*5.1. Planning for the One-Resource-One-Agent-Type*

The first experiment considered Wi-Fi as the only resource requested by agents (i.e., *A* = {*A*<sup>1</sup> : "*WiFi*"}). Only one agent type exists, and all agents use sigmoid-based function *f*<sup>1</sup> as the collision CDF.

We first studied the influence of the collision threshold *T* and the offset parameter *δ* on performance. Figure 5a shows the success rate of the one-resource-one-type SC-M\* with different thresholds *T* =0 (equivalent to the basic M\*), 0.2, 0.4, and 0.45, while the offset parameter is fixed to *δ* = 6.0. Table 1 (left) shows the run time in seconds for the experiment. The results clearly show that larger thresholds bring improvement in performance with a higher success rate and lower run time for a large system size (*m* > 50). The improvement in performance results from the property of SC-M\* that larger thresholds render more relaxed constraints, and thus, agents are less likely to collide on resources.

**Figure 5.** Impact of the collision threshold *T* (given *δ* = 6.0) and offset parameter *δ* (given *T* = 0.35) on one-resource-one-type SC-M\*.

Figure 5b shows the success rate of the SC-M\* with different offset parameters *δ* = 0, 3.0, 6.0, and 9.0, with fixed *T* = 0.35. Table 1 (right) shows the run time for the experiment. The results

illustrate that SC-M\* is sensitive to *δ* and can efficiently handle up to 100 agents with *δ* = 9.0. These results are reasonable because the sigmoid-based CDF is used in the experiments, featuring a surge in the collision probability at the experience value around the offset, and the offset parameter poses a cutoff value on the resource experience, with collision always announced once the resource experience is larger than the offset. The standard M\* (*T* = 0) can only scale to fewer than 30 agents. Taking advantage of this property, one can tune the parameters to trade off the scalability against the tightness of constraints.


**Table 1.** Run time of one-resource-one-type SC-M\* under different parameters.
