## 2 Methodology
### State Estimation of Communication Delay  
Let \(T_{\rm s}\) be the synchronized time of the slave which is the timestamp of the remote vehicle and \(T_{\rm m}\) be the time of the master which is the timestamp of the control center. Then,  
Figure 3: Example of one-way communication delay measurement using NTP server based on the GPS timestamp synchronization [38]. Figure 2: Diagram showing the procedure of the clock synchronization algorithm, where \(\delta\) is the round-trip delay, \(\theta\) is the offset, \(t_{0}\) is the time when a poll is requested from the client, \(t_{1}\) is the time the poll request is received from the server, \(t_{2}\) is the time of response from the server, and, finally, \(t_{3}\) is the time the response is received by the client. the one-way communication delay \(T_{\rm c}\) can be computed with synchronized time information as follows:  
\[T_{\rm c}=T_{\rm s}-T_{\rm m}. \tag{5}\]  
In this study, as shown in Figure 4, we categorize the communication delay into two types of modes which are passive communication delay (PD) and outlier communication delay (OD). In order to distinguish ODs from PDs, a Gaussian mixture model can be used to express the distributions using linear multiple Gaussians as follows:  
\[p(T_{\rm c})=\sum_{k=1}^{K}\pi_{k}\mathcal{N}(T_{\rm c}|\mathbf{\mu}_{k},\mathbf{\Sigma }_{k}), \tag{6}\]  
where \(k\) is the latent Gaussian variable, \(K\) is the number of Gaussian components, \(T_{\rm c}\) is the vector of the set of one-way communication delay samples, \(\pi_{k}\) is the mixture coefficient of the \(k\)th Gaussian, \(\mathcal{N}\) refers to the Gaussian, \(\mathbf{\mu}_{k}\) is the means of the latent Gaussian components, and \(\mathbf{\Sigma}_{k}\) is the variations of the latent Gaussian components. In order to derive (6), a discrete random variable \(z\) is introduced, which, having \(K\)-dimension \(z_{k}\in\{0,1\}\), can be defined as a one-hot encoded variable consisting of the specific element as \(1\) and the other as \(0\), which is \(\sum_{k}z_{k}=1\). The variable \(z\) can have a different \(K\) state depending on whether the specific element is \(0\) or not. The joint distribution \(p(x,z)\) can be defined with marginal distribution \(p(z)\) and conditional distribution \(p(x|z)\) as follows:  
\[p(z_{k}=1)=\pi_{k}, \tag{7}\]  
where \(\pi_{k}\) is a parameter which should meet the following condition:  
\[0\leq\pi_{k}\leq 1 \tag{8}\] \[\sum_{k=1}^{K}\pi_{k}=1. \tag{9}\]  
Finally, the upper distribution can be written as:  
\[p(z)=\prod_{k=1}^{K}\pi_{k}^{z_{k}}. \tag{10}\]  
Figure 4: Classification of communication delays based on the measured one-way communication delay \(T_{\rm c}\), where \(T_{\rm s}\) is the synchronized time of the time slave node, and \(T_{\rm m}\) is the synchronized time of the master node. where \(\hat{\pi}_{k}\) is the estimated weight of the \(k\)th Gaussian. Finally, the means and variances can be estimated as:  
\[\hat{\rho}_{k} =\frac{\sum_{i=1}^{N}P(k|\mathbf{T}_{\text{c}|i},\lambda)\mathbf{T}_{\text{ c}|i}}{\sum_{i=1}^{N}P(k|\mathbf{T}_{\text{c}|i},\lambda)} \tag{19}\] Figure 2:0  
where \(\hat{\mu}_{k}\) and \(\hat{\sigma}_{k}\) denote the expected mean and standard deviation of the \(k\)th Gaussian during \(K\) Gaussians. Figure 5 shows a schematic diagram of the proposed method to distinguish ODs from PDs. Furthermore, it can be written as the following equation:  
Figure 2:1  
where \(l\) is the sample interval. By using Equation (21), the communication delays can be estimated in real time to distinguish ODs and also can be used as a judgment criterion for the instant communication delay \(T_{\text{c}|i}\).