## 6 Simulation Results
### Simulation Results
We used the same experimental setup as in Figures 4 and 5, with the cost constraint set to be very small. From the experimental findings, it is evident that as the approximation ratio \(\epsilon\) gradually increases, the delay values of the offloading solutions produced by TUL also tend to increase. Meanwhile, the running time of TUL decreases. Additionally, the running time of TUL is significantly smaller than that of the brute force algorithm. However, it is important to note that, even with the increase in \(\epsilon\), the delay values of these solutions never exceed the bound of \((1+\epsilon)\) times the optimal solution. This behavior highlights the effectiveness of TUL in maintaining a close approximation to the optimal solution. Figure 8: Effect of \(\epsilon\) on TUL. Figure 6: Effect of task size on delay. Figure 7: Effect of task size on running time. In Tables 2 and 3, we compare the effects of the number of tasks, the number of execution nodes, and the value of \(\epsilon\) on the two algorithms to analyze their approximation ratio, time complexity, and overall differences. In this experiment, the task mapping is derived from a randomly generated DAG diagram. to represent the task dependency structures. From the results, it is observed that when the number of tasks, \(N^{\prime}\), increases while keeping other parameters constant, the delay and running time of each algorithm increase. However, due to the random generation of DAG, increasing the number of tasks may increase the width of the DAG rather than its depth, which sometimes results in a smaller delay value despite a larger number of tasks. When the number of execution nodes, \(M^{\prime}\), increases while keeping other parameters constant, the delay and running time of each algorithm decrease. This is because the increased number of execution nodes leads to a larger set of feasible solutions, thereby improving the chances of finding better solutions. When the values of \(N^{\prime}\), \(M^{\prime}\), and \(\epsilon\) are small, the running times and results of TUL and BFF show little difference. Since the time complexity of TUL and BFF are linearly related to \(N^{\prime}\) and \(N^{\prime 2}\), respectively, the running time difference between TUL and BFF is not significant when the number of tasks increases. However, as the number of execution nodes increases, the running time difference between BFF and TUL becomes much more significant. This is because the running time of BFF is proportional to \(M^{d_{in}}\), while the running time of TUL is proportional to \(M^{\prime 2}\). In a DAG, the maximum indegree, \(d_{in}\), is usually greater than 2, so the running time of BFF increases much more significantly with \(M^{\prime}\) than that of TUL. Additionally, we observe that when \(\epsilon\) decreases, TUL's running time difference becomes more pronounced with increases in tasks and execution nodes. This is because the running time of TUL is inversely proportional to \(\epsilon\), so for the same changes in the number of tasks and execution nodes, smaller values of \(\epsilon\) result in larger variations in running time. Furthermore, when the value of \(\epsilon\) decreases while keeping other parameters constant, TUL typically produces more accurate results but at the cost of longer running times. Although there are cases where a larger \(\epsilon\) yields better results, TUL guarantees a \((1+\epsilon)\)-approximate solution. Regardless of the value of \(\epsilon\), TUL will never exceed the optimal solution by more than a factor of \((1+\epsilon)\), further validating the truth of TUL's approximation ratio.