## 4 Improved Adaptive Backoff Algorithm (I-ABA)
### Markov-Based Mathematical Model for I-ABA  
A Markov chain is a mathematical model that describes the behavior of a system that transitions between a set of states such that the probability of moving from one state to another depends only on the current state and not on past states before it. The state space of a Markov chain is the set of all possible states that a system can visit at any given time. In Figure 5, we show the Markov chain model while highlighting the new expression for \(W\). It is essential to mention that modeling backoff algorithms of 802.15.4 using Markov chains are widely adopted in the literature. In creating the Markov chain in Figure 5, we highly benefited from that literature. The state transition and stationary probabilities associated with our Markov chain are similar to or borrowed from our previously published  
Figure 4: The I-ABA algorithm. works [11]. In Figure 5, each state in the Markov model is described by a pair (i, j), where i can be 0 (backoff/CCA states), \(-1\) (successful transmission states), or \(-2\) (collision states). States (0, j) (with j \(\in[1,\,W-1]\)) refer to the states wherein the node is in backoff (in which the node is involved in no activity). State (0, 0) corresponds to CAA1, and State (0, \(-1\)) corresponds to CCA2. States (\(-1,\,\text{j}\)), where j \(\in[0,\,\text{L}_{\text{s}}-1]\), correspond to the time period required to successfully transmit a packet. Finally, States (\(-2,\,\text{j}\)), where j \(\in[0,\,\text{L}_{\text{c}}-1]\), correspond to the time wasted due to a packet collision. The parameters shown in Figure 5 are defined in Table 1.  
For the performance parameters, we reuse the expressions that have been defined in previous works of ours. We note that the derivations are tedious, and to avoid redundancy, we refer the interested readers to the references mentioned above. Therefore, we \begin{table}
\begin{tabular}{c c c} \hline \hline \# & **Parameter** & **Definition/Mathematical Expression** \\ \hline
1 & \(N\) & The size of the network \\ \hline
2 & \(L_{s}\) & The time needed to transmit a packet successfully \\ \hline
3 & \(L_{c}\) & The time wasted due to a frame collision \\ \hline
4 & \(\alpha\) & The probability of finding the medium busy during CCA1 \\ \hline
5 & \(\beta\) & The probability of finding the medium busy during CCA2 \\ \hline
6 & \(\tau\) & The probability that a node initiates CCA1 \\ \hline
7 & \(P_{c}\) & The probability of collision \\ \hline
8 & \(\overline{T}_{BO}\) & The mean backoff time \\ \hline
9 & \(\overline{T}_{CCA}\) & The mean time spent in CCAs \\ \hline
10 & \(\pi_{SC}\) & The probability of a successful cycle \\ \hline
11 & \(\pi_{BC}\) & The probability of a busy cycle \\ \hline
12 & \(\pi_{CC}\) & The probability of a collision cycle \\ \hline
13 & \(p_{tx}\) & The average Power consumed during packet transmission \\ \hline
14 & \(P_{rx}\) & The average Power consumed during packet reception \\ \hline
15 & \(P_{ddle}\) & The average Power consumed during the backoff \\ \hline
16 & \(b_{ij}\) & The probability of being at state (i,j) in the Markov chain \\ \hline
17 & \(x\) & \(\alpha+(1-\alpha)\beta\) \\ \hline
18 & \(y\) & \((1-\alpha)(1-\beta)\)\(P_{c}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters associated with the Markov chain model. Figure 5: Markov-based model for I-ABA. Note that \(W\ =\ h(P_{C})W_{max}\). find it sufficient to provide the expressions without repeating the work that has been published before. The parameters that we will use to evaluate the performance of I-ABA are the probability of collision, channel utilization, channel idle time, channel collision time, delay, reliability, average energy consumption, average energy wasted in collisions, and fairness. In the following subsections, we define these parameters and represent them mathematically based on the Markov-based model shown in Figure 5. All mathematical expressions are generated under the assumption that we have unacknowledged saturated traffic conditions in the network. 1.