## 3 System Model and Performance Analysis  
Assume that in a single BSS, there are \(n\) STAs (\(n_{1}\) critical patient STAs and \(n_{2}\) mild patient STAs), each STA is saturated, and the STA always has at least one packet to transmit. To transmit packets, STA needs to continuously listen to the busy/idle state of the channel,  
Figure 5: Utilization of the spectrum resources by SU (\(a>1\)). Figure 6: Contrast CR-OFDMA with traditional OFDMA and DRA-OFDMA. and if STA continuously listens to the channel idle with the DIFS, let the backoff counter randomly select a value to start backoff. For each idle slot, the value of the backoff counter is subtracted by one until the value of the backoff counter is reduced to 0, and then the STA can send packets. The notations in the model are summarized in Table 2.  
In order to reduce the collision, we learned the backoff mechanism from the literature [27]. \(b(t)\) is stochastic process that presents the value of the backoff counter at time slot \(t\), and \(s(t)\) denotes stochastic process representing the backoff stage at time \(t\). The pair \((i,j)\) is used to represent the state \((s(t),b(t))\) of each STA. The value of the backoff counter is chosen randomly from \((w_{i-1},w_{i}-1)\) for the STA in backoff phase \(i\), where \(w_{i}\) denotes the size of the competition window of the \(i\)th backoff phase and \(w_{i}=2^{i}w_{0}\), \(w_{0}\) is the initial value of the competition window. When \(i=0\), the value of the backoff counter is chosen randomly from \((0,\ w_{0}-1)\). The backoff phase takes the value from \((0,\ 1,\ 2\ldots m)\), and if the packet of the STA is successfully transmitted, then the backoff phase is reduced by one; if the collision occurs, the backoff phase is increased by one. Since high-priority STAs can transmit on the remaining subchannels after random contention failure, \((i^{*},0)\) is introduced to represent the state of this part of the high-priority STAs. Thus, compared with the backoff mechanism in [27], the STA does not enter the next backoff phase directly after a collision occurs. Instead, it enters the state \((i^{*},\ 0)\) (at this time, the backoff phase is still \(i\)). The STA enters the next backoff phase \(i+1\) if the transmission on the remaining subchannel collides again. After the STA collided into state \((i^{*},0)\), if it successfully transmits on the remaining subchannel, the STA's backoff state will change to \(i-1\). If the STA successfully transmits the packet without collision, the STA's backoff state is minus one. STA in the maximum backoff stage \(m\) enters state \((m^{*},0)\) after a collision and will return to the \(m\) backoff stage without plus 1 if the collision occurs again. Assume that the probability of collision is \(p_{1}\) when critical patients with high-priority compete fairly with low-priority patients, and the probability of collision is \(p_{2}\) when the high-priority patients transmit on the remaining subchannel. We define \(d_{0}=w_{0}-1\), \(d_{i}=w_{i}-w_{i-1}\). The state transition diagram for the high-priority patients is given in Figure 7.  
The transition probabilities are listed as follows:  
1. The STA listens to the channel idle, and the value of the backoff counter is reduced by 1: \[P\{(i,j)|(i,j+1)\}=1;\ \ 0\leq i\leq m,\ \ w_{i-1}\leq j\leq w_{i}-2.\] (2)
2. The STA collides during fair competition, and STA enters state \((i^{*},0)\) from state \((i,0)\): \[P\{(i^{*},0)|(i,0)\}=p_{1};\ \ \ 0\leq i\leq m.\] (3)
3. The STA does not collide, and the backoff state subtracts 1: \[P\{(i-1,r)|(i,0)\}=\frac{1-p_{1}}{d_{i-1}};\ \ 0\leq i\leq m,\ w_{i-2}\leq r\leq w_{i-1}-1.\] (4)