# Power Consumption Modeling of Discontinuous Reception for Cellular Machine Type Communications
## 4 System Model
### Proposed Semi-Markov Model  
Since MTC covers a wide area of applications ranging from smart metering to emergency alerting, therefore, its traffic characteristics also vary. For instance, MTC devices mainly generate a huge traffic volume by accessing the network on a large scale. Therefore, the devices may attempt that several times to perform packet activities. Considering the random generation of MTC traffic, the input MTC traffic is modeled according to the Poisson process with a packet arrival rate of \(\lambda\). Thus, the packet inter-arrival time, \(t\) follows an exponential distribution with a common mean value equal to \(1/\lambda\). The proposed semi-Markov chain model for the DRX mechanism for MTC devices is depicted in Figure 3. Moreover, the time spent in one particular state (holding time) is also considered a random variable, according to References [27; 28]. In addition, the corresponding states in the semi-Markov chain model are described as follows:  
1. \(S_{1}\) represents the RRC connection setup state for the device to re-enter the network after releasing from \(RRC\_Connected\). 2. \(S_{2}\), \(S_{3}\), and \(S_{4}\) correspond to the low, high, and max power consumption states as explained in Section 4.1. 3. \(S_{5}\) is the RRC connection release state after the DRX inactivity timer expires and is called the tail time state [13]. 4. \(S_{6},S_{8},S_{10},\ldots,S_{2N+4}\) are the \(N\) short sleep states within \(N\) short DRX cycles. The short sleep mode is proposed particularly for delay-sensitive MTC traffic. 5. \(S_{7},S_{9},S_{11},\ldots,S_{2N+5}\) are the \(N\) on duration periods within \(N\) short DRX cycles and therefore correspond to \(N\) short sleep states. 6. \(S_{L}\) is the long sleep state within the long DRX cycle. 7. \(S_{L+1}\) is the on duration period within the long DRX cycle, which corresponds to the long sleep state. Figure 2: The power consumption curve, redrawn based on Reference [25]. In the proposed semi-Markov model, parameters such as the DRX inactivity timer \(T_{I}\), short DRX cycle \(T_{SC}\), long DRX cycle \(T_{LC}\), RRC release timer \(T_{tail}\), and number of short cycles (\(N\)) are specified. In addition, the state transition probability from a state \(j\) to a state \(k\) in the proposed model is denoted by \(p_{jk}\) with \(j,k\in\mathbb{N}^{+}\) and \(j,k\in\{1,2,3,4,5,\ldots,2N+4,2N+5,L,L+1\}\). Furthermore, the MTC device transmits and receives data in the active power modes, such as in states \(S_{2}\), \(S_{3}\), and \(S_{4}\), and tail state \(S_{5}\), as presented in Figure 3. The inactivity timer \(T_{I}\) is restarted whenever there are uplink/downlink transmissions. Moreover, if there is no further data activity within the time period \(T_{I}\), the device enters \(S_{5}\). Otherwise, it enters into the corresponding active power state. Therefore, the respective transition probabilities for the above case are given in Equations (2) and (3):  
\[p_{22}=p_{33}=p_{44}=1-e^{-\lambda T_{I}}, \tag{2}\]  
\[p_{25}=p_{35}=p_{45}=e^{-\lambda T_{I}}. \tag{3}\]  
In state \(S_{5}\), the device is still registered and attached to the network. Therefore, when a packet transmission/reception occurs during this state, the device directly enters into one of the corresponding active power states. Otherwise, the device is released from the network and thus enters into the short sleep cycles. The transition probabilities \(p_{52}\), \(p_{53}\), and \(p_{54}\) for the above cases are given in Equations (4)-(6), respectively:  
\[p_{52}=(1-e^{-\lambda T_{tail}})\theta_{2}, \tag{4}\]  
\[p_{53}=(1-e^{-\lambda T_{tail}})\theta_{3}, \tag{5}\]  
\[p_{54}=(1-e^{-\lambda T_{tail}})\theta_{4}, \tag{6}\]  
where \(\theta_{2}\), \(\theta_{3}\), and \(\theta_{4}\) represent the probability of corresponding power states when the device is in a specific cell environment. It is further noted that \(P_{TX}\) should remain updated according to the channel state information to ensure the packet activity during the tail state.