## 3 Delay and Energy Consumption Analysis
### Absorbing Markov Chain Model  
In the AMC model depicted in Figure 3, each state represents the number of end-devices that have successfully transmitted their data packet to the network coordinator since the contention process started. The AMC is characterized by a transition matrix \(\mathrm{P}=\left[p_{ij}\right]\), where each element \(p_{ij}\) represents the probability that the number of successful end-devices changes from \(i\) to \(j\) after one frame, with \(i,j=\left\{0,1,...,n\right\}\). The process will finish when all the end-devices have succeeded transmitting their data packet, i.e., \(i=n\). Once this absorbing state is reached, the probability that the process changes to any other state is zero. Figure 3: Generalized state transition diagram of the absorbing Markov chain that models the evolution of the numbers of end-devices that succeed in transmitting data using FSA-ACK, FSA-FBP and DFSA (Dynamic FSA)  
Figure 2: FSA-FBP operation example with four end-devices and three contention slots per frame. The transition probabilities between states depend on the probability \(P_{s}(m_{c},m_{1},n_{c})\) that there is a number \(m_{1}\) of successful slots in a given frame with \(m_{c}\) contention slots having \(n_{c}\) contending end-devices, as derived in [20]. According to [21], the probability \(P_{s}(m_{c},m_{1},n_{c})\) can be expressed as  
\[P_{s}\left(m_{c},m_{1},n_{c}\right)=\frac{\left(m_{c}\right)\prod\limits_{k=0} ^{m_{1}-1}\left(n_{c}-k\right)G\left(m_{c}-m_{1},n_{c}-m_{1}\right)}{{m_{c}}^{ n_{c}}}, \tag{1}\]  
where  
\[G\left(T,t\right)=T^{t}+\sum\limits_{k=1}^{t}\left\{\left(-1\right)^{k}\cdot \prod\limits_{j=0}^{k-1}\left\{\left(t-j\right)\left(T-j\right)\right\}\left(T -k\right)^{t-k}\frac{1}{k!}\right\}, \tag{2}\]  
with \(T=m_{c}-m_{1}\) and \(t=n_{c}-m_{1}\). Hence, the transition probabilities \(p_{ij}\) for FSA-ACK and FSA-FBP can be defined as  
\[p_{ij}=\left\{ \begin{array}{ll}P_{s}\left(m,j-i,n-i\right),&if\ \ \left(i<j\leq i+m\right)\\ 0,&if\ \left(j<i\right)\ or\ \left(j>i+m\right)\\ 1-\sum\limits_{k=i+1}^{i+m}p_{ik},&if\ \left(i=j\right)\ and\ \left(i<n\right)\\ 1,&if\ \left(i=j\right)\ and\ \left(i=n\right)\end{array} \right.. \tag{3}\]  
The rationale for the FSA-ACK and FSA-FBP transition probabilities is that in the first condition, \(j-i\) end-devices succeed and the number of contending end-devices is \(n-i\), where \(i\) the number of end-devices that have succeeded in previous frames. In the second condition, the number of successful end-devices decreases, or is greater than the number of slots, which are both difficult to achieve. In the third condition, no new end-devices succeed and the process is not yet absorbed. Finally, in the fourth condition the process is absorbed. In contrast, the transition probabilities \(p_{ij}\) for DFSA can be defined as  
\[p_{ij}=\left\{ \begin{array}{ll}P_{s}\left(m_{i},j-i,n-i\right),&if\ \left(i<j\leq i+m_{i}\right)\\ 0,&if\ \left(j<i\right)\ or\ \left(j>i+m_{i}\right)\\ 1-\sum\limits_{k=i+1}^{i+m_{i}}p_{ik},&if\ \left(i=j\right)\ and\ \left(i<n \right)\\ 1,&if\ \left(i=j\right)\ and\ \left(i=n\right)\end{array} \right.. \tag{4}\]  
The rationale for the DFSA transition probabilities is similar to FSA-ACK and FSA-FBP, but the number \(m_{i}\) of contention slots per frame changes dynamically as the number \(i\) of successful end-devices increases, i.e., \(m_{i}=\left\lceil\rho\ (n-i)\right\rceil\). Based on the AMC model, we define Q as a sub-matrix of the transition matrix P that contains the probabilities in the transient states (i.e., \(i<n\) and \(j<n\)). Then, the fundamental matrix N of the process can be defined as  
\[\text{N}=\left(\text{I}-\text{Q}\right)^{-1}, \tag{5}\]  
where each element \(n_{ij}\) of N is the average number of frames that the process remains in the transient state \(X_{j}\). This means that no new end-devices succeed for \(n_{ij}\) consecutive frames, given that it started in the transient state \(X_{i}\).