In wireless environments, beacon frames are periodically broadcast from the access point (AP) to multiple receiving nodes for one-way synchronization. However, due to the inherent nature of wireless propagation, several challenges arise in how these beacons are received across different nodes. Despite being transmitted simultaneously from the same AP, nodes may receive different beacon sequence numbers or, in many cases, even when the same sequence number is received, they experience heterogeneous delays caused by varying channel conditions, interference, or hardware-specific reception timing. This phenomenon is illustrated in Fig. 2, where the perceived beacon timing at each node differs from the reference due to stochastic transmission delay and reception offset. Such heterogeneity in beacon reception leads to inconsistencies in estimating the timing offset (DOFFSET) between each node’s local clock (TLOCAL) and the AP’s reference time (TAP). Without effective handling, these inconsistencies degrade synchronization performance and result in unstable system clock updates. Therefore, a robust preprocessing strategy is required to manage this temporal uncertainty prior to applying high-precision estimation techniques such as Maximum Likelihood Estimation (MLE).

This heterogeneity affects synchronization accuracy in two critical ways. First, when different nodes receive different beacon sequence numbers, each node references a different AP timestamp TAP,i, resulting in offset values computed as DOFFSET,i=TLOCAL,i−TAP,i that correspond to different reference points in time. Since each transmission from the AP may involve different transmission-side delays, these offsets do not align to a common clock model. Especially during the initial synchronization phase, before any statistical filtering is applied, this misalignment leads to increased uncertainty and wider offset variance across nodes, hindering early convergence of synchronization. Second, even when nodes receive the same beacon sequence number, the local reception timestamp TLOCAL,i can still vary significantly due to diverse wireless channel conditions, hardware interrupt latencies, and kernel-level processing delays. In such cases, the computed offset can be more accurately expressed as DOFFSET,i=(TLOCAL,i+δi)−TAP,i, where δi represents a stochastic delay term caused by reception-side variability. This δi is typically modeled as a zero-mean Gaussian random variable, introducing unpredictable noise into the offset measurement even under identical beacon conditions. The combination of these two issues—sequence misalignment and stochastic delay variation—leads to large offset dispersion during the early synchronization stage. This makes the system vulnerable to unreliable or unstable clock updates. To address this, a mechanism capable of absorbing such uncertainty and refining offset samples is essential prior to applying precise synchronization techniques.

The process of sampling and filtering this temporal data is implemented as shown in Algorithm 1. First, TLOCAL,i is obtained from the local time of the Linux kernel, while TAP,i is extracted from the received BF. By computing the difference between these timestamps as DOFFSET,i, the kernel system time deviation from the AP, which serves as a reference clock, is estimated. A sample is added to DOFFSET,i only if it is valid, ensuring that outlier samples do not introduce significant errors in the synchronization process. To maintain efficiency, a moving average characteristic is incorporated by replacing the oldest sample when a new one is added. Unlike a fixed sampling period, this algorithm dynamically determines the period length by calculating the period length every time a sample is added. Once DOFFSET,i accumulates period_size samples, a filtering process is applied to derive a robust time correction value. The filtering approach consists of two key steps (outlier sampling and median estimation). Outlier rejection is performed based on statistical thresholds to eliminate abnormal values. In particular, the greatest obstacle to synchronizing multiple clients is the common low—quality frames (i.e., heterogeneous receptions). Moreover, clients do not exchange information to confirm whether they have received the same sequence or to compare and evaluate each other’s sequence data. Because the client cannot assess frame quality immediately upon initial reception, we proactively discard any sample whose DOFFSET,i deviates from the running median by more than λ⋅σD, thereby suppressing abrupt fluctuations caused by stochastic delays and mitigating heterogeneous arrival effects. For median estimation, rather than performing a full sort operation, we employ Quickselect, an efficient selection algorithm that finds the median with lower computational overhead. This ensures that the filtering step remains stable under varying network conditions. By selecting the median from DOFFSET,i, the algorithm ensures that the time offset correction, denoted as DMIN, remains robust against extreme deviations. The system clock is then updated based on DMIN through update_clock, and the process is repeated continuously. This approach accelerates the synchronization process by directly applying the offset-based DMIN to the system clock rather than indirectly updating through skew or drift estimation.

To quantitatively evaluate the effectiveness of the proposed time synchronization method in reducing the time difference between two nodes, we compared it to a pure hard update approach. The analysis consists of three steps: first, we define the time model of the beacon frame transmitted from the AP; second, we construct a mathematical model of direct BF-based time updating, which serves as a baseline approach; and third, we analyze how the proposed method improves synchronization stability and precision. We define a random variable for the i-th beacon frame reception time from the AP as Xi. The node’s reception outcome is represented as a binary random variable Wi, where reception failure and success are defined as P(W=0)=p and P(W=1)=q, respectively. The received time is modeled as: (1)Υi=Xi−ti.

The most recent valid received time at the node, incorporating reception success probability, is defined as: (2)Υi∗=Φ(Υ1,W1,Υ2,W2,⋯,Υi,Wi).

For a given sequence of received time values, the expected latest valid time error is derived as: (3)E[Υi∗]=p(E[Υi]+qE[Υi−1]+q2E[Υi−2]+⋯+qi−1E[Υ1])=pE[Υ](1+q+q2+⋯+qi−1)≈p1−qi1−qE[Υ]=E[Υ].

The inter-node time difference is then given by: (4)Z=Xnode1−Xnode2=Υnode1−Υnode2.

To compare the variance reduction effect of the proposed method, we analyze the median-based filtering process. The median estimation follows: (5)u¯={Υm′,if n is odd,n=2m+112(Υj′+Υm+1′),if n is even,n=2m,where Υm′ is target time to synchronize, and leads to: (6)E[Υm′]=12E[(Υm′+Υm+1′)]=12(E[Υm′]+E[Υm+1′])=E[Υm′].

Given that the received time errors Υi and Υi′ are assumed to follow a normal distribution, the variance σ′ of the median-based filtering system is approximated as: (7)σ2 ′=18mf(u¯)2=πσ24m≈π32σ2.

As a result, the proposed system reduces time error variance to about fracπ32 of the baseline hard update with m=16 sample size. Retaining the offset sample set {DOFFSET,i}, the median-based correction Dmin (or u¯), and the filtering window m defined, we now incorporate these same filtered samples into our MLE approach for stable stage (Section 3.3). The adoption of Quickselect enhances filtering efficiency by significantly reducing computational complexity. Instead of sorting the entire dataset, Quickselect finds the median in expected O(N) time, allowing fast convergence to a stable offset estimation. This approach ensures that unstable samples are quickly discarded while still achieving a precise clock. Although this method does not achieve the ultra-low error margins of MLE-based methods (e.g., 10 µs error range), it provides a practical balance between fast synchronization and stability, typically achieving a 30 µs error margin while maintaining real-time adaptability.

To enhance synchronization accuracy in one-way communication environments, we apply Maximum Likelihood Estimation (MLE) to estimate both the clock offset and clock skew. In dynamic wireless networks, the observed timing offset between the access point (AP) and receiving node is influenced not only by the true clock difference but also by stochastic transmission delays, which vary due to channel conditions and system load.

Building on the median-filtered samples Dmin and sample indices i from Section 3.2, we refine the offset representation by modeling the network delay di as a Gaussian variable, ensuring a seamless transition to the MLE-based estimation stage. Let TAP,i denote the timestamp embedded in the i-th beacon frame from the AP, and TLOCAL,i the corresponding reception timestamp at the node. The observed offset is expressed as: (8)DOFFSET,i=TLOCAL,i−TAP,i.

This offset includes both the actual clock offset θ and a variable network-induced delay di, modeled as a Gaussian random variable: (9)DOFFSET,i=θ+di,di∼𝒩(µd,σd2).

Given a collection of such samples, we estimate the optimal offset θ^MLE by maximizing the likelihood function, which leads to the closed-form estimator: (10)θ^MLE=1N∑i=1NDOFFSET,i.

To further track the clock skew ϕ, we employ a time-differential method based on successive offset changes: (11)ϕ^=1N−1∑i=1N−1DOFFSET,i+1−DOFFSET,iTAP,i+1−TAP,i.

Unlike static averaging, this method allows for continuous adaptation, making it suitable for environments with fluctuating delay and mobility. The estimated θ^MLE and ϕ^ are then used to correct the local system clock with high precision. In practice, the MLE-estimated offset θ^MLE is applied in a single step or gradually as a slew correction with the estimated skew ϕ^ to adjust the system clock.

Once a rapid convergence is achieved through the initial hard update mechanism described earlier, the synchronization process transitions into this MLE-based estimation stage. In this stage, adaptive sampling and lightweight filtering are applied to mitigate transient delay fluctuations, and clock correction is performed gradually to ensure smooth tracking of the reference time. This approach contributes to overall stability and robustness of the synchronization system under real-world wireless dynamics.

The clock offset refers to the time difference between two devices for the same nominal second, expressed as Cc−Cs. However, in synchronization processes such as NTP, PTP, or Chrony, it is the client that corrects the skew and drift. Thus, the offset can be more precisely defined as the difference between the reception time at the client tirx,c and the transmission time at the server titx,s. Based on the timing messages received by the client, offset samples are collected and used as parameters to operate the schemes proposed in Section 3. Accordingly, the variation of the offset becomes one of the most critical values in the synchronization process. We compare the offset pattern of WMC, a one-way synchronization method, with Chrony, a state-of-the-art two-way synchronization technique, as shown in Fig. 3. We define the first 10 minutes after the execution of the synchronization driver as the initial stage, during which we log the offset values of both WMC and Chrony. Since each raw offset sample is discarded by filters or diluted into mean or variance, the tendency of offset is more informative than the raw values themselves. Thus, we apply logistic regression based on a sigmoid function to the offset pattern, as shown in Fig. 3. The y-axis represents the normalized offset from the regression fit. The WMC scheme shows a normalized mean offset of 357.41 µs, whereas Chrony yields a higher initial mean offset of 1558.92 µs. A noteworthy observation is that the offset level reverses significantly between 400 to 500 s after the synchronization process begins. While WMC consistently maintains a lower offset level, Chrony initially exhibits unstable offset values but surpasses WMC in accuracy after 500 s. This behavior indicates that traditional two-way synchronization methods, like Chrony, adopt a conservative correction strategy that is effective not only for direct links but also via large scale network or backbone network. The use of adaptive polling intervals further reflects the design philosophy of NTP-based synchronization. Therefore, in scenarios with direct WLAN AP connections, WMC demonstrates superior performance over the initial stage.

This subsection analyzes how accurately the two clients were synchronized to the server. For a fair comparison, the proposed WMC scheme is evaluated alongside a one-way synchronization method based on MLE with hard updates and a two-way synchronization method implemented by Chrony. The difference between the clocks of the two clients, Cc1 and Cc2, is defined as the synchronization error and is measured using an oscilloscope. The actual timing precision of the client clocks during the initial phase, corresponding to the offset trend analyzed in Fig. 3, is shown in Fig. 4. Similarly, the error observed among the stable stage is presented in Fig. 5. The synchronization error between the two clients during the initial stage is presented in Fig. 4. Due to the non-dynamic scale limitation of the oscilloscope, the maximum error values were capped at 1000 µs for MLE and 2000 µs for Chrony. As shown in Fig. 4a, Chrony exhibits significantly higher error values in the early part of the initial stage, with a maximum windowed average of 15,190.00 µs and a mean error of 2610.90 µs. In contrast, WMC maintains a much lower error level, with a mean of 47.00 µs and a maximum average of 65.64 µs. MLE also outperforms Chrony, with a mean error of 341.75 µs and a maximum average of 924.83 µs. The corresponding cumulative distribution functions (CDF) in Fig. 4b confirm that WMC achieves the most stable and consistent synchronization, followed by MLE. Chrony’s error distribution is heavily skewed toward higher values, indicating less reliable synchronization during the initial stage. After 400 s, all three methods converge to a similar level of synchronization performance. During the entire experimental period, the clock error results are shown in Fig. 5. The two methods maintain low error levels with reduced fluctuations. For the final 300 s, Chrony achieves a mean error of 46.07 µs and a maximum windowed average of 86.73 µs, while WMC shows the best performance with a mean error of 34.49 µs and a lower maximum average of 48.17 µs. Specifically, compared to Chrony’s stabilization performance (690 s to reach 90% of the steady-state offset), WMC reaches the same 90% level in only 35 s, i.e., 5.1% of the baseline time. These results suggest that switching to an WMC mechanism is also more effective than Chrony in the stable stage.

Therefore, the results in Figs. 4 and 5 demonstrate that, in vehicular networks where WLAN-based synchronization is often transient and unstable, a hard update-based multi-stage approach is effective in achieving rapid convergence toward operational synchronization accuracy.

This subsection presents a simulation-based evaluation examining how time synchronization accuracy influences position estimation performance in vehicular communication systems. The experiment was conducted using the simulation of urban mobility (SUMO) simulator on an urban canyon map with dense building structures, as shown in Fig. 6a. Six and sixty vehicles were deployed on a closed-loop road approximately 1 km in length. The vehicles were allowed to accelerate up to 33.3 m/s, and the loop path was repeated to maintain continuous motion, resulting in a consistent trajectory illustrated in Fig. 6b. Within a general cooperative localization scenario [19,20], local position estimation is modeled as a Gaussian distribution such that P(|X|>0.1 m)<0.01. Each vehicle’s position was recorded using ground truth (GT) data, which were not available to the vehicle themselves, and distributed localization was modeled by assuming that the position of a reference vehicle (vehicle A) was shared with other vehicles as an input parameter for global estimation. In this context, we analyze the impact of time synchronization at the infrastructure level on the performance of vehicular localization systems. Two different clock offset correction methods were applied: Chrony and WMC (vehicles B and C). Localization accuracy was evaluated by computing the Fréchet distance between the GT trajectory and the offset-corrected trajectories. As shown in Fig. 7, Chrony resulted in a maximum error of 0.5870 m, with a mean of 0.0697 m and a standard deviation of 0.0854 m. In contrast, WMC significantly improved the accuracy, achieving a maximum error of only 0.1516 m, a mean of 0.0429 m, and a standard deviation of 0.0235 m. To illustrate the performance differences under various synchronization stages, two sample time windows were analyzed. In the early stage, shown in Fig. 8a, the position error in Chrony exceeded 0.4 m, while WMC reduced this to less than 10% of that error. During the stable stage (Fig. 8b), however, the difference between the two methods became less dominant, suggesting that the benefit of WMC is most significant during initial synchronization periods. These results collectively confirm that time synchronization plays a critical role in high-precision distributed localization, particularly in the early phases of cooperative motion. Meanwhile, the case with 60 nodes is shown in Fig. 8c and 8d. Chrony experienced a maximum localization error of 1.5483 m due to the increased packet exchanges, whereas that of WMC remained low at 0.1282 m.