# Design and Analysis of a Low Latency Deterministic Network MAC for Wireless Sensor Networks
## 5 Analytical Model for Shared Slots in the LLDN Superframe  
In this section, we design one Markov model to analyze the network performance on the shared slots. We consider that the network traffic is unsaturated. Based on such unsaturated traffic and the notations given in Table 1, the analytical model can be designed as follows. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \(N\) & Total number of nodes in a system. \\ \(L\) & Length of a data packet. \\ \(GACK\) & Group acknowledgment. \\ \(\lambda\) & Packet arrival rate. \\ \(R_{backoff}\) & Random backoff. \\ \(p_{1}\) & Probability that a node has a packet to transmit. \\ \(p_{2}\) & Probability that remaining time slots are sufficient to transmit a packet. \\ \(p_{b}\) & Probability that the channel is busy during a CCA. \\ \(\phi\) & Probability that a node is sensing the channel first time. \\ \(\alpha\) & Probability that the channel is busy due to simultaneous CCA. \\ \(\beta\) & Probability that the channel is busy due to data transmission. \\ \(P_{b}(\xi)\) & Probability of channel bit error rate. \\ \(P_{TSSS}\) & Probability of transmission success in a shared slot. \\ \(P_{RS}\) & Probability of getting a retransmission slot. \\ \(P_{RTS}\) & Probability of retransmission success. \\ \(P_{success}\) & Probability of transmission success within an LLDN superframe. \\ \(L_{Decon}\) & Length of the beacon packet in bits. \\ \(L_{data}\) & Length of the data packet in bits. \\ \(L_{GACK}\) & Length of the group acknowledgment packet in bits. \\ \(T_{CCA}\) & Duration of CCA. \\ \(\overline{T}_{L}\) & Duration of data transmission. \\ \(SS_{duration}\) & Duration of one shared slot. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation table. Recall that \(N_{S}\) is the number of nodes associated with a shared slot, and each node has data arrival rate \(\lambda\). As shown in Figure 3, the stochastic processes \(s(t)\) and \(c(t)\) represent the backoff states for a given node, which is fixed to zero, and the backoff counter for the contention window for the shared slot, i.e., \(W_{S}\), respectively. Let, \(b_{0,i}=\lim_{t\rightarrow\infty}P\{s(t)=0,c(t)=i\}\), where \(i\in\{0,1,2,...,W_{S}-1\}\). The time \(t\) corresponds to the system time. \(b_{0,-1}\), \(b_{-1,1}\) and \(b_{-2,0}\) represent the states corresponding to the start of the channel access, transmission and idle state, respectively. Let us consider \(p_{1}\) as the probability that a node generates one packet for transmission. The node first goes to the \(b_{-1,0}\) state, chooses a random backoff \(b_{0,i}\) state, for \(i\in\{0,1,2,...,W_{S}-1\}\). Let, \(p_{b}\) be the probability of accessing the channel busy during the CCA and \(p_{2}\) be the probability that the remaining time in the shared slot is sufficient to complete the data transmission. Once the channel is found idle, the node goes to the transmission state. We have considered \(p_{s}\) as the probability of successful transmission. Based on the proposed Markov chain model given in Figure 3, the transition probabilities for deducing the steady state probabilities can be derived as follows. \[b_{0,i}|b_{0,i+1}=1\;\;for\;0\leq i\leq W_{S}-1. \tag{5}\]  
\[b_{-1,0}|b_{0,-1}=p_{b}. \tag{6}\]  
Equation (5) represents the decrement of the backoff counter, which occurs with a probability of one. Equation (6) represents the probability of finding the channel busy during the CCA, and thereafter, a node selects a state in the next backoff state. Based on these transition probabilities, we can derive the steady state probabilities as follows. A node normally goes for the CCA after the random backoff period whenever the remaining time in the shared slot is enough to transmit the data. The corresponding steady state probability can be deduced as follows. \[b_{0,-1}=(1-p_{2})b_{0,0}. \tag{7}\]  
If the remaining time in the shared slot is sufficient for the data transmission, the node goes for the CCA, and upon finding the channel idle, it starts the data transmission.