## 4 Performance Evaluation
\begin{table}
\begin{tabular}{l c c c} \hline \hline \(\mathbf{\lambda}\) & \(\mathbf{0.5}\) & \(\mathbf{0.7}\) & \(\mathbf{0.9}\) \\ \hline RR & 0.0247 & 0.0491 & 0.0792 \\ LMLBR & 0.0521 & 0.0605 & 0.0666 \\ Difference (\(\delta_{bp}\)) & 0.0274 & 0.0114 & \(-\)0.0126 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of RR and LMLBR in terms of the average task blocking rate (Difference (\(\delta_{br}\)) represents the average blocking rate obtained by LMLBR minus the average blocking rate obtained by RR). delay is proportional to the queue length, which is proportional to the MECS load. Figure 6 shows that the rate of tasks completed within their delay constraints is the smallest for all \(\lambda\)s when LMLBR is used. Since the MECS load vector obtained by LMLBR is leximax minimal, the queue length vector whose element is the queue length of each MECS is also leximax minimal, which results in the smallest total delay for a given \(\lambda\). Overall, the rate of tasks completed within their delay constraint for LMLBR is 10-42% better than that achieved by other methods. To inspect the influence of the number of redirection per task on the distribution of MECS load, we show in Figure 7 the loads of all 50 MECSs at the same time slot when \(\lambda=0.9\). As the number of task redirection increases, the load imposed on a MEC system is distributed among MECSs more fairly. However, the congestion level of a network connecting MECSs also increases with the number of task redirections. In addition, the time from when a task is first offloaded from a device to a MEC system to the time when it is finally served by a MECS also increases with the number of times that the task is redirected. Since the problem of determining the optimal number of task redirection in itself deserves to be investigated thoroughly, we set this problem as one of our future works. To investigate the sensibility of performance with respect to the simulation parameters, we evaluated the performance by varying the queue size. In each MECS, we set the size of the service queue to be the same as that of the waiting queue. We show the  
Figure 6: Rate of tasks completed within the delay constraints versus the task offload rate. Figure 7: Influence of the number of redirect per task on the load distribution among MECSs when \(\lambda=0.9\). results in Tables 4 and 5. In Table 4, as the queue size increases, the task blocking rate decreases, but the difference is too small to mean much in practice. However, in Table 5, the performance comparison is meaningful in terms of the rate of tasks completed within the delay constraints. This is attributed to the following facts. In LMLBR, a MECS \(i\) redirects tasks to the MECSs whose loads are lower than \(\rho_{i}\). In addition, the amount of tasks redirected from MECS \(i\) to MECS \(j\) is not excessive because LMLBR considers the load difference between \(i\) and \(j\). However, in RR, an overloaded MECS redirects the tasks to the randomly selected MECS until it becomes underloaded. If tasks are redirected to one MECS excessively, the service queue of the MECS increases sharply, which results in the large waiting time and the decreases in the rate of tasks completed within the delay constraints. Since the proposed method requires the exchange of information between MECSs, it incurs communication overhead. If a broadcast channel is used to exchange the load information among \(|N|\) MECSs, the communication cost is \(O(|N|)\). If an unicast channel is used to exchange the information, the overhead is \(O(|N|^{2})\). The factor determining whether the exchange of information between MECSs is required or not is the way that a MECS decides whether it is overloaded. If a MECS decides to redirect its tasks when its load is larger than a local threshold value, the communication cost of the method is zero. However, it is difficult to configure an optimal threshold value according to the dynamic network condition.