## 2 System Model
The waiting delay \(L_{wait,n}(t)\) is calculated as follows:  
\[L_{wait,n}(t)=max\bigg{(}\frac{queue_{k}(t)}{f_{k}}\bigg{)},\ \forall\ k\in\mathcal{K}, \tag{2}\]  
In Equation (2), \(f_{k}\) is the task processing capacity of MECS \(k\). And \(queue_{k}(t)\) is the total workload of tasks waiting in the queue of MEC server \(k\) at time \(t\). In this paper, the optimal task partitioning ratio of task \(W_{n}(t)\) to \(K\) MECSs among the total \(M\) MECSs is determined. The generated tasks are always divided according to the determined partitioning ratio considering the state of \(K\) MECSs. In this process, the optimal ratio is calculated by considering the current load status of each MECS and the task status generated from HSTs. When task \(W_{n}(t)\) is assigned to \(K\) MECS, each MECS forms a queue to process the task assigned to it. At this time, in order to evaluate the waiting delay of task \(W_{n}(t)\) that occurred in the current time slot \(t\), the longest waiting delay among the \(K\) MECSs is considered. That is, the total waiting delay \(L_{wait,n}(t)\) of task \(W_{n}(t)\) is calculated based on the longest value among the waiting delay of the queues from the \(K\) MECSs. In Equation (1), \(L_{exec,n}(t)\) is the delay to execute \(W_{n}(t)\) at time slot \(t\). The calculation equation is as follows:  
\[L_{exec,n}(t)=max\bigg{(}\frac{C_{n}(t)\cdot x_{k}^{n}(t)}{f_{k}}\bigg{)},\ \forall\ k\in\mathcal{K}, \tag{3}\]  
Figure 2: The proposed overall environment based on the HSR system with the MECS. \[\sum_{k=1}^{K}x_{k}^{n}(t)=1, \tag{4}\]  
In Equation (3), \(x_{k}^{n}(t)\) is the partition ratio of task \(W_{n}(t)\) to be executed to MECS \(k\). Therefore, \(x_{k}^{n}(t)\) must be a value between 0 and 1, and Equation (4) must be satisfied. If \(x_{k}^{n}(t)\) is 0, it indicates that task \(W_{n}(t)\) is not processed to MECS \(k\), and if \(x_{k}^{n}(t)\) is 1, it means that task \(W_{n}(t)\) is processed only on MECS \(k\). In addition, since the sub-tasks are executed simultaneously on MECSs, the execution delay of tasks processed on each MECS may be different. In this case, the subtask with the longest execution delay among the sub-tasks is selected as the final execution delay. Therefore, the value of \(x_{k}^{n}(t)\) is an important factor that determines the efficiency in task execution. In Equation (5), \(L_{tran,n}(t)\) is the transmission delay to send to the MECS closest to the HST that generated task \(W_{n}(t)\) at time slot \(t\). Since the result value after processing the task on the MECS is very small, the downlink transmission delay is not considered separately. The calculation formula for \(L_{tran,n}(t)\) is as follows:  
\[L_{tran,n}(t)=\frac{S(t)}{R_{n,m}(t)}, \tag{5}\]  
In Equation (6), \(R_{n,m}(t)\) is the uplink transmission rate at which HST \(n\) transmits a task \(W_{n}(t)\) to MECS \(m\), and the calculation formula is as follows:  
\[R_{n,m}(t)=B_{n,m}log_{2}\left(1+\gamma_{n,m}(t)\right), \tag{6}\]  
\(B_{n,m}\) represents the network bandwidth between HST \(n\) and MECS \(m\), and \(\gamma_{n,m}(t)\) represents the signal-to-noise ratio (SNR) between HST \(n\) and MECS \(m\). \[\gamma_{n,m}(t)=\frac{p_{n,m}(t)g_{n,m}(t)}{\sigma^{2}}, \tag{7}\]  
In Equation (7), \(p_{n,m}(t)\) is the transmission power of HST \(n\) toward MECS \(m\), \(\sigma^{2}\) represents the noise power, and \(g_{n,m}(t)\) is the path loss between HST \(n\) and MECS \(m\). Task throughput \(T_{n}(t)\) means the cumulative sum of the number of cycles when a task is completed. The research we propose aims to minimize the throughput and execution delay time of tasks generated in HST by optimizing the resources of the MEC system in the HSR environment. The objective function for this can be defined as Equation (8) by integrating the optimization of two objectives using the weight (\(\omega_{1}\in[0,1]\)). The cost function in Equation (8) is designed as a linear form (\(\omega_{1}\cdot T_{n}(t)-(1-\omega_{1})\cdot L_{n}(t)\)).