## 3 A Non-Real-Time DTN-Based Forwarding Algorithm  
### A New Multi-Objective Mathematical Model  
In this section we introduce a new mathematical model, built over our previous research [23], that will describe the message forwarding from sensor nodes, through Gateways (GWs) and Mobile Access Points (data mules or MAPs), to the Internet Connected Server (ICS) where the sensor data will be sent to the Central Management and Monitoring Platform (MMS). This mathematical model \begin{table}
\begin{tabular}{l l} \hline \hline
**Characteristics** & **Protocols** \\ \hline Flooding-based & Epidemic [11], Spray and Wait [12] \\ Social-based & SimBet [15], BUBBLE [13], and PeopleRank [16] \\ Hybrid protocols & PROPHET [9], MPAD [14] RAPID [17], LSF [19] and SMART [18] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related protocols. assumes that the communication will only take place from sensor nodes (source of information) to the MMS (the ICS will be the destination). The mathematical model is presented in Equations (2)-(8). We are proposing a model for a network where their nodes can change their availability at any given time. The availability changes for the sensor nodes come from the fact that the GOLDFISH project uses microwave communication (IEEE 802.11b) and antennas floating over water. The antennas movement by the river flow can cause the water to cover them or tilt them to an angle where communication with the GW is no longer possible. Also, communication from the GW to the MMS is not always possible through the mobile cell network, so we will instead use local transportation means like buses or boats to recover sensor data from the GW and take them to the nearest ICS where it could be transferred to the MMS. The availability of the MAPs will change depending on their physical closeness with the GW. We assume that these vehicles transporting the MAPs will periodically or eventually visit the site where the GW is located. Figure 4 shows a graphical representation of the problem, where the availability probability \(a_{ij}(t)\) between each pair of nodes depends on the existence (at any given time) of a physical link between them, and the delivery probability \(d_{ij}(t)\) depends on the availability probability and the knowledge of the next node about a path to the destination, see Equation (1). In Equation (1), node \(k\) is the destination node, node \(i\) is the source node, and node \(j\) is an intermediary node. If the node \(j\) can deliver the message to node \(k\) at time \(t\), the delivery probability \(d_{jk}(t)\) will be equal to \(a_{jk}(t)\). \[d_{ij}(t)=a_{ij}(t)d_{jk}(t) \tag{1}\]  
Our mathematical model will have a set of nodes (\(N\)) where some nodes will be information-originating ones (sensor nodes in the river), others will be transit nodes (GW and MAPs), and one will be the destination node (the ICS in contact with the MMS). Each of these nodes will have a demand or an offer modeled by the parameter \(b_{i}(t)\) (an integer vector) that can change over time. There is also a set of paths (\(P\)) that represent possible paths over the network through time. Paths are given by possible physical connections, so a path can be created through contacts of several nodes even if they happen at different times. The delivery probabilities \(d_{ij}(t)\) are each of the probabilities that a node \(i\) can reach its destination through a node \(j\). They depend on the link availability probability and the next node delivery probability to the destination. Equation (1) describes the delivery probability for a node \(i\) to reach a node \(k\) through a node \(j\). Links between nodes have a given capacity modeled by \(c_{ij}\), and nodes also have buffers to store data for some time, modeled by \(c_{ii}\). We are assuming these capacities to be constant over time. The decision to send data over a link or store it until later is modeled through the positive integer variable \(x_{ij}(t)\).