## 5 Numerical Simulation  
To validate the performance of our work, a numerical simulation is executed in this section. The common parameters in this simulation are shown in Table 2. If extra or varied parameters are used in a specific scenario, they will be seperately described in the following discussion. To display the interaction of ONUs, MPPs, and MAPs, two fiber networks are concurrently connected to the Internet, forming the simulation scenario. Each OLT has 16 ONUs, via a 1:16 splitter. The fiber backhaul is a WDM-based PON. In this PON, the bandwidth is allocated in a static way. This means that the bandwidth for each ONU is fixed. The OLT would allocate a fixed bandwidth for each ONU, according to the ONU's regular service-level agreement (i.e., bandwidth, latency, etc.) The user/human/machine connects to ONUs through the multi-hop wireless link. The data-computing speed of servers at ONUs, OLTs, and the Internet, are different. The values of some constant parameters, such as \(p_{0}\), \(\sigma\), and \(M\), are also shown in Table 1.  
The work of [20] is selected as a control group. In this control group, a three-tier network is also considered. The backhaul is not the fiber network with signal attenuations, but a fixed network with an ideal bandwidth (i.e., no signal attenuation). The wireless communication in the control group is completed via the single-hop wireless link, not the multi-hop wireless \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline \(p_{0}\), transmit power of users & 35 dBm \\ \(\sigma\), noise power density & \(-50\) dBm/Hz \\ M, the number of receiving antennas & 10 \\ \(f^{ONU}\), the data-computing seepd of servers at ONU & 3000 GHz \\ \(f^{OLT}\), the data-computing seepd of servers at OLT & 2000 GHz \\ \(f^{Int}\), the data-computing seepd of servers at Intnet & 1000 GHz \\ The number of OLTs & 2 \\ The number of ONUs for each OLT & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Common parameters. link used in our work. Note that the solution to the control group's formulated problem is realized by the commutation method, not the constraint condition relaxation. The data transmission rates of our work and the control group in the wireless sub-network are shown in Figure 3. The bandwidth of the wireless link is 1000 Hz. According to the 802.11 protocol family, the number of hops varies from 1 to 15. The traffic in this simulation is simplified as a data stream with a certain rate. For the control group, the data transmission rates with different \(\alpha\) values are almost the same. For our work, data transmission rates with different \(\alpha\) values are presented as three clearly differentiated curves. The signal attenuation in the control group only has an influence on the initial user/human/machine signal. The signal attenuation in our work affects the initial user/human/machine signal and noise at the same time. With the increase in hop number, the transmission rate of our work decreases. This is the result of the multiple hops' multiplication effect. For the same reason, the larger \(\alpha\) is, the stronger the downward trend of the data transmission rate. For each situation in Figure 3, the smallest value is 99.9999% of the largest value. The difference is negligible. The signal strength received at the place of OLTs is shown in Figure 4. The distance here is from an ONU to its connected OLT. Assume that the signal strength at the place of ONUs is 0.045 mW, according to the calculation in Figure 3. The signal strength is inversely proportional to the distance. The further the distance, the more the signal fades in the fiber network and the weaker the signal strength. This can be also extrapolated, based on Equation (4). Figure 4: The signal strength vs. the distance. Figure 3: The data transmission rate vs. the number of hops. For each distance in our work, the situation with the greatest \(\beta\) has the largest signal strength.