# Power Consumption Modeling of Discontinuous Reception for Cellular Machine Type Communications
## 4 System Model
### Power Consumption in Active/Non-Active Modes  
According to Reference [25], the power consumption curve of a device can be separated into two regions based on its uplink transmission power \(P_{TX}\) and the spatiotemporal variations of the radio channel, see Figure 2. Therefore, a power consumption threshold \((\gamma)\) is set for separating the curve into two regions based on their slopes. The low transmission power is characterized by slope \(\alpha_{L}\), whereas high power transmission is characterized by \(\alpha_{H}\) in Reference [25]. In addition to slope parameters, \(\beta_{L}\) and \(\beta_{H}\) represent the y-intercept of the curve for the low and high power consumptions, respectively. Moreover, this particular characteristic can be observed for all devices. The main motivation behind this categorization is the different stages of the power amplifiers used within the devices. Therefore, power consumption, \(P_{w}\) of a device can be independently approximated by the following linear functions:  
\[P_{w}=\begin{cases}\alpha_{L}\cdot P_{TX}+\beta_{L},\text{ if }P_{TX}\leq \gamma\\ \alpha_{H}\cdot P_{TX}+\beta_{H},\text{ if }P_{TX}>\gamma\end{cases}. \tag{1}\]  
Based on Equation (1), the empirical power consumption can be categorized into distinct modes. Therefore, a space of four distinct power states is introduced in this paper based on Reference [25]. Moreover, the power consumption values for the idle and the device transmission states were considered, which are described below.  
In the non-active state, the device is not in a continuous reception mode. However, it can be in on duration, short sleep, long sleep, tail, or RRC connection state. Moreover, the power consumption is reduced to zero when the device is in short/long sleep state. However, in active state, the device is in a continuous reception mode, where it can transfer data. Since the power consumption curves were observed to be linear for all tested devices in Reference [25], the following power consumption states were considered within the active mode.  
1. Low: In this state, the transmit power control (TPC) algorithm of LTE-A described in Reference [26] adjusts the transmission power in a way that a predefined target signal to noise ratio (SNR) at the eNB can be achieved. When the uplink transmission power \(P_{TX}\) is below a device-specific threshold \(\gamma\), \(P_{TX}=0\) dBm was found out to be a typical value for the low power state [25]. Thus, the average power consumption in the low power state is \(\bar{P}_{2}=P_{w}(P_{TX}=0\) dBm).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\lambda\) & Packet arrival rate in s\({}^{-1}\) \\ \(T\) & Packet inter-arrival time in s \\ \(\alpha,\beta\) & Slope parameters for \(P_{TX}\) line \\ \(\theta_{k}\) & Transition probability from RRC to active \\ \(P_{w}\) & Power consumption of device \\ \(S_{i}\) & State \(i\) where \(i\in[1,2,\ldots,2N+5,L,L+1]\) \\ \(p_{i,j}\) & State transition probability from \(i\) to \(j\) \\ \(\pi_{i}\) & Stationary probability of state \(i\) \\ \(H_{i}\) & Holding time of state \(i\) \\ \(P_{S}\) & Power-saving factor \\ \(\hat{d}\) & Wake-up latency \\ \hline \hline \end{tabular}
\end{table}
Table 3: List of symbols used for modeling a DRX mechanism.  
High: If the overall transmission power \(P_{TX}\) is higher than the device-specific threshold \(\gamma\), the device enters the high power state due to the use of different power amplifiers. Due to the linear behavior of power consumption in a high power state, as shown in Figure 2, the average power consumption in a high power state is set to be \(\bar{P}_{3}=P_{w}(P_{TX}=(P_{max}+\gamma)/2)\).
3. Max: If the TPC algorithm can no longer achieve the target SNR by compensating for the path loss, the device enters the max power state, in which the device power consumption has reached the maximum value. Therefore, the average device power consumption for the max power state is selected to be \(\bar{P}_{4}=P_{w}(P_{TX}=P_{max})\).