## 3 Problem Statement and Assumptions for the Localization Scheme  
We consider here a typical scenario of a wireless multi-hop network composed of both mobile nodes and nodes that are deployed at fixed positions. In this scenario, the position estimation of the mobile nodes should be updated frequently enough so that the nodes can be tracked as they move. Hence, each localization of these nodes should be performed within a given period \(T_{0}\), which depends on the mobility pattern of each node. In this work, we assume that \(T_{0}\) is known. In fact, there are several methods available in the literature (e.g., [15, 16, 17, 18]) to compute it in real time.  
As explained in Section 1, the position of a node in a wireless network is usually computed from the measurements of a parameter of the radio signal (RSS, TOA, _etc._). The accuracy of the localization will therefore depend on the accuracy of these measurements.  
The measured radio parameters are random variables, so their uncertainty can be reduced, in general, by averaging a number of measurements. In particular, both TOA and RSS measurements are usually described as random Gaussian variables [23], so if \(n\) measurements are averaged, the standard deviation can be reduced by a factor of \(\sqrt{n}\), provided the measurements are independent.  
Let us suppose that a node will average the RSS or TOA of all packets received during a period \(T_{0}\), which depends on how often the localization of the node must be performed, with the objective of achieving a given accuracy in the distance estimations. The number of received packets (\(n\)) during this localization period will determine the accuracy of the RSS or TOA measurements, and therefore, the accuracy of the distance estimation. Due to the nature of the radio link, the receiver node will not receive all the packets that are transmitted by the transmitter node. The probability of receiving a packet in a certain link, that is, the packet reception probability (\(PRP\)) or packet reception rate, is a function of the signal to noise ratio (\(SNR\)), (see Section 3.1). Therefore, in order to receive \(n\) packets during a time interval \(T_{0}\), packets can be sent either with a high transmission power (so that the \(SNR\) at the receiver is high and very few transmitted packets are lost) at a rate of \(n/T_{0}\), or with a lower transmission power (some packets will be lost) but at a higher rate (\(k/T_{0}\), with \(k>n\) being the number of transmitted packets during the time interval \(T_{0}\)).  
The goal is to optimize the transmission strategy at each node with the aim of obtaining a given accuracy in the RSS or TOA measurements, while minimizing the energy consumption in the global network. As the radio system is the most energy demanding part of a node (see Section 1) we will only take into account the energy consumption of the radio system. The transmission strategy will be defined by two parameters: the transmission power (\(P_{TX}\)) and the time between transmitted packets (\(T\)), that is, the inverse of the packet transmission rate (\(k/T_{0}\)).  
We will assume that all nodes in the network are synchronized; hence, one-way TOA measurements can be performed in the network. In addition, given that nodes are synchronized, they can turn off their radio when they are not transmitting or expecting to receive a packet from another node, so idle listening is avoided and the energy consumption of the radio system is only due to transmitting and receiving packets.  
In order to evaluate the energy consumption during the localization procedure, some assumptions about the receiver and radio channel behavior must be done. In the following, we propose to use simple models of the receiver and the channel.