## 3 System Model and Problem Description
### System Model  
As illustrated in Figure 1, in an VEC environment, ESs are deployed on signal Base Stations (BSs) near users. This paper uses the set \(B=\{b_{1},b_{2},...,b_{m}\}\) to represent \(m\) BSs deployed in an area, each BS is represented by a triple \((l_{i}^{b},f_{i}^{b},e_{i}^{b})\), where \(l_{i}^{b}\) represents the geographic location of the BS \(b_{i}\), \(f_{i}^{b}\) represents the signal coverage radius of the \(b_{i}\), \(e_{i}^{b}=(c_{i}^{b},sw_{i}^{b})\) represents the ES in the \(b_{i}\), where \(c_{i}^{b}\) represents the remaining computing power of the \(e_{i}^{b}\). \(sw_{i}^{b}\) means the task information performed on the \(e_{i}^{b}\) in the recent period. Use the set \(N=\{n_{1},n_{2},...,n_{k}\}\) to represent \(k\) Network Routers (NRs) deployed in an area; each NR is represented by a two-tuple \((l_{i}^{m},f_{i}^{m})\), where \(l_{i}^{m}\) represents the geographic location of the NR \(n_{i}\), \(f_{i}^{m}\) represents the failure rate of the \(n_{i}\), the failure rate is assumed to be constant. Use the set \(R=\{r_{1},r_{2},...,r_{n}\}\) to represent the services that need to be migrated of Edge Users (EUs). In this model, services are migrated in the form of tasks. Each task is defined by a quintuple \((a_{i}^{r},d_{i}^{r},td_{i}^{r},l_{i}^{r},p_{i}^{r})\), where \(a_{i}^{r}\) represents the arrival time of task \(r_{i}\), \(d_{i}^{r}\) represents the amount of data transmitted, \(td_{i}^{r}\) represents the maximum tolerable time, and \(l_{i}^{r}\) represents the user's position; \(p_{i}^{r}\) represents the amount of computation required to complete task \(r_{i}\). \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(B\) & The set of base stations \\ \(N\) & The set of network routers \\ \(R\) & The set of tasks \\ \(M^{r_{i}}\) & The set of Migration paths for task \(r_{i}\) \\ \(MT_{i}\) & The Migration time of task \(r_{i}\) \\ \(b_{i}\) & The \(i\)-th base station in \(B\) \\ \(n_{i}\) & The \(i\)-th network router in \(N\) \\ \(r_{i}\) & The \(i\)-th task in \(R\) \\ \(m_{i}^{r_{i}}\) & The \(j\)-th Migration path in \(M^{r_{i}}\) \\ \(l_{i}^{b}\) & The coordinates of \(b_{i}\) \\  
In the proposed system model, it is assumed that each BS is equipped with only one ES, and the performance of all ES is exactly the same. BS can forward and compute tasks (ES is responsible for the computation), while NR can only forward tasks and does not participate in the computation of tasks. The failure rate of the BS is assumed to change in real time, the failure rate of the NR remains constant all the time, and the EU always communicates with the nearest BS. This paper abstracts the transmission link between each node (ES and NR) in mobile edge computing network into an undirected graph. It transforms the problem into how to transmit tasks in the undirected graph efficiently. As illustrated in Figure 2, when EU moves from left to right, the possible service migration paths are: ES1-NR1-NR5-ES5, ES1-NR1-NR4-ES5, ES1-NR2-ES3-NR5-ES5, and so on. How to determine the direction of EU motion belongs to the problem of trajectory prediction, which is not the content of this paper. In this paper, we focus on the problem of task fault-tolerant migration when the origin and destination of the EU are known. We set \(M^{r_{i}}=\{m_{1}^{r_{i}},m_{2}^{r_{i}},...,m_{s}^{r_{i}}\}\) is used to represent \(s\) migration paths of a task \(r_{i}\), and each path \(m_{i}^{r_{i}}\) is composed of a multiple node. The performance of ESs in a real VEC environment is not constant. Still, it changes dynamically with the influence of the time and external factors (e.g., weather, temperature, etc. ), which leads to the traditional shortest path-based service migration strategy not always achieving better results. If the failure rate of NR1 in Figure 2 is much greater than that of ES3, the migration effect of selecting the path ES1-NR2-ES3-NR5-ES5 is significantly better than that of the paths ES1-NR1-NR5-ES5 and ES1-NR1-NR4-ES5.