As one of the most widely investigated studies in wireless sensor networks (WSNs), multihop networking is increasingly developed and applied for achieving energy efficient communications and enhancing transmission reliability. To accurately and realistically analyze the performance metric (energy efficiency), firstly we provide a measurement of the energy dissipation for each state and establish a practical energy consumption model for a WSN. According to the analytical model of connectivity, Gaussian approximation approaches to experimental connection probability are expressed for optimization problem on energy efficiency. Moreover, for integrating experimental results with theories, we propose the methodology in multihop wireless sensor networks to maximize efficiency by nonlinear programming, considering energy consumptions and the total quantity of sensing data to base station. Furthermore, we present evaluations adapting to various wireless sensor networks quantitatively with respect to energy efficiency and network configuration, in view of connectivity, the length of data, maximum number of hops and total number of nodes. As the consequence, the realistic analysis can be used in practical applications, especially on self-organization sensor networks. The analysis also shows correlations between the efficiency and maximum number of hops, that is the multihop systems with several hops can accommodate enough devices in ordinary applications. In this paper, our contribution distinguished from others is that our model and analysis are extended from experiments. Therefore, the results of analysis and proposal can be conveniently applied to actual networks.

Recently, multihop wireless sensor networks (WSNs) are widely developed and applied to energy efficient data collections from environments by establishing reliable transmission radio links and employing data aggregation algorithms, which can eliminate redundant transmissions and provide fusion information. In this paper, energy efficiency which consists of not only energy consumptions but also the amount of received data by the base station, as the performance metric to evaluate network utilities is presented for achieving energy efficient data collections. In order to optimize energy efficiency for improvements of network utilization, we firstly establish a graphical game theoretic model for energy efficiency in multihop WSNs, considering message length, practical energy consumptions and packet success probabilities. Afterwards, we propose a graphical protocol for performance optimization from Nash equilibrium of the graphical game theory. The approach also consists of the distributed protocol for generating optimum tree networks in practical WSNs. The experimental results show energy efficient multihop communications can be achieved by optimum tree networks of the approach. The quantitative evaluation and comparisons with related work are presented for the metric with respect to network energy consumptions and the amount of received data by the base station. The performances of our proposal are improved in all experiments. As an example, our proposal can achieve up to about 52% energy efficiency more than collection tree protocol (CTP). The corresponding tree structure is provided for the experiment.