# Intelligent Total Transportation Management System for Future Smart Cities
## 3 Results--Examples of Sub-Model Developments
### A New Car-Following Model  
One of the essential microscopic models applied by transportation management systems is the car-following model [56; 57; 58]. It describes the one-by-one following process of vehicles in the traffic flow. The first car-following models were developed in the 1950s [59; 60; 61]. Several concepts are behind the first applied, classic models [62; 63]:  
* stimulus-response model
* safe distance model
* psychophysical model
* cell-based model (cellular automata model)
* optimum velocity model
* trajectory-based model  
For the last decade, new transfer, stochastic, and intelligent models were developed. Generally, the vehicle's motion in traffic flows can be described with active dynamic nonlinear models. The vehicle motions are activated by the drivers' decisions based on the available traffic information, traffic conditions, situation awareness, and knowledge of the driven cars' performance, personal practice, skills, and available resources. Principally, the motion of vehicles is disturbed by stochastic effects and drivers reacting on the vehicles depending on their subjective decisions. After analyzing the drivers' decision-making processes in selecting their vehicles' appropriate velocity, we become convinced that the drivers calculate the required changes in speed based on the relative distance and speed between the vehicles. In addition, the driver should also consider the vehicles' performance. Altogether, these aspects result in a randomly composed real stochastic process that can be represented as a diffusion process:  
\[a=\dot{v}=f(v,t)+\sigma(v,t)\eta(t) \tag{5}\]  
This controlled stochastic process--as known--can be approximated by a Markov chain process that leads to the following simple model  
\[a[k+1]=b_{dv}(v_{n-1}[k]-v_{n}[k])+b_{dx}\Big{[}(x_{n-1}[k]-x_{n}[k])-\Delta x _{pdn}\Big{]}+\varepsilon[k] \tag{6}\]  
where \(a\) is the acceleration of the vehicle; \(\eta\) is the noise disturbing the process; \((v_{n-1}[k]-v_{n}[k])\) and \((x_{n-1}[k]-x_{n}[k])\) are the relative velocity and relative distance between the two \((n-1)\)-_th_ and \(n\)-_th_ vehicles, respectively; \(b_{dv}\) and \(b_{dx}\) are the sensitivity coefficient, which can depend on the time and given vehicle and driver behaviors; \(\Delta x_{pdn}\) is the predefined safe distance between the vehicles; \(k\) is the number of steps in the chain (\(t=k\Delta t\)); and \(\varepsilon(k)\) is the random value disturbing the speed decision and maintaining process. The introduced Markov model can be easily adapted to the real observed traffic by estimating the sensitivity coefficients. Different approaches can be applied to the calibration and validations [64]. In this study, the concept of the model was validated by comparison evaluation of the Markov car-following model (6) with the Gazis-Herman-Rothery (GHR) model (7) and the S-curve model (8) solved in a simulation environment. The GHR model was presented by [PERSON] et al. [65] as follows:  
\[a_{n}(t+T)=\lambda_{l,\;m}\frac{\left[v_{n}(t)\right]^{m}}{\left[x_{n-1}(t)-x _{n}(t)\right]^{l}}[v_{n-1}(t)-v_{n}(t)] \tag{7}\]  
where, \(m\), \(l\) are parameters for speed and distance headway and \(\lambda_{l,m}\) is a constant showing the characteristics of the drivers. Because the driver sensitivity curve is very similar to the Gompertz curve, we decided to use the modified Gompertz curve [63] to define the coefficient \(\lambda_{l,m}\) in Equation (8):  
\[a_{n}(t+T)=\left\{a_{1}e^{-a_{2}e^{-a_{3}\left[\frac{\left[v_{n-1}(t)-v_{n}(t )\right]}{\left[v_{n-1}(t)-v_{n}(t)\right]}\right]}}-a_{4}\right\}\frac{\left[v _{n}(t)\right]^{m}}{\left[x_{n-1}(t)-x_{n}(t)\right]^{l}}[v_{n-1}(t)-v_{n}(t)]^ {q} \tag{8}\]Equation (8) is called the S-curve model. The simulation results are demonstrated in Figures 5 and 6. Figure 5 demonstrates the simulation results, e.g., changes in acceleration, velocities, running distances, and relative distances of vehicles in cases of using the described three different models. The model parameters are given in the figures.