# An Energy-Efficient Approach to Enhance Virtual Sensors Provisioning in Sensor Clouds Environments
## 3 Problem Definition
### The Problem Definition  
A sensor cloud (\(SC\)) is a set of Wireless Sensor Networks (\(WSN\)) and can be defined as \(SC=\{WSN_{1},WSN_{2},...,WSN_{N}\}\). Each \(WSN_{i}\) is comprised of a set of sensors nodes \(WSN_{i}=\{SN_{i,1},SN_{i,2},...,SN_{i,N^{i}}\}\), where \(1\leq i\leq N\) and \(N^{i}\) is the number of \(SN\) in the \(WSN_{i}\), deployed in an area \(T_{i}=H_{i}\times W_{i}\), where \(H\) and \(W\) are the height and width respectively. Each sensor node comprises a set of sensors devices denoted by \(SN_{i,j}=\{S^{i,j}_{1},S^{i,j}_{2},...,S^{i,j}_{D^{ij}}\}\), where \(1\leq j\leq N^{i}\). A sensor device means the analog or digital circuit used to capture some physical or environmental conditions (temperature, light, etc.). Each sensor device \(S^{i,j}_{k}\), where \(1\leq k\leq D^{ij}\), has a type/dimension \(T^{ij}_{k}=Type_{t}\), which denotes the physical condition it can sensors. The set of all dimensions monitored by the \(SC\) is \(TYPE=\{Type_{1},Type_{2},...,Type_{N_{T}}\}\) (\(1\leq t\leq N_{T}\)). Given a sensor device, there is a set of measurements associated with it denoted by \((S^{i,j}_{k}=\{M_{1},M_{2},...,M_{M^{ik,ij}}\})\). The user's query is defined by \(Query_{id}=\{T,user,sample\_interval,time,R\}\), where \(id\) means the query identification, \(T\subset TYPE\) means the user's interested dimension, _user_ denotes the user identification, _sample_interval_ denotes how often data are physically sampled, _time_ represents how long the node has to sample the measurements, and \(R\) stands for _Region of Interest_, i.e., the geographical area in which the sensors nodes should act. Several queries can be requested. The set of all queries is \(QUERY=\{Query_{1},Query_{2},...,Query_{N_{Q}}\}\), where \(N_{Q}>0\). \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \multicolumn{2}{l}{**B. Dimensions**} \\ \hline \(N\) & Number of \(WSN\) in the \(SC\) \\ \(N^{i}\) & Number of \(SN\) in the \(WSN_{i}\) \\ \(D^{ij}\) & Number of sensors devices of the \(SN_{i,j}\) \\ \(N_{T}\) & Number of all types of physical variables the \(SC\) can monitor \\ \(M^{k,ij}\) & Number of measurements of the \(k_{th}\) sensor device of \(SN_{i,j}\) \\ \(N_{Q}\) & Number of triggered queries \\ \(N_{C}\) & Number of clusters created by ACASIMv2 \\ \(|C_{cJ}|\) & Number of sensor nodes inside the \(c-th\) cluster \\ \(N_{\Omega}\) & Number of selected sensor nodes \\ \(N_{A}\) & Number of Ants \\ \hline \multicolumn{2}{l}{**C. Functions**} \\ \hline \(\Phi\) & Function to compute the number of unique sensor types present in a set \\ \(Cv\) & Function to test if the sensors nodes in \(\Omega\) fully cover \(R\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
**Definition 1**. : _Sensors ratio (\(\kappa_{j,i,id}\)). It is defined by the ratio between the number of the type of sensors requested by the user's query (\(Query_{id}\)) and the number of the type of sensors provided by a particular sensor Node \(SN_{j,i}\). Values close to 0 mean that more of the type of sensors needed is provided by a specific sensor node. SR is expressed as:_  
\[\kappa_{j,i,id}=\frac{\Phi(SN_{i,j})}{\Phi(Query_{id})} \tag{1}\]  
_where \(\Phi(...)\) defines a function to count the number of unique sensor types presents in a set._  
**Definition 2**. : _Normalized residual energy (\(\lambda_{j,i}\)). It is defined as the ratio of the current energy level to the initial energy level, expressed as,_  
\[\lambda_{j,i}=(\frac{E_{SN_{j,i}}^{cur}}{E_{SN_{j,i}}^{ini}})^{-1} \tag{2}\]  
_where \(E_{SN_{j,i}}^{cur}\), and \(E_{SN_{j,i}}^{ini}\) are the current and the initial battery level, respectively._  
**Definition 3**. : _Proximity with BS (\(\chi_{i,j}\)). It measures how close a sensor Node \(SN_{i,j}\) is to its base station \(BS_{i}\) (based on the euclidian distance).