## 5 Energy-Efficient Dynamic Routing Using C-V2X Communication  
In this section, we describe the changes to the C-EEDR application to accommodate the limitations of using the realistic communication system. First, we describe the communication model used, then how this model is implemented in the INTEGRATION simulation tool. We used the cellular vehicle-to-everything (C-V2X) 4G LTE-V communication technology to model the communication system. We used a realistic communication simulation model of the C-V2X communication technology and integrated it with the traffic simulator INTEGRATION [2]. We used Packet Reception Ratio (PRR) curves provided from a communication company [20]. The PRR is the probability that a packet will be received by a vehicle using V2V/V2I C-V2X communication at a specific distance and density of vehicles. The company used its proprietary analysis to produce PRR curves at different distances (0 to 760 m with step 10 m) and vehicle densities (60 to 1200 vehicles with step 60 vehicles). Figure 1 shows the PRR curves produced, which represent a realistic approximation to the real world, as the company's simulator, used to produce the PRR curves, has been calibrated with the real word measurement data [20]. The curves were generated for vehicles moving on a highway with line of sight (i.e., no buildings or obstacles preventing sight). There is a PRR curve for each density index. The density values represent the number of vehicles around the receiving vehicle within 500 m. The x-axis represents the distance between the transmitting and receiving vehicles. The y-axis represents the probability that the transmitted packet will be received. As can be observed in Figure 1, as the density of the vehicles increases, the PRR decreases even at the same distance. The same behavior is observed with the distance, where the PRR value decreases with increasing the distance. To use these curves in our simulation tool, we could look up a 76 \(\times\) 10 \(\times\) 20 table of PRR values at a specific density and distance. If the required density or distance does not exist, then we can use the nearest values and interpolate for the required values. A faster approach is to fit a curve to those values. We fitted the PRR curves with 2 polynomial functions and an exponential function to come up with a formula that produces a PRR value given any distance between the receiving and the transmitting vehicles, and any number of vehicles around the receiving vehicle. Equation (5) shows the function used to fit the PRR curves. A two-step fitting process was performed to fit the PRR data. \[PRR=\frac{1}{\frac{a_{1}}{a_{2}*\exp{(\frac{d\mathit{sensity}}{a_{3}})}}+poly \gamma(density)*\exp{\frac{distance}{poly_{8}(density)}}}, \tag{5}\]  
where \(a_{1},a_{2},a_{3}\) are parameters calibrated using curve fitting. \(poly_{7}\) and \(poly_{8}\) are two polynomial functions of the 7 th and 8 th degrees, respectively. The variables \(density\) is the vehicular density and \(distance\) is the distance between the transmitting and the receiving vehicles. Equation (6) shows the polynomial function used to calculate \(poly_{7}\) and \(poly_{8}\) with \(n\) equal 7 and 8, respectively. \[poly_{n}(density)=\sum_{i=0}^{n}p_{i}*density^{i}, \tag{6}\]  
where \(p_{i}\) are the polynomial coefficients and are calculated using curve fitting. First, we fitted the PRR data (PRR vs. Distance) using an exponential three-parameters function. This fitting step was conducted for each density value (i.e., we had 20 \(\times\) 3 parameters). Using the fitted function, we can obtain the PRR value for any distance (between 0 and 760 m). Second, we used an exponential, 7 th degree polynomial, and 8 th degree polynomial functions to fit the three-parameter data produced from the first step. Using these fitted functions, we obtain the corresponding three parameters for any density value (between 60 to 1200 vehicles). Combining all four fitted functions, we obtain the function shown in Equation (5).