Explaining the Mean and Variability of Extra Travel Time in Traffic with Interacting Fast and Slow Vehicles

Abstract

In this work, the authors developed procedures to explain mean extra travel time (T) and extra travel time variability (V). This was carried out for situations (through simulations) where the fast vehicles’ travel time, whose speed tendency (sp₁) is the permitted speed limit, is negatively affected (i.e., increasing travel time) by the presence of slow vehicles, whose speed tendency (sp₂) is half the speed limit. The speed limit was set in the range of 60 km/h to 90 km/h, with seven cases, and every case had eight simulations, each with different p₁ (fast vehicles’ percentage) and p₂ = 100% − p₁ (slow vehicles’ percentage) values. p₂ ranged from 10% to 80% at intervals of 10%, for a total of 56 simulations. From the simulations’ data, we calculated the fast vehicles’ extra travel time, which is the additional time to traverse an avenue segment owing to the presence of slow vehicles. The fast and slow vehicles recreate heterogenous traffic in terms of speed. We developed procedures for modeling T and V with p₂, and V with T. For modeling, $~$71.42% of the data from simulations was used. We find that the models’ parameters values can be used for explaining the remaining data. In addition, we discovered that the pattern of p₂ vs. V, for p₂ ranging from 50% to 80%, is different among sp₁ cases and not linear.

1. Introduction

Bad traffic is experienced in most cities of developing countries. A current traffic problem, which is getting worse every day, is the increase in travel times (see the congestion levels in [1]). Thus, gaining insight into situations inducing higher travel times is of utter importance. One of those situations is when drivers, intentionally or owing to the operating limitations of the vehicle they drive, travel at a lesser speed than the posted speed limit. Therefore, in the same stretch of road, the drivers’ desired speed (or speed tendency) may vary and there is a likely chance of slow vehicles impeding fast vehicles. The traffic outcome when fast and slow vehicles interact is difficult to establish and is related with the drivers’ desired speed and the percentage of slow vehicles, among other variables.

The aim of the present research is, firstly, to perform simulations with different percentages of fast and slow vehicles for proposed values for each speed tendency. The output of these simulations is the mean extra travel time (T) and the extra travel time variability (V) of the fast vehicles. These data (i.e., T and V) are divided into two groups: the data for modeling and the data for validating. Later, procedures for modeling T and V are proposed with the intention to prove if the patterns in the data (for modeling) can be reproduced. In addition, we demonstrate that the data for validating can be explained with the proposed models, and we present how to calculate the models’ parameter values as part of the procedures.

In a traffic scenario with vehicles driving slower than the general traffic stream (thus generating a speed differential), when the faster vehicles are impeded by the slower vehicles, the former are tempted to perform overtaking maneuvers, if possible. In [2], it was proven that the vehicles’ travel speed (and thus also travel time) is affected, among others, by the traffic composition in terms of speeds. In [3], a relationship between speed variance and accidents was established, finding that an increase in average speed does not necessarily increase accident rates, but an increase in speed variance certainly increases them. In [4], the proposed algorithm triggers a need to overtake when the speed differential is 8 km/h. The work in [5] focuses on establishing (via simulations) the average travel time to complete a highway segment and the number of passenger cars’ lane changes with different percentages of heavy trucks. The results show that both variables increase as the percentage of heavy trucks increases, with the influence of heavy trucks on traffic being notorious in heavy traffic. In [6], speed indicators on sites with uniform speed limits (65 mph for passenger vehicles and trucks/buses) and with differential speed limits (70 mph for passenger vehicles and 60 mph for trucks/buses) were investigated. The results from the models were a mean speed of 65.3 mph for the uniform case (one) and 66.9 mph for the differential case (two). Similarly, the 85th percentile speed results were 70.7 mph for case one and 73.9 mph for case two. Finally, the standard deviation was 5.5 mph for case one and 6.8 mph for case two. Through simulations, the impact on traffic of differential speed controls was investigated in [7]. The speed control strategies were uniform posted speed limits (USLs), differential posted car–truck speed limits (DSLs), and differential mandated truck speed limits (MSLs). Considered positive for traffic safety, DSL and MSL reduce average travel speed, increase head-on time-to-collision, and reduce car–car overtaking. Considered negative, DSL and MSL increase percentage time spent following and total number of overtakes, decrease car–car overtaking, but increase car–truck overtaking. In [8], considering all types of vehicles, it was detected that the highest travel speed variability occur in freeways with DSL, followed by urban freeways with USL and a speed limit of 55 mph. For passenger cars, the speed was consistent when the speed limit was 70 mph, but at lower speed limits, more speed variation was detected.

Different factors are related to free-flow speed; one of those is differential speed. In [9], the free-flow speed was modeled considering as an explanatory variable the trucks’ percentage in order to test if the flow heterogeneity (the traffic composition in terms of vehicles’ type) impacted speed. The results showed that the trucks’ percentage is significant for the four models: daytime speed for cars and trucks, and nighttime for cars and trucks. In [10], with the aim of identifying explanatory factors for mean speed and speed dispersion considering two-lane rural highways on tangent segments and horizontal curves, free-flow speed models were developed.

The drivers’ travel speed is influenced by the posted speed limit. In [11], it was found that the posted speed limit is highly correlated with the average free-flow speed for urban streets, multilane highways, and freeways. In [12], the speed limits in Utah rural interstates were increased from 75 to 80 mph, finding that the travel speed and the probability to overcome the speed limit were then higher, and that by increasing the speed limit, the speed variance was not reduced. However, as the literature review presented in [13] suggests, the results concerning the effects of speed limit on speed variation do not support each other. The investigation in [14] centers on the local and global impacts of speed limits considering two groups of drivers—the compliant and the non-compliant drivers. When choosing speed, both consider the travel time cost, the perceived crash risk, and the perceived ticket risk.

The factors influencing the drivers’ desired speed on the road are numerous. There are multiple interacting factors that explain drivers’ behaviors [15]. A study on drivers’ speeding behavior is presented in [16]. The drivers’ characteristics and their relation to speed (as a risk of accidents level) are presented in [17]. In [18], the drivers’ speed choice (when there is at least a 6 s time gap to the next vehicle) at two different sites were compared, and at two different days for the same site. How a driver is influenced by the collective driving behavior is discussed in [19]. Empirical evidence suggesting that the speed choice is influenced by the pleasure of driving and other drivers’ behavior is presented in [20]. Preliminary data for supporting the notion that a driver adjusts their speed by comparing their own speed with that of others are introduced in [21]. A model including attitudes and perceptions of other drivers’ behavior explains approximately 15% of the variation in observed speed [22]. A speed choice prediction model was presented in [23]. It was found that the number of heavy vehicles (trucks and buses) and four-wheelers (informal para-transit) flowing in the same direction as the subject vehicle negatively influences the driver speed choice of the higher-speed class, and the same applies for trucks and motorcycles driving in the opposite direction. Buses, trucks, four-wheelers, and motorcycles negatively influence the speed of the lower-speed class (same traffic direction), and the same applies for non-motorized and buses driving in the opposite direction. The operating speed was modeled in [24], and the dependence of the drivers’ speed choice on individual driver’ behavior and the roadway geometry was demonstrated.

The necessity of vehicle class-specific modeling has been noted. Models to establish the free-flow speed of different types of vehicles are presented in [25]. In [26], it was found that the level-of-service (LOS) is different for each vehicle type. Besides, removing some vehicle types seems suitable to improve the system performance. Further, it was noticed that, in mixed traffic, the average stream speed better represents the dominant vehicle type, which is two-wheelers in [26]. The work in [27] uses percent speed-reduction and percent slower vehicles to define the LOS of a two-lane highway under heterogeneous traffic. It was concluded that it is impractical to evaluate the LOS of the entire traffic situation (with integration of motorized and non-motorized modes) with a common scale. In [28], two new methodologies are presented for estimating the percent time spent following (PTFS) on two-lane highways. The probabilistic method considers the slow-moving vehicles’ average speed and the distribution of the desired speeds to calculate the percentage of vehicles whose desired speed is superior to the slower vehicles’ average speed. This percentage (used to calculate the probability of a vehicle being impeded), along with the probability of a vehicle being part of a platoon, are used to calculate the percent following (PF), which is a surrogate measure for PTFS. In [29], it was noticed that the aggregation of faster and slower vehicles on a single class may not capture the interactions among the vehicles of the same class.

The dynamic of homogeneous and heterogeneous traffic is very different; in the latter, there are different types of vehicles, thus there are interactions (such as lane change maneuvers) between slow-moving and fast-moving vehicles. In [30], particle-hopping models for two-lane traffic were developed considering two types of vehicles: fast and slow vehicles, where the fast vehicles’ maximum allowed speed is considerable higher than that of the slow vehicles. The proposed models are different from each other because of the lane-changing rules, thus producing different results about the movement of the fast and slow vehicles. In [31], the heterogeneous traffic behavior in developing countries was reproduced with a discrete cellular automata (CA) model, modeling five types of vehicles: car, bus, truck, two-wheeler, and three-wheeler. In [32], a CA model was used to simulate traffic in a two-lane system; it was demonstrated that even small densities of slow vehicles induced the formation of platoons. Moreover, modifications were incorporated into the basic model, e.g., for allowing drivers to anticipate the speed of their predecessor, leading to diminishing the effect of slow vehicles. The multi-class traffic flow model presented in [33] incorporates heterogeneous drivers, i.e., drivers selecting a different speed to drive, thus there are faster vehicles trying to overtake slower vehicles. In addition, the model explains traffic flow theory phenomena: two-capacity regimes in the fundamental diagram, hysteresis, and platoon dispersion. In [34], a vehicle class was defined considering the desired speed in free-flow conditions, and a multiclass first-order simulation model was proposed that can replicate non-linear phenomena, including the dispersion of traffic platoons when a distribution of desired speeds exists. In [35], data were collected in several two-lane roads in India, and it was observed that the capacity decreases as the proportion of three-wheeler, tractor, or heavy vehicle increases, and that the capacity increases as the proportion of two-wheeler increases. In [36], through agent-based modeling, simulations representative of the traffic in the USA (with cars and trucks) and of the traffic in India (with different types of vehicles) were performed. It was found that the heterogeneity of vehicle types causes a high number of lane changes.

In this investigation, we simulate and analyze traffic, where the speed differential affects travel speed, thus affecting travel time. We were interested in finding a relation between the percentage of slow vehicles and two variables related with the fast vehicles: the mean extra travel time (T) and the extra travel time variability (V). We propose procedures for modeling two metrics of importance in traffic studies: T and V, which were calculated with the fast vehicles’ travel times from simulations. Our research is different from the current literature as we perform traffic simulations considering a wide range of fast and slow vehicles’ percentage values for a wide range of fast and slow vehicles’ speed tendency values, thus adding insight for this kind of traffic situation. Our proposed procedures have the advantage of including a method for calculating the parameters’ values of the models that describe the data (T and V) that were not used to calibrate the models. In this research, we have two general objectives:

(1)

For each fast vehicles’ speed tendency (sp₁) value, to run a simulation for each slow vehicles’ percentage (p₂) value, in order to acquire traffic information (the fast vehicles’ travel times) and calculate the mean extra travel time and the extra travel time variability. The data, i.e., T and V are divided in two groups: 1. the data for modeling and 2. the data for validating.

(2)

Establish procedures for modeling the first group of data and for explaining the second group of data. Calculate the models’ error and identify the procedures reporting the lower mean error, i.e., the suitable proposals to describe the T and V patterns.

2. Materials and Methods

2.1. Simulation Settings

As previously stated, in a simulation, we have fast and slow vehicles. The fast vehicles are represented by the p₁ percentage and their speed tendency is sp₁. The slow vehicles are represented by the p₂ percentage and their speed tendency is sp₂. The speed tendency (or desired speed) is the speed that a driver aims to reach and tries to maintain, without exceeding it. Moreover, the fast drivers’ desired speed is assumed to be the legal speed limit (or posted speed), and is selected in the range of 60 to 90 km/h. The slow drivers’ desired speed is the half of their counterpart (fast drivers), and is selected in the range of 30 to 45 km/h. The desired speed values are selected in a range that will provide meaningful results for different traffic scenarios.

The simulation scenario is a two-lane one-direction avenue segment of d = 500 m length, with constant flow and no entry points, and (looking for general results) without considering specific road geometry or environmental issues. The input of a simulation are fast and slow speed tendency, sp₁ and sp₂, respectively, and fast and slow vehicles’ percentage, p₁ and p₂, respectively. The output of a simulation are the fast vehicles’ travel times. This outcome depends on the percentage of fast and slow vehicles and their speed tendencies.

For simulating we use the simulator tool presented in [2], which integrates the microsimulation model for accelerating (decelerating) in free driving and following another vehicle, as well as for overtaking. Thus, the fast vehicles’ percentage and speed tendency can take a value from the sets $\left\{90\%,80\%,\dots ,20\%\right\}$ and {60 km/h, 65 km/h, 70 km/h,…,90 km/h}, respectively. p₂ can take a value from the set $\left\{10\%,20\%,\dots ,80\%\right\}$, while sp₂ can take a value from the set {30 km/h, 32.5 km/h, 35 km/h,…,45 km/h}. Recalling that p₁ = 100% − p₂ and $s{p}_2=s{p}_1/2$, a p₂ value fixes p₁, and a $s{p}_1$ value fixes $s{p}_2$. In a simulation, the fast vehicles’ travel time to complete the 500 m distance is being measured, which is influenced by the slow vehicles, because these force the fast vehicles to perform overtaking maneuvers. We have eight simulations (the possible values of p₂) for each of the seven sp₁ cases, thus 56 simulations were performed. We use the output data of 40 simulations (hereafter, data for modeling) for modeling, in these simulations sp₁ is a value from {60,65,75,85,90}. We use the output data of 16 simulations (hereafter, data for validating) for validating, in these simulations sp₁ is a value from $\left\{70,80\right\}$. The data for modeling and validating conform the total observed data. The simulation time of a single simulation is 600 s, thus the total simulation time was 33600 s, or 9.3333 h. For all simulations, we set the time lapse (f) between vehicles’ arrival per lane to be $f=6 s$, i.e., one vehicle arrives per lane each 6 s.

Table 1. Simulation settings.

Fast Vehicles’ Desired Speed (km/h)	Slow Vehicles’ Desired Speed (km/h)	Fast Vehicles’ Percentage (%)	Slow Vehicles’ Percentage (%)	Time Lapse between Vehicles per Lane (s)
A value of the set {60,65,…,85,90}	A value of the set {30,32.5,…,42.5,45}	A value of the set {90,80,…,30,20}	A value of the set {10,20,…,70,80}	6 s

shows the parameters’ values for setting a simulation. As the flow is constant, the time of day and day of week, which are variables related with demand, were out of the scope of this study. Instead, variables p₂ and sp₁ (and their counterparts, p₁ and sp₂, respectively) shape the traffic outcome (travel times).

When a vehicle is generated in a simulation, we need to establish if it will be a fast or a slow one. In any simulation, for each 10 vehicles generated (a group), it is guaranteed that the percentages of fast and slow vehicles of that group are equal to the percentages’ values (p₁ and p₂, respectively) of the simulation settings. This generation process seeks to avoid patterns between groups in the order by which the fast and slow vehicles are generated, but there is also randomness involved. For completeness, we repeated simulations for various selections of sp₁ and p₂ values, as the order of the generated vehicles will be different from one simulation to another and thus the outcome will be different too. We compare the T of simulations with same sp₁ and p₂ values and were not statistically different. The same applies for V.

2.2. Traffic Data

The travel time of the i-th vehicle (of those that count for p₁) is $t_i=t_i^{out}-t_i^{in}$, where $t_i^{out}$ is the timestamp (in seconds, the time starts as the simulation does) when the i-th vehicle leaves the avenue, and $t_i^{in}$ is the timestamp when the i-th vehicle enters the avenue. The expected travel time (t^sp) is calculated using Equation (1). As sp₁ is assumed to be the legal speed limit, it is also the maximum speed to travel along the distance d. Therefore, t^sp is the minimum time (without any delay) to traverse d.

$$t^{sp}=d/s{p}_1$$

(1)

The mean extra travel time (T) is calculated with Equation (2).

$$T={\displaystyle \sum }_{i=1}^N(t_i-t^{sp})/N$$

(2)

where i = 1…N is the vehicle index, and N is the total number of fast vehicles during the simulation.

Table 2. Mean extra travel time data.

${\mathit{p}}_2$	${\mathit{T}}_{60}$	${\mathit{T}}_{65}$	${\mathit{T}}_{75}$	${\mathit{T}}_{85}$	${\mathit{T}}_{90}$
10%	$T_{60}^{10}$ = 0.232 × 10−3 s	$T_{65}^{10}$ = 0.011688	$T_{75}^{10}$ = 0.066858	$T_{85}^{10}$ = 0.09583	$T_{90}^{10}$ = 0.052981
20%	$T_{60}^{20}$ = 0.255986	0.335886	0.321614	0.393612	0.315841
30%	$T_{60}^{30}$ = 0.915285	0.813281	0.75368	0.823075	0.784827
40%	$T_{60}^{40}$ = 1.944888	1.609535	1.718255	1.379915	1.400379
50%	$T_{60}^{50}$ = 3.237528	2.881251	2.453064	2.20284	2.190038
60%	$T_{60}^{60}$ = 4.812069	4.316362	3.70371	3.239808	3.135533
70%	$T_{60}^{70}$ = 6.774603	6.005609	4.949776	4.394159	4.239621
80%	$T_{60}^{80}$ = 9.029137	7.842978	6.649683	5.68031	5.467405

shows the T of the simulations with modeling purposes, so $T_{s{p}_1}^{p_2}=T_{60}^{10}=$ 0.232 × 10⁻³ s is the mean extra travel time from a simulation with $p_2=10\%$ and $s{p}_1=60 \mathrm{km}/\mathrm{h}$.

Extra travel time variability is calculated with Equation (3), similar as in the study presented in [37].

$$V=\frac{1}{N}{\displaystyle \sum }_{i=1}^N\left|t_i^e-T\right|$$

(3)

In Equation (3), V is the extra travel time variability and $t_i^e=t_i-t^{sp}$ is the extra travel time of the i-th vehicle. The calculated V values in

Table 3. Extra travel time variability data.

${\mathit{p}}_2$	${\mathit{V}}_{60}$	${\mathit{V}}_{65}$	${\mathit{V}}_{75}$	${\mathit{V}}_{85}$	${\mathit{V}}_{90}$
10%	$V_{60}^{10}$ = 5.28 × 10−5 s	$V_{65}^{10}$ = 0.011465	$V_{75}^{10}$ = 0.082488	$V_{85}^{10}$ = 0.119821	$V_{90}^{10}$ = 0.072548
20%	$V_{60}^{20}$ = 0.375195	0.452188	0.347738	0.409435	0.341801
30%	$V_{60}^{30}$ = 0.89158	0.681472	0.605966	0.583584	0.530159
40%	$V_{60}^{40}$ = 1.194243	0.888504	0.891808	0.768092	0.788808
50%	$V_{60}^{50}$ = 1.666407	1.207142	1.279672	1.250472	1.477645
60%	$V_{60}^{60}$ = 1.350847	1.152442	1.290058	1.167536	1.054867
70%	$V_{60}^{70}$ = 1.612525	1.612141	1.413731	1.062381	1.236766
80%	$V_{60}^{80}$ = 1.31128	1.257396	0.996451	1.23536	1.105068

, where $V_{s{p}_1}^{p_2}=V_{90}^{80}=1.105068 \mathrm{s}$, correspond to a simulation with $p_2=80\%$ and $s{p}_1=90 \mathrm{km}/\mathrm{h}$.

2.3. Modeling T

2.3.1. Procedure 1

T values corresponding to $s{p}_1=60 \mathrm{km}/\mathrm{h}$ and p₂ selections (second data column in Table 2 above) are stored in vector $A_{60}=\left[0.23\times {10}^{-3} 0.25\dots 6.77 9.02\right]$. The same procedure can be used to construct a vector for each sp₁ case. A vector content is accessed by position with index j = 1…8, for example $A_{65}\left(1\right)=0.0116$. We construct vector $B_{s{p}_1}$ with $A_{s{p}_1}$ in such a way that, for $j\ge 2$, $B_{s{p}_1}\left(j-1\right)=A_{s{p}_1}\left(j\right)-A_{s{p}_1}\left(j-1\right)$, because the values of vectors $B_{s{p}_1}$ might follow a linear pattern.

Table 4. Vectors $B_{s{p}_1}$ values.

${\mathit{B}}_{\mathit{s}{\mathit{p}}_1=60}$	${\mathit{B}}_{\mathit{s}{\mathit{p}}_1=65}$	${\mathit{B}}_{\mathit{s}{\mathit{p}}_1=75}$	${\mathit{B}}_{\mathit{s}{\mathit{p}}_1=85}$	${\mathit{B}}_{\mathit{s}{\mathit{p}}_1=90}$
$B_{60}\left(1\right) =$0.255754 s	$B_{65}\left(1\right) =$0.324197	$B_{75}\left(1\right) =$0.254756	$B_{85}\left(1\right) =$0.297783	$B_{90}\left(1\right) =$0.26286
$B_{60}\left(2\right) =$0.659299	0.477396	0.432067	0.429463	0.468985
$B_{60}\left(3\right) =$1.029603	0.796253	0.964575	0.55684	0.615552
$B_{60}\left(4\right) =$1.29264	1.271716	0.734809	0.822926	0.789659
$B_{60}\left(5\right) =$1.574541	1.435111	1.250645	1.036968	0.945494
$B_{60}\left(6\right) =$1.962534	1.689247	1.246066	1.154351	1.104088
$B_{60}\left(7\right) =$2.254534	1.837369	1.699908	1.286151	1.227784

presents the values of vectors $B_{s{p}_1}$. In the first column, we have the content of vector $B_{60}$; in the second, the content of $B_{65}$; and so on.

The Pearson’s linear correlation coefficient (r) between vector $B_{s{p}_1}$ and vector $J=\left[1 2\dots 6 7\right]$ (the vector containing the j = 1…7 index values) is expressed as $corr\left(B_{s{p}_1},J\right)$. As $corr\left(B_{s{p}_1},J\right)$ ≥ 0.95 for $s{p}_1$ values {60,65,75,85,90}, each $B_{s{p}_1}$ vector is modeled with a linear equation (Equation (4)). The equations presented in this section, and in the following ones, were solved with the least-squares technique.

$${\check{B}}_{s{p}_1}\left(j\right)=\left(a\cdot j\right)+b$$

(4)

With Equation (4) (with index j being the independent term), we calculate parameters a (slope) and b (y-intercept) for modeling vectors in Table 4, resulting in vectors ${\check{B}}_{s{p}_1}$.

Table 5. Vector ${\check{B}}_{s{p}_1}$ parameters’ values (procedure 1).

Parameter	${\check{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=60}$	${\check{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=65}$	${\check{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=75}$	${\check{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=85}$	${\check{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=90}$
a	0.3267	0.2715	0.2231	0.1748	0.1605
b	−0.0169	0.0327	0.0476	0.0984	0.1313

presents parameters a and b to obtain each vector ${\check{B}}_{s{p}_1}$. In the first column, the parameters’ values for obtaining ${\check{B}}_{s{p}_1=60}$ can be found; in the second, those for ${\check{B}}_{65}$; and so on. Parameter a values are in vector $P_a$, so $P_a =\left[0.32 0.27 0.22 0.17 0.16\right]$; while parameter b values are in vector $P_b$, so $P_b =\left[-0.01 0.03 0.04 0.09 0.13\right]$.

Consider now parameter a values in $P_a$ with position values $X_1=\left[1 2 4 6 7\right]$, which are the positions corresponding to the $s{p}_1$ values {60,65,75,85,90}. Position 1 corresponds to a parameter for describing $B_{60}$, and position 2 for $B_{65}$, thus missing positions 3 and 5, which correspond with $B_{70}$ and $B_{80}$, respectively. $X_1$ and $P_a$ are negatively correlated (r = −0.9849), while $X_1$ and $P_b$ are positively correlated (r = 0.9772). Then, by solving a linear equation relating $X_1$ and $P_a$, we calculate parameter $\tilde{a}$ values for $X_2=\left[3 5\right]$, which are the positions corresponding to the $s{p}_1$ cases $\left\{70,80\right\}$. The same path is taken for calculating parameter $\tilde{b}$ values through a linear equation, now relating $X_1$ and $P_b$. The models and their statistics are presented in

Table 6. Parameters’ models and statistics (procedure 1).

Model	Root Mean Squared Error	R-Squared	Adjusted R-Squared	p-Value
$\tilde{a}=-0.0266 x+0.3378$	0.0138	0.97	0.96	0.00223
$\tilde{b}=0.0222 x-0.0300$	0.0142	0.955	0.94	0.00412

. Thus, with the models in Table 6 and x = 3, we obtain $\tilde{a}$ and $\tilde{b}$ parameters’ values for estimating vector ${\widehat{B}}_{70}$, and with x = 5, we obtain the parameters’ values to estimate vector ${\widehat{B}}_{80}$.

Table 7. Vector ${\widehat{B}}_{s{p}_1}$ parameters’ values (procedure 1).

Parameter	${\widehat{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=70}$	${\widehat{\mathit{B}}}_{\mathit{s}{\mathit{p}}_1=80}$
$\tilde{a}$	0.2579	0.2047
$\tilde{b}$	0.0364	0.0808

presents $\tilde{a}$ and $\tilde{b}$ parameters’ values, used in ${\widehat{B}}_{s{p}_1} \left(j\right)=\left(\tilde{a}\cdot j\right)+\tilde{b}$.

A vector ${\widehat{B}}_{s{p}_1}$ is transformed to ${\widehat{A}}_{s{p}_1}$ in such a way that ${\widehat{A}}_{70}\left(1\right)$ and ${\widehat{A}}_{80}\left(1\right)$ are inferred through linear regression using values $[A_{65}\left(1\right) A_{75}\left(1\right) A_{85}\left(1\right)]$; for $k\ge 2$, ${\widehat{A}}_{s{p}_1}\left(k\right)={\widehat{A}}_{s{p}_1}\left(1\right)+{{\displaystyle \sum }}_{j=1}^{j=k-1}{\widehat{B}}_{s{p}_1}\left(j\right)$. A vector ${\check{B}}_{s{p}_1}$ is transformed to ${\check{A}}_{s{p}_1}$ such tat ${\check{A}}_{s{p}_1}\left(1\right)=A_{s{p}_1}\left(1\right)$; for $k\ge 2$, ${\check{A}}_{s{p}_1} \left(k\right)={\check{A}}_{s{p}_1} \left(1\right)+{{\displaystyle \sum }}_{j=1}^{j=k-1}{\check{B}}_{s{p}_1}\left(j\right)$.

2.3.2. Procedure 2

Now, vectors $A_{s{p}_1}$ for $s{p}_1$ values {60,65,75,85,90} are modeled with the exponential function presented in Equation (5), with $p_2$ (the slow vehicles’ percentage value) being the independent term, c and d being the model’s parameters, and index k = 1…8. The exponential function’s parameters’ values to obtain vectors ${\check{A}}_{s{p}_1}$ are presented in

Table 8. Vectors ${\check{A}}_{s{p}_1}$ parameters’ values (procedure 2).

Parameter	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=60}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=65}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=75}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=85}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=90}$
c	$c$ = 0.4167	0.3817	0.3612	0.3440	0.3379
d	$d$ = 0.0389	0.0384	0.0368	0.0356	0.0353

.

$${\check{A}}_{s{p}_1}\left(k\right)=c\cdot \exp \left(d\cdot p_2\right)$$

(5)

Parameter c values in vector $P_c$ = [0.4167 … 0.3379], and parameter d values in vector $P_d$ = [0.0389 … 0.0353]. The correlation coefficient between $X_1$ and P_c is r = −0.9567, and that between $X_1$ and $P_d$ is r = −0.9943. Thus, by means of a linear equation with independent values being X₁ and dependent values being P_c, we calculate parameter $\tilde{c}$ at X₂ values. The same method is used for calculating parameter $\tilde{d}$ at X₂ values with the derived linear model of X₁ versus P_d. Models and statistics for $\tilde{c}$ and $\tilde{d}$ are presented in

Table 9. Parameters’ models and statistics (procedure 2).

Model	Root Mean Squared Error	R-Squared	Adjusted R-Squared	p-Value
$\tilde{c}=-0.0120 x+0.4163$	0.0107	0.915	0.887	0.0107
$\tilde{d}=-0.0006 x+0.0396$	0.000199	0.989	0.985	0.000515

. The parameters’ values displayed in

Table 10. Vectors ${\widehat{A}}_{s{p}_1}$ parameters’ values (procedure 2).

Parameter	${\widehat{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=70}$	${\widehat{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=80}$
$\tilde{c}$	0.3803	0.3563
$\tilde{d}$	0.0376	0.0364

are used in ${\widehat{A}}_{s{p}_1}\left(k\right)=\tilde{c}\cdot \exp \left(\tilde{d}\cdot p_2\right)$.

2.3.3. Procedure 3

Now, vectors $A_{s{p}_1}$ for $s{p}_1$ cases $\left\{60,65,75,85,90\right\}$ are modeled with a polynomial equation, with $p_2$ being the independent term; see Equation (6).

$${\check{A}}_{s{p}_1}\left(k\right)=a_1{p}_2{}^n+a_2{p}_2{}^{n-1}+\dots +a_n{p}_2+a_{n+1}$$

(6)

We choose n = 2 (three parameters) for the polynomial order because the mean error $\overline{E}=mean\left(\left[E_{60} E_{65} E_{75} E_{85} E_{90 }\right]\right)=0.0356 \mathrm{s}$ is acceptable, where $E_{s{p}_1}$ is the error of comparing $A_{s{p}_1}$ and ${\check{A}}_{s{p}_1}$. Choosing n = 3 results in $\overline{E}=0.0295 \mathrm{s}$, which is not a significant improvement (a lower error). The parameters’ values of ${\check{A}}_{s{p}_1}\left(k\right)=a_1{p}_2{}^2+a_2{p}_2{}^1+a_3$ are presented in

Table 11. Vectors ${\check{A}}_{s{p}_1}$ parameters’ values (procedure 3).

Parameter	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=60}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=65}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=75}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=85}$	${\check{\mathit{A}}}_{\mathit{s}{\mathit{p}}_1=90}$
$a_1$	0.0016	0.0013	0.0010	0.0009	0.0008
$a_2$	−0.0159	−0.0127	−0.0029	−0.0010	0.0055
$a_3$	−0.0328	−0.0129	−0.0322	0.0278	−0.0991

. The $a_1$ parameter values in $P_{a_1}=\left[0.0016 \dots 0.0008\right]$, $a_2$ parameter values in $P_{a_2}=\left[-0.0159 \dots 0.0055\right]$, and $a_3$ parameter values in $P_{a_3}=\left[-0.0328 \dots -0.0991\right]$.

The linear correlation coefficient between $P_{a_1}$ and $X_1$ is r = −0.9860, that for $P_{a_2}$ and $X_1$ is r = 0.9788, and that for $P_{a_3}$ and $X_1$ is r = −0.2513. Considering r values, the ${\tilde{a}}_1$ and ${\tilde{a}}_2$ parameters were calculated using a linear equation, from $P_{a_1}$ versus X₁ and $P_{a_2}$ versus X₁, respectively. The ${\tilde{a}}_3$ parameter was calculated using a polynomial equation of order n = 4, from $P_{a_3}$ versus $X_1$. The parameters’ models and statistics are presented in

Table 12. Parameters’ models and statistics (procedure 3).

Model	Root Mean Squared Error	R-Squared	p-Value
${\tilde{a}}_1=-0.0001 x+0.0016$	6.58 × 10−5	0.972	0.00199
${\tilde{a}}_2=0.0033 x-0.0189$	0.00208	0.958	0.0037
${\tilde{a}}_3=-0.0027 x^4+0.0395 x^3-0.1907 x^2$ $+0.3565 x-0.2353$	0	1	---

, with these models we obtain ${\tilde{a}}_1$, ${\tilde{a}}_2$ and ${\tilde{a}}_3$ for each value in $X_2$. The parameters’ values are used in ${\widehat{A}}_{s{p}_1}\left(k\right)={\tilde{a}}_1{p}_2{}^2+{\tilde{a}}_2{p}_2+{\tilde{a}}_3$ and presented in

Table 13. Vectors ${\widehat{A}}_{s{p}_1}$ parameters’ values (procedure 3).

Explained Vector	$\mathbf{Parameter} {\tilde{\mathit{a}}}_1$	$\mathbf{Parameter} {\tilde{\mathit{a}}}_2$	$\mathbf{Parameter} {\tilde{\mathit{a}}}_3$
${\widehat{A}}_{s{p}_1=70}$	0.00129	−0.0087	−0.0370
${\widehat{A}}_{s{p}_1=80}$	0.0010	−0.0020	0.0087

.

2.4. Modeling V

2.4.1. Procedure 4

Mean extra travel time might explain extra travel time variability. The Pearson’s linear correlation coefficient (r) between T values (in vector $A_{s{p}_1}$, corresponding to $p_2=10\%$ to 80%) and V values (in vector $C_{s{p}_1}$) is expressed as $corr(A_{s{p}_1}$,$C_{s{p}_1})$. For the different $s{p}_1$ cases, $corr(A_{60},C_{60})$ = 0.7054, $corr(A_{65},C_{65})$ = 0.8334, $corr(A_{75},C_{75})$ = 0.7361, $corr(A_{85},C_{85})$ = 0.8268, and $corr(A_{90},C_{90})$ = 0.7543. There is a positive correlation between $A_{s{p}_1}$ and $C_{s{p}_1}$ for any $s{p}_1$ case, thus if T increases, so does V. Now, we use a polynomial equation to explain V with T, i.e., the independent values in $A_{s{p}_1}$ and the dependent values in $C_{s{p}_1}$. We propose that the choice of the polynomial equation degree should satisfy two conditions: the mean error $\overline{E}=mean\left(\left[E_{60} E_{65} E_{75} E_{85} E_{90} \right]\right)<0.1$, where $E_{s{p}_1}$ is the error between $C_{s{p}_1}$ and ${\check{C}}_{s{p}_1}$, and the correlation coefficient $\left|r\right|>0.60$ for each vector’s parameter values and $X_1$. A polynomial equation of order n = 3 satisfies both conditions. Then, ${\check{C}}_{s{p}_1}\left(k\right)=b_1{A}_{s{p}_1}{\left(k\right)}^3+b_2{A}_{s{p}_1}{\left(k\right)}^2+b_3{A}_{s{p}_1}\left(k\right)+b_4$; the parameters’ values are shown in

Table 14. Vectors ${\check{C}}_{s{p}_1}$ parameters’ values (procedure 4).

Parameter	${\check{\mathit{C}}}_{\mathit{s}{\mathit{p}}_1=60}$	${\check{\mathit{C}}}_{65 }$	${\check{\mathit{C}}}_{75 }$	${\check{\mathit{C}}}_{85}$	${\check{\mathit{C}}}_{90}$
$b_1$	0.0077	0.0031	0.0022	0.0221	0.0168
$b_2$	−0.1473	−0.0763	−0.0943	−0.2551	−0.2285
$b_3$	0.8393	0.5507	0.6649	0.9454	0.9467
$b_4$	0.1177	0.1706	0.0992	0.0200	0.0084

. Vector $P_{b_1}$ = [0.0077 … 0.0168] contains the $b_1$ parameter values for $s{p}_1$ cases {60,65,75,85,90}, $P_{b_2}$ = [−0.1473 … −0.2285], $P_{b_3}$ = [0.8393 … 0.9467], and $P_{b_4}$ = [0.1177 … 0.0084].

Thus, we obtain vectors ${\check{C}}_{s{p}_1}$ with parameters in Table 14. For modeling ${\tilde{b}}_1$, ${\tilde{b}}_2$, ${\tilde{b}}_3$, and ${\tilde{b}}_4$ we use a polynomial equation of order n = 3. For ${\tilde{b}}_1$, we have $P_{b_1}$ (dependent values) and $X_1$ (independent values), and so on for ${\tilde{b}}_2$, ${\tilde{b}}_3$, and ${\tilde{b}}_4$ considering $P_{b_2}$, $P_{b_3}$, and $P_{b_4}$, respectively. The parameters’ models and statistics are presented in

Table 15. Parameters’ models and statistics (procedure 4).

Model	Root Mean Squared Error	R-Squared	p-Value
${\tilde{b}}_1=-0.0006 x^3+0.0086 x^2-0.0305 x+0.0317$	0.00624	0.873	0.445
${\tilde{b}}_2=0.0062 x^3-0.0839 x^2+0.303 x-0.3799$	0.0313	0.961	0.249
${\tilde{b}}_3=-0.0161 x^3+0.2199 x^2-0.8209 x+1.452$	0.0192	0.997	0.0692
${\tilde{b}}_4=0.0038 x^3-0.0511 x^2+0.1695 x-0.0015$	0.0128	0.991	0.119

, where the models explain well (R-squared in the range $~$87% to $~$99%) the dependent variable (a parameter). The parameters’ values for estimating ${\widehat{C}}_{s{p}_1}\left(k\right)={\tilde{b}}_1{A}_{s{p}_1}{\left(k\right)}^3+{\tilde{b}}_2{A}_{s{p}_1}{\left(k\right)}^2+{\tilde{b}}_3{A}_{s{p}_1}\left(k\right)+{\tilde{b}}_4$ are presented in

Table 16. Vectors ${\widehat{C}}_{s{p}_1}$ parameters’ values (procedure 4).

Vector	${\tilde{\mathit{b}}}_1$	${\tilde{\mathit{b}}}_2$	${\tilde{\mathit{b}}}_3$	${\tilde{\mathit{b}}}_4$
${\widehat{C}}_{s{p}_1=70}$	0.0078	−0.1372	0.7466	0.1074
${\widehat{C}}_{80}$	0.0129	−0.1834	0.8322	0.0590

.

2.4.2. Procedure 5

The linear correlation between p = [10 20 … 70 80] (p₂ values) and $C_{s{p}_1}$ (V values) for $s{p}_1$ cases {60,65,75,85,90} is always positive and in the range of $~0.85$ to $~0.93$. As the percentage of slow vehicles increases, extra travel time variability increases. Now, with a polynomial equation, the dependent variable V is explained by the independent variable $p_2$. For the polynomial degree (starting with n = 1), we seek to meet two conditions (the same as in procedure 4): (1) $\overline{E}<$0.1 and (2) $\left|\mathrm{r}\right|>0.60$ for each $P_b$ vector. $\overline{E}=mean\left(\left[E_{60} E_{65} E_{75} E_{85} E_{90} \right]\right)$, where $E_{s{p}_1}$ is the error between $C_{s{p}_1}$ and ${\check{C}}_{s{p}_1}$, and $r=corr(P_b,X_1)$ is the linear correlation coefficient between $P_b$ (a vector containing the b parameter values corresponding with the $s{p}_1$ cases) and $X_1$. With n = 5, both conditions are met. The parameters’ values for modeling ${\check{C}}_{s{p}_1}\left(k\right)=b_1{p}_2{}^5+b_2{p}_2{}^4+b_3{p}_2{}^3+b_4{p}_2{}^2+b_5{p}_2+b_6$ are presented in

Table 17. Vectors ${\check{C}}_{s{p}_1}$ parameters’ values (procedure 5).

Vector	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	${\mathit{b}}_4$	${\mathit{b}}_5$	${\mathit{b}}_6$
${\check{C}}_{s{p}_1=60}$	−1.4070 × 10−8	3.3891 × 10−6	−0.3069 × 10−3	0.0122	−0.1633	0.6934
${\check{C}}_{65}$	−1.6574 × 10−8	3.2941 × 10−6	−0.2344 × 10−3	0.0069	−0.0506	0.0313
${\check{C}}_{75}$	−2.1557 × 10−9	3.4381 × 10−7	−2.7155 × 10−5	0.0012	0.0028	−0.0381
${\check{C}}_{85}$	3.1903 × 10−8	−6.8689 × 10−6	0.5373 × 10−3	−0.0188	0.3114	−1.5865
${\check{C}}_{90}$	1.8897 × 10−8	−3.9332 × 10−6	0.2878 × 10−3	−0.0090	0.1463	−0.7256

. The $b_1$ parameter values are in vector $P_{b_1}$, the $b_2$ parameter values are in vector $P_{b_2}$, and so on.

Each vector P_b was modeled with a linear equation, with X₁ as the independent values and P_b as the dependent values. The models and statistical results are shown in

Table 18. Models and statistics (procedure 5).

Model	Root Mean Squared Error	R-Squared	p-Value
${\tilde{b}}_1=7.5331\times {10}^{-9} x-2.6532\times {10}^{-8}$	1.02 × 10−8	0.825	0.0328
${\tilde{b}}_2=-1.6267\times {10}^{-6} x+5.7517\times {10}^{-6}$	2.12 × 10−6	0.836	0.0296
${\tilde{b}}_3=0.000128 x-0.00046068$	0.000166	0.838	0.0292
${\tilde{b}}_4=-0.0044404 x+0.016265$	0.0061	0.821	0.0341
${\tilde{b}}_5=0.063598 x-0.20505$	0.101	0.776	0.0485
${\tilde{b}}_6=-0.2882 x+0.82771$	0.529	0.72	0.627

.

With the models in Table 18, we calculate each parameter for $X_2$ values (corresponding to $s{p}_1$ cases $\left\{70,80\right\}$). The parameters’ values for describing ${\widehat{C}}_{s{p}_1}\left(k\right)={\tilde{b}}_1{p}_2{}^5+{\tilde{b}}_2{p}_2{}^4+{\tilde{b}}_3{p}_2{}^3+{\tilde{b}}_4{p}_2{}^2+{\tilde{b}}_5{p}_2+{\tilde{b}}_6$ are presented in

Table 19. Vectors ${\widehat{C}}_{s{p}_1}$ parameters’ values (procedure 5).

Vector	${\tilde{\mathit{b}}}_1$	${\tilde{\mathit{b}}}_2$	${\tilde{\mathit{b}}}_3$	${\tilde{\mathit{b}}}_4$	${\tilde{\mathit{b}}}_5$	${\tilde{\mathit{b}}}_6$
${\widehat{C}}_{s{p}_1=70}$	−3.9331 × 10−9	8.7164 × 10−7	−7.6672 × 10−5	0.0029	−0.0142	−0.0368
${\widehat{C}}_{s{p}_1=80}$	1.1133 × 10−8	−2.3817 × 10−6	0.1793 × 10−3	−0.0059	0.1129	−0.6132

.

2.4.3. Procedure 6

The linear correlation coefficient between P₁ = [10 20 30 40] ($p_2$ values from 10% to 40%) and $C_{s{p}_1}\left(1:4\right)$ is $\mathrm{corr}\left(P_1, C_{s{p}_1}\left(1:4\right)\right)\ge 0.98$ for all $s{p}_1$ cases {60,65,75,85,90}. Between P₂ = [50 60 70 80] ($p_2$ values from 50% to 80%) and $C_{s{p}_1}\left(5:8\right)$ are the following: $corr\left(P_2, C_{60}\left(5:8\right)\right)=$−0.5759, $corr\left(P_2, C_{65}\left(5:8\right)\right)=$0.3794, $corr\left(P_2, C_{75}\left(5:8\right)\right)=$−0.5309, $corr\left(P_2, C_{85}\left(5:8\right)\right)=$−0.2267, and $corr\left(P_2, C_{90}\left(5:8\right)\right)=$−0.6393. In $C_{s{p}_1}\left(1:4\right)$, the first four values, and in $C_{s{p}_1}\left(5:8\right)$, the last four values of $C_{s{p}_1}$. There is a strong positive correlation between $P_1$ and $C_{s{p}_1}\left(1:4\right)$ for all $s{p}_1$ cases. The correlations between $P_2$ and $C_{s{p}_1}\left(5:8\right)$ cases are not that strong. Thus, we modeled (independent variable $p_2$ and dependent variable V) a vector $C_{s{p}_1}$ in two parts: the first four points with a linear equation and the last four points with a polynomial equation. When modeling vectors $C_{s{p}_1}\left(1:4\right)$, we obtained parameter $b_1$ (in vector $P_{b_1}$) and $b_2$ (in vector $P_{b_2}$) values. The r between $X_1$ and $P_{b_1}$ is −0.8604, and that between $X_1$ and $P_{b_2}$ is 0.8160. $\overline{E}=0.0141 s$, with $E_{s{p}_1}$ expressing the error between $C_{s{p}_1}\left(1:4\right)$ and ${\check{C}}_{s{p}_1}\left(1:4\right)$. By modeling $C_{s{p}_1}\left(5:8\right)$, we obtained the $c_1$, $c_2$, and $c_3$ parameter’s values (in vectors $P_{c_1}$, $P_{c_2}$, and $P_{c_3}$, respectively). We selected for modeling $C_{s{p}_1}\left(5:8\right)$ a polynomial of degree n = 2, as $\overline{E}=0.0434<0.1$, and neither n = 2 nor n = 3 satisfy |r| < 0.60 for all correlations. Nevertheless, with n = 2, corr($P_{c_1},X_1)=$0.58, corr($P_{c_2},X_1)=$−0.59 and corr($P_{c_3},X_1)=$0.57 are good approximations. Henceforth, using parameters $b_1$ and $b_2$, we modeled a vector ${\check{C}}_{s{p}_1}\left(k_1\right)=b_1{p}_2+b_2$, with $k_1=1\dots 4$ and $p_2=10\dots 40$ (see

Table 20. Vectors ${\check{C}}_{s{p}_1}\left(1:4\right)$ parameters’ values (procedure 6).

Parameter	${\check{\mathit{C}}}_{60}\left(1:4\right)$	${\check{\mathit{C}}}_{65}\left(1:4\right)$	${\check{\mathit{C}}}_{75}\left(1:4\right)$	${\check{\mathit{C}}}_{85}\left(1:4\right)$	${\check{\mathit{C}}}_{90}\left(1:4\right)$
$b_1$	0.0409	0.0286	0.0268	0.0211	0.0233
$b_2$	−0.4094	−0.2066	−0.1895	−0.0595	−0.1509

). With parameters $c_1$, $c_2$, and $c_3$, we modeled a vector ${\check{C}}_{s{p}_1}\left(k_2\right)=c_1{p}_2{}^2+c_2{p}_2+c_3$, with $k_2=5\dots 8$ and $p_2=50\dots 80$ (see

Table 21. Vectors ${\check{C}}_{s{p}_1}\left(5:8\right)$ parameters’ values (procedure 6).

Parameter	${\check{\mathit{C}}}_{\mathit{s}{\mathit{p}}_1=60}\left(5:8\right)$	${\check{\mathit{C}}}_{65}\left(5:8\right)$	${\check{\mathit{C}}}_{75}\left(5:8\right)$	${\check{\mathit{C}}}_{85}\left(5:8\right)$	${\check{\mathit{C}}}_{90}\left(5:8\right)$
$c_1$	3.5789 × 10−5	−0.7501 × 10−3	−0.0010	0.6397 × 10−3	0.7276 × 10−3
$c_2$	−0.0126	0.1036	0.1317	−0.0846	−0.1039
$c_3$	2.1544	−2.1649	−2.6667	3.8998	4.8104

). Vector $P_{b_1}=\left[0.0409\dots 0.0233\right]$ contains $b_1$ parameter values, and vector $P_{b_2}=\left[-0.4094\dots -0.1509\right]$ contains $b_2$ parameter values. Vectors $P_{c_1}$= [3.5789 × 10⁻⁵ … 0.7276 × 10⁻³], $P_{c_2}$= [-0.0126 … −0.1039], and $P_{c_3}$= [2.1544 … 4.8104].

The derived linear models of X₁ vs. $P_{b_1}$ and X₁ vs. $P_{b_2}$ are presented in

Table 22. Models (for parameters ${\tilde{b}}_1$ and ${\tilde{b}}_2$ ) and statistics (procedure 6).

Model	Root Mean Squared Error	R-Squared	p-Value
${\tilde{b}}_1=-0.0026032 x+0.038616$	0.00454	0.74	0.0613
${\tilde{b}}_2=0.041151 x-0.36784$	0.0858	0.666	0.0921

. The parameters’ values (obtained with models in Table 22 at X₂ values) used in ${\widehat{C}}_{s{p}_1}\left(k_1\right)={\tilde{b}}_1{p}_2+{\tilde{b}}_2$ for $s{p}_1$ cases $\left\{70,80\right\}$ are displayed in

Table 23. Vectors ${\widehat{C}}_{s{p}_1}\left(1:4\right)$ parameters’ values (procedure 6).

Parameter	${\widehat{\mathit{C}}}_{70}\left(1:4\right)$	${\widehat{\mathit{C}}}_{80}\left(1:4\right)$
${\tilde{b}}_1$	0.0308	0.0256
${\tilde{b}}_2$	−0.2443	−0.1620

.

The parameter values in $P_{c_1}$, $P_{c_2}$, and $P_{c_3}$ were modeled with a polynomial equation of order n = 4 (for modeling five points). The models and statistics are shown in

Table 24. Models (for parameters ${\tilde{c}}_1$, ${\tilde{c}}_2$, and ${\tilde{c}}_3$ ) and statistics (procedure 6).

Model	Root Mean Squared Error	R-Squared	p-Value
${\tilde{c}}_1 = -1.8458\times {10}^{-5} x^4+0.00024889 x^3$ $-0.00088743 x^2+0.00041101 x+0.00028177$	0	1	---
${\tilde{c}}_2 = 0.0018896 x^4-0.02386 x^3+0.066804 x^2$ $+0.054576 x-0.1121$	0	1	---
${\tilde{c}}_3 = -0.04006 x^4 +0.42626 x^3 -0.22552 x^2$ $-6.0257 x +8.0195$	0	1	---

. The calculated parameters’ values (with models in Table 24 at X₂ values) used in ${\widehat{C}}_{s{p}_1}\left(k_2\right)={\tilde{c}}_1{p}_2{}^2+{\tilde{c}}_2{p}_2+{\tilde{c}}_3$ for $s{p}_1$ cases $\left\{70,80\right\}$ are shown in

Table 25. Vectors ${\widehat{C}}_{s{p}_1}\left(5:8\right)$ parameters’ values (procedure 6).

Vector	${\tilde{\mathit{c}}}_1$	${\tilde{\mathit{c}}}_2$	${\tilde{\mathit{c}}}_3$
${\widehat{C}}_{70}\left(5:8\right)$	−0.0012	0.1616	−3.8233
${\widehat{C}}_{80}\left(5:8\right)$	−0.2735 × 10−3	0.0293	0.4973

.

3. Results

3.1. Procedure 1 Results

With Equation (7), we establish the error between vectors $A_{s{p}_1}$ and ${\widehat{A}}_{s{p}_1}$, or between vectors $A_{s{p}_1}$ and ${\check{A}}_{s{p}_1}$, where q = 8 is the number of elements in $A_{s{p}_1}$. The error between vectors $A_{s{p}_1}$ and ${\widehat{A}}_{s{p}_1}$ is expressed as $E_{s{p}_1}(A_{s{p}_1},{\widehat{A}}_{s{p}_1})$, and that between vectors $A_{s{p}_1}$ and ${\check{A}}_{s{p}_1}$ as $E_{s{p}_1}(A_{s{p}_1},{\check{A}}_{s{p}_1})$. The errors between $A_{s{p}_1}$ and ${\check{A}}_{s{p}_1}$ (a vector directly modeled with the data for modeling) for $s{p}_1$ values {60,65,75,85,90} are as follows: $E_{60}(A_{60},{\check{A}}_{60})=$ 0.0222 s, $E_{65}(A_{65},{\check{A}}_{65})=$ 0.0519 s, $E_{75}(A_{75},{\check{A}}_{75})=$ 0.0550 s, $E_{85}(A_{85},{\check{A}}_{85})=$ 0.0238 s, and $E_{90}(A_{90},{\check{A}}_{90})=$ 0.0128 s. The errors between $A_{s{p}_1}$ and ${\widehat{A}}_{s{p}_1}$ (a vector indirectly modeled with the data for modeling, without the data for validating) for $s{p}_1$ values $\left\{70,80\right\}$ are as follows: $E_{70}(A_{70},{\widehat{A}}_{70})=$ 0.1817 s and $E_{80}(A_{80},{\widehat{A}}_{80})=$ 0.2505 s.

$$E_{s{p}_1}=\frac{{\sum }_{k=1}^{k=q}\left|A_{s{p}_1}\left(k\right)-{\widehat{A}}_{s{p}_1}\left(k\right)\right|}{q}$$

(7)

The mean error of the $A_{s{p}_1}$ vs. ${\check{A}}_{s{p}_1}$ cases is $\overline{E}=mean\left(\left[E_{60} E_{65} E_{75} E_{85} E_{90 }\right]\right)=0.0332 \mathrm{s}$, a lesser value than the mean error between the $A_{s{p}_1}$ vs. ${\widehat{A}}_{s{p}_1}$ cases, which is $\overline{E}=mean\left(\left[E_{70} E_{80} \right]\right)=0.2162 \mathrm{s}$. This is an expected result as vectors ${\check{A}}_{s{p}_1}$ were modeled directly with the data for modeling, i.e., with $A_{s{p}_1}$ for $s{p}_1$ values {60,65,75,85,90}, while vectors ${\widehat{A}}_{s{p}_1}$ were indirectly modeled with the data for modeling, but without the data for validating, i.e., without $A_{s{p}_1}$ for $s{p}_1$ values $\left\{70,80\right\}$. Figure 1 shows the graphs of the compared vectors in this section.

3.2. Procedure 2 Results

Vectors ${\widehat{A}}_{s{p}_1}$ for $s{p}_1$ values $\left\{70,80\right\}$ were obtained with the parameters’ values in Table 10 and Equation (5), and vectors ${\check{A}}_{s{p}_1}$ for $s{p}_1$ values {$60,65,75,85,90$} were obtained with the parameters’ values in Table 8 and Equation (5). The errors between the $A_{{\mathrm{sp}}_1}$ and ${\check{A}}_{s{p}_1}$ cases are as follows: $E_{60}(A_{60},{\check{A}}_{60})=$ 0.4151 s, $E_{65}(A_{65},{\check{A}}_{65})=$ 0.3944 s, $E_{75}(A_{75},{\check{A}}_{75})=$ 0.2957 s, $E_{85}(A_{85},{\check{A}}_{85})=$ 0.2390 s, and $E_{90}(A_{90},{\check{A}}_{90})=$ 0.2497 s. The errors between the $A_{s{p}_1}$ and ${\widehat{A}}_{s{p}_1}$ cases are as follows: $E_{70}(A_{70},{\widehat{A}}_{70})=$ 0.3582 s and $E_{80}(A_{80},{\widehat{A}}_{80})=$ 0.4095 s. The error of the directly modeled vectors ${\check{A}}_{s{p}_1}$ is comparable to the error of the indirectly modeled vectors ${\widehat{A}}_{s{p}_1}$. Figure 2 shows the graphs of the compared vectors in this section.

3.3. Procedure 3 Results

The errors between vectors ${\check{A}}_{s{p}_1}$ and $A_{s{p}_1}$ (directly estimated T vs. data for modeling) cases are as follows: $E_{60}(A_{60},{\check{A}}_{60})=$ 0.0226 s, $E_{65}(A_{65},{\check{A}}_{65})=$ 0.0552 s, $E_{75}(A_{75},{\check{A}}_{75})=$ 0.0617 s, $E_{85}(A_{85},{\check{A}}_{85})=$ 0.0264 s, and $E_{90}(A_{90},{\check{A}}_{90})=$ 0.0116 s. The errors between vectors ${\widehat{A}}_{s{p}_1}$ and $A_{s{p}_1}$ (indirectly estimated T vs. data for validating) are as follows: $E_{70}(A_{70},{\widehat{A}}_{70})=$ 0.1892 s and $E_{80}(A_{80},{\widehat{A}}_{80})=$ 0.2371 s. Figure 3 shows the graphs of the compared vectors in this section.

3.4. Procedure 4 Results

The errors between $C_{s{p}_1}$ and ${\check{C}}_{s{p}_1}$ for $s{p}_1$ values {60,65,75,85,90} are: $E_{60}(A_{60},{\check{A}}_{60})=$ 0.1165 s, $E_{65}(A_{65},{\check{A}}_{65})=$ 0.1117 s, $E_{75}(A_{75},{\check{A}}_{75})=$ 0.0620 s, $E_{85}(A_{85},{\check{A}}_{85})=$ 0.0616 s, and $E_{90}(A_{90},{\check{A}}_{90})=$ 0.1065 s. The errors between $C_{s{p}_1}$ and ${\widehat{C}}_{s{p}_1}$ for $s{p}_1$ values $\left\{70,80\right\}$ are: $E_{70}(A_{70},{\widehat{A}}_{70})=$ 0.1569 s and $E_{80}(A_{80},{\widehat{A}}_{80})=$ 0.1070 s. Figure 4 shows the graphs of the compared vectors in this section.

3.5. Procedure 5 Results

With the parameters’ values from Table 17, we modeled vectors ${\check{C}}_{s{p}_1}$, and with the parameters values from Table 19, we estimated vectors ${\widehat{C}}_{s{p}_1}$. The errors between the $C_{s{p}_1}$ and ${\check{C}}_{s{p}_1}$ cases are: $E_{60}(A_{60},{\check{A}}_{60})=$ 0.0753 s, $E_{65}(A_{65},{\check{A}}_{65})=$ 0.0637 s, $E_{75}(A_{75},{\check{A}}_{75})=$ 0.0362 s, $E_{85}(A_{85},{\check{A}}_{85})=$ 0.0376 s, and $E_{90}(A_{90},{\check{A}}_{90})=$ 0.1004 s. The errors between the $C_{s{p}_1}$ and ${\widehat{C}}_{s{p}_1}$ cases are: $E_{70}(A_{70},{\widehat{A}}_{70})=$ 0.1150 s and $E_{80}(A_{80},{\widehat{A}}_{80})=$ 0.1223 s. Figure 5 shows the graphs of the compared vectors in this section.

3.6. Procedure 6 Results

With parameters ${\tilde{b}}_1$ and ${\tilde{b}}_2$ (for a linear equation), we construct vectors ${\widehat{C}}_{s{p}_1}\left(1:4\right)$. With parameters ${\tilde{c}}_1$, ${\tilde{c}}_2$, and ${\tilde{c}}_3$ (for an order 2 polynomial equation), we construct vectors ${\widehat{C}}_{s{p}_1}\left(5:8\right)$. The errors between $C_{s{p}_1}$ and ${\check{C}}_{s{p}_1}$ for $s{p}_1$ cases {60, 65, 75, 85, 90} are: $E_{60}(A_{60},{\check{A}}_{60})=$ 0.0748 s, $E_{65}(A_{65},{\check{A}}_{65})=$ 0.0956 s, $E_{75}(A_{75},{\check{A}}_{75})=$ 0.0352 s, $E_{85}(A_{85},{\check{A}}_{85})=$ 0.0281 s, and $E_{90}(A_{90},{\check{A}}_{90})=$ 0.0534 s. The errors between $C_{s{p}_1}$ and ${\widehat{C}}_{s{p}_1}$ for $s{p}_1$ cases $\left\{70,80\right\}$ are: $E_{70}(A_{70},{\widehat{A}}_{70})=$ 0.1051 s and $E_{80}(A_{80},{\widehat{A}}_{80})=$ 0.1285 s. Figure 6 shows the graphs of the compared vectors in this section.

4. Discussion

4.1. Discussion for Explaining T

Table 26. Error between $A_{s{p}_1}$ and ${\check{A}}_{s{p}_1}$ (procedures 1, 2, and 3).

Procedure	${\mathit{E}}_{60}({\mathit{A}}_{60},{\check{\mathit{A}}}_{60})$	${\mathit{E}}_{65}({\mathit{A}}_{65},{\check{\mathit{A}}}_{65})$	${\mathit{E}}_{75}({\mathit{A}}_{75},{\check{\mathit{A}}}_{75})$	${\mathit{E}}_{85}({\mathit{A}}_{85},{\check{\mathit{A}}}_{85})$	${\mathit{E}}_{90}({\mathit{A}}_{90},{\check{\mathit{A}}}_{90})$	$\overline{\mathit{E}}$
Procedure 1	0.0222 s	0.0519	0.0550	0.0238	0.0128	0.0332
Procedure 2	0.4151	0.3944	0.2957	0.2390	0.2497	0.3188
Procedure 3	0.0226	0.0552	0.0617	0.0264	0.0116	0.0355

presents the error (obtained in procedure 1, 2, and 3) between a vector with data for modeling ($A_{s{p}_1}$) and the directly modeled vector (${\check{A}}_{s{p}_1}$). It also shows the mean of the errors $E_{s{p}_1}(A_{s{p}_1},{\check{A}}_{s{p}_1})$ for $s{p}_1$ cases {60,65,75,85,90}, i.e., $\overline{E}=mean\left(\left[E_{60} E_{65} E_{75} E_{85} E_{90 }\right]\right)$. The lower mean error is presented by procedure 1, followed closely by procedure 3.

Similarly,

Table 27. Error between $A_{s{p}_1}$ and ${\widehat{A}}_{s{p}_1}$ (procedures 1, 2, and 3).

Procedure #	${\mathit{E}}_{70}({\mathit{A}}_{70},{\widehat{\mathit{A}}}_{70})$	${\mathit{E}}_{80}({\mathit{A}}_{80},{\widehat{\mathit{A}}}_{80})$	$\overline{\mathit{E}}=\mathit{m}\mathit{e}\mathit{a}\mathit{n}\left(\left[{\mathit{E}}_{70} {\mathit{E}}_{80}\right]\right)$
Procedure 1	0.1817 s	0.2505	0.2162
Procedure 2	0.3582	0.4095	0.3839
Procedure 3	0.1892	0.2371	0.2132

presents the error between a vector with data for validating ($A_{s{p}_1}$) and the indirectly modeled vector (${\widehat{A}}_{s{p}_1}$). It also shows the mean of the errors $E_{s{p}_1}(A_{s{p}_1},{\widehat{A}}_{s{p}_1})$ for $s{p}_1$ cases {70,80}. The lower $\overline{E}$ is presented by procedure 3, followed closely by procedure 1. Thus, the techniques presented in procedures 3 and 1 are the most suitable for modeling and explaining T.

We notice that, for procedures 1, 2, and 3, $E_{80}>E_{70}$, meaning that ${\widehat{A}}_{70}$ is easier to explain than ${\widehat{A}}_{80}$. Moreover, for procedures 1 and 3, $E_{65}$ and $E_{75}$ are higher than $E_{60}$, $E_{85}$, and $E_{90}$. It follows that ${\check{A}}_{60}$, ${\check{A}}_{85}$, and ${\check{A}}_{90}$ are easier to model than ${\check{A}}_{65}$ and ${\check{A}}_{75}$. In procedures 1 and 3, $E_{90}$ presents the lowest error and $E_{75}$ presents the highest, thus ${\check{A}}_{90}$ is the most easily modeled vector and ${\check{A}}_{75}$ is the hardest to model.

In Figure 3, we observe that, for the $s{p}_1$ $\left\{70,80\right\}$ cases, the last two points of vectors ${\widehat{A}}_{s{p}_1}$ are clearly distant from the corresponding points of vectors $A_{s{p}_1}$ (in contrast with the other six points). The last two points are the T when the percentage of slow vehicles is 70% and 80%. Hereafter, ${\overline{E}}^{x,y}$ is the mean error of the x {1,2,…,5,6} procedure and the y {1,2} comparison, where y = 1 indicates a comparison between the directly modeled and observed data (Table 26 or

Table 28. Error between $C_{s{p}_1}$ and ${\check{C}}_{s{p}_1}$ (procedures 4, 5, and 6).

Procedure #	${\mathit{E}}_{60}({\mathit{C}}_{60},{\check{\mathit{C}}}_{60})$	${\mathit{E}}_{65}({\mathit{C}}_{65},{\check{\mathit{C}}}_{65})$	${\mathit{E}}_{75}({\mathit{C}}_{75},{\check{\mathit{C}}}_{75})$	${\mathit{E}}_{85}({\mathit{C}}_{85},{\check{\mathit{C}}}_{85})$	${\mathit{E}}_{90}({\mathit{C}}_{90},{\check{\mathit{C}}}_{90})$	$\overline{\mathit{E}}$
Procedure 4	0.1165 s	0.1117	0.0620	0.0616	0.1065	0.0917
Procedure 5	0.0753	0.0637	0.0362	0.0376	0.1004	0.0626
Procedure 6	0.0748	0.0956	0.0352	0.0281	0.0534	0.0574

), and y = 2 indicates a comparison between the not directly modeled and observed data (Table 27 or

Table 29. Error between $C_{s{p}_1}$ and ${\widehat{C}}_{s{p}_1}$ (procedures 4, 5, and 6).

Procedure #	${\mathit{E}}_{70}({\mathit{C}}_{70},{\widehat{\mathit{C}}}_{70})$	${\mathit{E}}_{80}({\mathit{C}}_{80},{\widehat{\mathit{C}}}_{80})$	$\overline{\mathit{E}}=\mathit{m}\mathit{e}\mathit{a}\mathit{n}\left(\left[{\mathit{E}}_{70} {\mathit{E}}_{80}\right]\right)$
Procedure 4	0.1569 s	0.1070	0.1320
Procedure 5	0.1150	0.1223	0.1187
Procedure 6	0.1051	0.1285	0.1168

). Then, ${\overline{E}}^{3,2}=mean\left(\left[E_{70}(A_{70},{\widehat{A}}_{70}\right) E_{80}(A_{80},{\widehat{A}}_{80})]\right)=0.2132 \mathrm{s}$ is the mean error of procedure 3 and $s{p}_1$ $\left\{70,80\right\}$ cases.

With procedure 1, the mean error between the directly modeled and observed data, i.e., ${\overline{E}}^{1,1}=0.0332 \mathrm{s}$, is lesser that the mean error between the not directly modeled and observed data, i.e., ${\overline{E}}^{1,2}=0.2162 \mathrm{s}$. This holds for the three procedures. Figure 1 shows that, for $A_{s{p}_1} \mathrm{vs}. {\widehat{A}}_{s{p}_1}$, the first six points of ${\widehat{A}}_{70}$ and ${\widehat{A}}_{80}$ are visible close to the corresponding points of $A_{70}$ and $A_{80}$, respectively. The last two points of ${\widehat{A}}_{70}$ are above those of $A_{70}$, and the last two points of ${\widehat{A}}_{80}$ are below those of $A_{80}$.

In procedure 2, $E_{s{p}_1}(A_{s{p}_1},{\check{A}}_{s{p}_1})$ varies between 0.2390 and 0.4151 s, and ${\overline{E}}^{2,1}=0.3188 \mathrm{s}$ is not distant from ${\overline{E}}^{2,2}=mean\left(\left[E_{70} E_{80}\right]\right)=0.3839$ s, so the error of the directly modeled data $E_{s{p}_1}(A_{s{p}_1},{\check{A}}_{s{p}_1})$ is similar to the error of the not directly modeled data $E_{s{p}_1}(A_{s{p}_1},{\widehat{A}}_{s{p}_1})$. Figure 2 shows that, for the directly modeled data (${\check{A}}_{s{p}_1}$), the first three points are above the observed data ($A_{s{p}_1}$), point four is close (from above or below), points five to seven are below, and point eight is above. Comparing the indirectly modeled with the observed data, we notice that points 1, 2, and 3 of ${\widehat{A}}_{s{p}_1}$ are slightly above the corresponding points of $A_{s{p}_1}$ (for $s{p}_1$ cases $\left\{70,80\right\}$). For ${\widehat{A}}_{70}$, the estimated points 4 and 7 are visibly accurate, the estimated points 5 and 6 are below the observed value, and the estimated point 8 is above the observed value. For ${\widehat{A}}_{80}$, the estimated points 4 and 5 are visibly accurate, and the estimated points 6, 7, and 8 are below the observed value.

With these results, we found procedures 1 and 3 to be suitable to model the data for modeling and to explain the data for validating.

4.2. Discussion for Explaining V

Table 28 presents the error between the directly modeled data and data for modeling, with $\overline{E}=mean\left(\left[E_{60} E_{65} E_{75} E_{85} E_{90 }\right]\right)$. Table 29 presents the error between the indirectly modeled data and data for validating, with $\overline{E}=mean\left(\left[E_{70} E_{80}\right]\right)$.

For modeling V, procedure 6 is the best option. Here, we have that ${\overline{E}}^{6,1}<{\overline{E}}^{5,1}<{\overline{E}}^{4,1}$ and that ${\overline{E}}^{6,2}<{\overline{E}}^{5,2}<{\overline{E}}^{4,2}$. Figure 6 shows that, in vectors ${\check{C}}_{s{p}_1}$ for the $s{p}_1$ {60,65,75,90} cases, points 6 and 7 are the least accurate (i.e., dissimilar with the observed), with ${\check{C}}_{85}$ being the most accurate ($E_{85}$ = 0.0281 s) and ${\check{C}}_{65}$ being the least accurate ($E_{65}$ = 0.0956 s). In vector ${\widehat{C}}_{80}$, points 7 and 8 are the least accurate; therefore, when $s{p}_1$ = 80 km/h and for $p_2$ = 70% and $p_2$ = 80%, it was difficult to explain V.

In Figure 5, and regarding the directly modeled data, points 5 and 6 of vectors ${\check{C}}_{s{p}_1}$ for the $s{p}_1$ $\left\{60,65,90\right\}$ cases are visibly inaccurate, and points 4 to 7 of vector ${\check{C}}_{90}$ are also rather imprecise. Additionally, ${\check{C}}_{90}$ has the highest error ($E_{90}=0.1004$ s) among ${\check{C}}_{s{p}_1}$ vectors of procedure 5, while ${\check{C}}_{75}$ along with ${\check{C}}_{85}$ present the lowest errors $E_{75}\cong E_{85}\cong 0.03 \mathrm{s}$. Regarding the indirectly modeled data, in vector ${\widehat{C}}_{70}$, points 5 to 7 are the least accurate, while in ${\widehat{C}}_{80}$, the least precise are points 5, 7, and 8. Hence, there are imprecisions when $p_2=50\%$ and $p_2=70\%$ for both of the $s{p}_1$ $\left\{70,80\right\}$ cases.

Procedure 4 presents the higher $\overline{E}$, as can be seen in Table 28 and Table 29. Table 28 shows that vectors ${\check{C}}_{s{p}_1}$ for the $s{p}_1$ cases {60,65,90} display $E_{s{p}_1}$> 0.1. Table 29 shows that vector ${\widehat{C}}_{80}$ is more accurate than ${\widehat{C}}_{70}$, because $E_{80}<E_{70}$ (0.1070 s < 0.1569 s).

With the results presented above, we consider procedures 5 and 6 as the best options for modeling and explaining the observed data. Besides, one important finding is that the pattern of V values, in the range $p_2=50\%$ to $p_2=80\%$, is not the same among $s{p}_1$ cases. It appears that a presence of about 50% of slow vehicles is the lowest threshold from where, and up to 80% in this study, the fast vehicles’ extra travel time variability follows a nonlinear pattern.

4.3. Contributions

The contributions of this study are as follows:

A better understanding of the traffic situations recreated through simulations, i.e., situations where vehicles traveling at different speeds interact with each other.

The developed procedures (and models within) for modeling and explaining the observed data from simulations. The best procedures in terms of a lower error were identified.

The patterns of T and V for each $s{p}_1$ case and $p_2$ range were identified. T exhibited an identifiable pattern for all $s{p}_1$ cases and $p_2$ ranges. For all $s{p}_1$ cases and the range $p_2=10\%$ to 40%, V shows a linear pattern. For $s{p}_1$ {60,65,75,80,90} cases and the range $p_2=50\%$ to 80%, V shows an unidentifiable pattern. For $s{p}_1$ {70} case and the range $p_2=50\%$ to 80%, V shows an inverted U-shaped pattern. For $s{p}_1$ $\left\{85\right\}$ case and the range $p_2=50\%$ to 80%, V shows a U-shaped pattern.

The findings of this work help to better understand heterogeneous traffic scenarios. The procedures can be used to describe the traffic of real locations and, with this information, authorities may decide which actions are required to alleviate traffic issues. Moreover, the traffic outcome of possible future scenarios can help to better plan public infrastructure that avoids congestions, low speeds, high noise levels, pollutions, and other problems that arise as a result of traffic problems.

4.4. Practical Application

A two-lane, one-direction avenue is a common infrastructure in street networks of heterogeneous traffic, i.e., where different types of vehicles circulate and where the vehicles’ desired speed may not be the same. Examples of this infrastructure are the avenues’ segments described in

Table 30. Avenues’ features.

Segment’ Name	Segment’ Location	Start GPS Coordinate	End GPS Coordinate	Segment’ Length
Estación Central (north)	Durango, México	24.036787, −104.666690	24.037343, −104.660777	$~600 \mathrm{m}$
Estación Central (south)	Durango, México	24.036681, −104.666694	24.037276, −104.660766	$~600 \mathrm{m}$

.

The two segments described in Table 30 are separated by a sidewalk and the posted speed limit for both is 60 km/h, despite that it is common that vehicles traveling at a lower speed than the speed limit slow down the traffic. The procedures introduced in the present study were developed with the intention of analyzing the traffic of segments with similar features to those presented in Table 30. After calibration, the models describe the average extra travel time and the extra travel time variability of the fast vehicles according to the percentage of slow vehicles circulating and the desired speeds. With this information, the corresponding authorities can adjust (or create) policies for restricting or penalizing the slow vehicles in order to keep a desired level of service, or to simply inform users of the current level of service.

5. Conclusions

Procedures 1, 2, and 3 model and explain the mean extra travel time (dependent variable) through the percentage of slow vehicles (independent variable). Procedures 1 and 3 accurately modeled the data for modeling, with procedure 2 being the least accurate. Moreover, procedures 1 and 3 are accurate for explaining the data for validating, both for $p_2=10\%$ to 60%, but less accurate for $p_2=70\%$ and 80%, concluding that, when the percentage of slow vehicles is 70% or more, it became difficult to explain the data for validating.

Procedures 4, 5, and 6 model and explain the extra travel time variability (dependent variable). In procedure 4, the independent variable is the mean extra travel time, and in procedures 5 and 6, it is the percentage of slow vehicles. Procedure 4 is the least accurate to model the data for modeling and to explain the data for validating, therefore $p_2$ is a better choice of independent variable than T. Procedures 5 and 6 modeled the data for modeling at approximately equal accuracy. Procedure 5 presents the higher error for ${\check{C}}_{90}$, being least precise in the range $p_2=40\%$ to 70%. Procedure 6 presents the higher error for ${\check{C}}_{65}$, being least precise at $p_2=60\%$ and 70%. Moreover, procedures 5 and 6 explain with approximately the same accuracy the data for validating. For procedures 5 and 6, we can say that, in ${\widehat{C}}_{70}$ and ${\widehat{C}}_{80}$, in the range $p_2=10\%$ to 40%, the V values (the corresponding observed values present a linear pattern) in general are better explained than in the range $p_2=50\%$ to 80%, although in the last range with $s{p}_1=70 \mathrm{km}/\mathrm{h}$, the observed V values likely present an inverted U-shaped form, and with $s{p}_1=85 \mathrm{km}/\mathrm{h}$, a U-shaped form.