Power Efficient Topology Control in Wireless Ad Hoc Networks * Chih-Cheng Tseng** Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. cctseng@mail.jwit.edu.tw Kwang-Cheng Chen Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. chenkc@cc.ee.ntu.edu.tw Abstract—In the wireless ad hoc networks, for prolonging the communication duration of the nodes, the transmission power is required to be minimized to conserve the limited battery life. In addition, for the wireless ad hoc networks to be applicable, the networks are required to be connected. Unfortunately, the two requirements are against each other. In this paper, by analyzing the probability of isolated node, we obtain the relationships between transmission range, service area and network connectedness. Furthermore, we propose a generalized (N,B) wireless ad hoc network model to investigate the impact of boundary nodes on the network connectivity, equivalent service area and average node degree. Keywords-ad hoc networks, connectivity, power conservation I. INTRODUCTION Wireless ad hoc network is a self-organizing network architecture that is rapidly deployable and adapts to the propagation conditions and to the traffic and mobility patterns of the network nodes. Possible examples of the wireless ad hoc networks are tactical military applications, disaster recovery operations, exhibitions or conferences. Due to the characteristics of lacking fixed infrastructure and all the communications are carried over the wireless medium, nodes are usually mobile. As a consequence, the power source is mainly supplied by batteries. To prolong the communication duration of the nodes, the transmission power must be minimized so as to conserve the limited battery life. However, for a wireless ad hoc network to be applicable, the network is required to be connected, i.e. given any two nodes in the network, there must exist at least one path to connect these two nodes. Unfortunately, it is contradiction between power conservation (shorter transmission range is preferred) and network connectivity (longer transmission range is preferred) in designing wireless ad hoc network. Our first objective, in this paper, is to propose a solution to compromise these two issues. One of the first researches on this topic is examined in the context of a “broadcast percolation” in [1]. For broadcast percolation in one spatial dimension, analytical expressions for the average extent of percolation are derived and a model for two-dimensional spatial percolation is presented along with related simulation results. They also suggested that the choice of optimal transmission radius might be bounded from below by the need to maintain desired network connectivity. In [2], if nodes are distributed according to a two-dimensional Poisson point process with density D, then, in an infinite area there exists an infinite connected component with nonzero probability if the expected number of nearest neighbors of a transmitter N 0 is bounded by 0 2.195 10.526 N < < . In addition, they also conjectured that transmission range for connectivity and covering are the same. However, Piret in [3] proved that in one-dimensional, the transmission range problem and the covering problem may exist at different range and, further, they conjectured that even in two-dimensional, these two problems are also different. Based on two different mobility models, Santi and Blough in [4] studied the relationship of the transmission range to maintain connectivity between the cases of stationary and mobility during some fraction of the operational time. They concluded that (i)it is not the mobility models but the “quantity of mobility”, which can be informally defined as the percentage of stationary nodes with respect to the total number of nodes influences the connectedness most and (ii)if brief periods of disconnection are allowed and/or only a significant fraction of the nodes being connected, the transmission range can be reduced largely. Gupta and Kumar [5] considered the problem of placing n nodes randomly and independently in a disc of unit area in R 2 and showed that if log ( ) 2 () n cn n r n π + = , then the resulting network is asymptotically connected with probability one if and only if () cn → +∞ . Similar results are obtained in [6]. Recently, Bettstetter [7] employed the nearest neighbor method to study the range assignment problem. He assumed the distribution of nodes as a homogeneous Poisson point process in R 2 and obtained the minimum transmission range for the network to be connected. Besides, [7] also employed the results on the geometric random graph [8] to obtain the transmission requirements to generate a k-connectivity wireless ad hoc network. Based on the work in [7], our approach uses the uniform node distribution to simplify the analysis of minimum transmission range and network connectivity. The second objective of this paper is to study the impact of boundary nodes on the network connectivity. Based on a proposed generalized ( , ) NB wireless ad hoc network model, the maximum number of boundary nodes for the network to maintain connectivity and the corresponding minimum average node degree are obtained. The rest of this paper is organized as follows: Section II provides criterion to achieve the optimum transmission range and network connectedness. In Section III, we generalize the * This research is supported in part by the National Science Council, Taiwan, R. O. C. under Contract NSC 92-2213-E-002-038. ** C-C Tseng is also with the Department of Electronic Engineering, Jin- Wen Institute of Technology, Xin-Dian, Taiwan, R. O. C.