Stochastic Optimization and Adaptive Control for Dynamic Bus Lane Management Under Heterogeneous Connected Traffic

Abstract

The efficiency of intelligent urban mobility increasingly depends on adaptive mathematical models that can optimize multimodal transportation resources under stochastic and heterogeneous conditions. This study proposes a Markovian stochastic modeling and metaheuristic optimization framework for the adaptive management of bus lane capacity in mixed connected traffic environments. The heterogeneous vehicle arrivals are modeled using a Markov Arrival Process (MAP) to capture correlated and busty flow characteristics, while the system-level optimization aims to minimize total fuel consumption through discrete lane capacity allocation. To support real-time adaptation, a Hidden Markov Model (HMM) is integrated for queue-length estimation under partial observability. The resulting nonlinear and nonconvex optimization problem is solved using Genetic Algorithm (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO), ensuring robustness and convergence across diverse traffic scenarios. Numerical experiments demonstrate that the proposed stochastic–adaptive framework can reduce fuel consumption and vehicle delay by up to 68% and 65%, respectively, under high saturation and connected-vehicle penetration. The findings verify the effectiveness of coupling stochastic modeling with adaptive control, providing a transferable methodology for energy-efficient and data-driven lane management in smart and sustainable cities.

1. Introduction

The rapid evolution of smart cities is transforming urban mobility through the integration of intelligent, connected, and sustainable transportation systems. As global urbanization accelerates, cities are increasingly challenged by congestion, excessive fuel consumption, and environmental degradation. Intelligent Transportation Systems (ITS) have therefore become a cornerstone for achieving efficient and sustainable mobility by leveraging data-driven and model-based control technologies to optimize traffic operations and energy utilization [1,2,3]. Public transportation, in particular, plays a pivotal role in promoting carbon neutrality and social equity, as demonstrated by recent smart transit initiatives in the United States, Australia, and China [4,5,6].

Among various ITS strategies, bus lane management is recognized as one of the most effective approaches to enhance transit efficiency and reliability in dense urban areas [7,8]. Conventional bus lanes (BLs), while guaranteeing priority for public transport, often suffer from inefficiencies such as unauthorized vehicle intrusion, underutilization during off-peak hours, and increased congestion in adjacent lanes during peak demand [9,10]. To overcome these limitations, Dynamic bus lane (DBL) strategies enable flexible lane capacity allocation in response to temporal traffic variations. Early models such as the Intermittent Bus Lane (IBL) [11] and Bus Lane with Intermittent Priority (BLIP) [12,13] attempted to optimize road-space usage through rule-based switching mechanisms. However, these methods exhibited limited adaptability to stochastic traffic variations and frequently failed to maintain operational stability under high saturation conditions [14,15].

Recent advances in connected vehicle (CV) technologies have created new opportunities for adaptive and cooperative management of DBL systems. Through vehicle-to-everything (V2X) communication, connected vehicles can share speed, position, and signal phase information in real time, enabling predictive traffic control and coordinated bus prioritization [16,17]. Previous studies, including those by Guler et al. [18] and Zhao and Zhou [19], demonstrated that cooperative signal control combined with dynamic bus lane activation can substantially improve intersection throughput and travel time reliability. Building upon these findings, further developments—such as dynamic or cooperative bus lane priority algorithms [20,21]—have shown the potential of CV-assisted control in enhancing efficiency and sustainability. Nevertheless, most of these studies assume deterministic or homogeneous traffic conditions, neglecting the stochastic interactions between connected and non-connected vehicles. This simplification restricts the scalability and robustness of existing models in realistic heterogeneous traffic environments.

Modeling traffic flow under mixed connected environments requires a stochastic mathematical framework capable of capturing both correlated arrivals and random variability. Queueing theory provides a rigorous foundation for describing such dynamic processes, particularly at signalized intersections [22,23]. The Markov Arrival Process (MAP), an extension of the Poisson process, is widely recognized for its ability to represent bursty and correlated traffic arrivals [24,25,26]. For example, Alfa and Neuts [25] and Mirzaeian et al. [26] verified that MAP-based models can effectively characterize vehicular arrival dependencies in intelligent transportation systems. More recent studies have successfully employed MAP structures to optimize traffic signal control [27] and highway toll operations [28,29]. However, their integration into bus lane capacity optimization frameworks remains largely unexplored. This research gap highlights the necessity of a stochastic–adaptive modeling approach that can represent time-varying heterogeneous flows while supporting analytical optimization and control.

In parallel, traffic state estimation under partial observability has been extensively investigated using the Hidden Markov Model (HMM) framework. HMMs are particularly suitable for reconstructing unobservable states—such as queue length and congestion level—based on incomplete probe-vehicle observations [30,31]. Hao et al. [32] and Zhao et al. [33] demonstrated that HMM-based inference can accurately estimate queue length distributions even under low connected-vehicle penetration. The fusion of HMM estimation with stochastic optimization therefore provides a promising basis for developing adaptive and data-driven control strategies. Despite these advances, only limited research has explored how real-time estimation mechanisms can be mathematically embedded within optimization frameworks for bus lane management in connected and heterogeneous environments.

While previous MAP- or HMM-based studies [27,28,29,32,33] have effectively modeled correlated traffic arrivals or estimated hidden queue states, they treated these two aspects separately—focusing either on stochastic arrival modeling or on real-time state estimation. In contrast, the present study integrates both MAP-based stochastic flow modeling and HMM-based adaptive estimation into a unified optimization–control framework. Moreover, unlike earlier deterministic or single-stage approaches, our model embeds metaheuristic optimization (GA/DE/PSO) directly into the stochastic–adaptive structure, allowing coordinated offline capacity allocation and online queue regulation under heterogeneous connected traffic. This coupling of stochastic theory, probabilistic estimation, and evolutionary optimization constitutes a key advancement beyond existing MAP/HMM-based traffic control studies.

To address these limitations, this study proposes an adaptive mathematical optimization–control framework for the Connected Dynamic Bus Lane (C-DBL) system at isolated intersections under heterogeneous traffic conditions. The framework integrates MAP-based stochastic queueing modeling, HMM-based real-time estimation, and metaheuristic optimization algorithms—including Genetic Algorithm (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO)—to achieve energy-efficient and adaptive lane management. Specifically, the model determines the optimal lane capacity allocation that minimizes total fuel consumption and vehicle delay while enabling online adjustments according to real-time traffic states. The proposed methodology contributes to the literature by combining stochastic modeling, adaptive control, and evolutionary optimization within a unified mathematical structure, offering a scalable solution for sustainable and intelligent urban mobility.

The main contributions of this research are fourfold:

• A stochastic queuing model for heterogeneous connected traffic. A MAP-based queuing framework is developed to capture dynamic arrival correlations between connected and conventional vehicles in mixed traffic flows, improving the realism of traffic representation.
• An intelligent optimization algorithm for lane capacity management. A non-linear optimization problem is formulated to minimize total fuel consumption, solved via metaheuristic techniques suited for complex and nonconvex systems.
• An adaptive control mechanism integrating real-time estimation. The HMM-based queue estimation model enables online adjustments of the C-DBL activation and capacity according to observed traffic conditions, ensuring responsive and stable operation.

2. C-DBL System and Control Strategy

2.1. System Description

As shown in Figure 1, the C-DBL system includes a dedicated bus lane alongside adjacent general-purpose lanes (for through, left, or right turns). Its primary purpose is to provide a partially dedicated lane for CVs. CVs are required to have mutual communication capabilities to collect real-time vehicle queue data. Buses utilize global positioning systems or wireless communication features to capture their status. Roadside devices acquire and forward real-time vehicle queue and signal phase and timing (SPaT) information to a cloud server for data analysis and planning. In the C-DBL system, the trajectory of CVs after entering the system can be divided into four main segments: (1) changing lanes to enter the C-DBL; (2) queuing within the C-DBL during the red signal phase; (3) waiting to depart; and (4) accelerating to a specified speed.

Based on the description, we establish a dual-layer optimization-control framework for C-DBL, as illustrated in Figure 2. The upper layer is an optimal model based on SQM, designed to determine the optimal capacity configuration of the C-DBL under varying traffic scenarios. The lower layer is a dynamic control decision model that estimates the traffic situation in the social lane (SL) using the characteristics of connected vehicles and indicates when CVs should enter the C-DBL based on the obtained optimal capacity. Specifically, the control strategy is triggered at the start of each intersection signal cycle ${(C}_i, i=\mathrm{1,2},\dots$), when the signal light turns red, the first action is to determine the number of queued vehicles $V$ in the SL during the current signal cycle ($C_i)$. If $V$ reaches the optimal configuration $L^{*}$, the second action is to determine whether the last bus is within the priority access range. If it is not, the CV will then enter the C-DBL.

In the proposed C-DBL system, the lane prioritization policy is designed as a hierarchical structure. Under normal conditions, the C-DBL maintains bus priority to ensure reliable public transport operation to ensure public transport reliability. However, during non-bus phases or when the bus headway exceeds a threshold, the C-DBL can be adaptively shared with connected vehicles (CVs) from adjacent social lanes. The access decision is governed by the real-time queue state estimated via the HMM-based module, ensuring that shared use by CVs does not interfere with scheduled bus operations. Therefore, the lane prioritizes buses first while conditionally admitting CVs when capacity permits.

2.2. Vehicle Queue Detection Using HMM

In connected vehicle environments, probe vehicle data can be used to estimate queue lengths at signalized intersections. This process is modeled using an HMM, where the queue lengths are treated as hidden states, and the stopping positions of probe vehicles serve as observations.

The sequence of queue lengths $V=\{V_1,V_2,\dots ,V_C\}$ follows a time-homogeneous Markov process, with the transition probabilities describing how queue lengths evolve between cycles. The observations $q=\{q_1,q_2,\dots ,q_C\}$ represent the partial queues observed by probe vehicles, with the emission probabilities relating these observations to the underlying queue lengths.

The emission probability $P(q_i\mid V_i)$ for a given cycle is defined by

$$P\left(q_i∣V_i\right)=\left\{\begin{array}{c}\begin{array}{ll}{p}^{n_i}{(1-p)}^{{V_i-n}_i}& if V_i\ge \left|q_i\right|\end{array}\\ \begin{array}{ll}0& if V_i\le \left|q_i\right|\end{array}\end{array}\right.$$

(1)

where $n_i$ is the number of probe vehicles in cycle $i$, and $p$ is the penetration rate of probe vehicles.

To estimate queue lengths cycle-by-cycle, we apply the Maximum Likelihood Estimation (MLE) using the Viterbi algorithm. This approach finds the most probable sequence of queue lengths given the observed data by maximizing the likelihood of the sequence. The transition probabilities, emission probabilities, and initial state probabilities are required for this estimation process. For details on the HMM, please refer to the literature [32,33].

2.3. Transit Priority Communications Range

A networked environment facilitates the direct transmission of real-time data to the traffic signal control system. To ensure seamless execution of prioritized access, the control system must determine the minimum upstream distance required to initiate prioritization decisions. This distance is referred to as the minimum communication range $D_{pr}$.

Unlike traditional fixed-position detection systems [18], this distance is variable within a connected environment. Therefore, the activation range for priority should be dynamically adjusted based on real-time bus kinematics and current traffic conditions. Let us assume that the signal cycle duration at the target intersection is $c$, with a green light duration of $g$, a red-light duration of $r$, and that the allowable throughput capacity of the C-DBL is $L^{*}$ (in pcu/cycle). According to traffic flow theory, the required activation distance can be expressed as follows

$$D_{pr}\left(i\right)\ge L^{*}\left(i\right)+\mathrm{m}\mathrm{a}\mathrm{x}\{r_{re}\left(i\right)L^{*}\left(i\right){\displaystyle \frac{k_{jam}g}{r}},v_{bus}{r}_{re}\}$$

(2)

where $k_{jam}$ denotes traffic density, $r_{re}$ denotes the remaining red-light duration after the last CV enters during the $i$-th signal cycle, and $v_{bus}$ indicates the speed of the connected bus. Additionally, the capacity $L^{*}\left(i\right)$ of the C-DBL during the i-th signal cycle will be discussed in the following section.

2.4. Traffic Flow Modeling in Heterogeneous and Connected Environments

We consider the arrival process of hybrid traffic flow, comprising both connected vehicles (CVs) and conventional vehicles (NCVs), within a networked environment. The arrival process of these vehicles is modelled using a MAP, which effectively captures the correlation and heterogeneity of vehicle arrivals in intelligent transportation systems [25,26].

Let the state of the underlying Markov process governing the vehicle arrivals be denoted as $v_{t, }t\ge 0$, and the state space is $\{\mathrm{0,1},\dots ,M\}$. The dynamics of the process $v_t$ follow an irreducible continuous-time Markov chain with transition rates defined by the infinitesimal generator $D$. For two vehicle types, the MAP is completely characterized by the state space of the underlying process and the corresponding transition matrices $D_k$ for each type of vehicle, where $k=\mathrm{1,2}$ with $D_0$ representing the rate matrix when no vehicle arrives.

The vehicle arrival rates for each type $k$ (with $k=1$ for CVs and $k=2$ for NCVs) are computed as

$${\lambda }_k=\vartheta D_{k}e, k=\mathrm{1,2}$$

(3)

where $\vartheta$ is the steady-state probability vector of the underlying Markov process $v_t$, and $e$ is a column vector of ones. The total arrival rate of vehicles, $\lambda$, is the sum of the individual rates

$$\lambda ={\lambda }_1+{\lambda }_2$$

(4)

The variance of the inter-arrival times for vehicles of type $k$ is expressed as

$$v\left(k\right)=2\vartheta {\left(-D_0-D_k\right)}^{-1}e-{\left({\displaystyle \frac{1}{{\lambda }_k}}\right)}^2, k=\mathrm{1,2}$$

(5)

The correlation coefficient $c\left(k\right)$ between consecutive inter-arrival times for vehicles of type $k$ is given by

$$cor\left(k\right)={\displaystyle \frac{[\vartheta {\left(D_0+D_k\right)}^{-1}{\lambda }_k{D}_k{\left(D_0+D_k\right)}^{-1}e-\frac{1}{{\lambda }_i^2}]}{v\left(k\right)}}$$

(6)

Detailed definitions of the abbreviations and mathematical symbols covered in this study can be found in the Abbreviations.

3. Stochastic Queuing Model and Performance Index

3.1. System State

To construct a theoretical model of the queuing process at a signalized intersection under the C-DBL system. Let us define two phase matrices, $H^G$ and $H^R$, representing the system states during the green and red phases, respectively. Each matrix encodes the active duration of its corresponding signal phase within one complete cycle. Specifically, $H^G\in {\mathbb{R}}^{\left(g+r\right)\times g}, { H}^R\in {\mathbb{R}}^{\left(g+r\right)\times r}$, where g and r denote the number of discrete time steps (or phase units) of the green and red lights, respectively. For the matrix $H^G$, the elements are defined as follows: if $i=\mathrm{1,2},\dots ,g-1$ or $i=r,j=1$, then $H_{(i,j+1)}^G=1$; otherwise, $H_{(i,j)}^G=0$. For the matrix $H^R$, the elements are defined as follows: if $i=\mathrm{0,1},\dots ,r-1$, then $H_{(g+i,j+1)}^R=1$; otherwise, $H_{(g+i,j)}^R=0$.

At time $n$, let $L_n$ represent the number of vehicles in the normal lanes and $K_n$ denote the number of vehicles in the C-DBL. Additionally, $H_n$ refer to the values of the green and red phases, respectively, and $J_n$ indicate the state of the MAP. Thus, the system state can be characterized by a Markov chain $\{L_n,K_n,S_n,J_n,n>0\}$ with a state space

Let $l$ and $k$ denote the number of social vehicles in the social lane and C-DBL, respectively, and $s$ denote the signal state at time $t_n$, where $l\ge 0, 0\le k\le V$. The state space $\mathrm{\varnothing }$ of the process $\{L_n,K_n,S_n,n>0\}$ can be represented as

$$\begin{array}{cc}\hfill \mathrm{\varnothing }=\{\left(l,0,s,j\right),l\ge 0& , 1\le s\le g+r,1\le j\le M\}\hfill \\ & \cup \left\{\begin{array}{c}\left(l,k,s,m\right),0\le l\le V,1\le k\le \mathit{min}\left(k,V\right),\\ V-l\le s\le V-k\end{array}\right\}\hfill \\ & \cup \left\{\begin{array}{c}\left(l,k,s,m\right),l\ge V,1\le k\le \mathit{min}\left(k,V\right),\\ 1\le s\le V-k\end{array}\right\}\hfill \\ & \cup \left\{\begin{array}{c}\left(l,k,s,m\right),l\ge V,1\le k\le \mathit{min}\left(k,V\right),\\ g+l\le s\le c\end{array}\right\}\hfill \end{array}$$

(7)

The state $\mathrm{\varnothing }$ is arranged in lexicographic order, and the infinitesimal generator of the Markov chain $\{L_n,K_n,S_n,J_n,n>0\}$ exhibits the following block structure

$$Q=\left[\begin{array}{cccccc}{B}_0& B_1& 0& \cdots & 0& 0\\ A_2& A_1& A_0& \cdots & 0& 0\\ & A_2& A_1& A_0& ⋮& 0\\ & & \ddots & \ddots & \ddots & ⋮\\ 0& 0& \cdots & A_2& A_1& A_0\\ 0& 0& 0& \cdots & A_1& A_0\end{array}\right]$$

(8)

where

$$B_{i,j}^0=\left\{\begin{array}{c}\begin{array}{ll}\left[0,H^R⨂D_1\right]& 0\le i\le V-1;j=i+1\\ {[H}^G⨂D_0,0]& 1\le i\le V;j=i-1\\ H^G⨂D_0,H^R⨂D_0]& i=j=0\\ \left[0{,H}^R⨂D_0\right]& 1\le i=j\le V\end{array}; \end{array}\right.$$

$$B_{i,i}^1=\begin{array}{cc}diag\left\{\left[H^G⨂\left(D_1+D_2\right),H^R⨂D_2\right]\right\}& 1\le i\le V\end{array}$$

$$A_{i,i}^0=diag\left\{\begin{array}{cc}\left[0,H^R⨂D_2\right]& 1\le i\le V\end{array}\right\}$$

$$A_{i,j}^2=\left\{\begin{array}{c}\begin{array}{ll}{[H}^G⨂D_0,0]& 1\le i\le V;j=i-1\\ {[H}^G⨂D_0,0]& i=j=0\end{array}\end{array}\right.$$

$$A_{i,j}^1=\left\{\begin{array}{c}\begin{array}{ll}[0,H^R⨂D_1]& 0\le i\le V-1;j=i+1\\ \left[H^G⨂(D_1+D_2) H^R⨂D_0\right]& 1\le i=j\le V\end{array}\end{array}\right.$$

3.2. Model Solution

Assume that the Markov process $\{L_n,K_n,S_n,J_n,n>0\}$ is irreducible and defined within a finite state space. Consequently, for any specified set of system parameters, there exists a unique stationary distribution $\mathrm{\hslash }$ associated with the process.

$$\hslash Q=0;\hslash e=1$$

(9)

where $e$ is the unit column vector, $\mathrm{\hslash }=({\mathrm{\hslash }}_0,{\mathrm{\hslash }}_1,{\mathrm{\hslash }}_{2,}\dots )$ and ${\mathrm{\hslash }}_j^i=({\mathrm{\hslash }}_{j1}^i,{\mathrm{\hslash }}_{j2}^i,\dots ,{\mathrm{\hslash }}_{jV}^i)$.

As shown in Equation (9), the stationary distribution $\mathrm{\hslash }Q=0$, which can be solved using standard linear algebra methods. However, solving high-dimensional systems becomes challenging due to constraints on computational memory and processing speed. To resolve these issues, a specialized stable algorithm is recommended. Algorithm 1 summarizes the computational procedure based on the matrix-geometric method to obtain the stationary distribution and key performance metrics.

The solution process for the stochastic queueing model is summarized in Algorithm 1, which is based on the matrix-geometric method for level-dependent quasi-birth-and-death (QBD) processes. The goal of this algorithm is to derive the stationary probability vector $\mathrm{\hslash }$ and the rate matrix $R$, which describe the steady-state behavior of the system under heterogeneous traffic arrivals modeled by the MAP structure. The main inputs, intermediate computations, and outputs are detailed step-by-step below.

Algorithm 1. Solving Process for Random Queueing Models
Step	Description
Input: Block matrices $\{B_0,B_1,A_0,A_1,A_2\}$ from the infinitesimal generator of the MAP/QBD system; convergence threshold $\epsilon$. Output: Steady-state probability vectors $\mathrm{\hslash }$ and the rate matrix $R$.
Step 1. Initialization	Import the block matrices $\{B_0,B_1,A_0,A_1,A_2\}$ representing transitions among boundary and internal levels. Set iteration counter $k=0$ and initialize $R^{\left(0\right)}=0$
Step 2. Ergodicity Check and Base Probability	Compute the stationary vector $\mathrm{\hslash }$ satisfying $\mathrm{\hslash }{\sum }_{i=0}^2{A}_i=\mathrm{\hslash };\pi e=1$. The stability (ergodicity) condition $\mathrm{\hslash }{A}_{0}e<\mathrm{\hslash }{A}_{2}e$ is verified to ensure that the system converges to a steady state.
Step 3. Computation of the Rate Matrix $R$	Iteratively solve $R$ from the nonlinear matrix equation $A_0+R{A}_1+R^2{A}_1=0$. Using the functional iteration $R\left[k+1\right]=-(A_0+R^2{A}_1){A_2}^{-1}$, repeat until $\left|\right|R(k+1)-R\left(k\right)\left|\right|<\epsilon$.
Step 4. Boundary Probability Computation	Form the boundary-level blocks $\left({\mathrm{\hslash }}_0,{\mathrm{\hslash }}_1\right)$ by solving $\left({\mathrm{\hslash }}_0,{\mathrm{\hslash }}_1\right)=\left(\begin{array}{cc}{B}_0& B_1\\ A_2& A_1+R{A}_2\end{array}\right)$, together with the normalization condition ${\mathrm{\hslash }}_{0}e+{\mathrm{\hslash }}_1{(I-R)}^{-1}e=1$
Step 5. Computation of Level Probabilities	Obtain the stationary probability vectors for higher levels as ${\mathrm{\hslash }}_n={\mathrm{\hslash }}_1{R}^{n-1},n\ge 1$ for, $n\ge 1$. This step yields the limiting distribution of queue states under steady conditions.
Step 6. Performance Metric Evaluation	Based on {${\mathrm{\hslash }}_n$}, compute expected queue length, average delay, and fuel consumption as given in Equations (10)–(15).

3.3. Performance Metrics

3.3.1. Vehicle Average Queue Length and Delay

The average queue length and delay are computed according to Equations (10) and (11), respectively, which link the expected number of vehicles to the system’s service rate.

$$\overline{L}={\sum }_{l+k=0}^{\infty }\left(l+k\right){\sum }_{s=1}^{g+r}{\sum }_{j=1}^M{\hslash }_{l,k,s,j}$$

(10)

$$W={\displaystyle \frac{E\left(L\right)}{\lambda }}={\displaystyle \frac{{\sum }_{l+k=0}^{\infty }\left(l+k\right){\sum }_{s=1}^{g+r}{\sum }_{j=1}^M{\hslash }_{l,k,s,j}}{{\sum }_{k=1}^2\vartheta D_{k}e}}$$

(11)

3.3.2. Queuing of Vehicles in SL

The number of queued vehicles in the SL can serve as a straightforward indicator of the effectiveness of C-DBL system, compared against the average total queue length. Let’s denote ${\overline{L}}_S$ as the number of vehicles in the social lane.

$${\overline{L}}_S={\sum }_{l=0}^{\infty }{\sum }_{k=0}^V{\sum }_{s=1}^{g+r}{\sum }_{j=1}^M{k\hslash }_{l,k,s,j}$$

(12)

3.3.3. Maximum Queue Length for Vehicles

Let us designate ${\overline{L}}_{max}$ as the maximum total number of social vehicles at signalized intersection and ${\overline{L}}_{sl-max}$ as the maximum number of vehicles in the social lane.

$${\overline{L}}_{max}={\sum }_{l+k=0}^{\infty }(l+k){\sum }_{j=1}^M{\hslash }_{l,k,(g+r),j}$$

(13)

$${\overline{L}}_{sl-max}={\sum }_{l=0}^{\infty }{\sum }_{k=0}^{V}k{\sum }_{j=1}^M{\hslash }_{l,k,(g+r),j}$$

(14)

3.3.4. Fuel Consumption of Vehicles in a Cycle

Depending on the state of travel of the vehicle at the intersection (acceleration, deceleration, idling, constant speed). The total fuel consumption of the vehicle can be calculated using the formula provided by Wu et al. [34].

$$F={\sum }_{i=1}^{\lambda T\left(g+r\right)}{F}_i=\lambda (g+r)({\overline{\mathcal{R}}}_a{\overline{t}}_a+{\overline{\mathcal{R}}}_d{\overline{t}}_d{+\overline{\mathcal{R}}}_i{\overline{t}}_i{+\overline{\mathcal{R}}}_c{\overline{t}}_c)$$

(15)

where $F$ represents the total fuel consumption of all vehicles passing through the intersection; $F_i$ represents the fuel consumption of vehicle $i$ passing through the signalized intersection; ${\overline{\mathcal{R}}}_a$, ${\overline{\mathcal{R}}}_d$, ${\overline{\mathcal{R}}}_i$, ${\overline{\mathcal{R}}}_c$ denote the average fuel consumption rate of the vehicle under acceleration, deceleration, idling, and constant speed, respectively; and ${\overline{t}}_a$, ${\overline{t}}_d$, ${\overline{t}}_i$, ${\overline{t}}_c$ represent the duration of the vehicle under acceleration, deceleration, idling, and constant speed, respectively.

Define the number of stranded vehicles ${\overline{L}}_g$ at the moment of the end of the green light by the following equation

$${\overline{L}}_g={\sum }_{l+k=0}^{\infty }(l+k){\sum }_{j=1}^M{\hslash }_{l,k,g,j}$$

(16)

Thus, the total number of vehicles stops under the C-DBL system is divided into the following two scenarios.

• Non-Congested $(L_g<V)$

In this scenario, the intersection is in a non-congested state. Social vehicles will not receive the information to enter the BL. The total number of stops $N_1$ can be calculated using the following equation

$$N_1={\overline{L}}_{sl-max}+min({\displaystyle \frac{{\overline{L}}_{sl-max}\lambda }{1-\lambda }},\lambda g)$$

(17)

• Congested $(L_g\ge V)$

In this situation, the signalized intersection is in a congested state, and the C-DBL provides passage for social vehicles. At this time, the total number of stops ${ N}_2$ can be expressed as

$$N_2=2\left({\overline{L}}_g-V\right)+min\{{\displaystyle \frac{\mathit{min}[\left({\overline{L}}_g-V\right),{\overline{L}}_{sl-max}]\lambda }{1-\lambda }},\lambda g\}$$

(18)

Therefore, the number of stops under non-congested and congested conditions are described by Equations (17) and (18), and the stopping rate per vehicle is calculated by Equation (19).

$$R_s={\displaystyle \frac{N_1+N}{(g+r)\lambda }}$$

(19)

The total vehicle fuel consumption through signalized intersection can be expressed as

$$F=\lambda \left(g+r\right)\left[\left({\overline{\mathcal{R}}}_a+{\overline{\mathcal{R}}}_d\right)·R_s·{\overline{t}}_d{+\overline{\mathcal{R}}}_i·W\right]$$

(20)

where ${\overline{t}}_d$ is the average time for each acceleration and deceleration of the vehicle.

4. Optimal Capacity Allocation

4.1. Problem Definition

The optimization problem is formulated in Equation (21), where the nonlinear objective function $F\left(L\right)$ in Equation (22) minimizes total fuel consumption derived from Equation (20).

$$\begin{gathered} F\left(L^{*}\right)=\min {\displaystyle \frac{F\left(L\right)-F\left(0\right)}{F\left(0\right)}} \\ s.t. 0\le L\le L_{max}, L\in N \end{gathered}$$

(21)

where $F\left(L\right)$ denotes the total fuel consumption of all vehicles under lane configuration $L$, and $F\left(0\right)$ represents the baseline fuel consumption when no C-DBL control is implemented. The normalization by $F\left(0\right)$ allows for a dimensionless performance index, facilitating comparison across different traffic demand levels or arrival modes.

The fuel consumption function $F\left(L\right)$ is a non-linear, non-convex function of $L$, where changes in the allocated bus lane volume result in variations in the traffic flow dynamics and fuel consumption of vehicles. The objective function is often expressed in the form of

$$F\left(L\right)=C_s·R_s{\left(L\right)·{\overline{t}}_d+C}_d·W\left(L\right)$$

(22)

where $C_s$ denotes the fuel consumption of the vehicle at the intersection due to acceleration and deceleration and $C_d$ denotes the fuel consumption of the vehicle at the intersection due to idling.

4.2. Solution Methodology

The high-dimensional stochastic system is solved using the matrix-geometric method summarized in Algorithm 1 to solve the optimization problem, three well-known optimization algorithms—Differential Evolution (DE), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO)—are employed. These algorithms are adapted for discrete optimization by ensuring that the bus lane allocation $V$ is treated as an integer variable., Algorithm 2 shows the pseudo code for solving the optimal C-DBL volume. The solution methodology is as follows:

• Initialization: Define a population of candidate solutions (individuals or particles), randomly initialized within the feasible solution space for $L$ (i.e., $L_{min}\le L\le L_{max}, L\in N$).
• Fitness Evaluation: Evaluate the fitness of each candidate solution by computing the objective function $F\left(L\right)$. The fitness is based on minimizing the fuel consumption over time, which depends on the lane allocation $L$.

$$Fitness \left(L_i\right)=F\left(L_i\right)$$

(23)

Algorithm 2. GA pseudo-code for solving the optimal C-DBL volume

INPUT: Objective function F(L)

OUTPUT: The optimal bus lane capacity L* that minimizes fuel consumption F(L)

Initialize:

Set the number of generations t = 0. Generate an initial population P(t) randomly within Lmin ≤ L ≤ Lmax.

Evaluate: Evaluate the population P(t) using the objective function F(L)

while termination criterion is not satisfied do

t = t + 1

Choose users to assemble P(t) from P(t − 1) based on their fitness.

Alter users of P(t) by applying crossover and mutation operations.

Apply mutation and crossover to generate new individuals (solutions for L) Ensure that L remains a discrete integer after mutation and crossover operations

end while

Return the optimal bus lane capacity L that minimizes the fuel consumption F(L), which corresponds to the best solution found during the evaluation.

5. Validation

5.1. Comparison with Existing Research

To demonstrate the feasibility of the C-DBL strategy, this paper conducted a numerical comparison with existing literature, including a DBL design proposed by Zhao and Zhou [19] to regulate the allowance of left-turning buses into the opposite dedicated bus lane, a pre-signal strategy proposed by He et al. [35] to reduce vehicle delay, and a DBL control strategy proposed by Shan et al. [36] in a partially connected vehicle environment, Wu et al. [17] developed and evaluated bus lanes with intermittent and dynamic priority in connected vehicle environments. All papers provide detailed data analysis of their respective systems.

As shown in Figure 3a, under varying traffic saturation levels (0.2–1.0), the proposed C-DBL strategy demonstrates significant improvement in reducing traffic delays compared to three state-of-the-art DBL methods. At low saturation levels (0.2–0.4), the C-DBL strategy achieves a 12–18% improvement over the model proposed by Wu et al. [17]. For medium saturation levels (0.6–0.8), it surpasses the rule-based strategy developed by Zhao and Zhou [19] by 22–27%. Under high saturation conditions (V/C = 1.0), the C-DBL strategy reduces delays by 35%, highlighting its advantage in congested scenarios compared to He et al.’s [35] traffic-responsive system. Figure 3b highlights the energy efficiency comparison between C-DBL and Shan et al.’s [36] hybrid DBL framework. Under peak hour conditions, the proposed strategy reduces fuel consumption by 40.55%, outperforming Shan et al.’s [36] benchmark by 19.13%. The comparative analysis confirms that the C-DBL strategy is a technically feasible and environmentally beneficial solution for modern transportation systems.

5.2. Simulation

To validate the effectiveness of the proposed methodology, we developed four representative traffic scenarios through the SUMO simulation platform. A comprehensive comparative analysis was conducted among three approaches: (1) the baseline SUMO simulation, (2) HMM with varying connected vehicle penetration rates (10%, 50%, 90%), and (3) the proposed method, with particular focus on vehicle queue length estimation performance.

As illustrated in Figure 4a,b, The proposed method demonstrates remarkable consistency with SUMO simulation results in vehicle queue length estimation, while exhibiting significantly lower estimation errors compared to traditional HMM approaches. Quantitative error metrics reveal that our method achieves an RMSE of 3.33, maintaining minimal deviation from the SUMO baseline. This performance substantially outperforms HMM implementations with 10% penetration rate (RMSE = 47.8981) and 50% penetration rate (RMSE = 5.4231), while approaching the accuracy level of HMM-90% implementation (RMSE = 2.8825). In terms of MAPE metric, the proposed method achieves 0.55%, demonstrating clear advantages over HMM-10% (6.08%) and HMM-50% (0.70%), with only marginal difference compared to HMM-90% (0.46%). These quantitative findings collectively demonstrate that the proposed MAP-based SQM effectively characterizes traffic flow performance in connected environments, showing strong agreement with simulation benchmarks. The results validate the method’s generalization capability and engineering applicability for real-world implementations.

6. Numerical Experiment

To ensure the reproducibility and transparency of the proposed experiments, the simulation configuration is summarized in

Table 1. Simulation Settings for the C-DBL Numerical Experiments.

Category	Parameter/Description	Value/Setting
Simulation Environment	Platform	SUMO 1.19.0 (for microscopic traffic simulation) + MATLAB 2023a (for optimization and analysis)
	Processor and Memory	Intel Core i5-9300H (2.4 GHz), 16 GB RAM (Intel Corporation, Santa Clara, CA, USA)
Signal Control Parameters	Signal cycle length (C)	100 s
	Green phase duration (g)	50 s
	Red phase duration (r)	50 s
Road Configuration	Intersection layout	1 through lanes + 1 C-DBL
Traffic Composition	Vehicle types	CVs, NCVs and Buses
	Proportion	30% CVs, 60% NCVs, 10% Buses
Arrival Scenarios	Traffic saturation levels	0.3 (low), 0.6 (medium), 0.9 (high)
	Arrival modes	(i) Steady (M); (ii) Busty NCV (MAP1); (iii) busty CV (MAP2)
Optimization Parameters	Algorithms	GA, DE, PSO
	Population size	50
	Maximum generations	100
	Mutation/Crossover settings	Empirically tuned for convergence stability
Validation Setup	Simulation runs	30 independent runs per scenario with random seeds

. The model integrates the MAP-based stochastic traffic flow generation, HMM-based queue estimation, and GA/DE/PSO under heterogeneous connected traffic environments.

6.1. Optimal Capacity Configuration

To analyze the numerical results of the C-DBL optimal capacity allocation experiments for different traffic saturation and arrival modes, we evaluate the performance of three optimization algorithms (GA, DE and PSO) in terms of the optimal capacity $L^{*}$ and the associated fuel consumption $F\left(L^{*}\right)$.

As shown in

Table 2. Experimental parameter.

	M	MAP1	MAP2
I	$D_0=-0.025$ $D=0.025$	$D_0=\left[\begin{array}{cc}-0.065& 0.024\\ 0.008& -0.028\end{array}\right]$ $D=\left[\begin{array}{cc}0.04& 0\\ 0& 0.02\end{array}\right]$	$D_0=\left[\begin{array}{cc}-0.065& 0.024\\ 0.008& -0.028\end{array}\right]$ $D=\left[\begin{array}{cc}0.04& 0\\ 0& 0.02\end{array}\right]$
II	$D_0=-0.1$ $D=0.1$	$D_0=\left[\begin{array}{cc}-0.542& 0.203\\ 0.027& -0.095\end{array}\right]$ $D=\left[\begin{array}{cc}0.339& 0\\ 0& 0.068\end{array}\right]$	$D_0=\left[\begin{array}{cc}-0.086& 0.032\\ 0.092& -0.323\end{array}\right]$ $D=\left[\begin{array}{cc}0.054& 0\\ 0& 0.231\end{array}\right]$
III	$D_0=-0.225$ $D=0.225$	$D_0=\left[\begin{array}{cc}-0.438& 0.164\\ 0.008& -0.281\end{array}\right]$ $D=\left[\begin{array}{cc}-0.274& 0\\ 0& -0.201\end{array}\right]$	$D_0=\left[\begin{array}{cc}-0.274& 0.104\\ 0.114& -0.399\end{array}\right]$ $D=\left[\begin{array}{cc}0.171& 0\\ 0& 0.285\end{array}\right]$

, the experiments consider three traffic saturation levels (I: 0.3; II: 0.6; III: 0.9) and three vehicle arrival modes: stable arrival (M), conventional vehicle burst arrival (MAP1) and CVs burst arrival (MAP2).

From

Table 3. Optimal capacity allocation results.

Saturation Level	Arrival Mode	Optimization Algorithm	Optimal Capacity ${\mathit{L}}^{\mathit{*}}$ (pcu/cycle)	FC Reduction (%)
0.3	Steady (M)	GA	2	0.143
		DE	2	0.143
		PSO	2	0.143
	Burst (MAP1)	GA	2	0.042
		DE	2	0.042
		PSO	2	0.042
	Connected Burst (MAP2)	GA	2	0.896
		DE	2	0.896
		PSO	2	0.896
0.6	Steady (M)	GA	2	2.322
		DE	2	2.322
		PSO	2	2.322
	Burst (MAP1)	GA	4	1.436
		DE	4	1.436
		PSO	4	1.436
	Connected Burst (MAP2)	GA	3	18.599
		DE	3	18.599
		PSO	3	18.599
0.9	Steady (M)	GA	5	27.858
		DE	5	27.858
		PSO	5	27.858
	Burst (MAP1)	GA	5	27.027
		DE	5	27.027
		PSO	5	27.027
	Connected Burst (MAP2)	GA	5	50.766
		DE	5	50.766
		PSO	5	50.766

, it can be observed that as the saturation level increases or as the arrival pattern transitions from steady to busty, the optimal capacity shows an increasing trend. The improvement in fuel consumption is particularly significant under high saturation or high variability arrival patterns. The optimization effects of the three algorithms are generally similar; however, slight differences exist in certain scenarios, indicating that the capacity configuration problem proposed in this study, C-DBL, exhibits good adaptability to various evolutionary algorithms.

It is noteworthy that under a saturation level of 0.9 and a bursty arrival pattern (especially MAP2), the maximum reduction in fuel consumption can exceed 50%. This finding validates that a reasonable configuration of lane capacity can effectively reduce queuing and idling energy consumption, thereby significantly alleviating the energy waste and emission pressure associated with congestion. Overall, the experimental results provide strong evidence for the feasibility and robustness of capacity optimization based on C-DBL across diverse traffic scenarios, offering substantial support for energy savings and emission reductions in intelligent transportation systems operating in high-traffic or highly variable environments.

6.2. Results of the C-DBL Control Strategy

This section analyzes the effect of adaptive C-DBL and fixed-capacity C-DBL strategies on vehicle fuel consumption reduction under different traffic saturation levels, based on experimental results with a penetration rate of 0.9 (Figure 5a) and 0.3 (Figure 5b) for connected vehicles, respectively. As shown in Figure 5, the adaptive C-DBL consistently outperforms the fixed-capacity C-DBL as the traffic saturation increases. Notably, at a traffic saturation of 1.2, the adaptive C-DBL achieves a fuel consumption reduction of about 67%, which is more than that of the fixed-capacity scheme (57%), demonstrating its excellent adaptability under high saturation and high penetration conditions.

In Figure 5, the adaptive C-DBL shows an advantage even though the overall fuel consumption reduction is small. At a traffic saturation of 1, the adaptive strategy reduces fuel consumption by 15%, while the fixed capacity approach reduces fuel consumption by 9%. Together, these results validate that the adaptive C-DBL control strategy can reduce fuel consumption more effectively than the fixed-capacity scheme at different traffic saturation levels, regardless of whether the penetration of connected vehicles is high or low.

7. Sensitivity Analysis

7.1. Sensitivity to Random Traffic Flows

This study evaluates the impact of the C-DBL strategy on traffic performance across varying saturation levels (0.1 to 0.9). Figure 6 illustrates the key effects on vehicle delay, fuel consumption, and the number of stops. The data presented in Figure 6a clearly demonstrate that C-DBL outperforms the C-DBL control strategy across all metrics at different saturation levels. At a saturation level of 0.1, the optimization improvements for C-DBL in terms of delay, fuel consumption, and the number of stops is 0.47%, 0.18%, and 0.10%, respectively. As saturation increases to 0.9, CDBL achieves a delay of 34.75%, showing significant optimization compared to DBL in terms of reduced delay, fuel consumption, and the number of stops.

C-DBL consistently exhibits stronger performance than DBL in alleviating congestion, reducing fuel consumption, and minimizing stops across the entire traffic volume range, with particularly pronounced improvements at higher traffic volumes. Figure 6b further indicates that the performance enhancements of C-DBL relative to DBL in terms of traffic intensity and vehicle queue length significantly increase with rising saturation levels. This suggests that C-DBL possesses greater adaptability and cooperative control advantages under high-flow and even oversaturated conditions, effectively reducing vehicle queuing and significantly alleviating congestion while lowering energy consumption.

7.2. Sensitivity to CVs Penetration

To evaluate the applicability and robustness of the proposed C-DBL strategy under varying CV penetration, we conduct numerical experiments at 10%, 20%, 40%, 60%, 80%, and 100% CV market shares. As shown in Figure 7a, when the CV penetration is only 10%, C-DBL reduces vehicle delays, fuel consumption, and number of stops by approximately 11.9%, 9.3%, and 9.3%, respectively, compared with the conventional DBL approach. As the penetration rate increases to 100%, these improvements further rise to 56.1%, 51.3%, and 59.3%, respectively, indicating that higher CV penetration enables C-DBL to leverage vehicle connectivity and real-time information sharing more fully for significant performance gains.

Figure 7b provides additional insight into C-DBL’s sensitivity to CV penetration by examining queue lengths. Specifically, as the CV penetration rate increases from 10% to 100%, the queue length of social vehicles (i.e., non-connected vehicles) decreases markedly from approximately 14.95 NCVs units (pcu) to 5.94 pcu. Meanwhile, the queue length of connected vehicles increases from near 0.0 pcu to about 2.37 pcu. This redistribution of queues highlights C-DBL’s ability to dynamically balance congestion between social and connected vehicles in mixed traffic settings. Overall, these findings confirm the robustness and adaptability of C-DBL across a wide range of CV penetration levels, demonstrating its potential to enhance network-wide efficiency in evolving connected and automated transportation systems.

7.3. Sensitivity to the C-DBL Capacity

To comprehensively assess the applicability of the C-DBL strategy under varying traffic congestion conditions, this study conducts numerical research based on four key performance indicators: vehicle delay, fuel consumption, number of stops, and queue length in SL. The analysis spans saturation levels from 0.1 to 0.9 and congestion levels from 1 to 7.

As illustrated in Figure 8a, the optimal lane-capacity configurations converge stably across increasing traffic saturation levels under GA, DE, and PSO. For instance, under low saturation (0.1) and mild congestion (congestion level 1), the optimization is only 0.54%. In contrast, at high saturation (0.9) and severe congestion (capacity level 7), the optimization reaches 41.81%. These results indicate that the C-DBL strategy is more effective at reducing vehicle delays when the road network is under high load. Furthermore, at the same saturation level (0.9), the optimization exhibits a nonlinear variation with increasing capacity of C-DBL (initially rising from 6.37% to a peak of 43.39%, then declining to 37.57%), suggesting that C-DBL requires the integration of other control mechanisms to maintain stability under moderate to high congestion conditions. Figure 8b shows that total fuel consumption decreases monotonically as the allocated capacity approaches its optimum, but improves significantly when saturation is ≥0.5 and congestion level is ≥3 (increasing from 5.32% to 31.41%).

The results presented in Figure 9a demonstrate a positive correlation between fuel consumption optimization and saturation levels. Under low congestion (congestion levels 1 to 2), the optimization of fuel consumption increases gradually with saturation, while under high congestion (congestion levels 6 to 7), the optimization effect improves significantly, indicating a strong dependency of C-DBL’s fuel consumption optimization performance on traffic volume thresholds. Figure 9b further highlights the optimization advantage of C-DBL regarding societal lane queue length, achieving an optimization rate of 50.48% under a saturation level of 0.9 and moderate congestion (congestion level 4), demonstrating the significant role of dynamic lane allocation in mitigating congestion spillover in bottleneck areas.

Overall, the response of the C-DBL strategy to congestion levels exhibits notable nonlinear characteristics. Under high saturation and severe congestion conditions, the strategy excels in reducing vehicle delays and alleviating societal lane queue lengths; however, its optimization effects are limited by activation thresholds in low saturation or mild congestion scenarios.

7.4. Sensitivity to Traffic Flow Heterogeneity

This section analyzes the sensitivity of C-DBL performance to traffic flow heterogeneity, considering Poisson arrivals, MAP I, and MAP II. As shown in Figure 10a, under high traffic saturation (0.9), system idleness is highly sensitive to traffic heterogeneity. For MAP I, SL exhibits a relatively low inter-arrival time variance (0.692) and moderate correlation (2.127), whereas CV presents a significantly higher variance (4.255) and stronger correlation (2.375). This stark contrast results in generally lower idleness, primarily due to the bursty arrivals of CV, which increase C-DBL utilization. In contrast, MAP II balances variance (SL: 1.257; CV: 3.121) while maintaining stable correlation (~1.9–2.3), leading to a relatively higher idleness compared to MAP I. The Poisson model demonstrates the least sensitivity, with idleness decreasing by only 1.5% (from 0.447 to 0.44), as its inherent randomness mitigates the impact of saturation-driven resource contention.

As shown in Figure 10b, at low saturation (0.3), reduced congestion leads to stable system idleness across all traffic models. Despite the higher variance of vehicle 2 (5.891), MAP I maintains a relatively low idleness (decreasing by 2.5%, from 0.161 to 0.166), as resource redundancy compensates for sporadic arrivals. MAP II further highlights its resilience, with moderate variance (SL: 2.161; CV: 2.337) and reduced correlation (~1.67–1.98), resulting in a slight idleness decrease (1.0%, from 0.312 to 0.309). The Poisson model remains stable (idleness: 0.447 to 0.44), underscoring its robustness under low-density conditions.

8. Discussion

This study developed a stochastic–adaptive optimization framework for Connected Dynamic Bus Lane (C-DBL) systems, integrating MAP-based stochastic queue modeling, HMM-based real-time estimation, and metaheuristic optimization to achieve coordinated lane capacity management in heterogeneous connected traffic. Compared with previous MAP-or HMM-based traffic control studies [27,28,29,32,33], the proposed framework establishes a closed-loop stochastic–adaptive mechanism that simultaneously captures correlated arrivals (via MAP) and real-time hidden-state feedback (via HMM), which were previously treated as independent sub-problems. This integration enables adaptive bus-lane capacity management under realistic heterogeneous connected environments.

The results reveal that the proposed C-DBL strategy effectively reduces intersection delay and fuel consumption—achieving up to 68.3% reduction in fuel use and 65.5% reduction in average delay—while maintaining system stability across varying traffic saturations and connected-vehicle penetration rates. The integration of MAP and HMM enables accurate representation of stochastic and partially observable traffic dynamics, while the application of GA, DE, and PSO ensures robust global optimization in nonconvex solution spaces.

From a broader perspective, this research contributes a generalizable stochastic–adaptive control paradigm applicable to other intelligent transportation systems. The combination of probabilistic modeling, real-time estimation, and evolutionary optimization offers a scalable approach for managing uncertainty, enhancing energy efficiency, and supporting sustainable urban mobility planning.