## 3 System Model
### Intra-Cluster Communication Session
\(P_{S_{1}S_{1}}\) is the probability of a node continuing in the sleep state. A node will continue to stay in the sleep state when its sleep timer has not expired; it will stay in that state for a random period of time. Therefore the probability \(P_{S_{1}S_{1}}\) is given by:

\[P_{S_{1}S_{1}}=1-1/u_{S_{1}} \tag{15}\]

A node will transit into the Active Tx/Rx state with a probability \(P_{S_{1}S_{2}}\) when its sleep timer expires and it is allocated a slot schedule to transmit its data:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]

where \(\Pr_{sch}\) is the probability that a node is scheduled within a communication frame. A node transits from the sleep state to the back-off state with probability \(P_{S_{1}S_{3}}\)if the sleep timer expires and there is data to transmit however there are no resources for a node to be scheduled:

\[P_{S_{1}S_{3}}=1/u_{S_{1}}*(1-\Pr_{sch}) \tag{17}\]

A node will exit from the active Tx/Rx only if it has completed its transmission hence the probability \(P_{S_{2}S_{3}}\) is defined as zero:

Figure 8: Node’s State transition Diagram. \[P_{s_{2}s_{3}}=0 \tag{18}\]

A node will continue to stay in the active Tx/Rx state with a probability \(P_{s_{2}s_{2}}\) if it has not completed its transmission:

\[P_{s_{2}s_{2}}=1-\sfrac{1}{u_{s_{2}}} \tag{19}\]

If transmission of packets is complete, the node transits to the sleep state from active Tx/Rx to conserve energy; hence the probability \(P_{s_{2}s_{1}}\) is given by:

\[P_{s_{2}s_{1}}=\sfrac{1}{u_{s_{2}}} \tag{20}\]

A node transits to the active Tx/Rx state from the active back-off state with a probability \(P_{s_{3}s_{2}}\) when its back-off timer expires and it is scheduled slot, given by the following:

\[P_{s_{3}s_{2}}=\Pr_{sch}*\sfrac{1}{u_{s_{3}}} \tag{21}\]

A node transits to the sleep state only when its buffer is empty. Therefore the node cannot transit from the back-off state to sleep state since in the back-off state it has not serviced all its packets, giving the probability \(P_{s_{3}s_{1}}\)as zero:

\[P_{s_{3}s_{1}}=0 \tag{22}\]

A node stays in the active back off state with a probability \(P_{s_{3}s_{3}}\) if its back-off timer has not expired or if its back-off timer expires and it is not scheduled for transmission, given by:

\[P_{s_{3}s_{3}}=\left(1-\sfrac{1}{u_{s_{3}}}\right)+\left(1/u_{s_{3}}*\left(1 -\Pr_{sch}\right)\right)=1-\Pr_{sch}*\sfrac{1}{u_{s_{3}}} \tag{23}\]

The mean termination rate from the active state \(u_{s_{2}}\) is determined from the average time a node spends in the active state. The time a node takes in the active state is a function of the available packets \(n\) and the time to transmit a single packet. Generally the number of packets for a node will be given as:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]0

where \(n_{s_{i}}=\frac{\lambda_{p_{i}}}{u_{s_{i}}}\), indicates the arrivals during each of the three node states. Based on the fact that the scheduler alternates the states of the nodes between active and back-off, the duration of time in these states will depend on the probability of the node being scheduled. Therefore the average time a node spends in active state \(\overline{T}_{s_{2}}\) is given by:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]1

where \(T_{ts}\) is the slot time to transmit a single packet. The steady state probabilities \(\rho_{s_{1}}\), \(\rho_{s_{2}}\) and \(\rho_{s_{3}}\) are obtained through solving for the equations below:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]2

### Performance Measures

#### 4.5.1 Energy Consumption Model

The energy consumed by radio communication at any time by a node is the function of the amount of energy it dissipates in each state, generally given as:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]3

where \(E_{s_{i}}\) is the amount of energy the sensor node dissipates in a state.