# Extending Wireless Reschargeable Sensor Network Life without Full Knowledge
## 5 Exact Probability of a Node's Existence at a Certain Power Level  
In this section, we compute the exact probability of a single node to be at a certain power level, by answering the following question: given a node with power level \(Y\), what is the probability of such a node, or one with less power, to exist in the network? We assume the following:  
* \(T\) time units have elapsed,
* the node was at power level \(W\) before \(T\) time units,
* discharge probability is \(p\) at each time unit.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \(B_{S}\) & base charging station, MCR recharge \& start point \\ \(N\) & set of all sensor nodes \\ \(n_{i}\) & sensor node at location \(i\) \\ \(N_{e}\) & set of all residual energy levels \\ \(e_{i}\) & residual energy of sensor node \(i\) \\ \(N_{d}\) & set of distances between nodes \\ \(d_{i}\) & distance from sensor node \(i-1\) to sensor node \(i\) \\ \(u\) & number of sensor nodes to charge \\ \(\lambda\) & cost of concentrically localizing MCR with a node \\ Y & optimal set of nodes to charge \\ \(\Psi\) & optimal WRSN time increase \\ \(\Gamma\) & optimal exploration termination point \\ \(v\) & set of nodes to charge with no knowledge \\ \(\psi\) & WRSN time increase with no knowledge \\ \(\gamma\) & exploration termination point with no knowledge \\ \(l\) & target power of a node to charge \\ \(E_{S}\) & battery capacity of a sensor node \\ \(E_{C}\) & battery capacity of a MCR \\ \(C\) & total power consumed for charging \\ \(M\) & total power consumed for moving \\ \(\Lambda_{m}\) & the MCR moving power consumption per move unit \\ \(r\) & sensor node discharge rate \\ \(p\) & sensor node discharge probability at each time unit \\ \(g(n)\) & node discharge function \\ \(k\) & number of sensor nodes that may reach zero power level \\ \(Y\) & power level of the currently explored sensor node \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations used in problem formulation.  
We start by identifying how many discrete steps \(t\) are needed for a node to reach power level \(Y\). Due to the discrete nature of discharge and knowing the discharge rate \(r\), we can define \(t\) as  
\[t=(W-Y)/r.\]  
Our problem is now simplified to finding the probability of \(t\) or more discharges to occur in \(T\) steps. We treat this as Bernoulli trials with success= \(p\) and failure= \(1-p\). The probability of exactly \(t\) discharges to occur will be  
\[\binom{T}{t}p^{t}(1-p)^{(T-l)},\]  
and the probability of a node with power less than or equal to \(Y\) will be  
\[\xi=\sum_{j=t}^{T}\binom{T}{j}p^{j}(1-p)^{(T-j)}.\]  
While the above formula can be used to find the total probability, we face two main problems when dealing with it. First, the numbers become significantly small with the increase of \(j\) (proportional to time elapsed). When using the parameters needed to simulate a large scale WRSN, the precision exceeds our computer's computational precision. Each computed probability component becomes zero. Even with an increase in computational power, we still face another problem. This calculated probabilistic value does not give us an exact prediction, increasing the chances of making mistakes if these values are used. Due to these limitations, we need to find a better and more scalable solution. We turn to utilizing an upper bound on our desired probability estimations instead of computing the exact probability. An upper bound will be more helpful in identifying highest possible values without having to consider exact probability and variance.