures, and improving vehicle design to accommodate different weather conditions [7].

This review paper is essential for comprehensively understanding how environmental factors, such as weather conditions, influence driver behaviour across different seasons. By synthesising studies from various climatic regions, the paper highlights key patterns and differences in driving behaviour, offering valuable insights for developing targeted traffic management strategies, informing policy decisions, and improving infrastructure planning. Furthermore, this review identifies significant gaps in the current literature and suggests directions for future research, aiming to advance our understanding of how environmental factors shape driver behaviour and impact road safety and efficiency.

The primary aim of this review is to analyse and integrate existing research on the impact of seasonal variations on driver behaviour, particularly focusing on car-following dynamics. This paper identifies common trends, discrepancies, and research gaps by examining the interplay between seasonal factors and driver behaviour, providing a foundation for future investigations. Understanding these dynamics enables policymakers and traffic authorities to design more effective interventions, improving road safety and traffic flow year-round. Ultimately, this review aims to contribute to developing strategies that enhance road transportation systems' safety and efficiency in response to seasonal challenges. This comprehensive analysis will help identify areas for further research and guide efforts to address the impact of environmental conditions on driver behaviour.

3.1 Overview of car-following dynamics

Road user errors significantly contribute to traffic accidents, often stemming from misperceptions, distractions, misjudgements of traffic flow and distance, improper overtaking or turning, running red lights, failing to yield, and following other vehicles too closely. Tailgating, a specific form of following too closely, is particularly dangerous as it significantly increases the risk of collisions. Factors such as drunk driving, speeding, and inadequate street lighting at night, combined with poor visibility, further amplify these risks. Specific driver characteristics, like being male, and road features like type and grade, also play a critical role in accident likelihood [8]. While human error is the leading cause of traffic accidents, other contributing factors include vehicle defects, road deficiencies, and adverse environmental conditions, although these are less significant. Table 1 illustrates the interrelationship between these factors, with human error being the primary cause of accidents in the UK [9]. Based on police reports from different countries, the influence of road users, vehicles, and road environments emerges as the dominant cause of accidents, as shown in Table 2. However, these studies often generalise findings, which may overlook nuanced behavioural variations across different driver demographics and specific roadway conditions.

Table ‎1. Contributing factors and their interaction in 2020 [9]

Table ‎2. Factors contributing to road accidents in different countries [12]

Car-following behaviour, a core aspect of traffic flow theory, focuses on the interaction between consecutive vehicles. This behaviour is essential for maintaining safe distances and ensuring smooth traffic flow, particularly on high-speed, multi-lane highways. Key parameters that define car-following behaviour include headway, speed, and acceleration. Headway is crucial for traffic safety and efficiency, whether measured by time or distance. Drivers often modify their headway according to their sense of safety and comfort, which fluctuates depending on traffic conditions [10]. However, research in this field frequently depends on simulations or controlled settings, which might not accurately reflect real-world complexities like driver distractions or the presence of various vehicle types. Several models have been developed to simulate car-following behaviour, with empirical and mathematical models being the most prominent. Empirical models rely on observed data, while mathematical models use equations to represent the relationships between variables. For example, Puan [11] demonstrated that empirical models of car-following behaviour are based on drivers' preferred following distances, as shown in Eq. (1):

H=A₀+A₁V     (1)

where, H denotes the distance headway (m), V refers to the vehicle's speed (m/s), A₀ represents the length of the car (m), and A₁ denotes the reaction time of the driver (s) [11].

However, Connolly et al. [13] developed a Route Evaluation by Vehicle Simulation (REVS) model based on a single-carriageway section of the road. This model assumes that the car-following distance remains constant regardless of speed and that all vehicles have the same lengths. Despite their utility, these models often oversimplify vehicle interactions by assuming uniform driver behaviour and road conditions, limiting their applicability to diverse real-world scenarios. The relationship between the car-following distance is described by Eq. (2) [13]:

H=L+10     (2)

where, H refers to the headway measured from rear to rear, and L denotes the length of the vehicle [13].

Further studies, like Hunt’s [14], proposed headway relationships for different vehicle types, such as cars and heavy goods vehicles (HGVs). Four car-following headway relationships were proposed [14]:

Car succeeding car, H_CC =2.124V−4.31     (3)

Car succeeding HGV, H_CH=2.052V+1.156     (4)

HGV succeeding HGV, H_HH=2.79V−3.997     (5)

HGV succeeding car, H_HC=2.854V−8.15     (6)

where, H is the measured distance from the forefront (m), V is the speed (m/s) [14].

H_CC refers to "Heavy Car Succeeding Car," a term used to describe the following of a heavier vehicle by a car in a traffic flow context.

H_GV stands for "Heavy Goods Vehicle," commonly used to refer to large vehicles such as trucks and lorries that transport goods.

H_CH denotes "Heavy Vehicles," a broader category encompassing various heavy-duty vehicles, including HGVs.

H_HH refers to "Heavy Hitter Haulage," a term sometimes used in freight transportation to describe particularly large and heavy cargo vehicles.

H_HC stands for "Heavy Hill Hold Control," a term used to describe a control mechanism or system designed to manage the behaviour of heavy vehicles when driving uphill, particularly to prevent rolling back or losing control.

Chen et al. [15] applied linear regression analysis to evaluate how road type and weather conditions affect perceived risk levels. Their findings revealed that various factors significantly impact the perceived risk index (PRI), with basic road segments and clear skies as reference points, represented by the constant with an impact value of zero. The results were validated by a high R-square value and a residual plot, which showed an unbiased and homoscedastic distribution. While regression models have effectively captured the relationship between headway (the distance between vehicles) and vehicle speed, the models used in prior research do not apply to this study. This is because previous observations were limited to two-lane single-carriageway roads and intersections, making them unsuitable for the current context.

In contrast, traffic flow theory is pivotal in understanding car-following behaviour by evaluating parameters such as traffic density, mean speed, and flow rate. These measurable parameters can be assessed using a variety of formulae. Immers and Logghe [16] employed a microscopic approach to analyse traffic flow, focusing on the trajectory of each vehicle as it moves within a lane. The vehicle's position over time defines the trajectory, with its rear bumper as the reference point. Figure 1 illustrates the outcomes of an experiment that applied car-following models. In this experiment, two vehicles started from a stationary position, with the trailing vehicle maintaining a 100-meter distance behind the lead vehicle. The model assumes a reaction time of one second and a sensitivity parameter of 5000 m²/s. Immers and Logghe [16] highlighted that these parameters are essential for understanding how the trailing vehicle responds to the lead vehicle's acceleration and deceleration. Their model effectively captures the dynamics of car-following behaviour, emphasising the critical role of reaction time and sensitivity in maintaining safe and efficient vehicle spacing within traffic flow.

Meanwhile, the impact of external environmental factors, particularly weather parameters, concerning traffic theory in studying driver behaviour is an area of research that has yet to be sufficiently explored. Rakha et al. [17] investigated the effect of inclement weather on the traffic stream behaviour on freeways regarding visibility and precipitation. Adverse weather was found to reduce operating traffic mean speeds, and the mobility and safety of drivers can be significantly impacted by reduced visibility caused by fog and rain. Furthermore, these models often overlook the combined impact of various environmental factors, including weather, road design, and traffic congestion, which can significantly alter driver perception, decision-making, and overall behaviour. This limitation underscores the need for more comprehensive models that integrate these factors to better understand driver behaviour under varying and complex conditions.

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Figure ‎1. Car-following model simulation results [16]

3.2 Impact of environmental conditions on driver behaviour

Weather conditions such as temperature, wind, and snowfall notably impact motorists' travel patterns. As temperatures rise, the frequency of trips tends to increase, while colder temperatures and snowfall lead to fewer trips [18]. Rainy days are associated with higher traffic volumes and accident rates than sunny days. Hjelkrem and Ryeng [19] found that the risk to drivers increases by factors of 1.3, 1.5, and 2.5 on wet, muddy, and snowy road surfaces, respectively. Figure 2 highlights how weather factors like rainfall, temperature, wind speed, visibility, and humidity affect travel behaviour. These weather-induced changes in travel patterns often alter the use of transportation systems, leading to increases or decreases in road traffic, depending on the prevailing conditions.

Figure ‎2. The interaction between weather conditions and travelling decisions [20]

Moreover, adverse weather conditions such as fog, dust storms, rainfall, and snow significantly impact driver behaviour and traffic safety. These conditions can severely impair visibility, alter road surface conditions, and affect vehicle handling, increasing the risk of traffic accidents. For example, fog and heavy rain can obscure a driver's view, making it difficult to judge distances and detect obstacles on the road [21]. Research shows drivers tend to increase their headways and reduce speeds during adverse weather to compensate for decreased visibility and longer stopping distances. However, these adjustments often fall short of fully mitigating the heightened risk of collisions, particularly rear-end crashes [22]. Rakha et al. [17] examined the effect of inclement weather on traffic flow on freeways, specifically focusing on visibility and precipitation. Their findings revealed that adverse weather conditions significantly reduce mean traffic speeds, underscoring the impact of weather on both traffic mobility and safety.

Tailgating, or following too closely behind another vehicle, becomes even more dangerous in adverse weather. Reduced visibility and slippery road surfaces significantly increase the stopping distance required to avoid collisions. Studies by Hamdar et al. [22] and Rawashdeh et al. [23] confirmed that adverse weather amplifies the risks associated with tailgating. In conditions like fog or rain, the likelihood of rear-end collisions increases due to delayed driver reactions and longer braking distances.

Additionally, Biswas et al. [24] highlighted the importance of maintaining appropriate headways, particularly in adverse weather conditions. Their study showed that tailgating contributes significantly to traffic accidents in Kuwait, stressing the need for public awareness and stricter enforcement of traffic regulations to prevent such behaviour. Environmental conditions are crucial in shaping driver behaviour, particularly within traffic flow theory. Adverse weather often forces drivers to adjust their speed and headway to maintain safety. The risks associated with these conditions can be mitigated through empirical research, mathematical modelling, and advanced technologies, ultimately enhancing overall traffic safety.

3.3 Seasonal variations in driving patterns

Seasonal variations play a significant role in influencing driving patterns and traffic flow. In regions with distinct seasons, drivers adjust their behaviour in response to changing weather, daylight hours, and road conditions. During the winter, conditions like shorter daylight hours, colder temperatures, and more precipitation, whether rain or snow, can significantly alter driving behaviours [25]. Drivers tend to adopt more cautious behaviour, including reducing speed and increasing headway to accommodate slippery road surfaces and decreased visibility [26]. In contrast, summer driving patterns are often more aggressive due to extended daylight hours and better road conditions. During this season, drivers are more likely to engage in risky behaviours such as speeding and tailgating, which increases the incidence of traffic violations and accidents [27]. This contrast underscores the importance of adaptive traffic management strategies tailored to seasonal changes.

For instance, Scandinavian countries, where harsh winters are common, have implemented traffic control measures such as mandatory use of snow tyres and dynamic speed limits that adjust to real-time road conditions. Similarly, traffic management systems in desert regions like the Middle East often incorporate heat-resistant road materials and weather alert systems to mitigate risks during extreme summer heat. These examples illustrate the critical role of localised strategies in addressing seasonal variations. Many studies have explored the impact of driver attitudes on traffic and accident rates. For example, Ma and Zhang [28] analysed how aggressive driving behaviours contribute to accidents at Highway-Rail crossings in the USA. They categorised drivers as aggressive if they engaged in risky actions such as "driving around or through the gate," "failing to stop," or "stopping on the crossing," based on data from the Federal Railroad Administration.

In contrast, drivers who "stopped and proceeded" were classified as calm. Their study found that aggressive drivers were linked to higher rates of injuries and fatalities at these crossings compared to calm drivers [28]. Such findings highlight the potential for targeted educational campaigns or stricter enforcement policies to address seasonal peaks in aggressive driving behaviours. Figure 3 illustrates the variation in driver behaviour across different weather conditions, showing that clear weather conditions increase the likelihood of aggressive driving, while adverse weather tends to curb such behaviours.

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Figure ‎3. Weather influences drivers’ behaviour [28]

Moreover, previous studies have extensively examined the influence of control variables on driver behaviour and traffic flow, focusing on the effects of weather conditions. When these variables are controlled for weather, they offer valuable insights into how different environmental scenarios impact traffic dynamics. For instance, Rämä and Kulmala [29] conducted a comprehensive study on the Finnish E18 motorway, where speed limit signs and variable message signs (VMS) were integrated with an automatic speed control system based on real-time meteorological data. This data was gathered from two unmanned road weather stations equipped with sensors that provided real-time information on road surface conditions. The study classified road and weather conditions into three categories: poor, moderate, and good, based on factors like rain intensity, snowfall, visibility, road surface condition, and wind velocity. The results showed that the weather-controlled VMS system positively impacted road safety, as it reduced mean traffic speeds and speed variations, indicating more consistent driving behaviour under changing weather conditions [29]. For example, in Finland, Rämä and Kulmala [29] studied the integration of VMS with real-time meteorological data on the Finnish E18 motorway. This system classified road and weather conditions into three categories: "poor, moderate, and good" based on factors like rain intensity, snowfall, visibility, road surface condition, and wind velocity. Their results demonstrated reduced mean traffic speeds and variations, contributing to more consistent and safer driving behaviour. Such adaptive systems, particularly in regions prone to sudden weather changes, can significantly improve road safety.

Similarly, Goodwin [30] reviewed research on the effects of weather on traffic flow, focusing on signalised arterial roadways. The study revealed that adverse weather conditions, such as rain and snow, increased driver headways, larger inter-vehicle spacing, and reduced speeds and acceleration rates. These changes in driving behaviour can significantly influence the efficiency of traffic signal timing plans, as modified driving patterns require adjustments to maintain smooth traffic flow. The findings highlight the need to consider weather conditions when designing and implementing traffic control systems. These findings are particularly relevant for regions with unpredictable weather, as they underscore the need for weather-responsive traffic signal timing plans. For instance, adaptive traffic signals that extend green light durations during heavy rain can help reduce congestion and ensure smoother traffic flow. Table 3 summarises various weather events and their effects on roadways and traffic operations, further demonstrating the complex interplay between weather and traffic flow [30].

Table ‎3. Impact of weather on roadways and traffic operations [30]

Future research could focus on the cost-benefit analysis of implementing advanced weather-controlled traffic systems in different regions to enhance the practical application of these findings further. Integrating real-time weather data into traffic management systems can help reduce the adverse effects of weather on driving behaviour and lower accident risks. In particular, regions with diverse seasonal climates, such as Scandinavia and desert environments, could serve as benchmarks for testing innovative strategies, offering valuable lessons for global implementation.

5.1 Data collection techniques

Data collection is a critical component of studying seasonal variations in driving behaviour. Accurate and comprehensive data allows researchers to identify patterns and draw meaningful conclusions about the impact of different seasons on driving. One of the primary techniques used in data collection is traffic surveillance systems, which include cameras and sensors installed along roadways to monitor traffic flow and gather data on vehicle speeds, headways, and traffic volumes. These systems provide continuous, real-time data invaluable for understanding how driving behaviours change with the seasons [49].

Another standard data collection method involves surveys and questionnaires distributed to drivers. These tools gather qualitative data on driver perceptions, attitudes, and self-reported behaviours in different seasonal conditions. For instance, surveys might ask drivers about their experiences driving in snow or extremely hot weather, their stress levels, and any changes they make to their driving habits during these times. This qualitative data complements the quantitative data collected from traffic surveillance systems, providing a more comprehensive picture of seasonal driving behaviours [50].

In addition to surveys and surveillance systems, researchers often use onboard diagnostics (OBD) devices installed in vehicles to collect detailed data on vehicle performance and driver behaviour. OBD devices can track various metrics, including speed, acceleration, braking patterns, and fuel consumption. This data is particularly useful for identifying how vehicle performance issues, such as engine overheating or reduced battery efficiency, correlate with different seasonal conditions. Figure 4 presents a streamlined process for handling test data gathered from the OBD systems of a Renault Meganne III before and after repair. The workflow starts with collecting torque data from two vehicles in `.csv` format and feeding it into 'OBD_T5c.exe', which extracts key parameters such as engine RPM, coolant temperature, and trip statistics. This data is then processed through various stages, including aggregation scripts and specific programs like `Coolant.exe` and `TimeShift.exe`, which analyse the engine's cooling behaviour and adjust cycle timings. The thermostat work cycle data is split and refined, followed by the generation of histograms to understand the performance metrics better. Finally, the processed data is input into Gnu plot and spreadsheet software for further analysis, creating detailed charts, tables, and statistical summaries of the vehicle's performance under different conditions [51].

Figure 4. Scheme for processing test data read from OBD systems [51]

Field and Naturalistic Driving Studies (NDS) are important data collection techniques. Ehsani et al. [52] emphasise the importance of Naturalistic Driving Studies (NDS) as a crucial method in transportation research. NDS collects real-world driving data, providing insights into drivers' behaviour in everyday driving situations. This approach involves monitoring participants in their vehicles under natural conditions without external intervention. The data collected is often extensive, including information on driver behaviour, vehicle performance, and the surrounding environment (e.g., road conditions, weather). The advantage of NDS is that it offers a more accurate representation of driving behaviour compared to simulated studies or crash reports, which might not capture the nuances of real-time decisions and reactions.

NDS has been particularly valuable in identifying the factors leading to crashes, near-crashes, and other risky driving behaviours. By capturing various variables, such as distraction, speed, and following distance, NDS can help researchers develop more effective safety interventions and policies. However, the paper also notes challenges, such as the vast amount of data that must be processed and concerns regarding privacy and consent. Despite these challenges, NDS remains a vital tool in advancing road safety research and understanding the complexities of driver behaviour in real-world scenarios. Figure 5 illustrates the visualisation of data from an Australian Naturalistic Driving Study (ANDS) conducted by Ehsani et al. [52]. The figure showcases a combination of multiple data sources collected during a driving session. These include vehicle telemetry (speed, acceleration, and gyroscopic data), a real-time map of the driver's route, and synchronised video feeds capturing the driver's behaviour and the external road environment. The graph tracks speed and acceleration, while the map plots the trip's trajectory. The video footage, divided into four quadrants, also captures different perspectives: the driver's face, the road ahead, and other in-vehicle angles. This comprehensive visualisation allows researchers to analyse driving behaviour, vehicle dynamics, and environmental factors in context, contributing to a deeper understanding of driver decision-making and road safety in natural conditions.

Figure 5. Visualisation of ANDS data [52]

5.2 Modelling approaches

Modelling approaches play a crucial role in understanding and predicting the effects of seasonal variations on driving behaviour. Traffic simulation models are one of the primary tools used for this purpose. These models simulate the movement of vehicles on a road network, taking into account various factors such as traffic volume, road conditions, and driver behaviour. By adjusting the parameters to reflect different seasonal conditions, researchers can predict how traffic flow and safety are likely to be affected in different seasons [53].

Headway modelling, whether measured as time or distance headway, refers to the separation between successive vehicles in a traffic lane. Time headway is the time interval between two vehicles passing the same point on the road, usually measured from a common feature like the front bumper [53]. In contrast, distance headway refers to the space between vehicles. This study specifically focuses on time headway, using it to develop an empirical car-following model based on headway and speed. The model aims to predict drivers' behaviour by analysing the relationship between headways and the associated speeds of vehicles, particularly those travelling with headways of 5 seconds or less, defined as impeded vehicles. The field data for headway and speed were collected and subjected to regression analysis, a statistical technique that explores the connection between dependent and independent variables. A simple linear regression model was chosen for this study because it effectively captures the relationship between headway and speed, aligning with previous research findings [54, 55].

The car-following models developed in this study were categorised based on vehicle size, focusing on the relationship between headway and speed. Four vehicle size categories were identified: small-size vehicles (≤ 6 m), medium-size vehicles (6-≤10 m), large-size vehicles (10-20 m), and truck-size vehicles (> 20 m). Each category was analysed within four car-following classes, such as small-size vehicles following small-size vehicles (small-small) or small-size vehicles following medium-size vehicles (small-medium). The study ensured a strong predictive relationship by developing the model with headway as the dependent variable and speed as the independent variable, achieving an R-squared value of 95%, with a 5% significance level (α = 0.05). While weather and driver response time were recognised as influential, they were excluded from the model due to measurement difficulties, making speed the sole predictor of headway. The resulting model offers a practical tool for understanding driver behaviour across different vehicle categories under varying traffic conditions.

Understanding how CRI evaluates a driver's perception of accident severity is crucial to developing the relationship between speed and the Chosen Risk Index (CRI). CRI is derived from the vehicle's speed, length (calculated from the wheelbase and weight), and the time gap (TG) between vehicles. The formula for calculating distance headway (H) is given in Eq. (7) [46]. This formula states that the distance headway is the product of the speed of the following vehicle (v) and the time headway (t), as shown:

H_distance = vt     (7)

As the speed of a vehicle increases, so does the potential severity of any ensuing accident, which is reflected in the CRI. Eq. (8) [46] calculates the CRI by multiplying the speed of the following vehicle (V) by its weight (W) and dividing the result by the time gap (TG), as follows:

$C R I=\frac{v \times w}{T G}$     (8)

In this formula, CRI is the Chosen Risk Index, V is the speed of the following vehicle, W is the vehicle's weight, and TG is the time gap in seconds. The CRI is highly relevant to accident analysis, as it highlights the influence of a vehicle's speed and weight on the perceived severity of a crash during car-following scenarios. Although advanced technologies like weight-in-motion sensors can measure a vehicle's weight, these technologies are costly and require installation beneath the road surface. Instead, the vehicle's wheelbase is used as a proxy for weight, assuming that vehicles of the same dimensions (width, height, and density) have similar weights [46].

The speed values for calculating CRI were derived from field data and represented the mid-points of speed class intervals. This allowed the development of a CRI-speed relationship during car-following, with data calculated for both winter and summer conditions. Furthermore, the study introduces the concept of relative CRI (CRIRelative), which evaluates the change in perceived accident severity between seasons. CRIRelative is calculated as the ratio of the CRI in winter to the CRI in summer, representing the relative risk over a given speed range during car-following.

Another important modelling approach is the speed-flow modelling. The speed-flow relationship quantifies how vehicle speeds relate to the flow rate on a highway section. This relationship is determined by correlating field-measured flow rates at various intervals with their corresponding speed values. Graphically plotting these values allows one to establish the maximum flow rate and associated speed.

This study considers four vehicle classes to analyse the data. However, these classes are converted to their equivalent Passenger Car Units (PCU) to standardise the analysis. This conversion translates the influence of different vehicle sizes—small, medium, large, and trucks—into the effect of passenger cars, thereby incorporating them into the overall flow rate. Consequently, the flow rate is expressed in Passenger Car Units per hour (pcu/h) rather than vehicles per hour (veh/h) [46].

The speed-flow relationship is derived from density plots of vehicles for each road under various weather conditions. Eq. (9) [46] represents this relationship:

Flow (v) = Density (D) × Velocity (S)     (9)

Typically, the linear relationship between speed and density can be calculated using Eq. (10) [46]:

$S=a\left(1-\frac{D}{b}\right)$     (10)

where, S = average speed (km/h), D = density (veh/km or veh/km/ln), a = free-flow speed (km/h), b = jam density (veh/km or veh/km/ln) [46].

Greenshields' model utilises this relationship to establish connections between flow-density and speed-flow. The flow-density relationship is articulated in Eq. (11) [46]:

$S=a\left(1-\frac{v / S}{b}\right)$     (11)

where, v = flow rate. The flow rate and density can be calculated using Eq. (12) [46]:

$v=b S-\left(\frac{b}{a}\right) S^2$     (12)