A Time-Dependent Dijkstra’s Algorithm for the Shortest Path Considering Periodic Queuing Delays at Signalized Intersections

Abstract

In urban road networks, queuing delays at signalized intersections often account for over half of the total travel time. The complexity of traffic signals and vehicle queuing makes traditional shortest path algorithms insufficient for real-time optimal path finding. This study proposes a Time-Dependent Dijkstra’s algorithm to address these challenges. The network topology is redesigned to model vehicle turning behaviors accurately. A periodic queuing delay parameter matrix for signalized intersections is introduced, storing traffic flow and signal phase parameters. Additionally, a time-varying weight matrix tracks the vehicle’s position in the signal cycle upon intersection arrival. Using cumulative curve theory, a periodic queuing-delay model is constructed to capture delays for vehicles arriving at different times. The algorithm updates the network weight matrix in real-time based on vehicle arrival times at intersections, enabling FIFO-consistent time-dependent shortest path computation for a given departure time. Numerical and SUMO simulations on a real-world road network in Suzhou Industrial Park (comprising 15 signalized intersections and 22 road segments) demonstrate the algorithm’s effectiveness. Results show a 25.36% reduction in travel time compared to the traditional Dijkstra’s Algorithm and a 10.46% reduction compared to an algorithm considering only signalized intersection waiting time when departure times vary. The results highlight the impact of periodic queuing delays, with the algorithm reducing travel time and improving path planning.

1. Introduction

With the rapid development of the Internet of Vehicles (IoV) and Cooperative Vehicle-Infrastructure Systems (CVIS), vehicle path navigation systems have become indispensable in modern urban traffic. However, as urbanization speeds up and traffic networks grow more complex, traditional path navigation systems increasingly show limitations in handling dynamic and time-varying traffic environments. The core function of modern vehicle path navigation systems is optimal path planning. At the heart of this planning is the Shortest Path Algorithm (SPA), which addresses the Shortest Path Problem (SPP) by identifying the path with the least travel time from start to end. Over the past decades, many scholars have proposed various SPAs [1,2,3,4]. However, existing SPAs primarily focus on path distance and time. In actual urban traffic networks, intersections are key nodes. The periodic changes in traffic lights and the presence of traffic flow can cause queuing delays at intersections. These delays vary with traffic light phases, making real-time and accurate calculation of the shortest path and travel time challenging. In most existing SPP research, scholars employ static traffic network models with fixed node distances or costs, which fail to accurately reflect actual vehicle turning movements at intersections [5,6]. Although many researchers have considered the impact of road segment impedance on optimal path costs in SPAs [7,8,9], these studies primarily focus on long-distance road network route calculations. In urban traffic networks, the prevalence of intersections, particularly signalized ones, along with short intersection spacing, means intersection related delays significantly affect overall path selection [10,11]. During traffic congestion, these delays can even exceed adjacent road segment travel times, highlighting static network models’ limitations in urban applications.

Existing SPAs mainly consider two types of signalized intersection delays: average delay and waiting time. Intersection average delay, caused by traffic flow and signals, is the average vehicle delay at intersections, manifesting as reduced speed and stops due to queue formation or upstream deceleration. Some studies have incorporated intersection average delay into SPA, showing its importance for path selection [12,13,14]. However, as traffic signals and flow are constantly changing, using average delay alone is insufficient for real-time path planning.

Signalized intersection waiting time is the duration an individual vehicle waits at the stop line due to traffic signals at an intersection in a zero-flow network. For navigation users, varying departure times from the origin mean vehicles encounter different signal phases at intersections. Given the fixed and periodic signal cycles with specific phase sequences over time, the waiting time at intersections changes with departure times, leading to varying optimal path results. However, few scholars have examined the impact of waiting time at signalized intersections in zero-flow networks on path selection and computational results in SPAs [15]. These studies have demonstrated that signal lights influence vehicles’ subsequent route choices and waiting times at intersections. Although some algorithms have improved the real-time and accuracy of the SPP by considering signalized intersection waiting times, they have not further integrated traffic flow at intersections to assess delay times.

Additionally, traffic simulation is important for verifying the feasibility of the method [16], but some researchers have only conducted numerical verification [17], which may lead to significant discrepancies from real-world results.

In this study, we redesigned the network model’s topology to accurately represent turning movements at intersections. By considering the Signalized intersection waiting time and the time-varying signal cycles in traffic flow networks, and integrating cumulative curve theory, we propose a method for real-time calculation of periodic queuing delays at signalized intersections. This allows us to express the real-time queuing delays for vehicles passing through intersections at different departure times. By appropriately storing various parameters, such as link and intersection distances, speeds, signal timings, and traffic flow, and incorporating these real-time data into Dijkstra’s algorithm for dynamic weight updates, we propose a time-dependent Dijkstra’s algorithm (TDSP). This improved algorithm better matches the actual urban road network and offers more precise optimal path results. To verify the effectiveness and accuracy of the proposed algorithm, we use both numerical simulations and SUMO 1.21.0 software simulations.

2. Literature Review

This study is closely related to the literature on optimization algorithms for the SPP and TDSP. As a classic problem in graph theory, the SPP has been extensively researched and improved upon by numerous scholars to address SPP in traffic applications. Dijkstra (1959) introduced the Dijkstra’s algorithm, a greedy based approach for solving the SPP, which is highly efficient and accurate in static networks [18]. Floyd (1962) presented the Floyd-Warshall algorithm, a dynamic programming-based method for computing shortest paths between all node pairs [19]. Hart et al. (1972) proposed the heuristic A* algorithm, which uses a heuristic function to accelerate path searching [20]. These classical algorithms, especially Dijkstra’s algorithm, have laid the foundation for subsequent research. With the advancement of computing technology, researchers have concentrated on enhancing the performance of SPAs in complex networks and dynamic environments to meet the demands of diverse scenarios. In the research on uncertain environments and complex networks, Xu et al. (2007) introduced an improved Dijkstra’s SPA for sparse networks, incorporating a weighted constraint function to better suit complex network environments [21]. Enayattabar et al. (2019) presented a fuzzy Dijkstra’s algorithm for uncertain environments, using fuzzy numbers to handle edge weights and address the traditional Dijkstra’s algorithm’s limitation in dealing with uncertainty [22]. Other studies integrate SPAs with reinforcement learning [23], SLAM-based vehicle planning [24], hybrid deep models [25], and multi-objective optimization frameworks [26,27], demonstrating the growing trend of SPA enhancement for diverse mobility scenarios. Although the aforementioned algorithms are significant in SPP research, most studies focus on static traffic networks and fail to adequately consider traffic flow impacts, resulting in algorithms that cannot reflect real-time traffic conditions. More importantly, in dynamic traffic networks, signalized intersection waiting times and intersection delays in urban traffic networks are not fully considered. This makes optimal path selection less precise during peak hours or under complex road conditions, limiting the algorithms’ practical application in urban traffic networks.

Our study also addresses intersection delays caused by traffic flow and signals. The signalized intersection delay model that divides delays into normal delays from fixed-mean vehicle arrivals and additional delays from random arrivals is a classic established foundation. Many subsequent studies, such as those solving time-window problems [28,29,30], have built upon this model. Several algorithms have considered intersection delays. Ziliaskopoulos and Mahmassani (1996) proposed an improved SPA that incorporated intersection turning delays and prohibited movements, though the delays were not real-time [31]. Chai et al. (2017) developed a model integrating dynamic traffic path planning and adaptive signal control, showing through simulations that it could reduce average delays and boost network performance under varying traffic demands [32]. Tu et al. (2020) studied the reliable SPP for electric vehicles in urban traffic networks and accounted for road speed correlations and intersection delays in their model [33]. Ju and Du (2022) presented a superpath search algorithm considering intersection delays, using an extended network representation based on signal timings to capture delays for different turning movements [34]. However, its delay values did not reflect traffic flow impacts, limiting its application. Despite these efforts, in urban traffic shortest path studies, intersection delays, influenced by signal lights, cannot be treated as average delays but should be regarded as periodic queuing delays, which are crucial for accurate shortest path solutions.

Our study also focuses on the impact of signalized intersection light states, yet few current studies address this aspect. In static road networks, Chen and Yang (2000) presented a novel SPA for urban traffic networks with signal constraints [35]. It enhanced path planning accuracy by considering signals but focused on static models. In addressing the aspect of signal light periodicity, Ahuja (2002) explored finding minimal-time and minimal-cost paths in street networks with periodic traffic signals [36]. Khanjary et al. (2011) investigated SPAs in synchronized traffic signal networks, optimizing path choices to cut waiting times by analyzing signal periodicity and phase info [37]. Hu et al. (2016) proposed a real-time SPA, enhancing the A* algorithm by considering traffic signal waiting times but not signal phase differences or traffic flow [38]. Sun et al. (2023) presented a reliable SPA for stochastic time-varying traffic networks, improving Dijkstra’s algorithm by incorporating signalized intersection waiting times [39]. However, the phase logic of waiting times in their study did not match reality, and intersection traffic flow was not considered. Shen et al. (2019) proposed a reliable energy saving path-planning algorithm for electric vehicles in stochastic road networks [40]. Based on Dijkstra’s and K SPAs, it incorporates road speed and signalized intersection waiting time. Although existing research has made progress in considering traffic light status impacts on path planning, most methods still have shortcomings. For instance, some studies only examine signalized intersection waiting times in zero-flow road networks, assuming fixed waiting times.

While TDSP algorithms have been extensively studied, few incorporate a physics-based, real-time delay model designed for the periodic queuing phenomena at signalized intersections, where delays vary predictably with both signal cycles and traffic flow. Specifically: (a) Network models often remain static, failing to explicitly represent turning movements at intersections where significant delays occur. (b) Intersection delay models frequently rely on average values (e.g., Webster’s model) or are calculated in zero-flow networks, neither of which captures the time-varying, periodic queuing delays experienced by individual vehicles under actual traffic flow. (c) Most importantly, few studies have integrated real-time signal phase states with actual traffic flow to compute dynamic, movement-specific delays, and subsequently embedded this real-time delay calculation into a dynamic shortest path algorithm. (d) Furthermore, validation often relies on numerical analysis alone, lacking verification through realistic microscopic traffic simulation.

3. Model Formulation

3.1. Network Topology Model

In urban traffic networks, vehicles approaching intersections typically face three turning options: going straight, turning right, or turning left. Conventional road network models struggle to effectively represent the turning information at intersection nodes. Additionally, the queuing delays at intersections account for a significant portion of the overall path traversal time, especially in high traffic flow networks. Some scholars have employed the dual graph method, treating dual arcs as turning behaviors in the network [41]. Nevertheless, this method involves a complex transformation of the road network and fails to explicitly reflect the queuing delays associated with different turning movements at intersections. The expanded road network approach requires an enlargement of the network scale, which can lead to increased complexity [42].

To better capture the real-time queuing situation of vehicles arriving at intersections, this study combines the ideas of the expanded road network method and the dual graph method to redesign the topology of the network model. Specifically, We split each physical intersection node into import sub-nodes (one per incoming lane) and export sub-nodes (one per outgoing lane). Edges between sub-nodes represent turning movements, enabling us to attach a unique queuing delay weight to each. This model not only explicitly represents the turning movements at intersections but also clearly shows the impact of each turning time on the overall path traversal time.

Take common cross-shaped and T-shaped intersections as examples. The division process and structure of the network model are shown in Figure 1a for the T-shaped intersection network topology and in Figure 1b cross-shaped intersection network topology diagram. for the cross-shaped intersection network topology. Figure 1 illustrates the process from a simple node diagram to a conventional road network and then to the global expanded road network of this study. For example, in the cross-shaped intersection shown in Figure 1a, there are four import nodes: $I_{1,a}^1$ represents the north import lane, $I_{1,a}^2$ the south import lane, $I_{1,a}^3$ the east import lane, and $I_{1,a}^4$ the west import lane. There are also four export nodes: $I_{1,e}^1$ represents the north export lane, $I_{1,e}^2$ the south export lane, $I_{1,e}^3$ the east export lane, and $I_{1,e}^4$ the west export lane. Through connection modeling, the straight-ahead, left-turn, and right-turn movements can be explicitly represented as edges, thereby clearly expressing the internal turning movements of the intersection in the network model.

The expanded network is formally defined as a directed graph $G=(V,E)$. The vertex set $V$ comprises mid-link nodes and intersection sub-nodes: import nodes $v_{i,d}^{in}$, and export nodes $v_{i,d}^{out}$, for each intersection I_i and approach direction d. The edge set Eincludes link edges (for road segments) and turning edges $(v_{i,d}^{in},v_{i,d}^{out})$ that explicitly represent turning movements. This structure combines the node expansion concept with the dual graph principle of representing turns as explicit edges.

3.2. Extended Network Weight Matrix

The traditional Dijkstra’s algorithm employs a fixed weight matrix for road network inputs, which fails to capture the dynamic nature of traffic. To better adapt to actual urban road conditions, this study expands the weight matrix by incorporating multiple actual parameters.

3.2.1. Basic Parameter Weight Matrices

(a) Distance Weight Matrix. The distance between nodes in a road network model is represented by a distance weight matrix D, which is of size $n\times n$. Here, $d_{uv}$ refers to the distance between nodes in the global expanded road network, as shown in Equation (1).

$$D=\left[\begin{array}{ccccc}{d}_{11}& d_{12}& d_{13}& \cdots & d_{1n}\\ d_{21}& d_{22}& d_{23}& \cdots & d_{2n}\\ d_{31}& d_{32}& d_{33}& \cdots & d_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ d_{n1}& d_{n2}& d_{n3}& \cdots & d_{nn}\end{array}\right]$$

(1)

(b) Speed Weight Matrix. In the road network model, the speed between nodes is represented by a speed weight matrix $\overline{V}$, as shown in Equation (2). The matrix has an $n\times n$ dimension, where ${\overline{v}}_{uv}$ denotes the speed between nodes in the global expanded road network.

$$\overline{V}=\left[\begin{array}{ccccc}{\overline{v}}_{11}& {\overline{v}}_{12}& {\overline{v}}_{13}& \cdots & {\overline{v}}_{1n}\\ {\overline{v}}_{21}& {\overline{v}}_{22}& {\overline{v}}_{23}& \cdots & {\overline{v}}_{2n}\\ {\overline{v}}_{31}& {\overline{v}}_{32}& {\overline{v}}_{33}& \cdots & {\overline{v}}_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\overline{v}}_{n1}& {\overline{v}}_{n2}& {\overline{v}}_{n3}& \cdots & {\overline{v}}_{nn}\end{array}\right]$$

(2)

3.2.2. Periodic Queuing Delay-Related Parameter Weight Matrices

In this study’s global expanded road network model, each edge within an intersection represents a specific turning movement. To reflect actual intersection conditions, we establish several weight matrices, including a signal cycle weight matrix, traffic flow weight matrix, green phase time per turning movement matrix, and time-varying weight matrix.

(a) Signal Cycle Weight Matrix. This matrix stores the signal cycle times for each intersection and is denoted as matrix $C$, as shown in Equation (3). For nodes within an intersection, the corresponding signal cycle value $c_{uv}$ is inputted, and the matrix has dimensions of $n\times n$.

$$C=\left[\begin{array}{ccccc}{c}_1& c_{12}& c_{13}& \cdots & c_{1n}\\ c_{21}& c_{22}& c_{23}& \cdots & c_{2n}\\ c_{31}& c_{32}& c_{33}& \cdots & c_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ c_{n1}& c_{n2}& c_{n3}& \cdots & c_{nn}\end{array}\right]$$

(3)

(b) Green Phase Weight Matrix. This matrix stores the green phase time per turning movement for each intersection signal cycle and is denoted as matrix $G$, as shown in Equation (4). For nodes within an intersection, the green phase time value $g_{uv}$ for the corresponding turning movement is inputted, and the matrix has dimensions of $n\times n$.

$$G=\left[\begin{array}{ccccc}{g}_1& g_{12}& g_{13}& \cdots & g_{1n}\\ g_{21}& g_{22}& g_{23}& \cdots & g_{2n}\\ g_{31}& g_{32}& g_{33}& \cdots & g_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ g_{n1}& g_{n2}& g_{n3}& \cdots & g_{nn}\end{array}\right]$$

(4)

(c) Traffic Flow Weight Matrix. The traffic flow weight matrix comprises an import flow matrix $Q$ and a saturation flow rate matrix $S$. Different flow matrices can be input for each time interval, as detailed in Section 3.3.

The import flow matrix $Q$ stores the traffic flow of vehicles arriving at different turning directions of an intersection within a specific time period, as shown in Equation (5). For each node within an intersection, the corresponding import flow value $q_{uv}$ is inputted, and the matrix has dimensions of $n\times n$.

$$Q=\left[\begin{array}{ccccc}{q}_1& q_{12}& q_{13}& \cdots & q_{1n}\\ q_{21}& q_{22}& q_{23}& \cdots & q_{2n}\\ q_{31}& q_{32}& q_{33}& \cdots & q_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ q_{n1}& q_{n2}& q_{n3}& \cdots & q_{nn}\end{array}\right]$$

(5)

Similarly, the saturation flow rate matrix $S$ stores the traffic flow of vehicles departing during the green phase at an intersection. The input method for matrix s is the same as that for matrix $Q$, as shown in Equation (6).

$$S=\left[\begin{array}{ccccc}{s}_1& s_{12}& s_{13}& \cdots & s_{1n}\\ s_{21}& s_{22}& s_{23}& \cdots & s_{2n}\\ s_{31}& s_{32}& s_{33}& \cdots & s_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ s_{n1}& s_{n2}& s_{n3}& \cdots & s_{nn}\end{array}\right]$$

(6)

(d) Time-Varying Weight Matrix. As shown in Equation (7), the time-varying weight matrix $T^t$ is used to dynamically store the real-time position in the signal cycle when vehicles arrive at each intersection node. This refers to the time interval from the end of the current green phase to the end of the signal cycle when the signal starts operating. The matrix is updated based on the system’s real-time operation to facilitate the calculation of periodic queuing delays at different times in Section 3.3.

$$T^t=\left[\begin{array}{ccccc}{t}_{11}^t& t_{12}^t& t_{13}^t& \cdots & t_{1n}^t\\ t_{21}^t& t_{22}^t& t_{33}^t& \cdots & t_{2n}^t\\ t_{31}^t& t_{32}^t& t_{33}^t& \cdots & t_{3n}^t\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ t_{n1}^t& t_{n2}^t& t_{n3}^t& \cdots & t_{nn}^t\end{array}\right]$$

(7)

In the initial stage, the initial time-varying matrix $T^0$ needs to be inputted. The initial time weight of $t_{uv}^0$ is determined by calculating the signal timing parameters of each intersection and analyzing the connections among the extended internal nodes of the intersection. The specific method is as follows. Suppose an intersection has $n$ phases and a signal cycle of $C_i$. The calculation formula is shown in Equation (8).

$$C_i={\displaystyle \sum }_n{P}_{n,i}$$

(8)

In Equation (8), $P_{n,i}$ represents the green phase time for the $n$ phase of the $i$ intersection node. Here, $t_s$ denotes the time when the first phase’s green light starts. Then, the $t_{uv}^{t_s}$ value corresponding to each phase is shown in Figure 2.

Thus, for each phase $m(1<m<n)$ of an intersection, the initial time-varying weight $t_{ij}^{t_s}$ is calculated using Equation (9).

$$t_{ij}^{t_s}=[t_s+{\displaystyle \sum _{i=m+1}^n{P}_i}]\mathrm{mod}(C)$$

(9)

Specifically, when m takes different values, the formulas are as shown in Equation (10).

$$\left\{\begin{array}{l}{t}_{ij}^{t_s}(m=1)=[t_s+{\displaystyle \sum _{i=2}^n{P}_i}]\mathrm{mod}(C_i),Phase 1\\ t_{ij}^{t_s}(m=2)=[t_s+{\displaystyle \sum _{i=3}^n{P}_i}]\mathrm{mod}(C_i),Phase 2\\ ⋮\\ t_{ij}^{t_s}(m=n-1)=[t_s+{\displaystyle \sum _{i=n}^n{P}_i}]\mathrm{mod}(C_i),Phase n-1\\ t_{ij}^{t_s}(m=n)=t_s\mathrm{mod}(C_i),Phase n\end{array}\right.$$

(10)

In summary, the initial time-varying weight matrix should be determined based on the number of phases at the intersection. Using Equation (9), calculate the initial value for each phase and input it into the time-varying weight matrix $T^t$ according to the corresponding phase of the extended sub nodes and edges.

In practical applications, different weight matrices are populated using different data sources. The distance matrix D and speed matrix $\overline{V}$ are obtained from static road network attributes or historical survey data. The signal cycle matrix C and green phase matrix G are derived from predefined signal timing plans. The traffic flow matrix Q and saturation flow matrix S can be updated using real-time traffic detector data or periodically updated historical observations.

3.3. Signalized Intersection Periodic Queuing Delay Calculation Model

Due to traffic signals at intersections, vehicles arriving there experience waiting times. Moreover, at actual urban intersections, queuing delays vary periodically with the traffic signal status. If the SPA could calculate real-time periodic queuing delays for each turning movement based on the actual traffic flow and signal status at intersections, path selection would better reflect actual urban road conditions.

3.3.1. Calculation Method Overview

In actual urban road networks, the periodic changes in traffic signals and fluctuations in traffic flow can cause queuing phenomena at intersections. Specifically, when the traffic light is red, vehicles slow down and stop due to the red signal, creating a stop wave starting from the first vehicle (#1), as shown in Figure 3a. As more vehicles arrive, the queue grows. When the signal turns green, vehicles begin to accelerate and depart from the intersection, starting with the first vehicle (#1), which creates a start-up wave and gradually dissipates the queue, as depicted in Figure 3b. Within the queuing section, vehicles are assumed to follow a non-overtaking (FIFO) discipline so that the cumulative-curve-based arrival-departure diagram yields a nondecreasing exit-time mapping with respect to arrival time.

To accurately capture the periodic queuing delays caused by preceding vehicles, this study uses the cumulative curve method [43] to build an Arrival-Departure Diagram for each vehicle. By showing the arrival and departure times at the intersection, it records the arrival time and models the periodic queuing delay for each vehicle. The diagram comprises an Arrival Cumulative Curve and a Departure Cumulative Curve, both affected by the signal, showing queue formation and dissipation. The queuing delay for each vehicle can be obtained from the diagram. Figure 4 illustrates the vehicle trajectories from Figure 3 and the corresponding Arrival-Departure Diagram derived from these trajectories and cumulative curves. The upper part of Figure 4 shows the spatiotemporal trajectory of nine vehicles entering and leaving the intersection’s stop line. From the trajectory, it can be seen that after vehicles enter the queuing section $l_q$, they will wait due to the signal. The time when vehicles leave the stop line is not simply the end of the red light.

The lower part of Figure 4 presents the Arrival-Departure Diagram based on cumulative curves. The x-axis denotes time, while the y-axis shows the cumulative number of vehicles. By projecting the times when nine vehicles enter the road section, we can derive the cumulative arrival flow curve. Similarly, projecting their entry times into the intersection’s queue segment $l_q$ yields the cumulative queue arrival curve $Q_1$. Projecting their departure times from the stop line gives the cumulative departure flow curve $S$. For each vehicle, the horizontal segment $t_t$ between $Q_1$ and $S$ represents the time from entering the queue section to departing from the stop line. This is the sum of the queue delay and the free-flow travel time through segment $l_q$. Note that vehicle #9 passes through without deceleration or stopping, indicating it traverses the queue section at the free-flow time $t_f$. By shifting $Q_1$ to the right by $t_f$, we obtain $Q_2$. The horizontal segment $t_q$ between $Q_2$ and $S$ then represents each vehicle’s queue delay.

Here, $t_f$ represents the effective queueing region for a specific turning movement (lane group), and the derived queue delay corresponds to the lane-group equivalent FIFO queue.

According to traffic wave theory, the cumulative queue arrival curve begins when the red light is on, and the cumulative departure curve starts when the green light is on. Moreover, if we can predict the arrival time of vehicles at the intersection and determine their arrival time on the diagram, we can calculate the corresponding queue delay for these vehicles.

3.3.2. Calculation of Periodic Queuing Delay at Signalized Intersections

Based on the above analysis, The signal status that a vehicle encounters upon arrival varies with the arrival time, directly affecting the periodic queuing delay. Thus, clarifying the vehicle’s arrival time at the intersection is crucial. As shown in Figure 5, the following relationships for intersection parameters are obtained, as shown in Equation (11).

$$\left\{\begin{array}{l}{t}_{a,j}=t_{d,i}+t_{ij}\\ t_{d,i}=t_{a,i}+t_{q,i}^t\end{array}\right.$$

(11)

In Equation (11), $t_{a,i}$ represents the vehicle’s arrival time at node i, $t_{d,i}$ denotes the vehicle’s departure time from node i, t₃ is the travel time from node i to node j, and $t_{q,i}^t$ represents the periodic queuing delay experienced by the vehicle when arriving at node i at time t.

(a) Intersection Parameter Relationships and Signal Phase Time Determination

In a connected vehicle environment, vehicles equipped with communication devices can access real-time information such as the distance to a signal and the signal status. If a vehicle departs intersection $i$ and the signal phase at downstream intersection $j$ is $t_{ij}^0$, the signal display time $t_{ij}^p$ when the vehicle arrives at intersection $j$ can be calculated using Equation (12), where mod denotes the modulo operation.

$$t_{ij}^p=\left[t_{ij}^0+t_{ij}\right]\mathrm{mod}\left(C_j\right)$$

(12)

(b) Periodic Queuing delay Calculation at Signalized Intersections

Using the approach in Section 3.3.1 and assuming single-point signal control at intersections, we can calculate the periodic queuing delay. As per Section 3.3.1, When a vehicle arrives at an intersection, the signal phase time is $t_{ij}^p$, the signal cycle is $C$, and there are $n$ phases. For each phase, then $n$ Arrival-Departure Diagrams need to be drawn and the periodic queuing delay for the corresponding phase of the subsequent road direction.

Figure 6 shows the Arrival-Departure Diagram for a specific phase. The specific calculation method is as follows: first, determine the time in the arrival-departure diagram. According to the signal phase time $t_{ij}^p$ (Equation (12)) when the vehicle arrives at the intersection, update the time-varying weight matrix $T^t$’s $t_{ij}^t$ to ${t_{ij}^t}^{\prime }$ using Equation (9). The new time-varying weight ${t_{ij}^t}^{\prime }$ moment is the vehicle’s arrival time in the arrival-departure diagram, and the calculation formula is shown in Equation (13).

$${t_{ij}^t}^{\prime }=[t_{ij}^p+t_{ij}^t]\mathrm{mod}(C_j)$$

(13)

Then, the periodic queuing delay time $t_{q,j}$ for vehicles arriving at intersection $j$ can be calculated using the Arrival-Departure Diagram. This is the time difference between the vehicle’s arrival time ${t_{ij}^t}^{\prime }$ and departure time $t_{l,j}$ from the stop line. The calculation formula for $t_{l,j}$ is shown in Equation (14).

$$t_{l,j}=t_{ry}+{\displaystyle \frac{{t_{ij}^t}^{\prime }{q}_{ij}}{s_{ij}}}$$

(14)

If the arrival time is greater than or equal to the queue dissipation point $t_{dq,j}$, which is the intersection of the cumulative arrival and departure curves, there is no periodic queuing delay. The calculation formula for $t_{dq,j}$ is shown in Equation (15).

$$t_{dq,j}={\displaystyle \frac{q_{ij}{t}_{ry}}{s_{ij}-q_{ij}}}$$

(15)

In summary, the periodic queuing delay $t_{q,j}$ for vehicles at intersections can be calculated using the formula shown in Equation (16).

$$t_{q,j}=\left\{\begin{array}{l}{t}_{ry}+{\displaystyle \frac{{t_{ij}^t}^{\prime }{q}_{ij}}{s_{ij}}}-{t_{ij}^t}^{\prime },{t_{ij}^t}^{\prime }<t_{dq,j}\\ 0,{t_{ij}^t}^{\prime }\ge t_{dq,j}\end{array}\right.$$

(16)

Assumption and scope. For the analytical expressions in Equations (14)–(16), the arrival rate and saturation flow are assumed constant within the evaluation interval (typically one signal cycle), which corresponds to a piecewise linear representation of the cumulative curves in the arrival–departure diagram. This assumption is used for computational tractability and is standard in cycle-based delay modeling. In practical deployment, non-uniform arrivals and time-varying discharge rates can be accommodated by updating the input matrices Q and S at a finer temporal resolution.

The calculation formula for the periodic queuing delay at signalized intersections is shown in Equation (17).

$$t_{q,j}=\max (t_{l,j}-{t_{ij}^t}^{\prime },0)$$

(17)

The model operates under the key assumptions of First-In-First-Out (FIFO) queue discipline and homogeneous vehicles for each turning movement. It considers a single, aggregated queue per approach. The presence of dedicated turning lanes, which can create lane-specific queues and alter discharge patterns, is a recognized complexity left for future model refinements.

(c) Calculation Example

To illustrate the calculation of periodic queuing delays at signalized intersections and the relationship between updating arrival times and delays in each direction, consider a four-phase intersection. Assume all directions have the same saturation and arrival flows. The green times are: Phase 1:15 s, Phase 2:20 s, Phase 3:10 s, and Phase 4:15 s, with a total cycle of 60 s. If a vehicle arrives when the signal display time is 20 s, and Phase 1’s green starts at 0 s, using Equation (13) gives arrival-departure times of 5 s, 45 s, 35 s, and 20 s for the four phases. Then, applying Equations (14)–(16) yields the periodic queuing delays, as shown in Figure 7.

The proposed periodic queuing delay model not only serves as the core component of the time-dependent Dijkstra’s algorithm but also provides a standalone theoretical framework for analyzing signalized intersection dynamics.

3.4. Total Path Weight Calculation

Path weight calculation includes link travel time and real-time periodic queuing delay at intersections. The link travel time weight matrix $T$ Equation (18) is calculated from the speed and distance weight matrices using Equation (19).

$$T=\left[\begin{array}{ccccc}{t}_{11}& t_{12}& t_{13}& \cdots & t_{1n}\\ t_{21}& t_{22}& t_{23}& \cdots & t_{2n}\\ t_{31}& t_{32}& t_{33}& \cdots & t_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ t_{n1}& t_{n2}& t_{n3}& \cdots & t_{nn}\end{array}\right]$$

(18)

$$t_{uv}={\displaystyle \frac{d_{uv}}{{\overline{v}}_{uv}}}$$

(19)

According to Section 3.3 and Equation (18), the final edge-weight matrix $w_{ij}^t$ for a path from node $i$ to node $j$ is calculated as the sum of the link travel time and the periodic queuing delay at intersections, as shown in Equation (20).

$$w_{ij}^t=t_{ij}+t_{q,j}$$

(20)

Finally, the final edge-weight matrix $W^t$ of the path is formed as shown in (Equation (21)).

$$W^t=\left[\begin{array}{ccccc}{w}_{11}^t& w_{12}^t& w_{13}^t& \cdots & w_{1n}^t\\ w_{21}^t& w_{22}^t& w_{33}^t& \cdots & w_{2n}^t\\ w_{31}^t& w_{32}^t& w_{33}^t& \cdots & w_{3n}^t\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ w_{n1}^t& w_{n2}^t& w_{n3}^t& \cdots & w_{nn}^t\end{array}\right]$$

(21)

On this basis, an time-dependent Dijkstra’s algorithm that accounts for periodic queuing delays at intersections can be constructed.

4. Time-Dependent Dijkstra’s Algorithm

4.1. Algorithm Concept Overview

The conventional Dijkstra algorithm assumes fixed edge weights, which makes it inadequate for urban signalized road networks where travel costs vary over time. The key enhancement of this study is to integrate the model developed in Section 3 into a TDSP framework, enabling dynamic updates of edge weights. Specifically, the network weight matrix is augmented with multiple traffic-flow-related parameters, and the edge costs are updated in real time to capture signal waiting times and traffic-flow-induced delays.

4.2. Algorithm Procedure

Step 1: Algorithm Initialization.

Step 1.1: Initialize data structures. For each node $u\in \{0,1,\dots ,n-1\}$, set the predecessor node array ${\mathrm{prev}}_u=-1$ to record the path predecessor nodes. The start time ${\tau }_s$ is set to the departure time $t_{depart}$.

Step 1.2: Set the source node parameters. Initialize the arrival time array $\tau =[{\tau }_u]$ with ${\tau }_u=+\infty$ for all nodes $u$. Use a priority queue $Q$, managed as a min-heap based on arrival time, and insert the source node $(t_{depart},s)$ into $Q$.

Step 1.3: Initial matrix processing. Compute the initial time matrix $T$ using Equation (18). Compute the initial time-varying weight matrix $T^0$ using Equation (9).

Step 2: Main Loop.

Step 2.1: Node selection and processing. Extract the node $u$ with the shortest arrival time from $Q$. If $u$ is the end node $e$, stop the loop and proceed to Step 3. If the current record time $curren{t}_{time}>{\tau }_u$, skip this node.

Step 2.2: Neighbor node iteration. For each neighbor node $v$ of $u$, check if there is an edge from $u$ to $v$.

Step 2.2.1: Signalized intersection check. Check if the edge from $u$ to $v$ is associated with a signalized intersection. If yes, proceed to Steps 2.2.2 and 2.2.3; otherwise, calculate $w_{uv}=t_{uv}$ and jump to Step 2.2.4.

Step 2.2.2: Update time-varying weight matrix. Update the time-varying weight matrix for each neighbor node $v$ of $u$ based on the arrival time ${\tau }_u$ and the formula from Section 3.3 (Equation (13)), considering the periodic signal cycle.

Step 2.2.3: Calculate periodic queuing delay. Calculate the periodic queuing delay $t_{q,u}$ using the updated time-varying weight matrix as per Section 3.3 and Equations (14)–(17).

Step 2.2.4: Total weight calculation and update. Calculate the total real-time weight $w_{uv}=t_{uv}+t_{q,u}$ by adding the periodic queuing delay from Step 2.2.3 to the initial time matrix.

Step 2.2.5: Calculate actual arrival time and perform relaxation. Compute the actual arrival time $t_{time}={\tau }_u+w_{uv}$. If $t_{time}<{\tau }_u$, update ${\tau }_v=t_{time}$, set ${\mathrm{prev}}_v=u$, and insert $(t_{time},s)$ into the priority queue $Q$, proceed to Step 2.1.

Step 3: Path Reconstruction and Result Output.

Initialize the path sequence ${\rm P}=\varnothing$. Starting from the end node $e$, trace back through the predecessor node array $\mathrm{prev}$ until the first node becomes the start node $s$. This generates the path sequence ${\rm P}$. Finally, calculate the total travel time as $T_{time}={\tau }_e-t_{depart}$.

The pseudocode for the time-dependent Dijkstra’s algorithm Considering periodic queuing delays at signalized intersections is shown in Algorithm 1.

Algorithm 1: An time-dependent Dijkstra’s algorithm accounting for periodic queueing delays at signalized intersections

4.3. FIFO Property and Algorithm Correctness

For label-setting algorithms like Dijkstra’s to guarantee optimality in time-dependent networks, the FIFO (First-In-First-Out) property must hold. This property, also known as the non-overtaking condition, states that an earlier departure from a node cannot result in a later arrival at the downstream node: if $t_1<t_2$, then $t_1+w_{uv}(t_1)\le t_2+w_{uv}(t_1)$. The edge weight function $w_{uv}(t)=t_{uv}+t_{q,uv}(t)$ defined by our periodic queuing delay model (Equations (16) and (17)) satisfies the FIFO property under the stable flow condition $q_{uv}\le s_{uv}$.

The free-flow travel time t is constant. The queuing delay $t_{q,uv}(t)$ is a piecewise function: it is either (i) a linear function $\alpha t+\beta$ with slope $\alpha =(q_{uv}/s_{uv}-1)$ when $t<{t_{d_i}}_j$, or (ii) zero when $t\ge {t_{d_i}}_j$. Given stable flow ($q_{uv}/s_{uv}\le 1$), the slope $\alpha$ is non-positive. Therefore, $t_{q,uv}(t)$ is a non-increasing function of $t$. Consequently, the composite function $f(t)=t+w_{uv}(t)=t+t_{uv}+t_{q,uv}(t)$ is non-decreasing. This satisfies the FIFO condition, ensuring that the standard Dijkstra’s procedure, when applied with our dynamically calculated weights, correctly finds the earliest-arrival path.

4.4. Algorithm Computational Performance

To evaluate the algorithm’s computational performance, we conducted tests using road networks of different scales, with the number of network nodes set at 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1000. For each network, we performed 100 runs to obtain the average computation time. The results are presented in Figure 8. Illustrating how the algorithm’s performance varies with the number of nodes. To accurately describe the relationship between runtime and node count, we fitted the test data. As shown in Figure 8a, the fitted curve shows a high degree of consistency with the actual data points.

The analysis in Figure 8b shows that the algorithm’s efficiency varies with the number of nodes. Below 500 nodes, the time growth rate is under 0.5 ms/node. Beyond this, the rate increases due to higher computational demands, reflecting the algorithm’s adaptive load-balancing mechanism. Similarly, the relationship between runtime and network nodes shows a fluctuating upward trend. The time-dependent Dijkstra’s algorithm is time-efficient for small and medium-scale road networks. However, its time growth rate increases significantly for large-scale networks. To meet practical time requirements, strategies like parallel computing or optimizing data storage methods can enhance its efficiency.

5. Numerical Experiments and Analysis

5.1. Traffic Network Description

We used a grid-based road network model for our experiments. As shown in Figure 9, the selected network is located in Suzhou Industrial Park, Jiangsu Province. Our field investigation collected parameters such as road lengths, intersection flows, average segment speeds, lane functions, and signal timings. The network comprises 15 intersections (3 T-shaped and 12 cross-shaped), 2 arterial roads, 6 secondary roads, and 22 road segments.

The turning movement traffic flows presented in

Table 1. Summary of turning movement flow at intersections.

Intersection	East Approach		West Approach		South Approach		North Approach
	Left Turn	Straight	Left Turn	Straight	Left Turn	Straight	Left Turn	Straight
I1	283	530	295	508	264	301	277	288
I2	245	199	233	190	134	581	160	602
I3	431	457	422	421	201	130	54	88
I4	325	344	315	366	367	287	376	302
I5	203	310	214	315	213	620	198	633
I6	332	312	-	400	340	-	-	-
I7	-	-	264	-	437	429	-	420
I8	210	-	-	-	-	497	343	464
I9	88	130	164	108	178	635	198	609
I10	230	210	225	241	204	612	233	623
I11	166	154	186	121	214	798	208	805
I12	154	320	133	303	288	350	267	355
I13	298	304	233	332	276	501	254	463
I14	204	418	221	429	301	410	297	422
I15	213	456	209	487	203	433	230	460

were obtained through video recording and manual counting at each intersection during the morning peak period (7:30–9:30 a.m.) on a typical weekday. The flows shown for 9:00 a.m. represent the average hourly volume derived from multiple signal cycles within a 15 min observation window centered at 9:00 a.m. This period represents stable, near-saturation traffic conditions commonly observed during the morning commute in Suzhou Industrial Park, providing a relevant scenario for testing the algorithm’s performance under congested, signal-controlled urban traffic.

The segment speeds in

Table 2. Summary of road segment weight information.

Road Segment Number	Node	Distance (m)	Average Speed (km/h)	Travel Time (s)	Road Segment Number	Node	Distance (m)	Average Speed (km/h)	Travel Time (s)
1	I1–I2	725	54	48.33	12	I7–I11	443	51	31.27
2	I1–I4	465	47	35.62	13	I8–I9	728	47	55.76
3	I2–I3	1300	57	82.11	14	I8–I12	270	48	20.25
4	I2–I5	523	53	35.52	15	I9–I10	613	46	47.97
5	I–I7	554	49	40.70	16	I9–I13	235	54	15.67
6	I4–I5	689	45	53.92	17	I10–I11	693	48	51.98
7	I4–I8	275	47	21.06	18	I10–I14	194	46	15.18
8	I5–I6	587	46	45.94	19	I11–I15	354	52	24.51
9	I5–I9	416	56	26.74	20	I12–I13	737	58	45.74
10	I6–I7	664	45	53.12	21	I13–I14	568	55	37.18
11	I6–I10	525	45	42.00	22	I14–I15	641	56	41.21

are space-mean speeds estimated from floating car surveys conducted during the same morning peak period. The signal timing plans in

Table 3. Summary of signal timing information at intersections.

Intersection	Cycle (s)	Phases	Phase 1 Green (s)	Phase 1 Movement	Phase 2 Green (s)	Phase 2 Movement	Phase 3 Green (s)	Phase 3 Movement	Phase 4 Green (s)	Phase 4 Movement	Phase 5 Green (s)	Phase 5 Movement
I1	120	4	44	E-W Through	27	E-W Left	25	N-S Through	24	N-S Left	-	-
I2	180	5	25	E-W Through	35	E-W Left	70	S Through	30	N Through	20	N-S Left
I3	180	4	55	E-W Through	57	E-W Left	48	S Through	20	N Through	-	-
I4	120	4	30	E-W Through	30	E-W Left	25	N-S Through	35	N-S Left	-	-
I5	160	4	35	E-W Through	25	E-W Left	70	N-S Through	30	N-S Left	-	-
I6	65	2	35	S Release	30	E-W Through	-	-	-	-	-	-
I7	65	2	25	W Release	40	N-S Release	-	-	-	-	-	-
I8	60	2	20	E Release	40	N-S Release	-	-	-	-	-	-
I9	180	4	30	E Release	35	W Release	85	N-S Through	30	N-S Left	-	-
I10	130	3	60	N-S Through	25	N-S Left	45	E-W Through	-	-	-	-
I11	130	3	80	N-S Through	20	N-S Left	30	E-W Through	-	-	-	-
I12	170	5	40	E Through	30	W Through	20	E-W Left	45	N-S Through	35	N-S Left
I13	160	4	37	E-W Through	38	E-W Left	54	N-S Through	31	N-S Left	-	-
I14	130	4	40	E-W Through	25	E-W Left	35	N-S Through	30	N-S Left	-	-
I15	130	4	42	E-W Through	25	E-W Left	38	N-S Through	25	N-S Left	-	-

are the fixed-time control schemes obtained from the local traffic management authority and were in operation during the data collection period.

Figure 10 illustrates the network model with 13 OD pairs and annotated link travel times. All intersections assume saturated flows of 1650 for through movements and 1550 for left turns. Numerical experiments were conducted on a PC running Python 3.11.5 on Windows 11 (22H2), with an Intel (R) Core (TM) i7-13700HX 2.10 GHz CPU and 16 GB memory.

5.2. Comparison of the Proposed Algorithm with Other Algorithms

In the case study, the shortest path is computed on the global expanded network rather than the physical-node graph. Each turning movement is represented as an internal edge whose weight is dynamically updated as the sum of link travel time and the phase-dependent periodic queuing delay.

5.2.1. Shortest Path at Different Departure Times

(a) Comparative Analysis of Computational Results from Different Algorithms

As shown in Figure 10, with $O_1$ as the start node (15 s from the intersection $I_1$) and $D_9$ as the end node (12 s from the intersection $I_{15}$), we used three algorithms to calculate the shortest path and travel time at different departure times: our proposed algorithm (considering real-time traffic and signal timing), an algorithm considering only signalized intersection waiting time, and the traditional Dijkstra’s algorithm (ignoring traffic and signal timing). Starting from time 0 with 50 s intervals over 15 runs, the results in

Table 4. Comparison of calculation results of three algorithms for $O_1-D_9$ at different departure times.

Departure Time	Proposed Algorithm		Considering Signalized Intersection Waiting Time		Traditional Dijkstra’s Algorithm
	Shortest Path	Travel Time (s)	Shortest Path	Travel Time (s)	Shortest Path	Travel Time (s)
0	O1-I1-I4-I5-I6-I7-I11-I15-D9	392.16	O1-I1-I4-I5-I6-I7-I11-I15-D9	379.00	O1-I1-I4-I8-I12-I13-I14-I15-D9	235.06
50	O1-I1-I4-I8-I9-I13-I14-I10-I11-I15-D9	462.16	O1-I1-I4-I5-I6-I7-I11-I15-D9	329.00
100	O1-I1-I2-I5-I6-I7-I11-I15-D9	421.89	O1-I1-I4-I5-I9-I13-I14-I15-D9	345.21
150	O1-I1-I2-I5-I6-I7-I11-I15-D9	371.96	O1-I1-I4-I5-I9-I13-I14-I15-D9	295.21
200	O1-I1-I2-I5-I6-I7-I11-I15-D9	322.28	O1-I1-I2-I5-I6-I7-I11-I15-D9	309.00
250	O1-I1-I2-I5-I6-I7-I11-I15-D9	403.75	O1-I1-I4-I5-I9-I13-I14-I15-D9	325.21
300	O1-I1-I2-I5-I6-I7-I11-I15-D9	353.92	O1-I1-I4-I5-I9-I13-I14-I15-D9	275.21
350	O1-I1-I2-I3-I7-I11-I15-D9	432.03	O1-I1-I2-I3-I7-I11-I15-D9	419.00
400	O1-I1-I2-I3-I7-I11-I15-D9	382.07	O1-I1-I2-I3-I7-I11-I15-D9	369.00
450	O1-I1-I2-I3-I7-I11-I15-D9	332.11	O1-I1-I2-I3-I7-I11-I15-D9	319.00
500	O1-I1-I4-I5-I9-I10-I14-I15-D9	330.17	O1-I1-I4-I5-I9-I10-I14-I15-D9	309.36
550	O1-I1-I4-I8-I9-I13-I14-I15-D9	382.07	O1-I1-I4-I8-I9-I13-I14-I15-D9	374.00
600	O1-I1-I2-I5-I6-I7-I11-I15-D9	436.96	O1-I1-I4-I5-I9-I13-I14-I15-D9	365.21
650	O1-I1-I2-I5-I6-I7-I11-I15-D9	387.32	O1-I1-I4-I5-I9-I13-I14-I15-D9	315.21
700	O1-I1-I2-I3-I7-I11-I15-D9	468.34	O1-I1-I4-I5-I9-I13-I14-I15-D9	395.21

show that the traditional algorithm is not affected by departure times, while the other two algorithms yield varying paths and travel times, highlighting the importance of traffic and signal considerations for accurate path planning.

As shown in Table 4, departure time significantly impacts travel time. The traditional Dijkstra’s algorithm yields the shortest travel times, which remain constant across different departure times. The algorithm considering signal light waiting times produces intermediate travel times that vary due to signal timing. The proposed algorithm, which accounts for queuing delays, results in the longest travel times, which are always greater than or equal to those from the other two algorithms. For instance, at a departure time of 300 s, the travel times are 235.06 s (traditional algorithm), 275.21 s (Considering Signalized Intersection Waiting Time), and 353.92 s (proposed algorithm). These results demonstrate that ignoring traffic signals or queuing delays can lead to inaccurate travel time estimates.

As shown in Table 4, the traditional algorithm yields the same path regardless of departure time. In contrast, the other two algorithms produce varying paths due to departure time differences. Specifically, at a departure time of 300 s (see Figure 11b), the three algorithms generate different paths: O₁-I₁-I₄-I₈-I₁₂-I₁₃-I₁₄-I₁₅-D₉, O₁-I₁-I₄-I₅-I₉-I₁₃-I₁₄-I₁₅-D₉, O₁-I₁-I₂-I₅-I₆-I₇-I₁₁-I₁₅-D₉. Similar variations are observed at departure times of 50 s (Figure 11a) and 700 s (Figure 11c). These results highlight that in practical applications, both signalized intersection waiting times and periodic queuing delays significantly affect path calculations, thereby greatly influencing users’ actual route choices.

(b) Comparative Analysis of Numerical Simulation Results from Different Algorithms

To further demonstrate the advantages of the proposed algorithm, we conducted numerical simulations comparing our algorithm with the Traditional Dijkstra’s Algorithm and the Considering Signalized Intersection Waiting Time algorithm. We calculated the actual travel time of the optimal paths derived from these algorithms under conditions with signalized intersections and traffic flow impacts.

In the tests, we assumed fixed-time signal plans, i.e., the cycle length and green splits remained constant. However, the instantaneous signal state (phase and remaining time) changes continuously with time within each cycle, and the turning movement arrival flows were held constant over the test horizon. The simulation results, tested over departure times from 0 to 2000 s in 10 s intervals, are shown in Figure 12. From Figure 12, it can be seen that travel time exhibits periodic variations due to different departure times. The proposed algorithm shows either shorter or equal travel times compared to the other two algorithms. Specifically, it outperforms the Considering Signalized Intersection Waiting Time algorithm and significantly outperforms the Traditional Dijkstra’s Algorithm. At certain departure times (e.g., 650, 900, and 1320 s), the Considering Signalized Intersection Waiting Time algorithm even shows much longer travel times than the Traditional Dijkstra’s Algorithm. Moreover, the proposed algorithm reflects a periodic variation in travel time with regular changes in departure time. This highlights its real-time adaptability and effectiveness in urban traffic path planning.

Additionally, the optimal path time from the simulation results and the total periodic queuing delay time at each intersection were extracted to obtain the ratio of total delay time to total travel time, as shown in Figure 13. The proposed algorithm consistently shows a lower or equal total delay time ratio compared to the other two algorithms. Specifically, the numerical simulation results are organized in

Table 5. Numerical simulation results of actual travel times and delay ratios for three algorithms at different departure times.

	Key Parameters	Average Travel Time (s)	Average Total Delay Time (s)	Average Ratio of Delay Time (%)	Average Improvement Percentage (%)
Different Algorithms
Traditional Dijkstra’s Algorithm		524.33	289.19	55	25.36
Considering Signalized Intersection Waiting Time Algorithm		437.10	182.71	40	10.46
Proposed Algorithm		391.38	126.35	32	-

.

As shown in Table 5, the proposed algorithm offers shorter and more stable travel times across different departure times, demonstrating significant advantages over the Traditional Dijkstra’s Algorithm and the Considering Signalized Intersection Waiting Time Algorithm. Specifically, it reduces travel time by 25.36% compared to the traditional algorithm (from an average of 524.33 s to 391.38 s) and by 10.46% compared to the considering signalized intersection waiting time algorithm (from an average of 437.10 s). It also achieves the lowest delay time ratio of 32%. In urban traffic networks with numerous signalized intersections and varying traffic flows, vehicles often encounter periodic queuing delays, which go beyond simple intersection waiting times or average delays. The proposed algorithm, by accounting for real-time periodic queuing delays at signalized intersections, demonstrates significant potential.

5.2.2. Shortest Path at Different OD Pairs

(a) Comparative Analysis of Computational Results from Different Algorithms

In this section, we analyze the shortest path travel time and route for different OD pairs at the same departure time. At a departure time of 1000 s, we selected five different OD pairs and calculated their travel times and shortest paths using three algorithms. The results are shown in

Table 6. Comparison of calculation results of three algorithms for different OD pairs at the same departure time (1000 s).

Departure Time	Proposed Algorithm		Considering Signalized Intersection Waiting Time		Traditional Dijkstra’s Algorithm
	Shortest Path	Travel Time (s)	Shortest Path	Travel Time (s)	Shortest Path	Travel Time (s)
O1-D9	O1-I1-I4-I8-I9-I13-I14-I15-D9	430.08	O1-I1-I4-I5-I9-I13-I14-I15-D9	314.00	O1-I1-I4-I8-I12-I13-I14-I15-D9	235.06
O1-D10	O1-I1-I4-I8-I9-I13-I14-I15-I11-D10	394.22	O1-I1-I4-I5-I6-I7-I11-D10	261.87	O1-I1-I2-I3-I7-I11-D10	233.41
O1-D7	O1-I1-I2-I5-I9-I10-I14-D7	290.31	O1-I1-I4-I5-I9-I13-I14-D7	214.13	O1-I1-I4-I8-I12-I13-I14-D7	192.85
O2-D7	O2-I1-I2-I5-I9-I10-I14-D7	388.29	O2-I1-I2-I5-I9-I13-I14-D7	272.59	O2-I1-I2-I5-I9-I13-I14-D7	190.44
O5-D13	O5-I12-I8-I4-I1-I2-D13	312.76	O5-I12-I8-I4-I5-I2-D13	203.52	O5-I12-I13-I9-I5-I2-D13	149.67

.

As shown in Table 6, for the five randomly selected OD pairs, the travel times calculated by our proposed algorithm are longer than those from the other two algorithms, indicating a more realistic reflection of actual conditions. Additionally, there are noticeable differences in the shortest path nodes calculated by the three algorithms for the same OD pair. For instance, for O₁-D₁₀, the paths derived by the three algorithms are as follows: O₁-I₁-I₄-I₈-I₉-I₁₃-I₁₄-I₁₅-I₁₁-D₁₀ (proposed algorithm), O₁-I₁-I₄-I₅-I₆-I₇-I₁₁-D₁₀ (Considering Signalized Intersection Waiting Time algorithm), and O₁-I₁-I₂-I₃-I₇-I₁₁-D₁₀ (Traditional Dijkstra’s Algorithm). Despite traversing more intersection nodes, our algorithm effectively avoids intersections with significant queuing delays, thereby reducing overall travel time and identifying shorter paths. Figure 14 illustrates that the proposed algorithm is applicable across different OD pairs and yields more realistic optimal path results compared to the other two algorithms.

(b) Comparative Analysis of Numerical Simulation Results from Different Algorithms

To further demonstrate the advantages of the proposed algorithm, we conducted numerical simulations on the optimal paths derived from the Traditional Dijkstra’s Algorithm and the Considering Signalized Intersection Waiting Time algorithm. We calculated their actual travel times considering waiting times and periodic queuing delays. At a departure time of 1000 s, we used all three algorithms to compute the travel time for 121 OD pairs (13 groups) shown in Figure 10. The results in Figure 15a indicate that our algorithm generally achieves shorter travel times than the other two, with extremely few cases of equal times. This proves that our algorithm, which accounts for periodic queuing delays, significantly outperforms the others and provides more realistic results.

From the CDF graph in Figure 15b, the proposed algorithm’s curve is consistently on the left indicating shorter travel times than the other two algorithms at the same cumulative probability. For instance, at a cumulative probability of 90%, our algorithm has a time threshold of 364.69 s, compared to 438.78 s for the Considering Signalized Intersection Waiting Time algorithm and 457.56 s for the Traditional Dijkstra’s Algorithm. This demonstrates the effectiveness of our algorithm. Furthermore, our algorithm shows less volatility and greater stability in actual travel time calculations.

Additionally, from the simulation results, we extracted the optimal path times and the total periodic queuing delay times at each intersection for different OD pairs. Figure 16 shows the ratio of total delay time to total travel time. Our proposed algorithm consistently achieves a lower or equal total delay time ratio compared to the other two algorithms. The specific results are summarized in

Table 7. Numerical simulation results of actual travel times and total delay-to-travel-time ratios for three algorithms across 121 OD pairs.

	Different Algorithms	Average Travel Time (s)	Average Total Delay Time (s)	Average Ratio of Delay Time (%)	Average Improvement Percentage (%)
Key Parameters
Traditional Dijkstra’s Algorithm		286.04	131.57	41	9.71
Considering Signalized Intersection Waiting Time Algorithm		271.07	104.89	35	5.13
Proposed Algorithm		250.67	82.88	31	-

.

As shown in Table 7, the proposed algorithm offers shorter and more stable travel times across different departure times, outperforming the Traditional Dijkstra’s Algorithm and the Considering Signalized Intersection Waiting Time Algorithm. Specifically, it reduces travel time by 9.71% compared to the traditional algorithm and by 5.13% compared to the considering signalized intersection waiting time algorithm, with an average travel time of 250.67 s. It also achieves the lowest delay time ratio of 31%. In urban driving scenarios with varying starting and ending points, the periodic queuing delays at intersections can significantly affect optimal path outcomes. Incorporating periodic queuing delays into the algorithm is crucial for accurate path planning.

5.3. Simulation Validation and Analysis

To validate the algorithm’s practicality, both numerical simulations and SUMO simulations are conducted to verify the shortest path results calculated in this study.

5.3.1. Simulation Environment Construction

To validate the algorithm’s practicality, we conduct both numerical and SUMO simulations. First, we use Open Street Map to extract electronic maps and generate road networks, which are then adjusted for lane settings. The traffic network information, flow rates, and signal timings provided in Section 4.1 are inputted into the simulation environment. The resulting simulation base map is shown in Figure 17. Finally, we specify the start and end points of the vehicle routes and set the departure time to complete the simulation. SUMO simulations used the default passenger-vehicle settings with the (default) Krauss car-following model. Key default vType parameters were accel = 2.6 m/s², decel = 4.5 m/s², sigma = 0.5, tau = 1 s, minGap = 2.5 m, and length = 5 m; the default lane-changing model (LC2013) and default step length (1 s) were used.

5.3.2. Analysis of Different Departure Times

As shown in Table 4 of Section 5.2, the optimal paths from three algorithms at different departure times were simulated. The results in Figure 18 and

Table 8. SUMO simulation results of three algorithms at different departure times.

	Different Algorithms	Proposed Algorithm	Considering Signalized Intersection Waiting Time Algorithm	Traditional Dijkstra’s Algorithm
Key Parameters
Average Calculated Travel Time (s)		391.95	341.58	235.06
Average Simulated Travel Time (s)		405.31	436.94	536.07
Average Difference in Travel Time (s)		13.36	95.36	301.01
Ratio of Calculated to Simulated Time (%)		96.70	78.18	43.85
Average Improvement Percentage (%)			7.80	32.26

indicate that the proposed algorithm’s average time difference from the simulation is only 13.46 s, much lower than 95.36 s from the Considering Signalized Intersection Waiting Time Algorithm and 301.01 s from the Traditional Dijkstra’s Algorithm. This demonstrates the proposed algorithm’s superiority and practicality in actual traffic conditions.

5.3.3. Analysis of Simulation Results for Different OD Pairs

As shown in Table 6 of Section 5.2, the paths from three algorithms for five OD pairs were simulated. The results presented in Figure 19 and

Table 9. SUMO simulation results of three algorithms for different OD pairs.

	Different Algorithms	Proposed Algorithm	Considering Signalized Intersection Waiting Time Algorithm	Traditional Dijkstra’s Algorithm
Key Parameters
Average Calculated Travel Time (s)		363.13	253.22	200.29
Average Simulated Travel Time (s)		381.085	468.24	471.8
Average Difference in Travel Time (s)		17.955	215.02	271.51
Ratio of Calculated to Simulated Time (%)		95.29	54.08	42.45
Average Improvement Percentage (%)		-	22.87	23.80

indicate that the proposed algorithm has the smallest average difference in travel time compared to the simulation results, at approximately 17.955 s. This is much lower than the Considering Signalized Intersection Waiting Time Algorithm (215.02 s) and the Traditional Dijkstra’s Algorithm (271.51 s). Thus, the proposed algorithm proves to be more accurate and practical under real-world traffic conditions.

The above SUMO simulation results provide a quantitative validation of the proposed method. This comparison also quantifies how the cycle-level piecewise approximation assuming uniform arrivals within each interval and a constant saturation flow within the same interval affects model accuracy. Overall, the proposed method achieves a mean travel-time error of 13.36 s (≈3.30%) and a calculated-to-simulated ratio of 96.70%.

From the two simulation scenarios above, it is evident that explicitly accounting for periodic queuing delays in the shortest-path algorithm is essential for realistic urban-road applications, and the simulation results of the proposed method are therefore more consistent with real-world conditions. Moreover, SUMO is a microscopic, lane-based simulation environment in which vehicle interactions such as car-following and lane-changing are explicitly modeled. Despite these complex behaviors, the proposed lane-group FIFO approximation still shows close agreement with the simulation outcomes.

6. Conclusions

This study proposes a time-dependent Dijkstra’s algorithm that incorporates a novel periodic queuing delay model for signalized intersections. The model calculates real-time, movement-specific delays by integrating traffic flow and signal phase data using cumulative curve theory. Implemented through an expanded network topology and dynamic weight matrix, the algorithm operates within a FIFO-compliant TDSP framework. It overcomes the limitations of static delay models, providing more accurate and realistic time-dependent routing guidance for urban networks.

Unlike most existing studies that rely solely on numerical analysis, this study validates the proposed algorithm through both numerical experiments and microscopic traffic simulation using SUMO, thereby providing a more realistic and comprehensive assessment of its performance. Simulation results from Suzhou Industrial Park’s road network model indicate that the proposed algorithm reduces average travel time by 25.36% compared to the traditional Dijkstra’s Algorithm and by 10.46% compared to an algorithm solely considering signal light waiting times. It also significantly reduces the proportion of delay time. These results, verified by SUMO simulations, demonstrate the algorithm’s advantages in travel time, delay time, and path selection, proving its effectiveness and practicality in urban traffic networks.

Despite these achievements, further improvements are possible in weight-matrix storage and overall computational efficiency. Future work will develop more efficient data structures and update mechanisms to support larger and more complex networks, and will explore integrating speed guidance into the time-dependent shortest-path framework to further reduce delays and improve routing accuracy. It should be noted that the proposed delay model currently assumes fixed-time signal control; however, it remains applicable if real-time signal phase information and remaining time, or a reliable short-term phase prediction, becomes available. Future studies will also consider uncertainty and diverse intersection scenarios to enhance estimation accuracy, and will further account for microscopic maneuvers such as lane-changing and overtaking. In the broader context of intelligent transportation systems, the proposed algorithm can serve as a building block for applications such as vehicle infrastructure cooperation, electric-vehicle routing, and autonomous driving.