## 2 Methods/Experimental Approach
### Power Consumption Model
#### 2.1.3 The Transceiver Energy Consumption Model  
Several research works have confirmed that radio communication dominates the power consumption in NB-IoT above all other functions [20; 21; 22]. The energy consumption model of the transceiver module follows the same analogy as one of the processors modeled in the previous sub-section. Like the processor, the transceiver also has different operation states. These operational states include the transmission, the reception, the idle, and the sleep states in general. The latter consume different levels of energy. The operation of the transceiver consequently also involves state transitions. By taking into account both the different operational states as well as the states transitions, the total energy consumption of a transceiver can be modeled as  
\[E_{Transceiver}=(E_{TX}+E_{RX}+E_{Idle}+E_{Sleep})+E_{Trans}, \tag{8}\]where \(E_{TX}\), \(E_{RX}\), \(E_{Idle}\) and \(E_{Sleep}\) represent the energy consumed during the transmission, the reception, the idle as well as the sleep states respectively and \(E_{Trans}\) is the total energy consumed during the transitions between the states. In most cases, there are nine possible state transitions from the four power mode states. These include a transition from the transmission to the idle state, and vice versa, the reception to the idle state and vice versa the transmission state to the sleep mode and vice versa; the reception state to the sleep mode and vice versa and the single transition from idle to sleep mode. Taking into account all the nine-state transitions mentioned above the energy consumption \(E_{Trans}\) for the probabilistic \(n\) possible transitions that can occur during a network operation are modeled as  
\[E_{Transitions}=\sum_{i=1}^{n}N_{st}(j)E_{st}(j) \tag{9}\]  
where \(N_{st}(j)\) is the frequency of a state transition \(j\) to occur. The value of this frequency varies with the type of operation that the NB-IoT node is carrying out (relaying information, transmitting or receiving) and \(E_{st}(j)\) is the energy consumed by every state transition \(j\). It is also very important to point out that the number of possible transitions between power modes is stochastic and can be modeled as given below. Given that  
* The state in which the processor of NB-IoT nodes normally remains in is mostly the sleep mode. This is the state in which it consumes the least of energy possible with most of its internal modules and clocks disabled. However, on event-driven instants, either time events or sensing events, the processor of the NB-IoT nodes will normally wake up and go from sleep mode to active mode to perform transmission or a reception handling activity. * Upon completion of the activity, it might go to another activity. If all activities are completed successfully, then it would normally go back directly to sleep mode or would momentarily go to the idle mode for a threshold amount of time and only later to sleep mode. This would mean that there would be 3 transitions on successful executions of a single activity and 4 transitions on failed execution of an activity (task) by the processor. * The number of possible transitions \(n\) between modes is therefore uncertain and can be modeled as follows
* Given a set of \(k\) activities to be carried out by the NB-IoT processor and given \(P_{s}\) the probability of successful execution of each activity, then the probability of failure of execution of an activity is \(1-P_{s}\), and thus \[n=3\times k\times P_{s}+4\times k\times(1-P_{s})=3\times k\times P_{s}+4\times k -4\times k\times P_{s}=k\times(4-P_{s})\] (10) Based on the above analysis, Equation (8) can be re-written in terms of power consumption as \[E_{Transceiver}=(P_{TX}+P_{RX})\times T_{d}+E_{Idle}+E_{Sleep})+E_{Trans},\] (11) where \(P_{TX}\) and \(P_{RX}\) represent the transmit and the receive powers respectively, and \(T_{d}\) is the transmission time duration on a single hop which is a probabilistic measure.