## 2 Methods/Experimental Approach
### Power Consumption Model
#### 2.1.2 A Stochastic Energy Consumption Model of the Processor Unit Using a Markov Model  
Consider a processor operation process as illustrated in Figure 1. The processor operates within three states in which it consumes different levels of power. As it operates in sleep mode, it consumes the least amount of energy. When it operates in idle mode, it consumes a medium level of energy. It consumes most of the energy when operated in active mode. Although the aim is to spend the least of time in the Active mode, the amount of time spent in each mode is unpredictable. The proposed Markov model depicts the various increasing states through which the processor passes during its most energy-consuming state (active). Let these states be named pa1, pa2, pa3, etc. Given a job arrival rate \(\lambda\) at which the different tasks are given to the processor, it services these tasks at a rate \(\mu\) while strives to move back to the lowest power mode (Sleep mode) or eventually to the idle mode. Statistically, it is proven that most NB-IoT nodes' processors are designed to spend almost 98% of their time in the sleep mode while they spend roughly 1.5% in the idle state and only 0.5% of their operational time in active mode. Therefore, although all the three operational modes' times are purely stochastic, the proposed model focuses on modeling the time spent in the active mode,  
Figure 1: Birth and death process of processor’s tasks. it however also assesses the probability of operational failure which therefore guides the entry into the idle state. In the proposed model, it is considered that the processor remains in the idle state for some time interval greater than some threshold. The processor then moves back to the sleep mode. In our proposed model, the following assumptions are made  
* The request arrivals follow a Poisson process with a mean rate of \(\lambda\). * The service time is exponentially distributed with a mean of \(1/\lambda\). * The CPU enters sleep mode if there are no more tasks to be executed for a time interval longer than an energy-aware defined threshold \(T\) that the network designer heuristically defines depending on the energy consumption objective of the design under consideration versus the network reliability objective. * The power-up processor of the processor takes a constant time \(D\) which depends on the type of processor being used. The operation process of the processor cannot naturally be modelled using the Markov model. Therefore, the method of supplementary variables as proposed in [19] is used to derive an alternative set of state equations to approximate the transitions for stationary analysis. The different mode transitions as highlighted in Figure 1 follow either of the two probabilistic distributions. As the processor transitions from the sleep state to the active mode, the processor goes through a power-up fixed duration \(D\) which then makes the probability for the processor to stay in each of the active states \(P_{a0}\) to \(P_{an}\) to exponentially decay by \(e^{-\lambda D}\). It is also directly proportional to the parameter \(\frac{\lambda D^{n-1}}{(n-1)! }\) for each active state \(n\). This implies that the more active states that the processor's operational process is involved in, the longer the duration it is in the active mode. On the other hand, the duration within the idle mode is any value from a few processor's machine cycles to a maximum threshold duration of \(T\). The duration of the transition from the active mode to the idle mode is directly proportional to the rate of arrival of tasks \(\lambda\) but inversely proportional to the rate at which tasks get services \(\mu\). This transition duration can be modeled as  
\[T_{ai}=\frac{\lambda\times e^{\lambda T}}{\mu} \tag{6}\]  
Considering that the NB-IoT nodes do not stay in any of the states permanently but rather move from one mode to the other, the sum of probability of staying in either mode is equal to unity (certainty).