## 4 Outage Probability Analysis
### Analysis Using Markov Chain  
We analyze the outage probability using Markov chain (MC), where the nodes of MC represent all the possible states of the buffers. The change of state will only occur if there is a successful transmission or reception of packets by the relay. Since this MC is stationary, irreducible, and aperiodic, a steady state \(\pi\) exists, i.e., \(\pi\times A=\pi\). The MC is constructed in such a way that an outage occurs only when there is no change in the state of the buffers  
While calculating the outage probability, we note that the scheme will be in the outage state under the following two conditions: (i) the buffers at receiving relays are full, and the buffers of the transmitting relays are empty; and (ii) the source or the transmitting relay, due to power temperature constraints at the primary receiver, cannot ensure the required SINR at their respective destinations. In the relay cluster, the number of relays are \(K\), each with a buffer size of \(L\). At any time, the number of data packets in each buffer determines the state. As there are \(K\) available relays and every relay is equipped with a buffer of size \(L\), so there are \((L+1)^{K}\) states in total. The \(l^{\text{th}}\) vector is defined as:  
\[S_{l}=[R(B_{1}),...,R(B_{k})],\qquad l=1...(l+1)^{K}. \tag{20}\]  
where \(R(B_{k})\) is the number of data packets in buffer \(B_{k}\) at state \(S_{l}\) and \(0\leq R_{l}(B_{k})\leq L\). Every state corresponds to one pair of \((K_{SI}^{S\to R},K_{SI}^{R\to D})\), corresponding to the number of available source-to-relay and relay-to-destination links, respectively. At state \(S_{l}\), the total number of available source-to-relay and relay-to-destination links is denoted as \(K_{SI}^{S\to R}\) and \(K_{SI}^{R\to D}\), respectively,  
It is clear that \(0\leq K_{SI}^{S\to R}\leq K\) and \(0\leq K_{SI}^{R\to D}\leq K\). If none of the buffers is full or empty, then all links are available such that \(K_{SI}^{S\to R}=K_{SI}^{R\to D}=K\); thus, the total number of links is 2K. However, in one time slot, by virtue of virtual duplex relaying, there could be two simultaneously active links. Assuming that at time \(t\), the state is \(S_{n}\), then \(S_{n+1}\) could be such that there is a relay receiving data packet, so that the number of packets in the corresponding buffer is increased, and at the same time, there is another relay transmitting data packet, while the number of packets in its buffer is decreased. It is also possible that all source-to-relay links are in outage or all relay-to-destination links are in outage. In this case, there can be only transmission or reception by a relay. Depending on which relay receives or transmits data, at time \(t+1\), the buffer may move from \(S_{l}\) to one of the several possible states \((S_{1},S_{2},S_{3},...,S_{(L+1)^{K}})\). We denote \(\cup_{l}\) as the set containing all states that can be moved from \(S_{l}\). Now, as the channels within secondary transmission and the secondary interfering and primary interfering channels are IID (independent and identically distributed) fading channels, we can assume that the SINRs for all channels are also IID. Therefore, the probability of selecting any pair of links is \(\frac{1}{K}\) or one of the \(S\to R\) links (when all \(R\to D\) links are in outage) is \(\frac{1}{K}\). Similarly, the probability of selecting any \(R\to D\) link (when all \(S\to R\) links are in outage) is again \(\frac{1}{K}\). Further, noting that the state remains unchanged if outage occurs (or decoding is not successful), the outage probabilities that \(S_{l}\) moves to a state in \(\cup_{l}\) is given by:  
\[P_{out}=\sum_{l=1}^{(L+1)^{K}}\pi_{l}\times p_{SI}^{out}, \tag{21}\]  
\[p_{SI}^{out}=p_{out}^{S\to R}\times p_{out}^{R\to D}, \tag{22}\]  
\[p_{out}^{S\to R}=(1-e^{-R/\gamma_{S\to R}})^{K_{SI}^{S\to R}}, \tag{23}\]  
\[p_{out}^{R\to D}=(1-e^{-R/\gamma_{R\to D}})^{K_{SI}^{S\to D}}, \tag{24}\]  
where, R is given by \(R=2^{\tau_{l}}-1\) and \(r_{t}\) is the rate in bps/Hz.