## 4 Numerical Results  
Let us consider the network configuration depicted in Figure 10, with 15 sensor nodes, one sink node and 5 capacity constraints that deploy a three-level cluster tree. Note that the following results can be extrapolated to WSNs with a larger number of nodes and a larger number of levels and thus, our simplified example has no loss in generality. Specifically, the Superframe (SF) is configured as follows: \(T_{BI}\) is set to 245.76 ms and is the same at all levels; the SF duration is 61.44 ms at the highest level, 30.72 ms at the intermediate level and 15.36 ms at the lowest level. According to [33], this configuration provides 9.38 kbps, 10.94 kbps and 13.02 kbps of data rate, respectively. Furthermore,if we consider 15 slots per BI dedicated to the Guaranteed Time Slot (GTS) allocation, the size of the slots in bits is 9, 21 and 50, respectively. Accordingly, the rate capacities of the clusters are set to \(\mathbf{c}=[3.0516,1.2820,1.2820,1.2820,0.5496]^{T}\) kbps. In our simulations, we compare our CDM-based approach to the widely-used First Come First Serve (FCFS) policy. We assume, without loss of generality, that the values of \(w_{j}\) and \(PDR_{j}\) are set to 1 and that all the sensors are of the same type and thus, all of them need to transmit \(n\) bits per BI. However, the rate requests are quantized by the time slot duration at each sensor so that each node asks for \(\bar{n}_{i}/T_{BI}\) kbps, where \(\bar{n}_{i}\) is the nearest multiple (above) of the packet size in bits at the i-th node. Figure 11 shows an allocation example for \(n=20\) bits. We notice that the CDM-based policy provides a rate allocation that is very close to the optimal, whereas the FCFS solution does not guarantee a fair resource allocation, even when the throughput is not penalized. Note that, in the example under consideration, there exists no trade-off between fairness and throughput because we have set error-free links. Note also that our proposed time slot allocation does not provide the optimum fair rate allocation due to the rounding effects. Figure 11: Allocation example for \(n=20\) bits. Figure 10: Network configuration used in the numerical results. In order to study the fairness degree provided by both solutions, we plot in Figure 12 the Fairness Index (FI) as a function of the requested bits \(n\). The FI takes values between 0 and 1 and measures how far is a certain resource allocation with respect to the one that is considered optimal or most fair. If \(FI=0\) the allocation is totally unfair and if \(FI=1\), the resulting allocation is completely fair. A value that is in-between such as 0.6 may be interpreted as follows: the 60% of allocations are considered fair in mean. The FI is computed as [34]  
\[FI=\frac{\sum_{j}z_{j}^{2}}{N(\sum_{j}z_{j})^{2}} \tag{29}\]  
where \(z_{j}=\frac{x_{j}}{r_{j}^{*}}\), being \(r_{j}^{*}\) the optimal fair allocation and \(x_{j}\) the allocation under test. Therefore, \(x_{j}\) represents the rate allocation that results from either our CDM-based time slot distribution or the standard-proposed FCFS approach. Furthermore, \(r_{j}^{*}\) is the real-valued optimal rate allocation, that is, without rounding. Note in the figure that the CDM-based approach provides a solution that is very close to the optimal whereas the FCFS allocation tends to be more unfair as the transmission requirements of the sensors grow until it stabilizes around 0.5, which is pretty unfair. Therefore, FCFS is an acceptable strategy for small traffic loads, that is, when the network resources suffice to satisfy all the sensor requirements. However, the allocation is unfair at high loads because the mechanism does not distinguish sensors that are parents of a sub-cluster tree from sensors that are not. Since the two receive the same amount of resources in mean, the sensors in the sub-cluster tree (including the parent node) are thus penalized.