## 3 MEC Network Model
### Network Model  
We consider a Heterogeneous Mobile Edge Computing Network, composed of \(N_{m}\) of MEC servers, deployed at small-cell access points (APs). An example of the network is given in Figure 1. We use \(k\) = {1, 2,..., \(N_{m}\)} as the index set of \(N_{m}\) tiers of MEC servers. MEC servers in different tiers have different transmission power, computing capabilities, number of CPU cores, and waiting queue length. The spatial locations of users and MEC servers are modeled as homogeneous Poisson point processes (PPP) with spatial density \(\lambda_{u}\) and \(\lambda_{m}\), respectively. The spatial locations of the \(k\)th-tier MEC servers are also modeled as homogeneous PPP with spatial density \(\lambda_{m,k}=p_{m,k}\lambda_{m}\), where \(p_{m,k}\) is the portion of the \(k\)th-tier MEC servers. The number of CPU cores and waiting queue length in \(k\)th-tier MEC servers are \(c_{k}\) and \(r_{k}\), respectively. In order to facilitate tracking and focusing on the core aspects of heterogeneous MEC network performance, we assume in this study that the waiting queues of MEC servers at different tiers are independent.  
\begin{table}
\begin{tabular}{l l l} \hline
**Reference** & **Research Scenarios** & **Evaluation Metrics** \\ \hline [PERSON] et al. [46] & single-server & response time, resource utilization \\ [PERSON] et al. [47] & single-server & response time, resource utilization, discard probability \\ [PERSON] et al. [48] & single-server & mean response time, resource utilization \\ [PERSON] et al. [49] & homogeneous multi-server & response time, mean length of busy, idle periods \\ [PERSON] et al. [21] & homogeneous multi-server & amount of data packets, waiting time \\ [PERSON] et al. [50] & heterogeneous multi-server & response time \\ [PERSON] et al. [51] & heterogeneous multi-server & response time \\ Ours & heterogeneous multi-server & response time, immediate service probability, blocking probability \\ \hline \end{tabular}
\end{table}
Table 1: Summary of Literature Review.  
\begin{table}
\begin{tabular}{l l} \hline
**Notation** & **Description** \\ \hline \(N_{m}\) & the max tier of MEC servers \\ \(k\) & index set of \(N_{m}\) tiers of MEC server \\ \(\lambda_{u},\lambda_{m}\) & spatial density of users and MEC server \\ \(p_{m,k},\lambda_{m,k}\) & portion and spatial density of the \(k\)th-tier MEC server \\ \(c_{k},r_{k},\mu_{k},\rho_{k}\) & number of CPU cores, waiting queue length, service rate and utilization factor in the \(k\)th-tier MEC server \\ \(T_{s}\) & time slot \\ \(p_{\rho,k}\) & probability that a user offloads to the \(k\)th-tier MEC server \\ \(B_{k}\) & bias factor of the user offloading to the \(k\)th-tier MEC server \\ \(p_{m,k}\) & transmission power of the \(k\)th-tier MEC server \\ \(\alpha\) & path loss exponent \\ \(v_{k}\) & arrival rate of user tasks to the \(k\)th-tier MEC server \\ \hline \end{tabular}
\end{table}
Table 2: Main Notations.  
In order to process the computation task, the user transmits the computation request message to the MEC server (i.e., uplink transmission) and receives the computation resulting data (i.e., downlink transmission). We assume each transmission occurs in a time slot \(T_{s}\) for simplicity. Users select the MEC server to process computation tasks based on association rules, which is based on the maximum biased received power. The probability that a user offloads to the \(k\)th-tier MEC server is indicated by \(p_{o,k}\), according to [52]:  
\[p_{o,k}=2\pi\lambda_{m,k}\int_{0}^{\infty}x\exp\Bigg{\{}-\pi x^{2}\sum_{j\in K }\lambda_{m,j}\bigg{(}\frac{P_{m,j}B_{j}}{P_{m,k}B_{k}}\bigg{)}^{\frac{2}{2}} \Bigg{\}}dx \tag{1}\]  
where \(B_{k}\) is the bias factor of the user offloading to the \(k\)th-tier MEC server, \(P_{m,k}\) is the transmission power of the \(k\)th-tier MEC server, and \(\alpha\) is the path loss exponent.