## 3 Preliminaries
### System Model  
Next, the system model is defined in order to describe and represent our system.  
* Mobile Data Foragers. The explorer entities in the system are modeled as MDF. Each MDF belongs to the set \(U=\{u_{1},u_{2},\ldots,u_{q}\}\). An MDF \(u\in U\) represents aerial vehicles flying over the operational environment. Each \(u\in U\) has a finite amount of steps it can make, limited storage and computational resources. Every time the MDF moves to an adjacent region, the number of steps of the MDF is reduced by one.
* Pheromones. Since there is no single reconnaissance route, each MDF \(u\in U\) is guided by a trail of pheromones. In this work, a pheromone is defined as an abstract data type as follows. A pheromone \(f\in F\) is represented as a tuple \(f=\{r,counter\}\), where \(r\) is the identifier of the region where the pheromone was placed, and \(counter\) is the number of pheromones placed in such region. There are two types of pheromones: food and travel pheromones. Food pheromones indicate that in a specific region there is something of interest to the application; food pheromones are denoted by the set \(F_{c}\). On the other hand, travel pheromones indicate that the region has been visited; they are denoted by the set \(F_{t}\). The set \(F\) of pheromones is the union of food and travel pheromones, that is \(F=F_{c}\cup F_{t}\). Each pheromone belongs to the set \(F=\{f_{1},f_{2},\ldots,f_{k}\}\). Each pheromone \(f\) has a lifetime associated to the maximum time a pheromone can be in a region.
* Operational environment. We represent the operational environment as a Hextille \(H\) of radius \(h_{ex}\) in the form of a set \(R=\{r_{1},r_{2},\ldots,r_{i}\}\), where each \(r\in R\) is a sampling region. The radius \(h_{ex}\) of the Hextille \(H\) is defined as the linear distance from the center of the Hextille to the farthest hexagon of the Hextille in any of the six directions. Figure 2 shows an example of the environment as a tilled hexagonal grid. It should be noted that the hexagonal grid is not restricted just to the specific radius shown in Figure 2 and can vary in radius. A two-dimensional space is considered along with the knowledge of the environment in the form of a map, but without the characteristic and conditions of the environment; that is, there is no information about where valuable data sources are located. Exploring the environment is done through one MDF. The MDF starts at the base station, explores the environment, and returns to the base station to refuel; this is called a trip. The sampling area is a subset \(S\subseteq R\), where each region \(r\in S\) is a hexagon with a diameter equal to the sensing range of an MDF \(u\in U\). Only one \(u\in U\) can be in the environment at any given time. It should be noted that each region \(r\in S\) has dynamic changing conditions, whichmeans that regions that are valuable do not necessarily remain valuable indefinitely. Each \(r\in R\) has a number of pheromones \(f\subseteq F_{r}\), where \(F_{r}\) is the set of pheromones present at region \(r\).
* Base station. The base station \(\zeta\) is a processing unit, associated to the physical place where each MDF lifts to explore the environment and drops the retrieved data after each expedition. It is assumed that the base station has enough resources to process, and send control messages to the MDFs. There is a unidirectional channel between the base station and the MDF present in the environment.
* Maximum reconnaissance time: This refers to the maximum time to cover the sampling area. It is denoted by \(T_{expMax}\). According to [PERSON] et al. (2017) the upper bound of exploration under energy constraints is \(\mathcal{O}(n^{2})\), where \(n\) is the number of regions.