# An Optimization Routing Algorithm for Green Communication in Underground Mines
## 3 GCUM and the Energy Consumption Model
### Energy Consumption Model  
The GCUM is a two-dimensional space, data are propagated in the clustering process in the following ways [34]:  
\(\blacklozenge\)\(E_{s}^{A}(\lambda_{1}t)\) represents the energy consumption of acquiring \(\lambda_{1}t\)_bits_ data.  
\(E_{s}^{P}(\lambda_{2}t)\) represents the energy consumption of processing \(\lambda_{2}t\)_bits_ data.
* \(E_{s}^{P}(\lambda_{2}t,d)\) represents the energy consumption of transmitting \(\lambda_{2}t\)_bits_ data over distance \(d\).
* \(E_{s}^{R}(\lambda_{3}t)\) represents the energy consumption of cluster head receiving \(\lambda_{3}\)_bits_ data from its member nodes.  
Where \(\lambda_{1}\) denotes the rate of acquiring data, \(\lambda_{2}\) denotes the rate of processing and transmitting data, \(\lambda_{3}\) denotes the rate of cluster head receiving data from its member nodes, and \(t\) is the time represented by the transmission rounds. In addition, \(E_{s}^{T}(\lambda_{2}t,d)\) and \(E_{s}^{R}(\lambda_{3}t)\) can be described in detail as following functions:  
\[E_{s}^{T}(\lambda_{2}t,d)=E_{i}(t)+\lambda_{2}td^{m}\]  
\[E_{s}^{R}(\lambda_{3}t)=E_{i}(t)+\lambda_{3}tE^{\prime}\]  
where \(E_{i}\) indicates the energy consumption of idle process. \(d\) indicates the communication distance between the two connected nodes, \(m\) is the energy consumption parameter of the transmitting distance, \(m=2\) is defined as the short distance of transmitting within the radius range of the cluster head, and \(m=4\) is defined as the long distance of transmitting data to the receiving node with multi-hops. \(E^{\prime}\) indicates the overhead of the beam forming to decrease the energy consumption. Therefore, the energy consumption function can be established as follows:  
\[E_{s}(t)=E_{s}^{A}(\lambda_{1}t)+E_{s}^{P}(\lambda_{2}t)+E_{s}^{T}(\lambda_{2 }t,d)+E_{s}^{R}(\lambda_{3}t) \tag{1}\]  
Thus, the residual energy of node \(s\) can be written as:  
\[\hat{E}_{s}(t)=E_{s}(0)-\sum_{i=1}^{t}E_{s}(i) \tag{2}\]  
where \(E_{s}(0)\) represents the initial energy of node \(s\) at time \(t\), \(\lambda_{1}\) represents the generation rate of data in the network due to the status of network, \(\lambda_{2}\) represents the throughput of data in channels or nodes, and \(\lambda_{3}\) represents the receiving data of member nodes decided by the number of member nodes.  
In addition, the energy consumption in the cluster will be also considered. The energy consumption of member nodes and cluster head nodes, as below:  
\[\begin{split}\widetilde{E}_{s^{\prime}}&=E+\lambda _{2}tD^{2}(s^{\prime},s)\\ \widetilde{E}_{s}&=E+N_{s}\lambda_{2}tE^{\prime} \end{split} \tag{3}\]  
where \(E\) denotes the inherent energy consumption of nodes by acquiring, processing and idle. \(s^{\prime}\) and \(s\) denotes member nodes and cluster head nodes, and \(D^{2}(s^{\prime},s)\) denotes the distance between member nodes and the cluster head node.  
In this paper, throughput \(\lambda_{2}\) can be calculated by the effective can be expressed as follows:  
\[\lambda_{2}=\frac{1}{V\lambda_{1}}\sum_{x=0}^{|V|}Q_{z}[z\frac{(1-(1-P_{a})^{ K})L}{T_{s}}] \tag{4}\]  
where \(V\) indicates the set of the node in the network and \(|V|\) indicates the number of nodes in this set, \(Q_{z}\) indicates that there are \(z\) nodes need to transmit data, \(1/T_{s}\) is the transmission rate of data packets, \(K\) is the retreat time upper limit and \(P_{a}\) is the probability of a channel being available. Specifically, \(Q_{z}\) and \(P_{a}\) can be formulated as follows:  
\[Q_{z}=\frac{|V|!}{z!}p^{z} \tag{5}\]  
\[P_{a}=P_{a|a}P_{a}+P_{a|b}(1-P_{a}) \tag{6}\]where \(P_{a|a}\) and \(P_{a|b}\) are conditional probability of channel availability based on whether the channel was available or not at the previous moment, and \(p\) is the probability that the data needs to be sent.