# Matching of Energy Provisions in Multihop Wireless Infra-Structures
## 4 Matching of Energy Provision: Basic Concepts  
The goal of efficient energy provisions in this paper is to enable matching between the energy supply and energy requirement at each BS in the network in order to maximize the network lifetime that the network can continue to operate. More specifically, this paper aims to match the energy provision differences with the energy differentiation among BSs in a WMI network. The proposed approach can be employed in practical scenarios given a WMI topology in the following two types of applications. (a) Provision of backup batteries for each BS according to the estimation of explicit energy differentiation among BSs. This application is essential for the sustaining a network in the no-charging period of emergency situations. As for user applications, we especially consider the main application of Internet access through the multihop wireless infrastructure; (b) Provisions of solar-rechargeable batteries at each BS for use at night during the no-charging period of solar energy systems. The application is essential for sustaining a network during the night time no-charging period of a solar-powered network.This application is useful for temporary low-cost networks used at temporary locations and events. As for user applications in night time, we especially consider the main applications of data collection from sensors, and IoT communications using multihop wireless infrastructure. Figure 3, illustrates the basic idea of efficient energy provisioning. Figure 3a shows a case in which all BS have equal energy provision, but run out of energy at different times. If the energy consumption rates of the BS are different, some BS will be able to operate for a longer time than others. In contrast, Figure 3b, shows the case where the BS have unequal energy provision and all BS run out of energy at the same time. The network may cease to operate if any BS runs out of energy. Energy will be wasted if some BS have energy remaining after the network ceases to operate. In this sense, the optimal provision of energy should result in all BS having the same lifetime. Let us describe this formally. Assume the BS have unique labels \(i\) (\(1\leq i\leq N\)). Denote the ith BS as BS(\(i\)). We define the lifetime of a BS as the time from starting operation until the time when all its  
Figure 3: The basic concept of energy provision for a given total resource of network energy. (**a**) Same energy provision; (**b**) Diversity of energy provision. energy is completely depleted. Denote the lifetime of BS(_i_) as \(T_{BS}(i)\), where \(1\leq i\leq N\). Denote the network lifetime as:  
\[T_{network}=\underset{1\leq i\leq N}{\text{minimum}}T_{\text{BS}}(i) \tag{1}\]  
**Proposition 1**. : _The maximum value of network lifetime \(T_{network}\) is achieved when the lifetime of each BS is identical: \(T_{BS}(i)=T_{BS}(j)=...T(N)\)_  
Proof. : We use the contradiction method to prove the proposition. Suppose the network lifetime is maximum but the BS lifetimes are not identical. Then there will be at least one BS(_i_) that has a longer lifetime compared with \(T_{network}\), which is the minimum lifetime among BSs, and such BS will have energy remaining, say \(\triangle\ E(i)\), at the time \(T_{network}\). Then an energy provision that re-assigns some of \(\triangle\ E(i)\) of BS(_i_) to all the BS with the shortest lifetime will achieve a longer value of \(T_{network}\). This is in contradiction with the assumption that the network lifetime is maximum. Next we describe how matching of energy provisioning to energy consumption leads to the maximum network lifetime. Suppose BS(_i_) has energy provision \(E_{0}(i)\) and average energy consumption rate \(E_{CR}(i)\) during its lifetime \(T_{BS}(i)\). By definition, \(E_{CR}(i)=E_{0}(i)/T_{BS}(i)\). Let the average of energy provision at all BSs in the network be \(E_{0-BS}\), and the average of average energy consumption rate at all BSs be \(E_{CR-BS}\).