# Cooperative Data Collection Mechanism Using Multiple Mobile Sinks in Wireless Sensor Networks
## 3 Network Environment and Problem Formulation
### Problem Formulation  
Energy conservation is the most important parameter that determines the lifetime of a wireless sensor network. In general, the lifetime of a given WSN is defined by the time length starting from the time point that the network operates to the time point that the first sensor node runs out its energy. This paper aims to develop a data collection mechanism which initially partitions the \(n\) sensor nodes into \(k\) subsets \(T=T_{1}\), \(T_{2},\ldots,T_{k}\) and assigns one mobile sink \(m_{i}\) to collect data generated by sensors in \(T_{i}\). Then the data collection mechanism aims to select \(k_{i}\) CPs from the locations of sensors in \(T_{i}\) and establish a route \(\pi_{i}\) which passes through the \(k_{i}\) CPs for data collection. Since the CPs are bottlenecks of network lifetime, saving their energy consumption can prolong the network lifetime. In the following, the energy consumption model considered in this paper is [21]. The energy consumption of wireless sensor nodes mainly happen on the operations including data sending and receiving. Let sender \(s_{i}\) and receiver \(s_{j}\) be a communication pair and \(s_{i}\) sends \(k\) bits to \(s_{j}\). Let \(d_{ij}\) be  
Figure 1: An example of the considered network environment. the distance between \(s_{i}\) and \(s_{j}\). The energy consumption for each sensor node receiving \(\delta\) bits can be measured by Equation (1). \[E_{R}=\delta\beta \tag{1}\]  
where \(\beta\) is the parameter indicating energy consumption for transmitting one bit. The energy consumption for sender \(s_{i}\) to transmit a packet to its parent \(s_{j}\) is expressed as Equation (2). \[E_{T}=\delta\varepsilon_{1}\ +\ \delta d_{i,j}^{Y}\varepsilon_{2} \tag{2}\]  
where \(\varepsilon_{1}\) is a parameter denoting the energy consumption for transmitting one bit, \(\varepsilon_{2}\) is the energy consumption factor of the amplifier circuit, \(d_{i,j}^{Y}\) is the distance between sensors \(s_{i}\) and \(s_{j}\), \(\gamma\) is the path-loss exponent, which usually ranges between 2 and 4, depending on the environment. The lifetime of sensors in \(T_{i}\) highly depends on the value of \(k_{i}\). A large value of \(k_{i}\) indicates that the mobile sink \(m_{i}\) can visit more CPs. This implies that all sensors in \(T_{i}\) can be further partitioned into \(k_{i}\) subsets. Each subset will be associated with a CP and all sensors in this subset should transmit their data to this CP. Consider each subset and the corresponding CP. Since the distance between each sensor and the CP might be larger than the communication range, a tree rooted by CP will be constructed for relaying the data from each sensor to CP along with the tree topology. Let subtree \(T_{ij}\) is rooted by CP \(p_{ij}\) and has \(u_{ij}+\ 1\) nodes. Sensor nodes of each subtree \(T_{ij}\) should upload their data to the root \(p_{ij}\). These roots will store the data and then transmit the data to mobile sink when the mobile sink visits them. A route that visits all CPs should be constructed for each mobile sink \(m_{i}\) to collect data from all CPs in \(T_{i}\). The energy consumption of sensor nodes mainly happen on the operations including data transmitting and receiving. Let \(E_{ij}\) denote the energy consumption of \(p_{ij}\) for receiving one packet from its children. Equation (3) evaluates the value of \(E_{ij}\) which equals to the sum of energy consumption for receiving \(u_{ij}\) packets and transmitting \(u_{ij}+\ 1\) packets in tree \(T_{i}\). \[E_{ij}=u_{ij}\ \times\ E_{R}\ +\ \left(u_{ij}\ +\ 1\right)\ \times\ E_{T} \tag{3}\]  
The network lifetime is determined by the lifetime of the sensor with the least energy. Since each CP consumes the highest energy in its tree, this paper aims to reduce the energy consumption of the CP which consumes maximal energy as compared with the other CPs. Let \(s_{ij}^{max}\) denote the bottlenecked CP in \(T_{i}\).