# A Jamming-Resilient and Scalable Broadcasting Algorithm for Multiple Access Channel Networks
## 5 Model of Injection
In comparison, for the LBIM, a single node had the highest injection rate, while all the other nodes had an almost 0% injection rate (see Figure 3). This situation is explained by how the LBIM works against a particular broadcast algorithm, for example, the Round Robin Withholding (RRW) broadcast algorithm. In the RRW algorithm, a single virtual token is passed in a round-robin fashion. The node holding the token will broadcast  
Figure 2: packets injection rates for the randomized injection model (RIM) for a 10 min interval. all packets from its queue and then passes the token to the next node in the cycle when it detects silence. The LBIM follows a special injection strategy to make queues grow typically large, beginning by choosing a node as the start of the cycle. Then, the LBIM begins injecting packets into the node preceding the one holding the token. When the token has completed a cycle, the LBIM will switch its injection behavior such that it injects packets into the node following the one holding the token. Due to this switching, one node's queue becomes flooded with injected packets compared to all other nodes, explaining why one node has the highest injection rate. For real-world network traffic, several charts were generated using analysis tools. For environments such as the school library and a hotel, the charts show that a typically low number of nodes generated most of the network traffic, with a relatively higher injection rate compared to all other nodes (see Figure 4). This trend was observed for all interval durations of 5, 10, and 15 min. On the other hand, environments such as residential apartment complexes (RAC) exhibited behaviors more consistent with the RIM (see Figure 5). In this type of network traffic behavior, the injection rate of packets was distributed almost uniformly among the nodes. To ensure that the observed traffic conforms to either of the traffic injection models, we calculated the distance between every collected traffic trace and both injection models. The distance between the two traffic traces is defined as the distance between their distributions of injection rates among nodes. To calculate this distribution, we classified nodes' injection rates into a predefined set of fixed-size bins, where each bin represents a range of injection rates. The probability of each bin was then calculated as the number of nodes in each bin divided by the total number of nodes. We generated this probability distribution for every collected and simulated network traffic trace. To calculate the distance between these two distributions, we used a divergence metric. KL-divergence is a method used for measuring the distance between two probability distributions [49]. However, KL-divergence fails when compared probability distributions do not have absolute continuity, i.e., some bins are empty. Therefore, we used a modified version of KL-divergence with no requirement of absolute continuity to compute the  
Figure 3: Packets injection rates for the LBIM for a 10 min interval. difference between these injection rates [50]. The modified KL-divergence is computed based on the following equation,  
\[D(p,j)=\sum_{x\in X}\frac{p(x)}{(0.5\times p(x)+0.5\times j(x))} \tag{1}\]  
where \(p\) and \(j\) are the two compared probabilities. The probability \(p\) belongs to the main probability distribution that is being compared, which is the real-world network traffic. The probability \(j\) is obtained from the probability distribution of the simulated network traffic. Figure 4: Packets injection rates of captured network traffic in a hotel. Figure 5: Packets injection rates of captured network traffic in a residential apartment complex. We computed the divergence between simulated traffic of both injection models and each collected traffic using the modified KL-divergence (see Table 1). The lower divergence value for each network environment is underlined.