# Energy Efficient Policies for Data Transmission in Disruption Tolerant Heterogeneous IoT Networks
## 3 System Model and Definitions
We also consider the pair of integer random variables \(W_{k}^{\rm{Ar}}\) and \(W_{k}^{\rm{Ay}}\) that represent the random displacement of the mobile node along the \(x\) and \(y\) axis respectively. We assume that the system's random parameters are independent of their values in previous stages. Furthermore, we assume that \(W_{k}^{m}\) is uniformly distributed between \(0\) and \(W_{m}^{\max}\) for all cells within the surveillance area. On the other hand, the probability distributions of \(W_{k}^{\rm{Ar}}\) and \(W_{k}^{\rm{Ay}}\) are not the same for all \((x_{k},y_{k})\). This dependence on \((x_{k},y_{k})\) stems from the fact that the mobile node will follow a different moving pattern in case it reaches a cell that lies on the border of the surveillance area. More specifically, the node will either remain in the same cell or move to a neighboring one as long as this transition does not cross the boundaries of the surveillance area. However, if the mobile node is in a cell that lies on the boundary of the surveillance area it will either remain on the boundary or bounce back into the surveillance area. Thus, for all \(k=0,1,\ldots\), we have that \(W_{k}^{\rm{Ar}}\) is uniformly distributed in the set \(\{-1,0,1\}\) given \(1<x_{k}<X\), uniformly distributed in the set \(\{0,1\}\) given \(x_{k}=1\) and uniformly distributed in the set \(\{-1,0\}\) when \(x_{k}=X\). \(W_{k}^{\rm{Ay}}\) is identically distributed to \(W_{k}^{\rm{Ar}}\). Given the current state of the system \(s_{k}\) and the control \(u_{k}\) the state of the system at the beginning of the \((k+1)\)-th stage will be,  
\[s_{k+1}= \begin{bmatrix}x_{k+1}\\ y_{k+1}\\ m_{k+1}\end{bmatrix} = \begin{bmatrix}x_{k}+W_{k}^{\Delta x}\\ y_{k}+W_{k}^{\Delta x}\\ m_{k}-u_{k}^{m}+W_{k}^{m}\end{bmatrix} . \tag{2}\]  
We note in Equation (2) that in order to derive \(m_{k+1}\) we subtract from \(m_{k}\) all \(u_{k}^{m}\) packets selected for transmission. This is equivalent to stating that all transmissions attempted within stage \(k\) are successful or considered successful by the mobile node. A mobile node will always consider a transmission successful, i.e., it will not attempt to retransmit a packet, when it does not require an acknowledgment of reception from the base station. This latter assumption regarding the configuration of the mobile node is justified by the fact that our focus is on IoT applications where energy efficiency is the major priority. To this end we consider the most energy efficient configuration for both the LP-WAN and LP-PAN protocols. This partially accounts to selecting the Aloha protocol for medium access control (MAC) and transmitting packets without requiring an acknowledgment of reception. The proposed configuration is supported by the 802.15.4 standard ([28], Section 4.5.4.2-3) for LP-PANs and is a typical configuration for LP-WANs [29]. With every state transition, as driven by control \(u_{k}\), there is an associated cost. In the case of packet transmissions the associated cost is the amount of energy required to transmit \(u^{m}\) packets over the selected interface. In the case of packet drops we assume a virtual energy cost per packet to prevent the mobile node from dropping packets freely. We denote with \(E_{l}\) and \(E_{w}\) the energy cost for the transmission of a single packet by use of the LP-PAN and LP-WAN interfaces, respectively, and with \(E_{d}\) the virtual energy cost related to a packet drop. We assume that \(E_{l}\) and \(E_{w}\) are fixed for all cells and equal to the average energy consumption induced to the mobile node for the transmission of a packet for a given interface. This assumption is justified by the fact that \(E_{l}\) and \(E_{w}\) are uncontrollable system parameters within the context of our work.