## 4 Change Detection Mechanism
### Detectability Analysis  
An arbitrary node \(i\) can detect such changes when there is a significant increase or decrease on \(\beta\). The sensitivity of this detection mechanism is first analyzed. Any change on the number of nodes will first affect the value of \(\sum_{j\
eq i}(1/\lambda_{i})\). The sensitivity of \(\beta\) in terms of \(\sum_{j\
eq i}(1/\lambda_{i})\) can be derived by finding the derivative of \(\beta=g^{-1}(\sum_{j\
eq i}(1/t_{j}))\). Since the expression of \(g^{-1}(\sum_{j\
eq i}(1/t_{j}))\) can hardly be explicitly found, its inverse function \(g(\beta)\) is instead analyzed. It can be shown that its derivative \(g^{\prime}(\beta)=\frac{1+mg^{m+1}}{(1-\beta^{m+1})^{2}}\frac{1}{1+L}\) is a monotonically decreasing function and when \(\beta\) is small, \(g^{\prime}(\beta)\) can be approximated as \(g^{\prime}(\beta)\approx\frac{1}{1+L}\). According to the property of the derivative of inverse function, the derivative of \(g^{-1}(\sum_{j\
eq i}(1/\lambda_{i}))\) is approximately \((1+L)\) when \(\beta\) is small. When the busy channel probability is high, the slope of \(g^{-1}(\sum_{j\
eq i}(1/\lambda_{i}))\) will decrease significantly. Figure 2 shows the sensitivity of \(\beta\)_versus_ the busy channel probability for different combinations of \(m\) and \(L\). It can be shown that when \(\beta<0.4\), the sensitivity is close to \(1+L\). It drops quickly when the \(\beta>0.5\) and the sensitivity is close to \(1\) when \(\beta=0.8\). Note that for control and monitoring applications, the traffic load is relatively low and \(\beta\) rarely exceeds \(0.5\). Thus, every unit of change in \(\sum_{j\
eq i}\big{(}1/t_{j}\big{)}\) will in general be amplified to around \(1+L\) units of change in \(\beta\). Furthermore, \(\beta\) is obtained by collecting the CCA information and is also subject to randomness. The randomness may also result in fluctuations in \(\beta\) even when there is no change on the number of active nodes. Thus, it is necessary to quantify such fluctuations and evaluate its impact on the proposed detection mechanism. Each CCA can be considered as an independently identically distributed (_i.i.d_) process with a mean busy channel probability \(\overline{\beta}\). In other words, each CCA is a Bernoulli trial and when \(T\) CCAs are performed for a single update interval, the number of busy results follows binomial distribution \(B(T,\ \overline{\beta})\). Furthermore, when \(T\overline{\beta}>5\) and \(\big{(}1-\overline{\beta}\big{)}>5\), the binomial distribution can be approximated as the normal distribution \(N(T\overline{\beta},T\overline{\beta}\big{(}1-\overline{\beta}\big{)})\)[35]. Thus, \(\beta\) also follows the normal distribution \(N(\overline{\beta},\frac{\overline{\beta}\big{(}1-  
Combining the results of the sensitivity analysis and the fluctuation analysis, it can be concluded that the changes on the traffic, namely \(\sum_{l\inchange}(1/t_{l})\), can be detected when the change on \(\beta\) caused by the traffic exceeds the maximal fluctuation on \(\beta\). In other words, \(\sum_{l\inchange}(1/t_{l})\) can be detected with \(\alpha\%\) confidence if:  
\[\sum_{l\inchange}(1/t_{l})\geq\frac{2\epsilon\sqrt{\frac{\overline{\beta}(1- \overline{\beta})}{T}}(1+m\overline{\beta}^{m+1})}{\left(1-\overline{\beta}^{ m+1}\right)^{2}(1+L)} \tag{15}\]  
In actual implementations, as \(\overline{\beta}\) is unavailable, the measured \(\beta\) can be used to approximate \(\overline{\beta}\). It is also more convenient to use the update interval \(I\) (sec) instead of \(T\). They can be easily related as \(I=\frac{\tau(1-\overline{\beta})\lambda_{l}}{(1-\overline{\beta}^{m+1})RB_{slot}}\), where \(R\) is the data rate (bps) at PHY layer and \(B_{slot}\) is the number of bits in a slot.