## 4 Performance Evaluation  
In this section, we verify the proposed method by comparing its performance with that of conventional schemes under the same environment. Henceforth, we call the proposed method, leximax minimal load balancing redirection (LMLBR). We selected the following three representative schemes for performance comparison: (1) **Random redirection (RR). ** In RR, the overloaded MECS redirects its tasks to a randomly selected MECS; (2) **Nearest redirection (NR). ** In NR, the overloaded MECS redirects its tasks to the MECS that is geographically closest; (3) **Least-loaded redirection (LR). ** In LR, the overloaded MECS redirects its tasks to the least-loaded MECS. We distributed 50 MECSs in an area of 1000 \(\times\) 1000 m. According to [20; 21; 22], we set the simulation parameters as shown in Table 2. The capacity of each MECS (\(C_{i}\)) was randomly selected in [9 GHz, 11 GHz], according to a uniform distribution. We set the task arrival process at a MECS to follow a Poisson point process (PPP) with rate \(\lambda\). When task \(x\) is generated, its \(a_{x}\), \(w_{x}\), and \(T_{x}\) are randomly chosen from the given ranges in Table 2 according to a uniform distribution. For example, the data size of a task is randomly selected in [3 kbits, 6 kbits]. We set the length of a time slot to 100 ms, which is the frame time between a device and a BS in an LTE system. We set the size of waiting queue and service queue of a MECS so that the number of offloaded tasks does not exceed the queue capacity of each MECS, when the number of devices associated with each MECS is evenly distributed and the number of offloaded tasks for each time slot is distributed uniformly. We first investigate the load variance of a MECS in Figure 2. The figure presents a box plot for the load changes of the 32 nd MECS during the simulation time when the task offload rate is 0.5, 0.7, and 0.9. In NR and LR, all the overloaded MECSs select the best MECS in terms of the distance (NR) or the load (LR) as their target MECS and redirect their tasks to the best MECS. Therefore, the load of the best MECS increases sharply after it accepts the tasks redirected to it within a time slot. Therefore, the load of a MECS varies substantially depending on whether the MECS is chosen as the best MECS. However, in the proposed method, the variance is relatively small because tasks are redirected from a congested MECS to more than one target MECS according to the load difference between the congested MECS and the target MECS. \begin{table}
\begin{tabular}{l l} \hline \hline
**System Parameters** & **Values** \\ \hline Subcarrier bandwidth & 15 kHz \\ Background noise & 10\({}^{-13}\) W \\ Data size of task \(x\) (\(d_{x}\)) & 3000–6000 bits \\ Workload of task \(x\) (\(w_{x}\)) & 0.1–1.0 GHz \\ Maximum delay allowed for task \(x\) (\(T_{x}\)) & 0.1–0.2 s \\ CPU frequency of MECS \(i\) (\(C_{i}\)) & 9–11 GHz \\ Waiting queue size of a MECS & 5 tasks \\ Service queue size of a MECS & 5 tasks \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 2: _Cont_. To scrutinize the load variance among MECSs, in Figure 3, we present the loads of all 50 MECSs at a given time slot when the task offload rate is 0.5, 0.7, and 0.9. LMLBR minimizes the MECS load vector lexicographically. In NR and LR, since the redirected tasks are absorbed into the best MECS at once, the loads of some MECSs are very high, while the loads of the other MECSs are low. However, in LMLBR, tasks are assigned to multiple MECSs, which makes the load difference among MECSs the smallest. To compare the four methods in terms of the quality of service provided by the MEC system, we inspected the average task blocking rate and the rate of tasks completed within their delay constraints. When a task is offloaded to a MECS whose queue is full, the task is blocked by the MECS. The average task blocking rate is defined as the fraction of tasks that are blocked by the MECSs in the system. In Figure 4, the average task blocking rate is depicted with varying \(\lambda\).