# Research on Self-Balancing System of Autonomous Vehicles Based on Queuing Theory

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## Abstract

**:**

## 1. Introduction

- (1)
- Define a traffic system with self-balancing of autonomous vehicles;
- (2)
- Based on the queuing theory, a road network model with self-balancing characteristics is established;
- (3)
- The balance strategy reduces the waiting time of the system;
- (4)
- While increasing service efficiency, the service intensity of the system is reduced;
- (5)
- Improve the operation efficiency of the transportation system.

## 2. Modeling Basis and Research Methods

#### 2.1. Queuing Network of Autonomous Vehicles

- (1)
- The service rate of node i is related to its queue length. When there are n
_{i}customers in the queue of node i, the service rate is ${u}_{i}\left(n\right)$, and the service time of each node is independent and follows a negative exponential distribution; - (2)
- Node i receives Poisson flow with a rate of ${\lambda}_{i}$;
- (3)
- Passengers transfer to node j with probability ${p}_{ij}$ after completion of service at node i, and the transfer probability has Markov characteristics;
- (4)
- The network is closed, with a fixed number of passengers K.

**R**: $\mathit{R}=({p}_{ij}|i,j=1,\cdots ,M)$; then, it can be known from the linear algebra homogeneous linear equation definition that Equation (2) satisfies: $\mathit{\lambda}(\mathit{I}-\mathit{R})=0$;

**I**is the identity matrix, $\mathit{\lambda}=\left\{{\lambda}_{i}\right\}$. As $\left|\mathit{I}-\mathit{R}\right|=0$, $\mathit{\lambda}=\left\{{\lambda}_{i}\right\}$ has an infinite number of solutions. The different solutions differ by a multiplier factor. Let $({e}_{1},{e}_{2},\cdots ,{e}_{M})$ be any set of non-zero solutions; then, ${e}_{i}$ is proportional to ${\lambda}_{i}$: that is, ${e}_{i}=\epsilon {\lambda}_{i}$ and $\epsilon $ is constant. In general, the element of $({e}_{1},{e}_{2},\cdots ,{e}_{M})$ could be fixed as a convenient value, such as ${e}_{1}=1$; that is, once there is access to node 1, the average access to node i is ${e}_{i}$ times, so ${e}_{i}$ is also called the relative access rate of node i. We can also form a probability distribution $\sum _{i=1}^{M}{e}_{i}}=1$ by normalizing $({e}_{1},{e}_{2},\cdots ,{e}_{M})$; thus, a network with only one customer could be considered to walk randomly in a node set. The probability of ${e}_{i}$ customer at node i is MC in state i. If $({e}_{1},{e}_{2},\cdots ,{e}_{M})$ satisfies Equation (2), we do not need to know the specific value. The definition of ${\sigma}_{i}={e}_{i}/{u}_{i}$, which is proportional to the service intensity ${\rho}_{i}={\lambda}_{i}/{u}_{i}$ of node i.

#### 2.2. Probability Distribution in Steady State

#### 2.3. State Parameter Determination

- (1)
- Queue length of node i:$$\begin{array}{ll}P({n}_{i}\ge k)& =\frac{1}{G(M,K)}{\displaystyle \sum _{\begin{array}{l}n\in \mathsf{\Phi}(M,K)\\ {n}_{i}\ge k\end{array}}{\displaystyle \prod _{j=1}^{M}{\sigma}_{j}^{{n}_{i}}}}\\ & =\frac{{\sigma}_{i}^{k}}{G(M,K)}{\displaystyle \sum _{\begin{array}{l}{m}_{j}={n}_{j}(i\ne j)\\ {m}_{i}={n}_{i}-k\\ n\in \mathsf{\Phi}(M,K)\\ {n}_{i}\ge k\end{array}}{\displaystyle \prod _{j=1}^{M}{\sigma}_{j}^{{m}_{j}}}.}\end{array}$$

- (2)
- The probability of node i being idle is:$$\begin{array}{l}{Q}_{i}(k=0)=P({n}_{i}\ge k)-P({n}_{i}\ge k+1)\\ =\left(\frac{G(M,K)-{\delta}_{i}(M,K-1)}{G(M,K)}\right)\end{array}.$$

- (3)
- The average queue length of node i:

## 3. Self-Balancing System for Fully Autonomous Vehicles

#### 3.1. Self-Balancing System Framework

#### 3.2. Queue Network Model of Self-Balancing System

#### 3.3. Establishment of Self-Balancing Optimization Model

_{i}is a function of H, which is the expected queue length in a steady state. Muntz and Wong confirmed the asymptotic behavior of Equation (21).

#### 3.4. Self-Balancing System Performance Index Calculation

_{wi}(0) = q

_{i}(0) = 0, the values of T

_{wi}(1) and q

_{i}(1), 1 ≤ i ≤ S can be obtained; then, the value when k = 2 can be obtained, and so on, until the iteration reaches the expected value. In each iteration, 2S + 1 is calculated. In an urban transportation system serving customers, the average waiting time and queue length could measure the operation efficiency of an urban travel service system. For customers, the shorter the waiting time, the better and the shorter the queue length, and the better for a transportation system. For an operating system, the shorter the average waiting time, the better, and the smaller the operation intensity of the system nodes, the smaller the operation burden of the system would be. Therefore, this paper selects these commonly used measurement indicators in queuing theory and introduces the MVA method to simplify the calculation; then, it iteratively calculates the indicators by setting the initial value.

## 4. Experimental Analysis

**θ**is

#### 4.1. Comparative Analysis before and after Self-Balancing

_{3}= 0.85, then e

_{1}= 0.8 and e

_{2}= 0.58.

**θ**and on the basis of the flow equation, the system equation can be iterated through the MVA method to obtain the average waiting time, the queue length, the actual throughput, and the service intensity at the node in the road network, as shown in Figure 3, Figure 4, Figure 5 and Figure 6 below.

#### 4.2. Comparative Analysis before and after System Self-Balancing

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Li, H.; Wang, J.; Bai, G.; Hu, X.
Research on Self-Balancing System of Autonomous Vehicles Based on Queuing Theory. *Sensors* **2021**, *21*, 4619.
https://doi.org/10.3390/s21134619

**AMA Style**

Li H, Wang J, Bai G, Hu X.
Research on Self-Balancing System of Autonomous Vehicles Based on Queuing Theory. *Sensors*. 2021; 21(13):4619.
https://doi.org/10.3390/s21134619

**Chicago/Turabian Style**

Li, Huanping, Jian Wang, Guopeng Bai, and Xiaowei Hu.
2021. "Research on Self-Balancing System of Autonomous Vehicles Based on Queuing Theory" *Sensors* 21, no. 13: 4619.
https://doi.org/10.3390/s21134619