## 2 Methods/Experimental Approach
### Power Consumption Model
#### 2.1.3 The Transceiver Energy Consumption Model
The transmission time \(T_{d}\) is a random variable which models the time duration for successful transmission on a single link. \(T_{d}\) is random due to the unpredictable nature of the wireless channel. However \(T_{d}\) remains a function of a few calculated parameters. Some of the main parameters include data size, data rate, and packet collision. * _Data size (M)_: represents the size of the data being transmitted over the link. This size is variable depending on the type of message being transmitted which could be the interest message, data message, or a control message such as acknowledgment messages (ACK). This quantity is usually measured in bits. * _Data rate (B)_: This quantity represents the bit rate and is usually measured in bits per second. Considering an ideal channel condition the transmission time duration can be modelled as \[T_{d}=\frac{M}{B}\] (12)  
However, in a practical situation, the time duration could vary from its ideal value as modeled in Equation (12). One of the main random parameters is _packet collision_ which can be described as follows: In a MAC protocol that makes use of a carrier sense multiple access, after a successful carrier sense has taken place, a collision could occur if another node transmits during the vulnerable period (\(\tau\)). In a network with a total generated traffic \(\lambda\), the probability of collision can, therefore, be modeled in terms of Poisson's distribution with \(\lambda\) being the rate of successful transmission. It can be represented as  
\[P_{Coll}=1-e^{-\lambda\tau}. \tag{13}\]  
At the network layer, the causes of the difference between the ideal transmission time (minimum transmission time) and the random transmission time duration can be looked at as being caused by re-transmission attempts. In a very broad sense and with some level of abstraction, a random parameter defined as \(P_{st}\) (probability of successful transmission) can be defined as  
\[T_{d}=\frac{M}{B\times(P_{st})} \tag{14}\]  
The probability of successful transmission is inversely proportional to the transmission time duration because the higher (\(P_{st}\approx 1\)), the closer it is to the ideal transmission time duration. In the proposed approach this probability value is statistically modeled as given below. Due to the stochastic nature of the wireless propagation channel, the probability of successful transmission is uncertain and therefore can be considered to be stochastic. However, in the proposed algorithm, it is computed dynamically and statistically based on the previous transmission success rate. The assumption is that all the uplink transmissions are followed by a downlink packet (either an ACK or NACK downlink packets for the successful and not successful transmission of the packet). The cumulative accounting of ACK and NACK packets on previous transmissions is used to categorize the state of the channel as either bad, medium or good conditions as well as to define the probability of successful transmission \(P_{st}\) which is given as follows  
\[P_{st}(m)=\frac{\sum\limits_{1}^{m-1}(ACKs)}{\sum\limits_{1}^{m-1}(ACKs+NACKs)} \tag{15}\]  
The energy consumption on a single link can, therefore, be modelled as  
\[E_{Transceiver}=\frac{(P_{TX}+P_{RX})\times M}{B\times P_{st}}. \tag{16}\]  
The energy consumption in the sleep mode (\(E_{Sleep}\)) and the energy consumption in the idle mode (\(E_{Idle}\)) can also be modelled in terms of power consumed in each of the two modes multiplied by the duration that the transceiver spends in each of them respectively. The duration of the NB-IoT node in sleep mode is a random variable that can be considered to be exponentially distributed with a mean value of \(\beta\).