Signal Timing Optimization Method for Intersections Under Mixed Traffic Conditions

Abstract

The increasing proliferation of new energy vehicles and autonomous vehicles has led to the formation of mixed traffic flows characterized by diverse driving behaviors, posing new challenges for intersection signal control. To address this issue, this study proposes a multi-class customer feedback queuing network (MCFFQN) model that incorporates state-dependent road capacity and congestion propagation mechanisms to accurately capture the stochastic and dynamic nature of mixed traffic flows. An evaluation framework for intersection performance is established based on key indicators such as vehicle delay, the energy consumption of new energy vehicles, and the fuel consumption and emissions of conventional vehicles. A recursive solution algorithm is developed and validated through simulations under various traffic demand scenarios. Building on this model, a signal timing optimization model aimed at minimizing total costs—including delay and environmental impacts—is formulated and solved using the Mesh Adaptive Direct Search (MADS) algorithm. A case study demonstrates that the optimized signal timing scheme significantly enhances intersection performance, reducing vehicle delay, energy consumption, fuel consumption, and emissions by over 20%. The proposed methodology provides a theoretical foundation for sustainable traffic management under mixed traffic conditions.

1. Introduction

According to relevant studies [1], the penetration rate of new energy vehicles in China is projected to exceed 50% by 2030. Meanwhile, with advancements in vehicular intelligence technologies, the proportion of Level 3 and Level 4 autonomous vehicles is also expected to rise accordingly. Studies [2] suggest that by 2050, Level 4 autonomous vehicles could account for approximately 65% of the vehicle fleet in the United States. Different vehicle types—such as conventional fuel-powered vehicles and autonomous vehicles—exhibit distinct traffic flow characteristics due to variations in driving behavior and automation capabilities [3,4,5]. Prior to the full deployment of intelligent connected and autonomous driving technologies, roads will continue to operate under mixed traffic conditions. In this context, traffic signal control remains a critical measure for ensuring road safety [6,7,8]. Therefore, investigating the properties of mixed traffic flow is essential for developing effective traffic management strategies.

Extensive research has been conducted on mixed traffic flow, primarily focusing on fundamental diagram models and stability analysis. The fundamental diagram model (FDM) describes the relationships among traffic flow volume, density, and speed, serving as a cornerstone of traffic flow theory. To address the limitations of deterministic FDMs in capturing traffic uncertainty, Lei et al. [9] introduced a non-parametric Gaussian process model to establish a stochastic fundamental diagram. Han et al. [10] proposed a macroscopic fundamental diagram with a volume–delay relationship (MFD-VD) using data from license-plate cameras and road congestion indices. Maiti et al. [11] derived the shape and properties of FDMs based on area-occupancy-normalized flow and speed models. Cheng et al. [12] examined scattering effects in fundamental diagrams and developed two stochastic models using lognormal and skew-normal distributions. Ahmed et al. [13] constructed fundamental diagrams for heterogeneous traffic using UAV-captured data from Karachi, Pakistan.

Stability analysis is another key research focus. Yao et al. [14] evaluated the linear stability of mixed traffic considering CAV degradation and reaction time diversity. Zeng et al. [15] developed a cellular automata model for heterogeneous traffic incorporating CAVs and human-driven vehicles (HDVs), integrating the Gipps safety model and various car-following modes to simulate platoon effects under different penetration rates. Cui et al. [16] enhanced the Intelligent Driver Model (IDM) by incorporating position, velocity, and acceleration data from multiple preceding vehicles, with stability validated via linear analysis. Montanino and Punzo [17,18] emphasized the importance of driver and vehicle heterogeneity in string stability modeling. Cheng et al. [19] proposed an Intelligent Driver Model (IDM) variant accounting for cyberattacks and dynamic communication topology. Li et al. [20] introduced a unified potential field-based car-following model and established stability conditions using characteristic equations. Ren et al. [21] experimentally evaluated the impact of mixed platoons on stability, efficiency, and ecology under varying penetration rates.

Despite these advancements, several research gaps remain. Most existing studies either focus on microscopic car-following behaviors or macroscopic flow relationships, while mesoscopic modeling approaches that can efficiently capture the stochastic, dynamic, and heterogeneous nature of mixed traffic are still underdeveloped. In particular, there is a lack of queueing-theoretic frameworks that integrally incorporate multi-class vehicle interactions, state-dependent road capacity, and congestion propagation under signal control. Such a framework is essential for evaluating intersection performance and optimizing signal timing in a computationally tractable manner. To bridge this gap, this study proposes a novel Multi-Class Customer Feedback Queueing Network (MCFFQN) model. Unlike conventional queueing models that assume homogeneous customers and fixed service rates, the MCFFQN model explicitly accounts for: (1) multiple vehicle classes with distinct driving behaviors (e.g., IDM for HDVs, CACC for CAVs), (2) state-dependent service rates that reflect congestion-sensitive travel speeds, and (3) a feedback mechanism that captures queue spillback and network-wide congestion propagation. By integrating these features, the MCFFQN model provides a unified mesoscopic representation of mixed traffic flow, enabling accurate performance evaluation and signal optimization under realistic, time-varying demand.

Building on this model, the present work develops a signal-timing optimization framework that minimizes total costs—including vehicle delay, energy consumption of NEVs, and fuel consumption and emissions of conventional vehicles—while satisfying practical signal constraints. The proposed approach is validated through simulation under various traffic demand scenarios, demonstrating its effectiveness in improving intersection performance under mixed traffic conditions.

The remainder of this paper is structured as follows: First, a node feedback queuing model is established, accounting for dynamic traffic demand, state-dependent capacity, and congestion propagation. Next, an area-wide road queueing network model is constructed, incorporating network topology and signal control. Key performance indicators—including average vehicle delay, fuel consumption, pollutant emissions, and the energy consumption of new energy vehicles—are analyzed. The accuracy of the proposed models is validated via simulation. Finally, a signal optimization model under mixed traffic conditions is developed to demonstrate the method’s effectiveness.

2. Intersection Queuing Network Model

2.1. Model Description

Queuing theory, as a mesoscopic model, offers advantages such as high computational efficiency and strong analytical tractability, and is widely applied in the field of transportation [22,23,24,25,26,27,28,29,30].

The process of vehicles passing through a road section can be described as a process of queuing to receive road section services; that is, considering vehicles as customers and road sections as service counters, the relationship between vehicles and road sections can be described as a queuing system. Each position on the road section that can accommodate vehicles is considered a service counter, and the number of service counters in the road section queuing system is the road capacity.

$$C=k_{jam}lw,$$

(1)

where $C$ is the capacity of the road system, which is also the number of service counters in the road queue system; $k_{jam}$ is the blockage density of the road section [22,23,24], usually taken as 115~165 vehicles/(km·lane); $l$ is the length of the road section (km); and $w$ is the width of the road section (lanes).

Based on our previous research [22,23], the time that it takes vehicles to pass through the road section is the service time, and its service rate $r$ can be represented as:

$$r={\displaystyle \frac{v}{l}},$$

(2)

where $v$ is the average speed of vehicles. When the number of vehicles is $x$, the road section service rate $u$ can be expressed as:

$$u=x\times r={\displaystyle \frac{x\times v}{l}}.$$

(3)

The road network can be formed by connecting multiple road sections in a queue system based on their topological structure. Therefore, the approach in this section is as follows: (1) Construct a road section node queuing system model. (2) Considering that when the traffic is congested, vehicles must wait for downstream road sections to become available before entering the system, after which we construct a road section node feedback queuing model. (3) Considering future traffic conditions with multiple types of vehicles driving on the roads, construct a multi-class customer queuing network model. (4) Considering the impact of traffic signals on traffic flow, construct a regional road queuing network model. (5) Study the key performance indicators focused on in regional road control.

2.2. Lane Queuing System

2.2.1. Road Section Node Queueing Model

The parameters of the queuing system considered include road section capacity, arrival rate, and service rate.

Road Capacity

The road capacity can be calculated by Equation (1).

Arrival rate

The arrival rate at a road section is time-varying, can be described as ${\lambda }_i\left(t\right)$. The arrival rate can be obtained through surveys.

Road service rate

The number of vehicles in the road section is defined as the system state, after which the average vehicle speed on the section is related to the system state, that is, it is state-dependent.

This study adopts an exponential speed model [31] to describe the average speed,

$$v_{x_i}=v_{i0}\exp \left(-{\left({\displaystyle \frac{x_i}{{\beta }_i}}\right)}^{{\gamma }_i}\right),$$

(4)

where $v_{i0}$ is the free flow speed, which is the speed limit of the road; the parameter $x_i$ is calibrated for the system state (i.e., the number of vehicles); and ${\gamma }_i$, ${\beta }_i$ are system calibration parameters of exponential speed model [22,23,31],

$${\gamma }_i=\mathrm{l}\mathrm{n}({\displaystyle \frac{\mathrm{l}\mathrm{n}\left(v_{ia}/v_{i0}\right)}{\mathrm{l}\mathrm{n}\left(v_{ia}/v_{ib}\right)}})/\mathrm{l}\mathrm{n}({\displaystyle \frac{a}{b}}),$$

$${\beta }_i={\displaystyle \frac{a}{{(\mathrm{l}\mathrm{n}(\frac{v_{i0}}{v_{ia}}))}^{1/{\gamma }_i}}}={\displaystyle \frac{b}{{(\mathrm{l}\mathrm{n}(\frac{v_{i0}}{v_{ib}}))}^{1/{\gamma }_i}}}.$$

Among these, $a$,$b$ are the feature points, which represent the system state as $a$,$b$; $v_{ia}$, $v_{ib}$ are the average speed of vehicles when the system state is $a$,$b$. The relevant literature provides the related parameters of the feature points [23,24].

When the speed of vehicles is considered state-dependent, the service rate of the road section also exhibits state-dependence, which is expressed as $r_i\left(x_i\left(t\right)\right)$ in this article.

2.2.2. Road Section Feedback Queueing Model

When the road system is relatively complex, directly applying queuing theory for modeling may lead to highly intricate analytical results. Therefore, this paper further adopts the fluid queuing theory [32], which offers greater computational simplicity.

If $f_{\mathrm{i}\mathrm{n}}^i\left(t\right)$, and $f_{\mathrm{o}\mathrm{u}\mathrm{t}}^i\left(t\right)$ represent the inflow rate and outflow rate of vehicles, respectively, and $x_i\left(0\right)$ represents the system state at time 0, then according to the law of conservation of flow, the system state at time $t$ can be described as:

$$x_i\left(t\right)=x_i\left(0\right)+\int f_{\mathrm{i}\mathrm{n}}^i\left(t\right)dt-\int f_{\mathrm{o}\mathrm{u}\mathrm{t}}^i\left(t\right)dt,$$

(5)

Taking the derivative of Equation (5) with respect to time at both ends, we have

$$x_i^{\prime }\left(t\right)=f_{\mathrm{i}\mathrm{n}}^i\left(t\right)-f_{\mathrm{o}\mathrm{u}\mathrm{t}}^i\left(t\right),$$

(6)

This is the fluid queuing equation, where $x_i^{\prime }\left(t\right)$ represents the rate of change in the system state. $f_{\mathrm{i}\mathrm{n}}^i\left(t\right)$ is not only related to the arrival rate of vehicles ${\lambda }_i\left(t\right)$ but also to the congestion probability of the road section ${PB}_i^L\left(x_i\left(t\right)\right)$.

$$f_{\mathrm{i}\mathrm{n}}^i\left(t\right)={\lambda }_i\left(t\right)\times \left(1-{PB}_i^L\left(x_i\left(t\right)\right)\right),$$

(7)

In Equation (7), the congestion probability is defined as in [23].

In Equation (6), $f_{\mathrm{o}\mathrm{u}\mathrm{t}}^i\left(t\right)$ is not only related to the overall service capacity of the road section $u_i^L\left(t\right)$, but also to the idle probability of the road section ${PE}_i^L\left(x_i\left(t\right)\right)$:

$$f_{\mathrm{o}\mathrm{u}\mathrm{t}}^i\left(t\right)=u_i^L\left(x_i\left(t\right)\right)\times \left(1-{PE}_i^L\left(x_i\left(t\right)\right)\right),$$

(8)

where $u_i^L\left(x_i\left(t\right)\right)$ represents the service rate, which can be calculated via Equation (3).

Based on the above analysis, the queuing system for road sections can be described as:

$${x_i^{L^{\prime }}}^{\left(t\right)}={{\lambda }_i\left(t\right)\times \left(1-{PB}_i^L\left(x_i\left(t\right)\right)\right)-u}_i^L\left(x_i\left(t\right)\right)\times \left(1-{PE}_i^L\left(x_i^L\left(t\right)\right)\right),$$

(9)

denoting the traffic capacity of the road section, where $u_i^L\left(x_i\left(t\right)\right)\times \left(1-{PE}_i^L\left(x_i^L\left(t\right)\right)\right)$ represents the output rate of the road section (i.e., traffic capacity). In Equations (7) and (8), the calculations of congestion probability ${PB}_i^L\left(x_i\left(t\right)\right)$ and idle probability ${PE}_i^L\left(x_i\left(t\right)\right)$ are described in detail in references [23,32].

It should be noted that, in the above queuing system, when an upstream vehicle reaches the section and the section becomes congested, it will leave the system directly, which is the loss of the queuing system. The superscript “$L$” in Equations (7)–(9) represents the loss of the queuing system. To accurately depict the waiting behavior of vehicles in the road network, it is necessary to further improve the above queuing system as a feedback queuing system. According to the research in [23], the feedback queuing system can be described as:

$${x_i^{F^{\prime }}}^{\left(t\right)}={\lambda }_i\left(t\right)\times (1-{PB}_i^L(x_i^F(t)\left)\right)-u_i^F\left(x_i^F\left(t\right)\right)\times \left(1-{PE}_i^L\left(x_i^F\left(t\right)\right)\right)+{\sigma }_i^{\prime }\left(t\right)\rho \left(t\right),$$

(10)

where the superscript “F” indicates the feedback queue, and ${\sigma }_i^{\prime }\left(t\right)\rho \left(t\right)$ represents the number of vehicles that need to queue when the road section is congested. The idea is to virtually create an artificial virtual space with infinite capacity in the road section $i$ to store the vehicles that need to queue. Furthermore, ${\sigma }_i^{\prime }\left(t\right)$ is the service rate after feedback elimination for the artificial virtual space, and $\rho \left(t\right)$ is the utilization rate of the artificial virtual node. According to existing research,

$${\sigma }_i^{\prime }\left(t\right)=\left(1-{PB}_i^F\left(x_i^F\left(t\right)\right)\right){\sigma }_i,$$

(11)

where ${PB}_i^F\left(x_i^F\left(t\right)\right)$ is the blocking probability of the system when considering artificial virtual space, and ${\sigma }_i$ is the service rate for artificial virtual space.

$${\sigma }_i={\displaystyle \frac{2{\mu }_i\left(C_i\right)}{1+{cs}_i^2}},$$

(12)

where ${\mu }_i\left(C_i\right)$ represents the service rate when the system is blocked. When the service time of the section follows an exponential distribution, ${cs}_i^2=1$; after which it becomes ${\sigma }_i={\mu }_i\left(C_i\right)$.

The utilization rate of artificial virtual space $\rho \left(t\right)$ is expressed as

$$\rho \left(t\right)={\displaystyle \frac{y_i\left(t\right)+1-\sqrt{y_i^2\left(t\right)+{2cs}_i^2{y}_i\left(t\right)+1}}{1-{cs}_i^2}},$$

(13)

where $y_i\left(t\right)$ is the system state of the artificial virtual space; namely, the overflow of vehicles, which can be described by the following difference equation:

$${y\prime }_i\left(t\right)=-{\sigma }_i^{\prime }\left(t\right)\rho \left(t\right)+{\lambda }_i^L\left(t\right){PB}_i^L\left(x_i^L\left(t\right)\right),$$

(14)

The output rate of the feedback queue system ${\theta }_i^F\left(t\right)$ can be described as

$${\theta }_i^F\left(t\right)=u_i^F\left(t\right)(1-{PE}_i^F\left(x_i^F\left(t\right)\right)),$$

(15)

and the road service rate considering feedback $u_i^F\left(t\right)$ is represented as

$$u_i^F\left(t\right)={\left[{\left(u_i\left(x_i^F\left(t\right)\right)\right)}^{-1}+{PB}_j^L\left(x_j^L(t)\right){{\sigma }_j^{\prime }\left(t\right)}^{-1}\right]}^{-1}.$$

(16)

The detailed calculation and description of the parameters in the above formulas can be found in References [23,32].

2.3. Regional Road Queue Network Model

The regional queueing network is essentially composed of multiple road sections combined according to a certain topological structure. Vehicles are transferred between different sections; meanwhile, traffic signals also have a significant impact on the transfer of vehicles between sections. This study aims to describe a regional queueing network model considering road topology and traffic signals, taking signalized intersections as an example.

The intersection is composed of road sections with different functions, generally classified into inlet road sections (represented by a set ${\mathcal{N}}_1$), outlet road sections (represented by a set ${\mathcal{N}}_2$), and turning road sections (represented by a set ${\mathcal{N}}_3$); in addition, road sections directly connected to the external system are represented by a set ${\mathcal{N}}_4$. If the set $\mathcal{N}$ represents the road sections composing the intersection, then, using ${pc}_{ij}\left(t\right)$ to describe the inherent transfer requirements between paths (i.e., turning proportion of vehicles at road sections), and using a 0–1 variable $a_{ij}\left(t\right)$ to indicate whether nodes $i$ and $j$ are connected (i.e., whether the current phase is green), the actual transition probability from node to node can be represented as

$${{pr}_{ij}\left(t\right)={pc}_{ij}\left(t\right)a}_{ij}\left(t\right),\forall i,j\in \mathcal{N},$$

(17)

For the actual arrival rates of each node in the queue network at intersections, the following rules apply.

First, for the nodes reaching the system from outside (i.e., entrance sections), their arrival rates can be obtained through methods such as video monitoring, and can be described as

$${\lambda }_i\left(t\right)={\lambda }_{0i}\left(t\right),i\in {\mathcal{N}}_4,$$

(18)

While, for other sections of the road except for the import section,

$${\lambda }_i\left(t\right)={\sum }_{j\in \mathcal{N}}{\theta }_j^F\left(t\right){pr}_{ji}\left(t\right).$$

(19)

3. Intersection Queuing Model with Multiple Classes

3.1. Multi-Class Customer Queueing Model for Road Section

Different types of vehicles have different characteristics. For example, fuel-powered vehicles have relatively low levels of intelligence, while new energy vehicles have a higher level of intelligence and the potential for coordinated operation. At the same time, new energy vehicles also have different characteristics from fuel-powered vehicles. Assuming that vehicles of different types arriving at a section all follow a Poisson process with parameters ${\lambda }_{\mathrm{T}i}\left(t\right)$, the arrival rate ${\lambda }_i\left(t\right)$ of the road section $i$ can be represented as

$${\lambda }_i\left(t\right)=\sum {\lambda }_{\mathrm{T}i}\left(t\right),$$

(20)

Among them, $\mathrm{T}$ represents the number of vehicles of different types. ${\lambda }_{\mathrm{T}i}\left(t\right)$ is the arrival rate of type $\mathrm{T}$ vehicles. And ${\lambda }_i\left(t\right)$ represents the overall arrival rate of all vehicle types. At this time, the number of vehicles in the road section $x_i\left(t\right)$ can be represented as the sum of the number of vehicles of different types $x_{\mathrm{T}i}\left(t\right)$; that is,

$$x_i^F\left(t\right)=\sum x_{\mathrm{T}i}^F\left(t\right),$$

(21)

Additionally, according to the principle of traffic conservation, the rate of change in the number of vehicles of type $\mathrm{T}$ on the road section should be equal to the sum of the rates of change in different types of vehicles.

$$x_i^{F^{\prime }}\left(t\right)=\sum {x_{\mathrm{T}i}^F}^{\prime }\left(t\right),$$

(22)

where ${x_i^F}^{\prime }\left(t\right)$ denotes the system state change rate of the feedback network system for road section $i$ at time $t$. And ${x_{\mathrm{T}i}^F}^{\prime }\left(t\right)$ represents the system state change rate for vehicle type T in the feedback network of road section $i$ at time $t$.

According to Equation (6), the system state change rate for vehicle type $\mathrm{T}$ on the road section is

$${{x_{\mathrm{T}i}^F}^{\prime }}^{\left(t\right)}=f_{\mathrm{i}\mathrm{n}}^{Ti}\left(t\right)-f_{\mathrm{o}\mathrm{u}\mathrm{t}}^{Ti}\left(t\right),$$

(23)

where the vehicle flow-in rate and flow-out rate of the road section are represented by $f_{\mathrm{i}\mathrm{n}}^{Ti}\left(t\right)$ and $f_{\mathrm{o}\mathrm{u}\mathrm{t}}^{Ti}\left(t\right)$, respectively.

In Equation (23), $f_{\mathrm{i}\mathrm{n}}^{Ti}\left(t\right)$ not only depends on the arrival rate of vehicles but also on the congestion probability of the road section, ${PB}_i^L\left(x_i^F\left(t\right)\right)$:

$$f_{\mathrm{i}\mathrm{n}}^{Ti}\left(t\right)={\lambda }_{\mathrm{T}i}\left(t\right)\times \left(1-{PB}_i^L\left(x_i^F\left(t\right)\right)\right),$$

(24)

The relationship with the road section output rate $f_{\mathrm{o}\mathrm{u}\mathrm{t}}^{Ti}\left(t\right)$ is included in Equation (23).

$$f_{\mathrm{o}\mathrm{u}\mathrm{t}}^{Ti}\left(t\right)=u_i^F\left(x_i^F\left(t\right)\right)\times \left(1-{PE}_i^L\left(x_i^F\left(t\right)\right)\right)\times {\displaystyle \frac{x_{\mathrm{T}i}^F\left(t\right)}{x_i^F\left(t\right)}}.$$

(25)

In Equation (23), ${\displaystyle \frac{x_{\mathrm{T}i}^F\left(t\right)}{x_i^F\left(t\right)}}$ is the penetration rate of type $\mathrm{T}$ vehicles.

The MCFFQN framework developed above is inherently generic and compositional. It is designed to accommodate any number of vehicle classes, where a “class” can be defined by any combination of distinctive features such as propulsion type, automation level, or even routing behavior. The key model components—the class-specific arrival rate ${\lambda }_{Ti}\left(t\right)$, the state $x_{Ti}^F\left(t\right)$, and the proportion $p_T$—provide the necessary hooks to integrate diverse vehicle types. The aggregate traffic dynamics and performance metrics are then derived from the interaction of these classes through the shared road network and state-dependent service rates. Therefore, the model’s core theoretical value lies in its ability to describe the operating mechanisms of mixed traffic flow at an abstract level, regardless of the specific definitions of the vehicle classes.

3.2. Service Rate of Multi-Class Customer Queueing System

In the above multi-class customer queuing system, the average speed $v_i\left(x_i\right(t\left)\right)$ of vehicles is a key determinant of their service rate. However, when different types of vehicles are mixed in traffic, the average speed differs from that of a single vehicle type due to mutual interactions between vehicles. The average speed of vehicles in a mixed state can be obtained through a basic diagram model of mixed traffic flow. In traffic flow studies, different types of vehicles driving behaviors can be simulated through different car-following models. For example, the IDM can be used to describe the driving behavior of regular fuel-powered vehicles [33,34], and the Cooperative Adaptive Cruise Control (CACC) model can be used to describe the driving model of vehicles [35] with a higher level of intelligence. Due to the different performances of different types of vehicles, different car-following models can be selected for description.

Different types of vehicles choose corresponding car-following behavior description models. Assuming that the speed function of different types of vehicles is $f_v^T\left(v\right)$ and the stable headway function of different types of vehicles is $f_h^T\left(v\right)$, as stated in the previous text, the proportion of different types of vehicles can be represented as

$$p_{\mathrm{T}}={\displaystyle \frac{x_{\mathrm{T}i}^F\left(t\right)}{x_i^F\left(t\right)}},$$

(26)

The average headway of the mixed traffic flow when stable is

$$f_h^{*}\left(v\right)=\sum p_{\mathrm{T}}\times f_h^{\mathrm{T}}\left(v\right),$$

(27)

Based on the headway, the traffic flow density on the road in a stable state can be determined.

$$f_k^{*}\left(v\right)={\displaystyle \frac{1000}{f_h^{*}\left(v\right)}},$$

(28)

Following this, the traffic flow is

$$f_q^{*}\left(v\right)=f_k^{*}\left(v\right)\times v,$$

(29)

The speed model of mixed traffic flow can be obtained based on Equation (41)

$$f_v^{*}\left(v\right)=f^{-1}\left(f_q^{*}\left(v\right)\right).$$

(30)

Based on Equation (30), the average speed of the multi-class customer queueing system can be determined.

It should be pointed out that, due to their different driving characteristics, IDM and CACC models are selected to describe different types of vehicles. However, as the proposed model is a mesoscopic model, it is inevitable that the influence of different drivers’ driving habits on driving behavior will be overlooked. This is an inherent flaw of the mesoscopic model; however, when there are a large number of vehicles, such a model can better describe the average state of system vehicles.

Mixed traffic flow is expected to be a common state of road sections for a long time to come. Under mixed traffic flow conditions, traffic signal control remains an important means of ensuring traffic safety. This study evaluates the operational performance of intersections under mixed traffic flow conditions, and the results can be used in the design of intersection signal control schemes to ensure traffic safety.

4. Performance Indicators and Solution Algorithms

4.1. Performance Indicators

4.1.1. Delay Time

According to existing research, the average delay of vehicles for any queue node can be described [36,37,38] as

$$D_i\left(t\right)=\left({\int }_{t_0}^{t_0+T}{x}_i^F\left(t\right)\mathrm{d}t\right)/\left({\int }_{t_0}^{t_0+T}{\theta }_i^F\left(t\right)\mathrm{d}t\right),$$

(31)

The delay of vehicles in the intersection is

$$\overline{D}={\sum }_{i\in \mathcal{N}}{x}_i^F\left(t\right)\times D_i\left(t\right).$$

(32)

4.1.2. Electric Energy Consumption

The instantaneous energy consumption of electric vehicles is related to their power and speed.

$$f_i^E=Fv,$$

(33)

Based on existing research, the driving equation of electric vehicles can be represented as

$$F=mgf+\delta m{\displaystyle \frac{dv}{dt}}+{\displaystyle \frac{\rho C_{D}A}{2}}{v}^2,$$

(34)

where $m$ is the average weight of the car, unit kg; $g$ is the gravitational acceleration; $\delta$ is the conversion factor of rotational mass, taken as between 1.1 and 1.4; $f$ is the rolling resistance coefficient, generally taken as between 0.010 and 0.018; $\rho$ is the air density, generally taken as 1.2258 kg/m³; the air resistance coefficient $C_D$ is taken as between 0.18 and 0.40; and A is the vehicle’s frontal area, generally taken as 0.5 m².

After analyzing the above, the average electric energy consumption of internal nodes in electric vehicles is

$$F_i^E={\int }_{t_0}^{t_0+T}{f}_i^D\left(t\right)x_i^F(t)dt,$$

(35)

and the electric energy consumption in the intersection is

$$\overline{F^E}={\sum }_{i\in \mathcal{N}}{x}_i^F\left(t\right)\times F_i^E.$$

(36)

4.1.3. Fuel Consumption and Emissions

The average fuel consumption and emissions are important considerations in the field of signal control, and this study adopts the fuel consumption and emissions estimation model proposed by [38].

$$f_i^y\left(t\right)=T_s{x}_i^F\left(t\right)\exp \left(v_i\left(t\right)P_y{a}_i\left(t\right)\right),$$

(37)

where $y\in \{\mathrm{C}\mathrm{O},\mathrm{H}\mathrm{C},{\mathrm{N}\mathrm{O}}_x,\mathrm{F}\mathrm{C}\}$ represents the average emission rate ($\mathrm{C}\mathrm{O},\mathrm{H}\mathrm{C},{\mathrm{N}\mathrm{O}}_x$) or fuel consumption (FC) rate of queuing nodes $i$ at time $t$; $T_s$ represents the discrete computational time step—in this paper, $T_s=1 \mathrm{s}$; $x_i^F\left(t\right)$ is the system state calculated by the queuing node feedback model at time $t$; $v_i\left(t\right)$ represents the average velocity of queuing nodes; and $a_i\left(t\right)$ represents the average acceleration of queuing nodes, which can be obtained from Equation (30).

$$a_i\left(t\right)={\displaystyle \frac{\mathrm{d}{v}_i\left(t\right)}{\mathrm{d}t}},$$

(38)

To estimate the calibration values for emission variables ($y\in \left\{\mathrm{C}\mathrm{O},\mathrm{H}\mathrm{C},{\mathrm{N}\mathrm{O}}_x\right\}$) or fuel consumption variables ($y=\mathrm{F}\mathrm{C}$), ref. [38] provided the following calibrated values of $P_y$:

$$P_{\mathrm{C}\mathrm{O}}=0.01\left[\begin{array}{cccc}-1292.81& 48.8324& 32.8837& -4.7675\\ 23.2920& 4.1656& -3.2843& 0\\ -0.8503& 0.3291& 0.5700& -0.0532\\ 0.0163& -0.0082& -0.0118& 0\end{array}\right],$$

$$P_{\mathrm{H}\mathrm{C}}=0.01\left[\begin{array}{cccc}-1454.4& 0& 25.1563& -0.3284\\ 8.1857& 10.9200& -1.9423& -1.2745\\ -0.2260& -0.3531& 0.4356& 0.1258\\ 0.0069& 0.0072& -0.0080& -0.0021\end{array}\right],$$

$$P_{{\mathrm{N}\mathrm{O}}_x}=0.01\left[\begin{array}{cccc}-1488.32& 83.4524& 9.5433& -3.3549\\ 15.2306& 16.6647& 10.1565& -3.7076\\ -0.1830& -0.4591& -0.6836& 0.0737\\ 0.0020& 0.0038& 0.0091& -0.0016\end{array}\right],$$

$$P_{\mathrm{F}\mathrm{C}}=0.01\left[\begin{array}{cccc}-753.7& 44.3809& 17.1641& -4.2024\\ 9.7326& 5.1753& 0.2942& -0.7068\\ -0.3014& -0.0742& 0.0109& 0.0116\\ 0.0053& 0.0006& -0.0010& -0.0006\end{array}\right].$$

The detailed derivation and calculation process of this model can be derived from the research results of [38,39]. Combining the above analysis, the average emissions or fuel consumption of nodes $i$ at $\left[t_0,t_0+T\right]$ are

$$F_i^y={\int }_{t_0}^{t_0+T}{f}_i^y\left(t\right)x_i^F(t)\mathrm{d}t,$$

(39)

and the emissions or fuel consumption in the intersection are

$$\overline{F^y}={\sum }_{i\in \mathcal{N}}{x}_i^F\left(t\right)\times F_i^y.$$

(40)

4.2. Solution Algorithms

Based on the above model, a recursive algorithm was developed to solve the multi-class customer feedback queuing network model, as detailed in Algorithm 1.

Algorithm 1: Recursive algorithm for solving multi-class customer feedback queuing network at intersections.

Input: System parameters of multi-class feedback queuing network New energy vehicle penatration: p Initial system state: $x_i^F\left(t_0\right)=x_{i0};x_{\mathrm{E}i}^F\left(t_0\right)=x_{i0}p;x_{\mathrm{H}i}^F\left(t_0\right)=x_{i0}\left(1-p\right)$ Path transition probability: ${pc}_{ij}\left(t\right)$ External time-varying arrival rate: ${\lambda }_{0i}\left(t\right)$ System parameters of queuing node: $l_i;w_i$ Traffic signal control scheme: $\left[ g_k;{yg}_k;r_k;ar;CL\right];\forall i,j\in \mathcal{N};k\in \mathcal{K};t\in \left[t_0,t_0+T\right]$

Output: Performance parameters of the concerned feedback queueing network at any time system state $-x_i^F\left(t\right);x_{\mathrm{H}i}^F\left(t\right);x_{\mathrm{E}i}^F\left(t\right)$

Recursive solution of feedback queuing network

For $i\in \mathcal{N}$

Calculate: $C_i;r_i\left(C_i\right);v_i\left(C_i\right);{\mu }_i\left(C_i\right);{\sigma }_i$

End For

For $t=t_0,t_0+1,\dots ,t_0+T-1$

For $i\in \mathcal{N}$

For $j\in \mathcal{N}$

Calculate $a_{ij}\left(t\right);{pr}_{ij}\left(t\right)$

End For

Calculate: ${PB}_i^L; {PE}_i^F;{PB}_i^F$

End For

For $i\in \mathcal{N}$

Calculate $u_i^{\prime }\left(t\right);{\theta }_i^F\left(t\right)$

End For

For $i\in \mathcal{N}$

Calculate: $x_i^L\left(t+1\right)$        Calculate: $x_i^F\left(t+1\right)$        Calculate: $x_{\mathrm{H}i}^F\left(t+1\right)$        Calculate: $x_{\mathrm{E}i}^F\left(t+1\right)$

End For

4.3. Model Verification

4.3.1. Parameter Setting

Traffic Survey

We use the intersection of Changjiang Middle Road and Qianjin Middle Road in Kunshan City as an example for our numerical experiments. Its channelized form is shown in Figure 1.

The intersection-related survey data are shown in

Table 1. Main survey data of the Changjiang Middle Road–Qianjin Middle Road intersection.

		Import Road Section	Turn Left	Straight	Turn Right	Export Road Section
East	road length (m)	285	50	50	50	335
	number of lanes	4	1	2	1	4
	ratio of turning (%)	N/A	16	70	14	N/A
South	road length (m)	483	70	70	70	553
	number of lanes	3	1	1	1	3
	ratio of turning (%)	N/A	47	38	16	N/A
West	road length (m)	253	50	50	50	303
	number of lanes	4	1	2	1	4
	ratio of turning (%)	N/A	26	65	9	N/A
North	road length (m)	280	70	70	70	350
	number of lanes	3	1	1	1	3
	ratio of turning (%)	N/A	42	35	23	N/A

.

In addition, referring to the study by [24], the road section blockage density $k_{jam}$ is taken as 160 vehicles/(km∙lane). In this paper, the three characteristic parameters of the exponential speed model are $v_0=60\mathrm{k}\mathrm{m}/\mathrm{h}$, $v_a=20\mathrm{k}\mathrm{m}/\mathrm{h}$, and $v_b=5\mathrm{k}\mathrm{m}/\mathrm{h}$.

This experiment actually investigates the traffic data during the evening peak period (17:00~18:00), as shown in

Table 2. Traffic volume survey form for Changjiang Middle Road–Qianjin Middle Road intersection (pcu).

Time	East	South	West	North
17:00 to 17:15	129	121	101	54
17:15 to 17:30	209	116	114	32
17:30 to 17:45	155	122	87	70
17:45 to 18:00	177	72	161	71

. The actual investigation counts the traffic data at 15 min intervals. To obtain real-time traffic flow data, this study uses a third-order polynomial interpolation method based on the 15 min traffic data to obtain the traffic curve during the evening peak period, as shown in Figure 2.

In this section, the effectiveness of the model in different demand scenarios is assessed. This study takes the transportation demand shown in Figure 2 as the standard demand scenario, and considers three transportation demand scenarios (describing different traffic demand scenarios of low saturation, high saturation, and oversaturation, respectively), as shown in

Table 3. Simulation verification of feedback queueing network model for different traffic demand scenarios during evening peak hours (17:00~18:00).

Number	Multiple	$\mathbf{Maximum} \mathbf{Traffic} \mathbf{Intensity} {\mathit{\rho }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$				$\mathbf{Average} \mathbf{Traffic} \mathbf{Intensity} {\mathit{\rho }}_{\mathbf{a}\mathbf{v}\mathbf{e}}$
		East	South	West	North	East	South	West	North
1	1	0.5198	0.5926	0.2487	0.5479	0.4184	0.4574	0.1851	0.3109
2	2	1.0396	1.1852	0.4973	1.0957	0.8368	0.9148	0.3702	0.6218
3	3	1.5594	1.7778	0.7461	1.6437	1.2552	1.3722	0.5553	0.9327

.

The first column in Table 3 represents the demand scenario number; the second column multiplier indicates the traffic demand relative to the standard demand scenario; columns 3 to 6 show the maximum traffic intensity ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{a}\mathrm{x}(\lambda \left(t\right)/{\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}),0\le t\le T$ at different intersection approaches during the evening peak period; columns 7 to 10 display the average traffic intensity ${\rho }_{\mathrm{a}\mathrm{v}\mathrm{e}}={\int }_0^T\lambda \left(t\right)\mathrm{d}\mathrm{t}/{(T\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}),0\le t\le T$ during the evening peak period in different directions.

Determination of car-following models for different types of vehicles

To demonstrate and validate the proposed MCFFQN framework with concrete, relevant examples, this study instantiates the vehicle classes based on prominent and synergistic technological trends observed in current transportation systems. We define two representative classes:

(1)

Conventional Vehicles (CV): Combining Internal Combustion Engine (ICE) propulsion with human-driven (HDV) characteristics, modeled by the IDM.

(2)

New Energy and Connected Vehicles (NECV): Combining electric propulsion with a higher level of automation/connectivity, modeled by the CACC.

This instantiation serves a proof-of-concept purpose. It reflects a common real-world scenario where technological advancements in powertrain and automation are often correlated. The choice of these two specific classes does not limit the generality of the MCFFQN framework; rather, it provides a concrete case study to verify that the model can accurately capture the dynamics arising from the interaction of two distinct, representative vehicle types. Investigating all possible cross-combinations of propulsion and automation (e.g., ICE-CACC, EV-HDV) is a valuable application of this framework, which we designate for future research.

The vehicle-following models describe the behavior of a vehicle following another vehicle, and offer rich results. The research on the vehicle-following behavior of fuel-powered vehicles often uses the IDM [33,34]. The research into the vehicle-following behavior of new energy vehicles often uses the Cooperative Adaptive Cruise Control (CACC) model [39,40].

The IDM proposed by [33] is widely used in the research on autonomous driving vehicles. The IDM is formulated as follows:

$$\dot{v}=a[1-{\left({\displaystyle \frac{v}{v_0}}\right)}^{\delta }-({\displaystyle \frac{s_0+v\times t_0-\frac{v\times ∆v}{2\sqrt{a\times b}}}{h-L}}{)}^2].$$

(41)

Here, $v$ is the desired speed; $\dot{v}$ is the desired acceleration; $v_0$ is the free flow speed; $∆v$ is the speed difference between the front and rear vehicles; $L$ is the length of the vehicle; $t_0$ is the safe headway time; $s_0$ is the safe headway distance; $a$ is the maximum acceleration; $b$ is the comfortable deceleration; and $\delta$ is the model’s estimated parameters. According to the research of [33,34], the recommended values for the parameters in IDM are shown in

Table 4. Parameters and recommended values of IDM.

Parameter	Value	Unit
$v_0$	120	km/h
$T$	1.5	s
$a$	1	s−1
$b$	2	s−1
$s_0$	2	m
$L$	5.0	m
$\delta$	4	N/A

.

Based on the tested vehicle data, the CACC model can better describe the car-following behavior of new energy vehicles [39,40]. The CACC model is shown in the equation

$$\left\{\begin{array}{l}v=v_p+k_{p}e+k_d\dot{e}\\ e=∆x-s_0-L-t_{c}v\end{array}\right..$$

(42)

Here, $v$ is the speed of the following vehicle at the current moment; $v_p$ is the speed of the following vehicle at the previous control moment; $e$ is the error between the actual headway and the desired headway, with its derivative form as $\dot{e}$; $s_0$ is the desired headway time; $L$ is the safe headway distance; is the length of the vehicle; and $∆x$ is the distance between the preceding and following vehicles.

After differentiating the equation, we obtain

$$\dot{v}={\displaystyle \frac{k_p\left(∆x-L-s_0\right)-k_p{t}_{c}v+k_d∆v}{k_d{t}_c+∆t}}.$$

(43)

According to the studies of [40], the parameters and recommended values of the CACC car-following model are shown in

Table 5. Parameters and recommended values of the CACC model.

Parameter	Value	Unit
$∆t$	0.01	s
$k_p$	0.45	N/A
$k_d$	0.25	N/A
$t_c$	0.6	s

.

According to the IDM and CACC models, the headway of fuel-powered vehicles and new energy vehicles can be calculated under stable conditions. In the stable state, the vehicle travels at a constant speed following the preceding vehicle with an acceleration of 0. Therefore, by setting the acceleration to 0 in Equations (41) and (42), i.e., $\dot{v}$= 0, the headways of fuel-powered vehicles and new energy vehicles under stable conditions can be obtained, as shown in Equation (44).

$$\left\{\begin{array}{l}{h}_{\mathrm{H}}^{*}=L+{\displaystyle \frac{s_0+vT}{\sqrt{1-{(v/v_0)}^{\delta }}}}\\ h_{\mathrm{C}}^{*}=L+s_0+v{t}_c\end{array}\right..$$

(44)

Here, $h_{\mathrm{H}}^{*}$ is the headway between fuel-powered vehicles under stable conditions and $h_{\mathrm{C}}^{*}$ is the headway between new energy vehicles under stable conditions. Assuming the total number of vehicles in mixed traffic flow is $Num$, the average headway in a stable mixed traffic flow is $h^{*}$,

$$h^{*}={\displaystyle \frac{Num\left(1-p\right)h_{\mathrm{H}}^{*}+Nump{h}_{\mathrm{C}}^{*}}{Num}}=\left(1-p\right)h_{\mathrm{H}}^{*}+p{h}_{\mathrm{C}}^{*}.$$

(45)

After simplification, we obtain

$$h^{*}=L+\left(1-p\right){\displaystyle \frac{s_0+vT}{\sqrt{1-{\left(v/v_0\right)}^{\delta }}}}+p\left(s_0+v{t}_c\right).$$

(46)

The penetration rate of new energy vehicles is $p$, i.e., the ratio of the number of new energy vehicles to the total number of vehicles in mixed traffic flow.

Traffic flow density can be represented as

$$k={\displaystyle \frac{1000}{h^{*}}}.$$

(47)

Here, $k$ represents the density of mixed traffic flow. By substituting Equations (45) and (46) into Equation (47), the relationship between traffic flow density and the average vehicle speed under stable conditions is obtained, as

$$k=1000{\times \left\{L+\left(1-p\right){\displaystyle \frac{s_0+vT}{\sqrt{1-{\left(v/v_0\right)}^{\delta }}}}+p\left(s_0+v{t}_c\right)\right\}}^{-1}.$$

(48)

The relationship between traffic flow and average speed is similar,

$$q=3600\times v·{\left\{L+\left(1-p\right){\displaystyle \frac{s_0+vT}{\sqrt{1-{\left(v/v_0\right)}^{\delta }}}}+p\left(s_0+v{t}_c\right)\right\}}^{-1}.$$

(49)

According to Equations (48) and (49), the relationships between traffic flow, density, and average speed under different penetration rate conditions are shown in Figure 3.

Figure 3 shows the following: (1) With the increase in the penetration rate of intelligent connected vehicles, the road capacity increases significantly. When the penetration rate is 0—i.e., all vehicles are manually driven—the capacity of a single lane is about 1600 vehicles/h, while when the penetration rate increases to 1, the capacity of a single lane reaches 3500 vehicles/h. (2) The speed curve of vehicles under different penetration rate conditions is significantly different, as shown in Figure 3b.

After determining the average speed of the mixed traffic flow, we can use the model in Section 3 to describe the behavior of the mixed traffic flow through the intersection.

It should be pointed out that for different types of vehicles, due to their different driving characteristics, IDM and CACC models are selected to describe their driving characteristics, respectively. However, as the model is a mesoscopic model, it is inevitable that the influences of different drivers’ driving habits on driving behavior are overlooked. Although this is an inherent flaw of the mesoscopic model, the mesoscopic model can better describe the average state of system vehicles when there are a large number of vehicles.

Linkage Between Car-Following Interactions and Queuing Dynamics

The interaction between different vehicle types—such as human-driven vehicles (HDVs) modeled with the Intelligent Driver Model (IDM) and connected automated vehicles (CAVs) modeled with the Cooperative Adaptive Cruise Control (CACC)—fundamentally reshapes the macroscopic traffic flow characteristics, particularly the average speed-density relationship $f_v^{*}(v)$ derived from Equations (44)–(49). In the proposed mesoscopic Multi-Class Customer Feedback Queuing Network (MCFFQN) framework, this interaction is not simulated directly at the microscopic vehicle-to-vehicle level; instead, it is analytically approximated and encapsulated within the state-dependent service rate $u_i^F(t)$. Specifically, the car-following models determine the stable headway $h^{*}$ for each vehicle type at a given speed $v$ (Equation (44)). The weighted mixture of these headways, based on the penetration rate $p$, yields the aggregate traffic density $k$ and flow $q$ (Equations (48) and (49)). The resulting speed-density relationship $f_v^{*}(v)$ directly feeds into the state-dependent service rate $u_i^F(t)=x_i^F(t)\cdot v_i(x_i(t))/l_i$ (Equation (3)), where $v_i(x_i(t))$ is obtained from $f_v^{*}(v)$. Consequently, the microscopic interactions govern the macroscopic flow properties, which in turn determine the queue service rate and thereby drive the queuing dynamics—such as queue formation, dissipation, and spillback—within the feedback network model. To validate this analytical approximation, the macroscopic behavior of the queuing network is compared against high-fidelity microscopic simulations performed in SUMO (Section 4.3.2), where vehicle-type interactions are explicitly simulated. The close agreement between the MCFFQN predictions and the microscopic simulation results (see

Table 6. Average error of queueing network model and discrete event simulation model in different demand scenarios.

Demand Scenario Number		1	2	3	Average
Absolute error (pcu)	Overall system state	0.2301	0.8690	0.8296	0.6429
	Intelligent connected vehicle system state	0.0690	0.2607	0.2489	0.1929
	Human-driven vehicle system state	0.1611	0.6083	0.5807	0.4500
Relative error (%)	Overall system state	6.4241	7.8402	5.0747	6.4463
	Intelligent connected vehicle system state	6.4581	7.7736	5.1623	6.4647
	Human-driven vehicle system state	6.4097	7.8690	5.0380	6.4389

and Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12) confirms that the proposed approach effectively captures the aggregate effect of mixed-traffic interactions on queuing dynamics while maintaining high computational efficiency.

4.3.2. Verification

We evaluate the effectiveness of the multi-class customer queueing network model under different demand scenarios. Accurate intersection performance evaluation depends on precise system state estimation, which is especially important for differentiating between new energy and human-driven vehicles. Validation is conducted using three traffic saturation levels (low, high, oversaturation), with the scenario in Figure 3 as the baseline.

This study was conducted based on the SUMO version 1.22.0. simulation software, which is used to construct a simulation environment for mixed traffic flow. SUMO can set different car-following rules for different types of vehicles, which can effectively simulate the mixed driving conditions of different types of vehicles. To prevent the random factors in a single simulation from affecting the results, this study performed 200 simulations with random seeds for each demand scenario, and takes the average value of the 200 simulations as the simulation result.

The simulation results are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. The solid red line in the figures represents the solution results of the feedback model proposed in this paper, the dashed blue line represents the average results of 200 simulations, and the dashed green line represents the 95% confidence interval of the simulation results. Running the simulation model once takes about 2.5 min, while solving the model in this paper only takes 0.2 s.

Figure 4, Figure 5 and Figure 6 show the simulation results of demand scenario 1; Figure 7, Figure 8 and Figure 9 show the simulation results of demand scenario 2; Figure 11, Figure 12 and Figure 13 show the simulation results of demand scenario 3. In addition, Figure 4, Figure 7 and Figure 10 describe the simulation results of the overall system state for each demand scenario; Figure 5, Figure 8 and Figure 11 show the simulation results of the intelligent connected vehicle system state for each demand scenario; Figure 6, Figure 9 and Figure 12 show the simulation results of the intelligent autonomous vehicle system state for each demand scenario.

The results show that the multi-class customer queueing network model proposed in this paper can accurately describe the dynamic queueing performance of each queueing node in different demand scenarios. The error statistics between the multi-class customer queueing network model and the discrete event simulation model under different demand scenarios are shown in Table 6. The average absolute error of the overall system state under three demand scenarios does not exceed 0.8690 vehicles, with an average absolute error of 0.6429 vehicles for the three demand scenarios; the relative error does not exceed 7.8402%, with an average relative error of 6.4463% for the three scenarios. Specifically, the average absolute error of the intelligent connected vehicle system state does not exceed 0.2607 vehicles, with an average absolute error of 0.1929 vehicles under the three demand scenarios. The relative error of the intelligent connected vehicle system state does not exceed 7.7736%, with an average relative error of 6.4647% for the three scenarios; the average absolute error of the human-driven vehicle system state does not exceed 0.6083 vehicles, with an average absolute error of 0.4500 vehicles under the three demand scenarios. The relative error of the human-driven vehicle system state does not exceed 7.8690%, with an average relative error of 6.4389% for the three scenarios.

The relative error in scenarios 1 and 3 is small, while the relative error in scenario 2 is large. This is because of the following: (1) when the traffic demand is low, it is difficult for the system to experience congestion; thus, the parameters related to blocking probability have a small impact, allowing the unified parameter setting to be effectively applied to this traffic demand scenario and resulting in simulation results that are close to the model’s results. (2) When the traffic demand is high, traffic congestion frequently occurs in the traffic system, the unified probability calibration parameters can adapt to all traffic facilities, leading to an increase in relative error. (3) As the traffic demand further increases, traffic congestion is further intensified, resulting in a larger average system state than in scenario 2, thereby causing a smaller relative error.

4.4. Potential of the Model for Handling Anomalous Traffic Conditions

The proposed MCFFQN model is inherently a dynamic, state-dependent feedback system. Its core strength lies in the ability to respond in real-time to changes in traffic state ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{F}}\left(\mathrm{t}\right)$ and adjust the service rate ${\mathrm{u}}_{\mathrm{i}}^{\mathrm{F}}\left(\mathrm{t}\right)$ and network flow allocation accordingly. This characteristic provides a fundamental capability to describe and respond to unexpected traffic disturbances. The potential is analyzed from the perspective of the model’s mechanism as follows:

(1)

Coping with demand spikes: A key input to the model is the time-varying arrival rate ${\mathrm{\lambda }}_{\mathrm{i}}(\mathrm{t})$. In the current study, ${\mathrm{\lambda }}_{\mathrm{i}}(\mathrm{t})$ is derived from survey data. To simulate a demand surge, one would simply replace ${\mathrm{\lambda }}_{\mathrm{i}}(\mathrm{t})$ with a spike function (e.g., a step or impulse function) during the incident period ${\mathrm{t}}_{\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}}$. The model would dynamically incorporate the increased vehicle count into the system state ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{F}}(\mathrm{t})$ via Equations (10) and (18). The resulting queue propagation and dissipation would be captured through the congestion probability ${\mathrm{P}\mathrm{B}}_{\mathrm{i}}^{\mathrm{L}}$ and the feedback mechanism ${\mathrm{\sigma }}_{\mathrm{i}}^{\prime }\left(\mathrm{t}\right)\mathrm{\rho }\left(\mathrm{t}\right)$. Performance metrics such as average delay $\overline{\mathrm{D}}$ would automatically reflect the impact of the surge.

(2)

Simulating local incidents (lane closures): This is equivalent to a temporary reduction in the service capacity ${\mathrm{C}}_{\mathrm{i}}$ of the affected road section (node $\mathrm{i}$). In our model, ${\mathrm{C}}_{\mathrm{i}}$ is a fundamental parameter for calculating the service rate ${\mathrm{u}}_{\mathrm{i}}^{\mathrm{F}}(\mathrm{t})$ (see Equations (1) and (16)). By setting ${\mathrm{C}}_{\mathrm{i}}$ to a lower value during the incident period, the model can automatically simulate the input-output imbalance, upstream congestion (manifested as an increase in ${\mathrm{P}\mathrm{B}}_{\mathrm{i}}^{\mathrm{L}}$), and potential vehicle rerouting (via adjustments to the transfer probability ${\mathrm{p}\mathrm{r}}_{\mathrm{j}\mathrm{i}}(\mathrm{t})$.

(3)

Tolerating sensor failures (data loss): In practical deployment, if certain detectors fail, ${\mathrm{\lambda }}_{\mathrm{i}}(\mathrm{t})$ may not be directly available. The upstream-downstream flow relationship embedded in the model (Equation (18)) can then be leveraged for data imputation or state estimation. For instance, if an entry link detector fails, the estimated output rate ${\mathrm{\theta }}_{\mathrm{j}}^{\mathrm{F}}(\mathrm{t}-1)$ from upstream links and historical turning ratios ${\mathrm{p}\mathrm{c}}_{\mathrm{j}\mathrm{i}}$ could be used to reconstruct ${\mathrm{\lambda }}_{\mathrm{i}}(\mathrm{t})$, demonstrating the model’s robustness in a data-driven context.

In summary, while this study focuses on validating signal optimization under regular mixed traffic conditions, the analysis above demonstrates that the proposed modeling framework provides a solid theoretical foundation for assessing system resilience (performance degradation) and recovery in stress scenarios. Conducting comprehensive resilience stress tests would require constructing a more complex library of anomaly scenarios and defining detailed “recovery” metrics (e.g., time to recover from the performance nadir to baseline levels), which constitutes a vital direction for our future research.

4.5. Discussion on Model Uncertainty and Robustness

The performance of the proposed MCFFQN model and the ensuing optimization results are influenced by several key input parameters and assumptions. While a comprehensive global sensitivity analysis (e.g., variance-based Sobol method) and robust optimization under uncertainty (e.g., chance-constrained programming) are beyond the scope of this foundational study, we provide a preliminary discussion on uncertainty sources and the model’s inherent robustness features.

4.5.1. Key Sources of Uncertainty

(1)

Input Parameters: The external arrival rates ${\lambda }_i\left(t\right)$ and the new energy vehicle penetration rate p are subject to forecasting errors and temporal fluctuations.

(2)

Model Parameters: Calibrated parameters such as the free-flow speed v_i0, the jam density $k_{jam}$, and the exponents in the speed model (${\beta }_i$, ${\gamma }_i$) have inherent estimation variances.

(3)

Behavioral Models: The car-following parameters (e.g., T in IDM, $t_c$ in CACC) represent average behaviors and exhibit inter-driver variability.

4.5.2. Inherent Robustness Features of the Framework

(1)

The feedback mechanism in the queuing network (Equation (10)) allows the model to dynamically absorb fluctuations in demand by adjusting virtual queues, providing a form of built-in disturbance rejection.

(2)

The state-dependent service rate $u_i^F\left(t\right)$ automatically adjusts section capacity based on congestion levels, making output metrics less sensitive to fixed parameter errors under congested conditions.

The scenario analysis in Section 4.3.2 (Table 4), where traffic demand is scaled, provides a basic one-dimensional sensitivity test. The results show that while absolute performance degrades with increased demand, the relative superiority of the model’s predictive accuracy is maintained across different saturation levels, indicating a degree of robustness.

5. Intersection Traffic Signal Optimization Under Mixed Traffic Flow

5.1. Optimization Model

5.1.1. Objective Function

Based on the performance indicators established in Section 4.1 of this paper, the primary objectives considered in the single-intersection optimization process include minimizing vehicle delay, reducing the energy consumption of electric vehicles, and lowering fuel consumption and pollutant emissions from fuel-powered vehicles.

Minimization of Vehicle Delay

During the process of intersection signal optimization, priority is given to traffic efficiency; i.e., minimizing the delay time for all vehicles,

$$\min Z_1=\overline{D}.$$

(50)

The delay time can be obtained from Equation (32).

Minimization of Electric Vehicle Energy Consumption

For electric vehicles, the energy consumption when passing through the intersection is minimized. Thus,

$$\min Z_2=\overline{F^E}.$$

(51)

The energy consumption can be obtained from Equation (36).

Minimization of Fuel Consumption for Fuel-Powered Vehicles

For fuel-powered vehicles, the fuel consumption when passing through the intersection is minimized. Thus,

$$\min Z_3=\overline{F^{FC}}.$$

(52)

The fuel consumption can be obtained from Equation (40).

Minimization of Emissions from Fuel-Powered Vehicles

For fuel-powered vehicles, the emissions when passing through the intersection are minimized. Thus,

$$\min Z_4=\overline{F^{CO}}.$$

(53)

$$\min Z_4=\overline{F^{HC}}.$$

(54)

$$\min Z_5=\overline{F^{{NO}_x}}.$$

(55)

The emissions can be obtained from Equation (40).

Transformation of Multi-Objective into Single-Objective

The aforementioned optimization problem is a multi-objective optimization problem. This study proposes converting it into a single-objective problem through cost integration. The cost of delay time can be quantified based on the average regional income $w_D$, the cost of fuel consumption can be calculated using fuel prices $w_{FC}$, the cost of energy consumption for electric vehicles can be derived from residential electricity rates $w_E$, and the cost of pollutant emissions can be assessed via emission taxes $w_{CO}$, $w_{HC}$ and $w_{{NO}_x}$. Thus, the multi-objective optimization problem can be transformed into a single-objective optimization problem as follows

$$\min Z=w_D\overline{D}+w_E\overline{F^E}+w_{FC}{F}^{FC}+w_{CO}\overline{F^{CO}}+w_{HC}\overline{F^{HC}}+w_{{NO}_x}\overline{F^{{NO}_x}}.$$

(56)

The objective function $Z$ represents the total cost incurred by all vehicles when passing through the intersection. The constructed objective of minimizing total cost not only improves regular traffic efficiency and environmental benefits but also indirectly enhances the intrinsic resilience of the intersection to minor traffic fluctuations by reducing the average system load and queue lengths. Dedicated optimization for large-scale disruptions remains a subject for future investigation.

5.1.2. Constraints

For signal control at an isolated intersection, the primary constraints include green time constraints, cycle length constraints, and integer variable constraints.

Green Time Constraint

To ensure the safe passage of pedestrians and other road users through the intersection, the green time $g_k$ for each phase should not be less than the minimum green time $g_k^{\mathrm{m}\mathrm{i}\mathrm{n}}$. To prevent excessive delays in other phases caused by prolonged green time in one phase, the green time $g_k$ for each phase should not exceed the maximum green time $g_k^{\mathrm{m}\mathrm{a}\mathrm{x}}$. Therefore, the green time for each phase must satisfy the following constraint:

$$g_k^{\mathrm{m}\mathrm{i}\mathrm{n}}\le g_k\le g_k^{\mathrm{m}\mathrm{a}\mathrm{x}},\forall k\in \mathcal{K}.$$

(57)

Cycle Length Constraint

The signal cycle length is equal to the sum of the green time, yellow time, red time, and all-red time for each phase. Furthermore, the signal cycle length must be constrained within the minimum and maximum cycle limits,

$${\sum }_{k\in \mathcal{K}}\left(g_k+y_k+r_k\right)+ar=CL,$$

(58)

$${CL}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le CL\le {CL}^{\mathrm{m}\mathrm{a}\mathrm{x}}.$$

(59)

In the equation, $g_k,{ y}_k$, and $r_k$ represent the green time, yellow time, and red time for phase $k$, respectively; $ar$ denotes the all-red time within the signal cycle, which is used to clear the intersection and ensure safety; $CL$ indicates the signal cycle length of the intersection, while ${CL}^{\mathrm{m}\mathrm{i}\mathrm{n}} \mathrm{a}\mathrm{n}\mathrm{d} {CL}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ represent the minimum and maximum allowable cycle lengths for the intersection, respectively.

Integer Variable Constraint

In practical applications, both the green time and cycle length of signalized intersections are set to integers; i.e.,

$$g_k,CL\in \mathbb{Z},\forall k\in \mathcal{K}.$$

(60)

Therefore, the optimization model in this study is a pure integer programming model.

The optimization model presented is deterministic. Its solutions are optimal for the given input parameters and expected demand patterns. For real-world implementation, a robust or stochastic version of this model that explicitly accounts for parameter uncertainty (e.g., in p or ${\lambda }_i\left(t\right)$) and demand volatility (e.g., via chance constraints on queue lengths) would be required to ensure consistent performance. The development of such a robust formulation is an essential direction for future work.

5.2. Solution Algorithm

5.2.1. Mesh Adaptive Direct Search Algorithm

To optimize signals for intersections incorporating a feedback queuing network, this study designs a corresponding optimization model. This model constitutes a non-convex integer programming problem, which is inherently NP-hard. Furthermore, the requisite performance metrics (e.g., system state $x_i^F\left(t\right)$, output rate ${\theta }_i^F\left(t\right)$) are nonlinear and solved recursively, lacking analytical solutions. Consequently, even without integer constraints, it remains a challenging global optimization problem. To address this complexity, the Mesh Adaptive Direct Search (MADS) algorithm [41] is employed, due to its ability to handle global nonlinear, non-convex mixed-integer problems with numerous variables [42].

The MADS algorithm, first introduced by Audet et al. [43] and originating from the generalized pattern search algorithm, operates on a discretized mesh of the variable space. Each iteration consists of two steps, namely, the search step and the poll step. The search step performs a global exploration by evaluating a finite set of trial points on the mesh to identify promising regions containing local optima. Subsequently, the poll step conducts a localized search around the current best solution using specific poll directions to refine the solution accuracy. The workflow of the MADS algorithm is illustrated in Algorithm 2.

Algorithm 2: MADS Algorithm Workflow

Initialization: Set the initial solution $x_0$ for the optimization problem and initialize the iteration counter $k=0$ Iterative Optimization While the stopping criteria are not met Search on the mesh via the search step to find a solution superior to $x_k$ If The search step fails to find a superior solution Search via the poll step on the mesh for a solution superior to $x_k$ End If A solution superior to $x_k$ is found through the search step and the poll step. Record the incumbent solution $x_{k+1}$ and coarsen the mesh Else Set $x_{k+1}=x_k$ and update the mesh End Set the iteration counter $k=k+1$ and update the solution to $x_{k+1}$ Check if the stopping criteria are met End

5.2.2. Preliminary Benchmark

To provide an initial validation of the MADS algorithm’s efficacy for the proposed optimization problem, a comparative experiment against a standard Genetic Algorithm (GA) was conducted. The experiment was designed to ensure a fair comparison under a constrained computational budget.

(1)

Experimental Setup:

A comparative experiment was conducted under the low-saturation condition (#1) to ensure a fair evaluation. Both algorithms optimized the same 5 integer decision variables (cycle length and four green times). The MADS algorithm was implemented using NOMAD 3.9.1 with stopping criteria set to a maximum of 200 iterations, a mesh size tolerance of 1 × 10⁻⁴, or minimal objective improvement, while the GA used MATLAB version R2023b’s toolbox with a population size of 50, 100 maximum generations, and standard crossover/mutation probabilities. To ensure a fair comparison of computational budget, a common limit of 1500 function evaluations (each corresponding to one full MCFFQN model run) was set as the primary termination condition for both solvers. Each algorithm was executed 20 times with random seeds to account for stochastic variability.

(2)

Results and Analysis:

The computational results of the different algorithms are presented in

Table 7. Performance Comparison Table of Different Algorithms.

Algorithm	Mean Best Total Cost (CNY)	Standard Deviation (CNY)	Average Function Evaluations to Converge	Average Runtime (s)
MADS	905.7	12.4	1120	38.5
GA	918.3	25.8	1500 (budget used)	45.8

.

The results indicate that:

(1)

Solution Quality and Robustness: MADS found better solutions on average (1.4% lower total cost) with significantly lower standard deviation (12.4 vs. 25.8 CNY), demonstrating superior solution quality and stability.

(2)

Computational Efficiency: MADS converged using fewer function evaluations on average (1120 vs. 1500), indicating higher search efficiency within the given budget.

This preliminary benchmark confirms that MADS is a well-suited and effective solver for the proposed MCFFQN-based signal optimization problem.

5.3. Numerical Example Analysis

5.3.1. Parameter Setting

Consistent with Section 4.3.1, this study selects the intersection of Changjiang Middle Road and Qianjin Middle Road in Kunshan City as a case for numerical experiments. The basic information of the intersection is provided in Figure 1 and Figure 2 and Table 1 and Table 2. The overall traffic flow characteristics of the intersection are summarized in Table 2.

This study solves the proposed signal timing optimization model for intersections using MATLAB. In this paper, it is assumed that the penetration rate of electric vehicles is 0.4. With reference to the disposable income of residents in Sichuan Province, the value of parameter $w_D$ can be set to 4.19 CNY/h. With reference to the residential electricity price in Chengdu, parameter $w_E$ is assigned a value of $0.6$ CNY/kWh. With reference to the price of 92# gasoline, parameter $w_{FC}$ is assigned a value of 9.47 CNY/L. With reference to the environmental protection tax schedule, parameters $w_{\mathrm{C}\mathrm{O}}$, $w_{\mathrm{H}\mathrm{C}}$, and $w_{{\mathrm{N}\mathrm{O}}_x}$ are assigned values of 21.55 CNY/kg, 14 CNY/kg, and 25.26 CNY/kg, respectively.

5.3.2. Comparative Validation Against a Steady-State Benchmark Model

To explicitly evaluate the efficacy of the proposed mixed-traffic signal optimization framework, this section systematically compares the performance of our scheme with that of Synchro (version 8.0), an industry-standard steady-state optimization tool. As a classic commercial software grounded in steady-state traffic flow theory, Synchro’s optimization logic represents mainstream practice in current engineering. To ensure a fair comparison, identical intersection geometry, traffic demand (Figure 2 and Table 3), and vehicle composition were used. First, the Synchro model was constructed and optimized based on field survey data to obtain its recommended “optimal” signal timing plan (cycle length, green splits), which served as the Baseline Scheme. The performance metrics (total cost, delay, energy consumption, emissions) of this baseline were then computed under five congestion scenarios (#1 to #5) using our MCFFQN simulation framework. Finally, the performance of our MADS-optimized scheme based on the MCFFQN model (the Proposed Optimized Scheme) was compared against this baseline. As shown in

Table 8. Performance comparison: proposed model vs. synchro.

Scenario	Performance Metric	Synchro	Proposed Model	Improvement
#1	Avg. Delay (s)	628.8585	392.138	−37.64%
	Total Energy (kWh)	57.0512	40.4244	−29.14%
	Total Fuel (L)	188.7654	86.5224	−54.16%
#3	Avg. Delay (s)	1731.934	1348.9405	−22.11%
	Total Energy (kWh)	122.7422	104.707	−14.69%
	Total Fuel (L)	547.8256	346.3006	−36.79%
#5	Avg. Delay (s)	3584.8697	3123.3264	−12.87%
	Total Energy (kWh)	254.2722	227.0482	−10.71%
	Total Fuel (L)	1195.8823	957.1958	−19.96%

, the proposed optimized scheme comprehensively outperforms the Synchro-based baseline across all tested scenarios.

As summarized in Table 8, our scheme achieves coordinated improvements in delay, energy consumption, and emissions across all congestion scenarios, effectively overcoming the limitation of conventional models that tend to focus on a single objective. Specifically, in the low-saturation Scenario #1, the average delay is reduced by 37.64%, while energy consumption and fuel consumption decrease significantly by 29.14% and 54.16%, respectively. These improvements are primarily attributed to the core mechanisms of the dynamic modeling approach: the state-dependent service rates capture the real-time relationship between speed and density, and the feedback mechanism effectively characterizes the propagation effects of queue spillback—both of which are key capabilities lacking in steady-state tools such as Synchro. Therefore, the comparative analysis demonstrates that the proposed framework can generate signal strategies that better adapt to the stochastic and dynamic nature of mixed traffic flows. It not only enhances traffic efficiency but also synergistically reduces energy consumption and environmental impact, reflecting its capability for multi-objective coordinated optimization.

5.3.3. Analysis of Results

The differences in various performance metrics under different congestion conditions were examined. These conditions are labeled as #1 to #5, representing initial traffic volumes of 0.1, 0.3, 0.5, 0.7, and 0.9 times the road capacity, respectively. Specifically, #1 corresponds to a low traffic volume state, while #5 represents a congested state. The operational performance of the intersection under the baseline and optimized schemes is summarized in

Table 9. Intersection performance statistics under the baseline signal timing scheme.

	#1	#2	#3	#4	#5
Vehicle Delay (s)	628.8585	1023.9897	1731.934	2507.5614	3584.8697
Energy Consumption (kWh)	57.0512	87.4397	122.7422	179.4114	254.2722
Fuel Consumption (L)	188.7654	294.3058	547.8256	836.4006	1195.8823
$\mathrm{Emissions} \mathrm{of} \mathrm{C}\mathrm{O}$ (g)	1772.5272	2636.041	4423.3931	6981.6978	9312.854
$\mathrm{Emissions} \mathrm{of} \mathrm{H}\mathrm{C}$ (g)	1675.2486	2509.3512	4166.5697	6710.571	8685.2047
$\mathrm{Emissions} \mathrm{of} {\mathrm{N}\mathrm{O}}_x$(g)	219.3792	325.0354	559.2722	815.0247	1120.1315
Intersection Total Cost (CNY)	1929.2419	3005.1623	5540.0784	8453.6878	12,057.3681

,

Table 10. Intersection performance statistics under optimized timing scheme.

	#1	#2	#3	#4	#5
Vehicle Delay (s)	392.138	771.4562	1348.9405	2151.1903	3123.3264
Energy Consumption (kWh)	40.4244	68.5363	104.707	157.2438	227.0482
Fuel Consumption (L)	86.5224	143.172	346.3006	633.7863	957.1958
$\mathrm{Emissions} \mathrm{of} \mathrm{C}\mathrm{O}$ (g)	907.0668	1570.4222	3429.9555	5016.2112	6665.4561
$\mathrm{Emissions} \mathrm{of} \mathrm{H}\mathrm{C}$ (g)	1202.4486	1922.2762	3521.6897	5797.071	7868.4847
$\mathrm{Emissions} \mathrm{of} {\mathrm{N}\mathrm{O}}_x$(g)	134.3934	189.2664	398.5358	597.0636	868.3775
Intersection Total Cost (CNY)	908.4718	1511.8237	3561.8294	6438.1922	9676.3169

and

Table 11. Degree of improvement in intersection performance indicators.

	#1	#2	#3	#4	#5
Vehicle Delay (s)	−37.64%	−24.66%	−22.11%	−14.21%	−12.87%
Energy Consumption (kWh)	−29.14%	−21.62%	−14.69%	−12.36%	−10.71%
Fuel Consumption (L)	−54.16%	−51.35%	−36.79%	−24.22%	−19.96%
$\mathrm{Emissions} \mathrm{of} \mathrm{C}\mathrm{O}$ (g)	−48.83%	−40.42%	−22.46%	−28.15%	−28.43%
$\mathrm{Emissions} \mathrm{of} \mathrm{H}\mathrm{C}$ (g)	−28.22%	−23.40%	−15.48%	−13.61%	−9.40%
$\mathrm{Emissions} \mathrm{of} {\mathrm{N}\mathrm{O}}_x$(g)	−38.74%	−41.77%	−28.74%	−26.74%	−22.48%
Intersection Total Cost (CNY)	−52.91%	−49.69%	−35.71%	−23.84%	−19.75%

Note: Negative values indicate reduction.

.

The results in Table 9 indicate that the initial road traffic congestion level significantly impacts intersection operational performance. As the degree of intersection congestion increases, traffic efficiency markedly declines, fuel consumption rises substantially, and pollutant emissions increase significantly. However, in the composition of the average total cost per vehicle at the intersection, the average vehicle delay still accounts for an absolute majority proportion, while the average fuel consumption cost and average pollutant emission cost constitute relatively minor shares. Consequently, their influence on traffic signal control strategies for the intersection remains limited.

Table 10 presents the optimized performance metrics, from which it can be seen that all indicators showed significant improvement compared to those before optimization.

The results in Table 10 and Figure 13 demonstrate that the proposed optimization model effectively improves the operational performance of the intersection. This is because while the Synchro method can optimize signal timing plans, it inadequately accounts for the impact of traffic congestion on road service rates (i.e., state-dependent service rates). Consequently, it estimates congestion dissipation processes using fixed service rates, leaving significant room for further optimization. In contrast, the proposed model incorporates state-dependent service rates, enabling it to accurately capture how changes in traffic congestion affect section service rates. This approach better aligns with actual traffic flow characteristics and thus more effectively enhances intersection traffic efficiency.

Figure 13. Comparison of indicators before and after optimization under varying congestion conditions. (a) Congestion conditions #1. (b) Congestion conditions #2. (c) Congestion conditions #3. (d) Congestion conditions #4. (e) Congestion conditions #5.

Figure 13. Comparison of indicators before and after optimization under varying congestion conditions. (a) Congestion conditions #1. (b) Congestion conditions #2. (c) Congestion conditions #3. (d) Congestion conditions #4. (e) Congestion conditions #5.

The degree of improvement in intersection performance indicators is shown in Table 10. Based on the optimization results, the following findings are noted:

(1)

The optimization method proposed in this study effectively reduces the average total cost for vehicles at the intersection. The results in Table 9, Table 10 and Table 11 demonstrate that by accounting for the impact of congestion on road service rates, the signal control scheme better adapts to traffic flow characteristics, unleashes the potential of signal timing optimization, and minimizes the total vehicle cost by up to 50%. Furthermore, as traffic congestion intensifies, the optimization efficacy of the model, while slightly diminished, remains above 20%;

(2)

The multi-objective optimization method proposed in this study effectively coordinates intersection performance metrics, including average vehicle delay, average fuel consumption, and average pollutant emissions. The optimized scheme demonstrates significant reductions in all these metrics—average vehicle delay, average fuel consumption, and average pollutant emissions—with most indicators decreasing by over 20%.

After analyzing the experimental results above, we can infer that the potential reasons may include the following:

(1)

State-Dependent Service Rate Modeling. The proposed model incorporates state-dependent service rates, dynamically capturing the impact of traffic congestion on road capacity. Unlike traditional methods (e.g., Synchro) that assume fixed service rates, this approach allows the signal control strategy to adapt in real-time to fluctuating traffic conditions, thereby significantly reducing delays and costs;

(2)

Multi-Objective Coordination Mechanism. By integrating multiple optimization objectives (delay, energy consumption, emissions) into a unified cost function with scientifically calibrated weights (e.g., based on disposable income, energy prices, and environmental taxes), the model effectively balances conflicting goals. This ensures synergistic improvements across all performance metrics rather than isolated gains;

(3)

Adaptability to Congestion Levels. Although optimization efficacy slightly decreases under severe congestion (e.g., from 50% to 20% improvement), the model maintains robust performance. This is attributed to its ability to prioritize critical cost components (e.g., delay costs dominating total costs) while still addressing secondary factors (e.g., emissions);

(4)

Algorithmic Advantages of MADS. The Mesh Adaptive Direct Search (MADS) algorithm efficiently handles the non-convex, integer-constrained optimization problem. Its global search capability (via search and poll steps) enables escape from local optima, ensuring the solution closely approximates the global optimum even for complex nonlinear systems;

(5)

Traffic Flow Physics Alignment. The feedback queuing network model accurately reflects fundamental traffic flow characteristics (e.g., speed-density relationships, acceleration-deceleration patterns), allowing the optimized signal timing to align with actual vehicle behaviors. This reduces unrealistic assumptions and enhances practical applicability;

(6)

Economic Incentive Integration. Quantifying delays and emissions into monetary costs (e.g., using disposable income for delay valuation and environmental taxes for emissions) creates a unified optimization framework. This directs the model toward societally optimal outcomes beyond purely engineering metrics.

These factors collectively explain why the proposed method outperforms conventional tools like Synchro, particularly in mixed traffic environments with high congestion variability.

5.3.4. Sensitivity Analysis

To quantify the trade-offs inherent in the multi-objective optimization problem and to assess the sensitivity of the solution to policy priorities, we conduct a weight sensitivity analysis under the different congestion scenario. The analysis focuses on three key cost weights: the value of time ($w_D$), the price of electricity ($w_E$), and the price of fuel ($w_{FC}$).

We employ a one-factor-at-a-time approach. Each weight is individually varied across a range from 0.25 to 4.0 times its baseline value (see Section 5.3.1), while all other weights are held constant. For each weight configuration, the optimization model is resolved using MADS, and the resulting optimal total cost, its component breakdown, physical performance metrics, and signal timing parameters are recorded.

Table 12. Total Cost Table under Different Weights.

Key Weight	Weight Factor	#1	#2	#3	#4	#5
$\mathrm{Value} \mathrm{of} \mathrm{Time} (w_D$)	0.25	1504.81	2434.18	4598.27	7101.10	10,248.76
	0.5	1620.56	2554.39	6260.29	7523.78	10,851.63
	1	1929.24	3005.16	5540.08	8453.69	12,057.37
	2	2102.87	3365.78	6315.69	9721.74	14,107.12
	4	2237.92	3576.14	6814.30	10,567.11	15,433.43
$\mathrm{Price} \mathrm{of} \mathrm{Electricity} (w_E$)	0.25	1888.73	2948.06	5390.50	8233.89	11,563.02
	0.5	1900.30	2963.09	5423.74	8318.43	11,792.11
	1	1929.24	3005.16	5540.08	8453.69	12,057.37
	2	1944.68	3044.23	5606.56	8597.40	12,382.92
	4	1973.61	3092.31	5739.52	8800.29	12,599.95
$\mathrm{Price} \mathrm{of} \mathrm{Fuel} (w_{FC}$)	0.25	1884.87	2848.89	5168.89	7752.03	10,972.20
	0.5	1906.09	2927.03	5312.94	8064.82	11,333.93
	1	1929.24	3005.16	5540.08	8453.69	12,057.37
	2	1979.40	3128.37	5911.26	9062.35	13,021.96
	4	1987.12	3125.37	5983.28	9383.59	13,624.83

synthesizes the Total Cost change in key metrics when each weight is quadrupled. And

Table 13. Sensitivity Analysis of Key Metrics to a Fourfold Increase in Individual Cost Weights.

Varied Weight (4× Baseline)	Δ Total Cost	Δ Average Delay	Δ Total Electricity Consumption	Δ Total Fuel Consumption
$w_D$ (Prioritize Efficiency)	28.1%	−7.9%	5.1%	10.3%
$w_E$ (Prioritize Electrification)	3.6%	1.8%	−14.4%	0.7%
$w_{FC}$ (Prioritize Fuel Economy)	12.8%	2.4%	1.4%	−13.7%

shows that the asymmetry in trade-offs is evident: heavily prioritizing delay reduction ($w_D\times 4$) increases total cost substantially (+28%) for a moderate delay gain (−7.9%), while significantly worsening energy and emission outcomes. Prioritizing a specific energy type ($w_E$ or $w_{FC}\times 4$) effectively curbs its consumption with a relatively minor impact on total cost and other objectives.

6. Discussion

This study proposed a Multi-Class Customer Feedback Queuing Network (MCFFQN) model and a corresponding signal optimization framework for intersections under mixed traffic conditions. The results presented in Section 5 demonstrate significant performance improvements. This section discusses the implications of these findings, links them to the original research questions and existing literature, and clarifies the contributions and boundaries of our work.

This study successfully achieved two core objectives. First, it developed a tractable mesoscopic mixed traffic flow model (MCFFQN). Validation results demonstrate that the model accurately captures system dynamics, such as queue lengths for different vehicle classes, with low error margins (Table 6). This success stems from its effective integration of state-dependent service rates (capturing congestion effects) and a feedback mechanism (capturing queue spillback), thereby overcoming the oversimplifications common in existing aggregate models. Building on this, the model was further applied to signal timing optimization, achieving a coordinated multi-objective optimization of efficiency, energy consumption, and environmental impact. A comparative analysis with Synchro (Section 5.3.2) shows that the proposed framework not only reduced delay in Scenario #1 by 37.64% but also concurrently cut energy use by 29.14% and fuel consumption by 54.16% (Table 11). This substantively shifts the optimization focus from a single objective (efficiency) to the multi-objective synergy required for sustainable traffic management, thereby affirming and advancing the research direction emphasized in recent literature [44,45].

The superior performance relative to the steady-state benchmark (e.g., Synchro) can be attributed to two fundamental modeling advances: first, the dynamic versus static representation—while traditional tools assume fixed saturation flow rates, the state-dependent service rate in our model dynamically adjusts section capacity based on real-time density, allowing the optimization to respond to actual fluctuating traffic conditions rather than average states. This explains why our approach maintains significant improvements even under oversaturated conditions (#5), where static models often fail. Second, the network effects enabled by feedback—unlike isolated intersection models, our feedback queuing mechanism accounts for the upstream propagation of congestion. This prevents the optimizer from generating locally efficient timing plans that merely shift queues to adjacent links, thereby yielding more robust and network-aware solutions.

This study bridges multiple research areas. While the use of queueing theory in traffic studies has a well-established foundation [22,23,25,46,47], existing models often treat vehicles as homogeneous. Recent research on mixed traffic has primarily focused on microscopic car-following stability [15,48] or macroscopic fundamental diagrams [17,18]. The novelty of the MCFFQN lies in its mesoscopic, multi-class queueing-theoretic abstraction, which is specifically reflected in two aspects: first, it operationalizes findings on stability/heterogeneity—the model translates mechanistic insights from microscopic car-following models (e.g., IDM, CACC) into mesoscopic service rates, thereby enabling these findings to be applied to network-level control; second, it provides a new optimization variable—it explicitly treats state-dependent capacity as an optimizable element, which is not addressed in conventional signal optimization tools or most macroscopic approaches.

7. Conclusions and Future Work

7.1. Conclusions

A mixed traffic flow with different types of vehicles is expected to be a common scenario for the foreseeable future. Understanding the characteristics of mixed traffic flow offers important support for managing and controlling mixed traffic flow. This study describes the dynamic nature of traffic flow through time-varying arrival rates; characterizes the randomness of traffic flow through queuing theory; describes the state-dependence (i.e., the impact of congestion on road service capacity) through feedback queuing theory; and describes the mixed traffic flow conditions through multi-class customer queuing theory, thereby developing a mixed traffic flow queuing theory model. To ensure that the model can be effectively applied to traffic management, this study also assessed indicators such as average vehicle delay, the average fuel consumption of fuel-powered vehicles, the pollutant emissions of fuel-powered vehicles, and the energy consumption of new energy vehicles based on queuing models. As the results obtained with the abovementioned models are closely related to the number of vehicles on the road section, it was necessary to first ensure the accuracy of the evaluation of the number of vehicles on the road section. Therefore, a microsimulation model of mixed traffic flow based on the SUMO simulation software was constructed to verify the effectiveness of the developed model. The results show that the multi-class customer feedback queuing network model developed in this article can effectively characterize the operating conditions of mixed traffic flow.

Furthermore, based on the multi-class customer queuing network model, this study developed a signal timing optimization scheme for intersections under mixed traffic flow conditions. This scheme aims to minimize vehicle delay, reduce the energy consumption of electric vehicles, decrease the fuel consumption of fuel-powered vehicles, and lower pollutant emissions from fuel-powered vehicles. Using a single intersection as a case study, the effectiveness of the optimized scheme was validated. The results demonstrate that the proposed signal control strategy simultaneously reduces all the aforementioned metrics, indicating its strong practical applicability.

It should be noted that the limitations of this article mainly relate to the fact that the developed MCFFQN model does not consider a situation with no signal control; therefore, in the actual application process, further improvements could be made to the MCFFQN model by applying it under conditions of no signal control. In addition, the MCFFQN model developed in this article does not take into account sloping urban roads, so there may be deviations in areas characterized by special terrains such as mountains. In the future, the team will continue to improve the model.

7.2. Future Work

This study has certain limitations, which also point to directions for future extension:

(1)

System Resilience and Robustness Testing: As discussed in Section 4.4, a crucial future direction is to apply the proposed model to explicit resilience stress-test scenarios. This includes simulating traffic incidents causing sudden local capacity drops, unforeseen demand pulses during peak hours, or sensor failures. We will define specific performance degradation (e.g., delay growth rate) and recovery metrics (e.g., number of cycles required to return to baseline performance) to quantitatively evaluate the resilience of different signal strategies under extreme conditions.

(2)

Fine-Grained Analysis of Vehicle Heterogeneity: While the current study validates the MCFFQN framework with two broadly defined yet representative vehicle classes, the model itself is capable of incorporating a more granular classification. A natural and important extension is to fully decouple attributes like propulsion type and automation level and to systematically evaluate traffic flow performance under all their cross-combinations. This would allow for the isolation of individual technological factors’ contributions to overall system efficiency, energy consumption, and emissions, providing deeper insights for policy and technology deployment.

(3)

Uncertainty Quantification and Robust Optimization: Future research will rigorously address parameter and demand uncertainty. This will involve: (i) conducting global sensitivity analysis (e.g., Sobol indices) to identify and rank the most influential parameters on key performance indicators; and (ii) formulating and solving a stochastic or robust optimization version of the signal timing problem, potentially incorporating chance constraints to limit the probability of excessive queue lengths or delays, thereby enhancing the reliability of control strategies under real-world variability.