# Energy Efficient Policies for Data Transmission in Disruption Tolerant Heterogeneous IoT Networks
## 3 System Model and Definitions
At the beginning of the \(k\)-th stage the mobile node identifies its current cell \((x_{k},y_{k})\) through some localization mechanism that we assume to be exact and the number of packets \(m_{k}\) in its backlog. We assume that the node's backlog is finite with a maximum capacity of \(M\) packets. Values \(x_{k}\), \(y_{k}\) and \(m_{k}\) constitute the state of the system at stage \(k\), which we denote with the column  
Figure 1: Example of the network availability map for a rectangular surveillance area divided in square-shaped cells. vector \(s_{k}=[x_{k},y_{k},m_{k}]^{T}\). The state of the system summarizes all past information that is relevant for future optimization. Given its current state the mobile node must make a decision regarding the wireless interface to use and the number of packets to transmit or drop. We denote the set of all possible decisions, termed controls, with \(U=\{(u^{w},u^{p},u^{m}):u^{w}\in\{0,1\},\ u^{p}\in\{0,1\},\ u^{w}+u^{p}\
eq 2,\ u^{m} =1,2,\ldots,M\}\), where \(u^{w}\), \(u^{p}\) are binary variables indicating the selection of the LP-WAN and LP-PAN interface respectively and \(u^{m}\) is the number of packets to transmit or drop from the backlog. Equation \(u^{w}+u^{p}\
eq 2\) indicates that the sensor is not allowed to use both interfaces at the same stage by prohibiting both \(u^{w}\) and \(u^{p}\) to be equal to one at the same stage. On the other hand, if both \(u^{w}\) and \(u^{p}\) are zero we distinguish two cases based on whether \(u_{m}\) is zero or a positive integer. In the first case the mobile node will not transmit any packets while in the latter it will drop \(u_{m}\) packets. For each system state only a subset of the decisions in \(U\) will be available to the mobile node, unless of course both types of network are available in the current cell and the backlog is full. To determine the set of available controls in each state we note that given the current state of the system \(s_{k}\), and the network availability map, the mobile node can determine the available types of network in the current cell and restrict the range of values for \(u^{w}\) and \(u^{p}\) accordingly. Furthermore, the number of packets in the backlog \(m_{k}\) is an upper bound for the number of packets the mobile node can transmit or drop, i.e., \(u^{m}\leq m_{k}\). Additionally, we assume that the mobile node preemptively transmits or drops packets so that the backlog can always accommodate the maximum number of new packets \(W_{m}^{\max}\) that may be generated during the current stage. This queue management strategy, although atypical, results in a significant state space reduction since dropped packets are not explicitly accounted for in the state vector. To enforce the latter constraint on the set of controls we require that \(u^{m}\geq\max\{0,\ m_{k}+W_{m}^{\max}-M\}\). Finally, to simplify notation, we note that the set of available controls in any state \(s_{k}\) remains unchanged across stages, i.e., it is independent of index \(k\). Thus we may drop index \(k\) from vector \(s_{k}\) and its elements and summarize the constraints discussed above for any state \(s=[x,y,m]^{T}\in S\) as follows,  
\[U(s)=\{(u^{w},u^{p},u^{m}):u^{w}\in\{0\}\cup\{a^{w}_{x,y}\},u^{p} \in\{0\}\cup\{a^{p}_{x,y}\},\\ \max\{0,\ m+W_{m}^{\max}-M\}\leq u^{m}\leq m,\ u^{m}\in\mathbb{N}\}, \tag{1}\]  
where \(u^{m}\) represents packet drops only when no network is available, i.e., when both \(u^{w}\) and \(u^{p}\) are equal to zero and \(u^{m}>0\). At the beginning of the next stage the state of the mobile node will change as a result of the control \(u_{k}=(u^{w}_{k},u^{p}_{k},u^{m}_{k})\in U(s_{k})\) applied at the \(k\)-th stage and the random variables of the system. A random variable of the system is \(W_{k}^{m}\), which is an integer random variable indicating the number of new packets added in the backlog during the \(k\)-th stage by the sensory equipment of the mobile node.