## 3 Network Model
### Channel Sensing Model  
For secondary users to begin transmitting, they need to first sense the spectrum for possible PU transmission on the channel. Thus, for channel sensing, we use compressive sensing techniques. This reduces the internal sensing power consumption problem since the sensing nodes are not mobile and may thus not necessarily be powered by batteries; it also increases spectrum efficiency since the cognitive radios do not do sensing and overcomes the hidden primary user problem and uncertainty brought about by shadowing and fading [2]. In order to model the channel, we need to first compute the probability of false alarm, which is the probability that a free channel is incorrectly sensed as having a PU on it, and the probability of mis-detection, which is the probability of incorrectly concluding that a channel is free, yet a PU exists on it. For external sensing, we adopt the model proposed by [34]. Since the external node does the sensing on behalf of the SUs, the SUs will be in wake up mode if and only if they have data to transmit, otherwise, they are in sleep mode which helps them save the energy which would be needed in listening to the external network. For energy detection, it is hypothesized that the signal at the external node at any time \(t\in[0,T]\) is given by Equation (4):  
\[y(t)=\left\{ \begin{array}{l l}x(t)+n(t)&:&\text{channel busy, }H_{1}\\ n(t)&:&\text{channel idle, }H_{0}\end{array} \right. \tag{4}\]  
where \(y(t)\) is the signal received by the external node, \(n(t)\sim N(0,\sigma_{n}^{2})\) is the Additive White Gaussian Noise (AWGN) described by a normal distribution with mean 0 and variance \(\sigma_{n}^{2}\), \(x(t)\) is the transmitted PU signal, assumed to be Circularly Symmetric Complex Gaussian (CSCG) distributed with variance \(\sigma_{x}^{2}\), \(H_{0}\) and \(H_{1}\) are the hypotheses that the PU is OFF the channel and ON the channel, respectively. Channel sensing is done by energy detection. The probability of false alarm \(p_{f}\) and the probability of mis-detection \(p_{m}\) are given as [1]:  
\[p_{f}=Q\left(\frac{\lambda_{T}-N\sigma_{n}^{2}}{\sqrt{2N\sigma_{n}^{2}}}\right) \tag{5}\]  
\[p_{m}=1-Q\left(\frac{\lambda_{T}-N(\sigma_{n}^{2}+\sigma_{x}^{2})}{\sqrt{2N( \sigma_{n}^{4}+  
We adopt a decentralized dynamic spectrum access for SUs, where each SU senses for channels separately. We assume that an SU can only transmit one packet of data in every round. Whenever an SU has data packets to transmit, any of the following states may occur on a given channel. We adopt the states developed in [4]:  
1. Detection: Under this state, the SU senses the presence of a PU on the channel. The probability of detection \(p_{d}\) is given as \[p_{d}=p_{on}-p_{m},\] (8) where the probability of a PU being ON the channel \(p_{on}\) is given by (2), and the mis-detection probability \(p_{m}\) is given by (6). 2. Mis-detection: Under this state, the channel is occupied by a PU but the SU incorrectly senses it to be a free channel. The probability of mis-detection is given by Equation (6). 3. False Alarm: The channel is free and would be available for transmission usage by the SU but the SU misreads the channel to be occupied. This means that this particular SU cannot transmit on this channel at that instant. The probability of false alarm \(p_{f}\) is given by Equation (5). 4. SU channel co-existence: If the channel bandwidth \(W\) is high enough for two or more SUs to transmit on a single channel, then transmission can be done for all SUs that select that particular channel. The probability that two or more SUs select one channel to transmit on \(p_{co}\) is given by the following equation: \[p_{co}=(p_{off}-p_{f})\left[1-\left(1-\frac{1}{I}(1+\frac{p_{off}}{p_{on}}) \right)^{K-1}\right].\] (9)
5. Collision: Since a PU is the first priority user of a channel, if a PU appears and needs to transmit on a channel which is already occupied by a SU, collision happens and the SU would be interrupted and have to vacate the channel for the PU.