# Analytical Solution of the Mixed Traffic Flow Cellular Automaton FI Model with the Next-Nearest-Neighbor Interaction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. NIFI Model

^{max}. The short vehicle and the long vehicle take up l

_{S}cells and l

_{L}(l

_{L}> l

_{S}) cells, respectively. At each evolutionary time step, all vehicle states are updated in parallel according to evolution rules (3) and (4).

#### 2.2. Analytical Solution of NIFI Model

_{i}is the length of the ith vehicle. When the vehicle density approaches the critical density ${\rho}_{\mathrm{c}}$, traffic flow changes from free flow to congestion. Under the condition of high vehicle density, the evolutionary step (3) of the NIFI model manifests that the motion of the ith vehicle must be accompanied by the motion of the (i + 1)th vehicle and the (i + 2)th vehicle in front considering the effect of the next-nearest-neighbor (i + 2)th vehicle in front, To put it another way, under the condition of high vehicle density, the ith vehicle can move forward at an average velocity of twice the average gap due to the movement of the next nearest neighbor. The average velocity V satisfies the following equation:

_{i}= 1). According to Equation (9), the following average velocity is acquired:

_{max}, there is the following equation:

_{max}in free flow into consideration, the relation of the average velocity with vehicle density is obtained:

_{c}, all vehicles can also travel at the maximum velocity ${V}_{\mathrm{max}}$. By substituting Equation (17) into Equation (18) and considering Equation (6), we can, thus, derive the following equation:

_{c}, of mixed traffic flow is obtained as follows:

_{c}, of mixed traffic flow also has a similar mathematical expression. When the length of a long vehicle is much higher than that of a short one, (l

_{L}> l

_{s}), Equation (20) can be simplified to the following equation:

_{S}= 1), $C=\rho ,{C}_{c}=\frac{2}{{V}_{\mathrm{max}}+2}$. The corresponding equation of the fundamental diagram is:

_{c}= 0, it corresponds to a one-dimensional (1D) single traffic flow composed of long vehicles with the maximum velocity, ${V}_{L}^{\mathrm{max}}$, $C=\rho {l}_{L},{C}_{c}=\frac{2{l}_{L}}{{V}_{L}^{\mathrm{max}}+2{l}_{L}}$. The corresponding equation of the fundamental diagram is:

_{L}= 1), Equation (27) is in agreement with Equation (26).

## 3. Simulation

^{4}cells under periodic boundary conditions. All vehicles were randomly distributed along the lattice chain with a random velocity at the beginning. The simulation was performed with 50 independent runs in different initial configurations. Each run for 3 × 10

^{4}iterations and the first 2 × 10

^{4}iterations were discarded in measuring the quantities of interest.

#### 3.1. Numerical Simulation for 1D Single Traffic Flow

#### 3.2. Numerical Simulation of the Mixed Traffic Flow

#### 3.2.1. Effect of Mixing Ratio on Mixed Traffic Flow

_{n}, the increase of short vehicles will lead to traffic jams easily. Moreover, when the occupancy is small, Figure 3 displays that the flow rate has different slopes due to the mixing ratio, C

_{n}. In the case of traffic congestion, the flow rate slope was the same and had nothing to do with the mixing ratio, C

_{n}. The curves in Figure 3 merged into a line whose slope is −2. These numerical simulations were in good agreement with Equation (25) obtained from the theoretical analysis.

#### 3.2.2. Effect of Vehicle Length on Mixed Traffic Flow

_{S}= 1). The maximum velocities of the long vehicle and short vehicle were, respectively selected as ${V}_{L}^{\mathrm{max}}$ = 10 and ${V}_{S}^{\mathrm{max}}$ = 5 and the mixing ratio C

_{n}= 0.5. Figure 5 and Figure 6 show the fundamental diagrams and average velocity for long vehicle length l

_{L}= 2, 3, 5, 8, and 10 cells, respectively. All figures show that the numerical simulation was in agreement with the analytical solution.

#### 3.2.3. Effect of the Maximum Velocity on Mixed Traffic Flow

_{L}= 2 cells and ${V}_{L}^{\mathrm{max}}$ = 10. Accordingly, the mix ratio was ${C}_{\mathrm{n}}$ = 0.5. In the same way, Figure 7 and Figure 8 show the fundamental diagram and average vehicle velocity vs. occupancy for different maximum velocities of the short vehicle, ${V}_{S}^{\mathrm{max}}$ = 1, 2, 5, 8, and 10.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Fundamental diagram for different mixing ratios (l

_{S}= 1 cell, l

_{L}= 2 cells, ${V}_{L}^{\mathrm{max}}$ = 10 and ${V}_{S}^{\mathrm{max}}$ = 5).

**Figure 4.**Average velocity vs. Occupancy, C, for different mixing ratios (l

_{S}= 1 cell, l

_{L}= 2 cells, ${V}_{L}^{\mathrm{max}}$ = 10 and ${V}_{S}^{\mathrm{max}}$ = 5).

**Figure 5.**Fundamental diagram for different lengths of vehicle (l

_{S}= 1 cell, C

_{n}= 0.5, ${V}_{L}^{\mathrm{max}}$ = 10 and ${V}_{S}^{\mathrm{max}}$ = 5).

**Figure 6.**Average velocity vs. Occupancy, C, for different lengths of vehicle (l

_{S}= 1 cell, C

_{n}= 0.5, ${V}_{L}^{\mathrm{max}}$ = 10 and ${V}_{S}^{\mathrm{max}}$ = 5).

**Figure 7.**Fundamental diagram for different maximum velocities (l

_{S}= 1 cell, l

_{L}= 2 cells, C

_{n}= 0.5 and ${V}_{L}^{\mathrm{max}}$ = 10).

**Figure 8.**Average velocity vs. Occupancy for different maximum velocities (l

_{S}= 1 cell, l

_{L}= 2 cells, C

_{n}= 0.5 and ${V}_{L}^{\mathrm{max}}$ = 10).

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**MDPI and ACS Style**

Zhang, Y.; Xue, Y.; Qiao, Y.; Cen, B.
Analytical Solution of the Mixed Traffic Flow Cellular Automaton FI Model with the Next-Nearest-Neighbor Interaction. *Sustainability* **2022**, *14*, 7127.
https://doi.org/10.3390/su14127127

**AMA Style**

Zhang Y, Xue Y, Qiao Y, Cen B.
Analytical Solution of the Mixed Traffic Flow Cellular Automaton FI Model with the Next-Nearest-Neighbor Interaction. *Sustainability*. 2022; 14(12):7127.
https://doi.org/10.3390/su14127127

**Chicago/Turabian Style**

Zhang, Yanxin, Yu Xue, Yanfeng Qiao, and Bingling Cen.
2022. "Analytical Solution of the Mixed Traffic Flow Cellular Automaton FI Model with the Next-Nearest-Neighbor Interaction" *Sustainability* 14, no. 12: 7127.
https://doi.org/10.3390/su14127127