# Power Consumption Modeling of Discontinuous Reception for Cellular Machine Type Communications
## 4 System Model
### Proposed Semi-Markov Model
In addition, in case of any uplink or downlink packet activity during the short sleep mode, the packets remain in buffer at the device or eNB, respectively, for the remaining duration of the short sleep period until the next corresponding on duration state. Therefore:  
\[p_{2i+4,2i+5}=1, \tag{7}\]  
with \(i\in\mathbb{N}^{+}\) and \(i\in[1,N]\). However, if there is no packet activity during the complete \(i^{th}\) short cycle, the device then enters the \((i+1)^{th}\) short cycle. The corresponding transition probabilities are given in Equations (8) and (9):  
\[p_{2i+5,1}=1-e^{-\lambda T_{SC}}\text{ for }i\in[1,N], \tag{8}\]  
Figure 2:0  
Figure 2:1 The proposed semi-Markov-based DRX model for MTC in LTE-A. \(T_{SC}\) represents the duration of a short cycle. If there is no packet activity during all \(N\) DRX short cycles, the device enters the long cycle. The respective transition probabilities are given in Equations (10)-(13):  
Figure 2:2  
Figure 2:3  
Figure 2:4  
Figure 2:5  
However, if the device has to wake up to perform any activity in order to send or receive data, it again establishes the RRC connection after being released from RRC_Connected state. Therefore, it goes through the RRC connection state from on duration in order to enter the active modes. The corresponding power state probabilities during active mode are \(p_{12}=\theta_{2}\), \(p_{13}=\theta_{3}\), and \(p_{14}=\theta_{4}\). In addition, let \(\pi_{j}\) denote the probability that the device stays in state \(S_{j}\) during the steady state. Then, the following equations hold for the stationary probability distribution, as given in Equation (14):  
\[\left\{ \begin{aligned} \pi_{j}&=\sum_{k=1}^{L+1}\pi_{k} \cdot p_{kj}..\text{ for }j\in\{1,2,3,4,...L+1\}\\ &\sum_{j=1}^{L+1}\pi_{j}=1\end{aligned} \right.. \tag{14}\]  
Thus, we can obtain the stationary probability distributions which are given in Equation (15):  
\[\prod=\left\{ \begin{aligned} \pi_{1}&=\frac{e^{- \lambda T_{tail}}}{K}\\ \pi_{2}&=\frac{e^{-\lambda T_{I}}\theta_{2}}{K}\\ \pi_{3}&=\frac{e^{-\lambda T_{I}}\theta_{3}}{K}\\ \pi_{4}&=\frac{e^{-\lambda T_{I}}\theta_{4}}{K}\\ \pi_{5}&=\frac{1}{K}\\ \pi_{2i+4}&=\pi_{2i+5}=\frac{e^{-\lambda[T_{SC}(i-1 )+T_{tail}]}}{K}\\ \pi_{L}&=\pi_{L+1}=\frac{e^{-\lambda T_{tail}}e^{- \lambda NT_{SC}}(1-e^{-\lambda T_{LC}})}{K}\end{aligned} \right., \tag{15}\]  
where \(K\) is denoted by:  
\[K=1+e^{-\lambda T_{tail}}+e^{-\lambda T_{I}}+2e^{-\lambda T_{tail}}\Big{[} \frac{1-e^{-\lambda NT_{SC}}}{1-e^{-\lambda T_{SC}}}+\frac{e^{-\lambda NT_{SC}} }{1-e^{-\lambda T_{LC}}}\Big{]}. \tag{16}\]  
Let \(H_{j}\) denote the state holding time of state \(S_{j}\), whereas \(T_{tail}\), \(T_{SC}\), \(T_{LC}\), and the RRC Connection Time \(T_{RC}\) are known, then the holding time can be expressed as \(H_{5}=T_{tail}\), \(H_{2N+4}=T_{SS}\), where \(T_{SS}\) is defined as \(T_{SC}-T_{ON}\), \(H_{L}=T_{LS}\), where \(T_{LS}=T_{LC}-T_{ON}\) and \(H_{1}=T_{RC}\), respectively. Moreover, when the device is in the on duration state of the short DRX cycle, there can be three cases. For instance, a packet arrives after the expiry of the on duration timer with the probability \(p_{2i+5,2i+6}\) or \(p_{2N+5,L}\)In the second case, a packet arrives at the \(l\)th subframe of the on duration period with the probability of \(p_{j}^{ON}\), which can be given as follows:  
\[p_{j}^{ON}=e^{-\lambda(T_{SC}-T_{ON}+j-1)}-e^{-\lambda(T_{SC}-T_{ON}+j)},\]  
whereas, in the third case, a packet arrives during the short sleep period with the probability \(p_{s}=1-e^{-\lambda(T_{SC}-T_{ON})}\). Moreover, for the third case, we assume that the device enters the RRC connection state as soon as it enters the on duration state. Therefore, the state holding time will be \(T_{sh}=0\).