## 3 System Model
### Intra-Cluster Communication Session
The number of packets arrivals \(X_{t}\) at the \(t\)th frame of period \(T\) is a Poisson process of arrival rate \(\lambda_{p_{i}}\) with a probability distribution given as [21,



in state \(x=j\) where there are \(j\) number of active nodes in a frame, this probability is derived by applying the stationary distribution to the transition matrix \(P_{ij}\) and solving the below equations:

\[\pi_{j}=\sum_{t=0}^{K}\pi_{i}\,P_{ij}\,,\sum_{j=0}^{K}\pi_{j}=1 \tag{9}\]

The probability that out of the \(j\) active sensor nodes in a network of \(z\) clusters, \(k\) nodes belong to a cluster is a binomial process given by [23, 24]:

\[P(k\leq CN-1|j)=\sum_{k=0}^{CN-1}{j\choose k}\big{(}1/_{2}\big{)}^{k}\big{(}1 -1/_{2}\big{)}^{j-k}\pi_{j} \tag{10}\]

where \(CN\) is the number of sensor nodes in a cluster. #### 4.3.2 Node Scheduling

A sensor node is scheduled within a cluster if it has large number of packets due for transmission and low energy resources in its battery. The scheduler is cross-layer based as it involves the packets and energy level at the MAC and physical layer respectively. It is represented by a medium access decision parameter \(\varphi_{i}\) for all contending nodes. The medium access decision parameter is defined as the ratio of the number of available packets (\(n_{i}\)) to the remaining energy level \(Er_{i}\) for each node. It is used to decide the probability of a node to win the medium. The probability that a node \(w\), out of the \(k\) contending nodes, wins the medium for the transmission slot \(P(w|k)\) is defined by the fact that a node has the maximum value of the access decision parameter among its contending members and is defined in simulation as [25]:

\[P(w|k)=\prod_{t=1}^{k}P(\varphi_{i}\leq\varphi_{w}) \tag{11}\]

The probability of winning the contention is the probability that a node is scheduled in a slot, that is \(P(w|k)=\Pr_{sch}\), where \(\varphi_{i}\) for each individual node in a cluster is a function of the number packets arrivals and the energy level in the sensor nodes. A node that spends the most time in the active state, transmits the most packets, and has the highest utilization factor \(\tau_{i}\)[26, 27], hence the lowest remaining energy. Therefore the remaining energy level \(Er_{i}\) is proportional to the time a node spends in the active state. The number of packets \(n_{i}\) is also proportional to the arrival rate and the frame period \(T\), giving \(\varphi_{i}\) is given as:

\[\varphi_{i}=\frac{\lambda_{\varphi_{i}}*T}{u_{s_{2}}} \tag{12}\]

where \(\lambda_{\varphi_{i}}\) is the arrival rate of each node type defined in Equation (7). In a cluster normal nodes are scheduled based on the medium access decision parameter while CH and RCH are given priority of access as they route their packets and those of other nodes. For \(k\) normal nodes, with the same arrival statistics in a frame the probability of the nodes being scheduled is given as:

\[\Pr_{sch}=1/_{k} \tag{13}\]where \(k\) is the total number of active nodes in a cluster. ### Node State Analysis

The sensor node state behavior is modeled as a Markov process of state space S = {Sleep (\(S_{1}\)), Active Tx/Rx (\(S_{2}\)), Active back-off (\(S_{3}\))} as illustrated in Figure 8. The state transition vary depending on the type of node due to their variable arrivals, however they generally follow similar behavior. A node is in the sleep state if it has no data to transmit or not receiving any data. It is in the active back-off state when it has data packets in its buffer but it is not scheduled for transmissions. When scheduled for communication it transmits its data packets in the active Tx/Rx state. The transition probability matrix P for the sensor node state behavior is:

\[P= \begin{pmatrix}P_{S_{1}S_{1}}&P_{S_{1}S_{2}}&P_{S_{1}S_{3}}\\ P_{S_{2}S_{1}}&P_{S_{2}S_{2}}&P_{S_{2}S_{3}}\\ P_{S_{3}S_{1}}&P_{S_{3}S_{2}}&P_{S_{3}S_{3}}\end{pmatrix}  \tag{14}\]

The transition probabilities are determined from the transition events.