# Energy Efficient Policies for Data Transmission in Disruption Tolerant Heterogeneous IoT Networks
## 3 System Model and Definitions  
We consider a rectangular surveillance area that is subdivided into smaller square-shaped cells as shown in Figure 1. Each cell within the surveillance area is identified by the Cartesian coordinates of its center, i.e., by a pair \((x,y)\), where \(x=1,2,\ldots,X\) and \(y=1,2,\ldots,Y\). Furthermore, each cell _may_ host a Low Power Wide Area Network (LP-WAN) base station [18], featuring wide-area connectivity for relatively low power and low data rate devices that is not provided by legacy wireless technologies. LP-WAN base stations may use technologies such as LoRa [19; 20], SigFox [21] or NB-IoT [22]. What is more, each cell may also host a Low Power Personal Area Network (LP-PAN) base station, i.e., a base station that supports higher bit rates and lower power consumption albeit at a much sorter range compared to an LP-WAN base station. An example of a LP-PAN base station could be an 802.15.4 base station [23]. Finally, cells may host both types of base stations. Let \(A^{p}\) and \(A^{w}\), be integer matrices whose elements take values in the set \(\{0,1\}\) to indicate that a specific cell lies within the communication range of at least one LP-PAN and LP-WAN base station, respectively, e.g., \(a^{w}_{xy}=1\) (\(a^{w}_{xy}=0\)) indicatesthat cell \((x,y)\) lies (doesn't lie) within the communication range of one or more LP-WAN base stations. We note here that the optimal way to choose the base station to associate with, among many base stations of the same type, is beyond the scope of this work. Given \(A^{p}\) and \(A^{up}\), we may compute a network availability matrix as \(A^{n}=A^{p}+2A^{up}\) which enables us to present network availability in a single figure (see Figure 1). More specifically, cells that lie exclusively within the communication range of at least one LP-PAN or at least one LP-WAN base station are respectively marked with 1 and 2, while cells that lie within the communication range of both types of base station are marked with a 3. Cells that lie outside the communication range of all base stations lack network connectivity and are marked with a 0. We note here that in this work we do not consider the temporal variation of the network availability map, however, this does not necessarily render our proposed scheme inapplicable. The key idea here is that the efficiency, and not the applicability, of the derived packet transmission policy depends on the accuracy of the network availability map, which in turn depends on the process and specifications of its construction. For example, to mark a specific network type as available in a specific cell of the network availability map one could pose the most stringent constraint on its temporal availability, thus producing a map that rarely changes as time passes by. What is more, temporal changes in network availability can be dealt with other network mechanisms, such as packet retransmissions, although we do not consider such mechanisms in this work. We do expect though that, even in the absence of such mechanisms, the proposed packet transmission policy will not result in a complete breakdown of network service although it will definitely be suboptimal. We emphasize here that utilization of mobile nodes that may offer opportunistic connectivity, as well as the creation of the network availability map is beyond the scope of our work. For recent work, related to the subject of network availability map creation, such as network topology inference and spatial availability and reliability of wireless links the interested reader may refer to [24; 25; 26; 27]. Next, we consider a mobile node that is appropriately equipped with a LP-PAN interface, a LP-WAN interface and sensors. The mobile node performs a two-dimensional random walk within the surveillance area collecting data either through its sensory equipment or by reception of packets from its neighboring nodes. We assume that the random walk evolves in stages of equal duration indexed by \(k=0,1,\dots\).