# A Novel IEEE 802.15.4e DSME MAC for Wireless Sensor Networks
## 4 Analytical Models for DSME MAC  
In this section, we propose the Markov model to analyze the channel access scheme of the devices in the shared slots. It is assumed that the traffic condition is unsaturated. Based on such unsaturated traffic condition and the notation given in Table 1, the analytical model can be designed as follows. We assume that \(M\) devices are allotted to one superframe for data communication with the coordinator. For a given device, the stochastic processes \(s(t)\) and \(c(t)\) represent the backoff stage for \(NB\) and backoff counter for \(CW\), respectively as shown in Figure 5. Let  
\[b_{i,x}=\lim_{t\rightarrow\infty}P\{s(t)=i,c(t)=x\},\]  
where \(x\in\{-1,...,W_{i}-1\}\)\(W_{i}=2^{min(i+macMINBE,macMAXBE)}\)\(i\in\{0,...,m\}\). The time \(t\) corresponds to the beginning of the slot time and is directly related to the system time. \(b_{i,-1}\) represents the states corresponding to the start of the channel access, where \(i\in\{0,...,m\}\). In our our proposed model, since a device has to wait for two backoff slots before performing the second CCA, we represent the states \(b_{i,-2}\) and \(b_{i,-3}\) corresponding to the waiting states prior to the second CCA, where \(i\in\{0,...,m\}\). \(b_{i,-4}\) represents the states corresponding to the second CCA before start for transmission, where \(i\in\{0,...,m\}\). \(b_{-1,l}\), represents the transmission state, where \(l\in\{1,2,...,L\}\). \(L\) is the number of transmission slots. The state \(l\) stands for the idle state when a device has no request packet to transmit. \(q\) is defined as the probability that the device has a packet to transmit. After generating the request packet, a device goes to the state \(b_{-1,0}\). Upon reaching at the state \(b_{-1,0}\), the device chooses a random backoff and goes to the state \(b_{i,0}\), where \(i\in\{0,1,2...,W_{0}-1\}\). The state \(b_{i,j}\) represents the backoff states, where \(i\in\{0,...,m\}\), \(j\in\{-1,...,W_{i}-1\}\). Let, \(\alpha\) and \(\beta\) be the probability of accessing channel busy during the first and second CCA, respectively. Let \(p\) be the probability that the remaining time in the CAP period is sufficient to complete the data transmission. A device goes to the transmission state, if the channel is idle and attempts to transmit requests. Let \(P_{S}\) be the probability that the transmission is successful. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \(M\) & Total number of devices assigned to one superframe; \\ \(L_{D}\) & Length of a data packet; \\ \(L_{ACK}\) & Length of an acknowledgement packet; \\ \(q\) & Packet generation probability; \\ \(R_{b}\) & Duration of random backoff; \\ \(p\) & Probability that remaining time slots are sufficient to transmit a packet; \\ \(\phi\) & Probability that a device is sensing the channel first time; \\ \(\alpha\) & Probability that the channel is busy during first CCA; \\ \(\beta\) & Probability that the channel is busy during second CCA; \\ \(P_{b}(\zeta)\) & Probability of channel bit error rate; \\ \(P_{S}\) & Probability of transmission success; \\ \(P_{C}\) & Probability of transmission collision; \\ \(P_{tr}\) & Probability of transmission; \\ \(T_{CCA}\) & Duration of CCA; \\ \(T_{D}\) & Duration of data transmission; \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation table. Based on the proposed Markov chain model given in Figure 5, the steady state probabilities can be derived as follows. \[P\{b_{i,k}|b_{i,k+1}\}=1\ \ for\ 0\leq i\leq m\ and\ 0\leq k\leq W_{i}-1. \tag{3}\]  
\[P\{b_{i,k}|b_{i-1,0}\}=\frac{p+(1-p)\alpha+(1-p)(1-\alpha)\beta}{W_{i}} \tag{4}\]  
\[for\ 0\leq i\leq m\ and\ 0\leq k\leq W_{i}-1.\]  
\[P\{I|b_{m,0}\}=(p+(1-p)\alpha+(1-p)(1-\alpha)\beta)+(1-p)(1-\alpha)(1-\beta)P _{S}. \tag{5}\]  
Equation (3) represents the decrement of backoff counter, which occurs with probability 1. Equation (4) represents the probability of finding channel busy during CCA and thereafter a device selects uniformly a state in the next backoff state.