Literature review
2.1 Microscopic modeling of traffic stream
2.1.1 Conventional car-following models
Car-following models describe the processes due to which drivers follow other vehicles in the traffic stream. In developed world, car-following term was used since the traffic stream consisted of mostly cars. However, in the developing world, the term can be replaced by, and hereafter referred as, ‘vehicle-following’ or simply ‘following behavior’. These models have been studied for seven decades (for e.g., Pipes, 1953, stating that safe headway increases with speed). Since car- following attempts to understand the interplay between phenomena at the individual level and global behavior on a more macroscopic scale, it forms as one of the chief processes of modern- day traffic theory. It is clear that a detailed understanding of this key process is now becoming increasingly important as modeling traffic behavior has significant impact on tackling traffic problems. Many old car-following models are developed by now.
2.1.1.1 GHR model and linear (Helly) model
The research team of the General Motors (GM) Company produced five generations of their car following models, all having the basis that the response of the following driver (acceleration or deceleration) is a function of the sensitivity of this driver and the stimulus. Sensitivity-stimulus models were introduced by GM research laboratories and represent the basis of older models. The GHR model represents this group. In this model, acceleration of the follower is a function of follower’s speed, relative speed, spacing, and driver reaction time delay. This model was also developed by Chandler et al. (1958). The authors (Gazis et al., 1961) also performed several tests in a tunnel in New York for validating the model. Since the GHR model presumes that the acceleration is only proportional to the relative speed, it cannot ensure the achievement of the driving objective to keep a safe distance from vehicles. (Bevrani et al., 2012). When the relative speed approaches zero, there is no response for the following vehicles no matter how far or close it is from the leading vehicle. This is the most well-known model which relates dependency of deceleration of following vehicle (FV) onto the relative speed of leading vehicle (LV) and FV and the current speed of FV. It proposes a stimulus-response type function. The safety distance or collision avoidance model (1959) seeks to specify a safe following distance within which a collision would be unavoidable if FV suddenly decelerates.
Linear (Helly) model (1959) includes the acceleration/braking of the LV into the GHR model. A linear model was proposed, which stated that the acceleration of FV is related to the relative speeds between LV and FV, speed of the FV, the deviation of the relative spacing and the desired following distance, and driver’s reaction time. A combination of both the Helly and GHR models was developed by Xing (1995). This model was also calibrated. It contains terms describing relationship of acceleration with relative speed and spacing (GHR model), acceleration/braking from rest (linear model), and behavior under free-flow conditions.
Forbes (1953) was the pioneer in considering human factors in traffic flow theory. The author concluded that, when cars are following each other on a densely packed road, drivers become more alert and aggressive, and a sudden change in driver response time is reflected with the change of headways before and after a leading car aggressively slows down. The study stated that the time
headways of the experimental platoon were in the 1 to 1.5 second range while following at constant speed before the slow-down, but about twice as large after the experimental and unexpected deceleration by the lead car.
Newell (1961) investigated car-following model which gave a governing equation for vehicle speed other than acceleration. The inclusion of the functional speed–headway relation entails the Newell model to achieve the objective of driving behavior in a car following. It is regarded as the beginning of study into car-following models with speed–headway relations. Wang et al. (2011) also developed three car-following models based on Newell model; by making certain assumptions for perceiving ability of drivers. Treiber et al. (2000) proposed the intelligent driver model (IDM).
Here, acceleration of FV is related to the speed of FV, maximum acceleration, relative speeds, desired gap and actual gap available. based on this, the model decides the position of FV which is an outcome of intelligent acceleration strategy. The model combines both the free road acceleration and an intelligent braking strategy.
2.1.1.2 Gipps’ car following model
Gipps (1981) proposed a car-following model for the response of the following vehicle based on the assumption that each driver sets limits to his/her desired braking and acceleration rates. The model calculates the speed of FV at the next time-step if the speed and conditions at the previous time-steps are input. However, it does not take into account the acceleration or deceleration of vehicles at different stages, and thus may not present realistic reaction of the driver of FV. Based on the presence or absence of an LV, the driver of FV may either (i) accelerate to its desired speed with an increasing acceleration rate; or (ii) select his/her speed to ensure that safe stopping on sudden stopping of LV. The first condition, known as accelerating to desired speed at the free- flow condition, can be adopted for weak lane disciplined traffic too if there is the absence of any interaction of vehicles.
Gunay (2007) have modified the equation for considering the presence of veering in weak lane discipline traffic. In such traffic, the driver also has the opportunity to veer to a safe speed to move parallel to the LV, if it suddenly brakes. This safe speed depends on the allowed width of the corridor between several other LV’s, explained further in Section 2.2.1.
2.1.1.3 Fuzzy logic model
The major advancement for the rule-based models is the fuzzy logic model (Kikuchi and Chakroborty, 1992) where such models typically divide their inputs into some overlapping `fuzzy sets’ each one describing how adequately a variable fits the description of a `term’. This approach considers that human perceives the environment, uses his/her knowledge and experience to infer possible actions, and responds approximately. Precise (input) variables consist of distance headway, relative speed and acceleration. The consequence of the rules is reaction and position updation of leading vehicle. For example, the rules are of the form ‘If at time t, the distance headway is very large, AND relative speed is moderately negative, AND acceleration of leading vehicle is negative, THEN at time (t+dt) follower should accelerate mildly. This model represents
the approximate behavior and not by quantitative evaluation. All these models are described in detail by Brackstone and McDonald (1999).
2.1.1.4 Psychophysical car-following model
The psychophysical car-following model (Michaels, 1963) was based on the assumption that drivers can sense the changes in the relative speeds of the leading and following vehicles by analyzing the change in apparent size of the leading vehicles and react according to the visual angle changes subtended by the vehicle ahead. These changes are modelled based on certain thresholds, and quantifying these thresholds and calibrating them was a difficult task, and till today the calibration of threshold parameters vary as per traffic conditions. Calibration was conducted by the visual angle models (Ferrari, 1989) or threshold requirements (Evans and Rothery, 1973).
Several studies (Wiedemann, 1974; Bekey et al. 1977; Lee, 1976; Wiedemann and Schwerdtfeger, 1987; Wiedemann and Reiter, 1992; Reiter, 1994; Krauss et al., 1999) were conducted to investigate the thresholds and the following vehicle acceleration. Wiedemann (1974) considered four car-following regimes in the psychophysical model- 1. free flow behavior, 2. Closing in behavior, 3. Car-following behavior, and 4. emergency braking situations. The psychophysical model approach is helpful to accurately describe the car-following behavior, but its calibration remains a major issue. Further, none of these models represents car-following in weak lane discipline conditions, wherein vehicles can close in more than the expected safe distance headway at higher amounts of staggering because they have the option to veer if the LV suddenly brakes.
2.1.1.5 Vehicle-following models pertaining to heterogeneous traffic
Earlier studies described heterogeneity in simulation models by converting the vehicle into a standard vehicle using various equivalency factors (Khan and Maini, 2000), and then the usual modeling approach of homogeneous streams is applied. However, this approach may not incorporate vehicle type-wise interactions and their characteristics. Thus, these modeling approaches cannot be used. A framework of traffic simulation model, which can replicate the movement of heterogeneous traffic, has been developed by Marwah and Singh (2000) to analyze the various environment of the road system. Traffic studies have been conducted on different roads of Kanpur metropolis. It includes parameter estimation of different sub-models such as sub-model each for roadway features, traffic input and field data. Different vehicle types, including a majority (up to 65%) of non-motorized vehicles were included in the analysis with different free speeds.
Levels of service (LOS) experienced by different vehicles was calculated. This basic structure of simulation model is adopted in several types of research.
A two-dimensional car following model was also proposed by Xie et al. (2008). It is based on an optimum speed model for pedestrian flow in lateral and longitudinal directions (Nakayama et al., 2005). Acceleration/deceleration models are considered using the speed differences, and the interactions with various boundaries (including vehicle edges) are considered. If adequate data collection techniques are available, the validity of this model in mixed traffic stream can be explored.