362 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 5, MAY 2008 Optimal Placement of Wireless Nodes for Maximizing Path Lifetime Enrico Natalizio, Member, IEEE, Valeria Loscr´ ı, Member, IEEE, and Emanuele Viterbo, Senior Member, IEEE Abstract— In this work we investigate the deployment of wireless nodes in order to maximize the lifetime of a data ﬂow. We develop a mathematical model for determining the best placement of nodes by taking into consideration the energy of each node involved in the data ﬂow. By using our mathematical model we achieve two major objectives: the maximization of the shortest node’s lifetime and the convergence of all the nodes’ lifetime to a unique value. Index Terms— Wireless sensor networks, lifetime. I. I NTRODUCTION T HE placement of nodes in a wireless network is an important and growing research ﬁeld since the energy consumption and the lifetime of a network rest on the power used in the transmission and reception. This power usage, in turn, depends on the mutual position of the pair of communi- cating nodes. Toumpis and Tassiulas found a scalar nonlinear partial differential equation for determining the optimal positions of nodes in a massively dense sensor network, so that a minimum number of nodes is needed for a data ﬂow [1]. The objective of their work is to ﬁnd a tradeoff between the length of routes and the number of nodes in each route, without taking into consideration the energy consumption. In [2] the authors explore the problem of the optimal placement of Wireless Sensor Networks (WSN) devices, but they do not investigate the scenarios where the transmission range of nodes is adaptive nor do they take into account the optimal positions of the nodes in relation to their residual energies. Moreover, we know from [3] that a straight path, between source and destination, is most energy efﬁcient and there is also a unique hop count for any distance that minimizes the cost of communications. Goldenberg et al. show in [4] that the optimal positions of the relay nodes must lie entirely on the line between the source and the destination, and these nodes must be evenly spaced along the line. Therefore, from now on, we shall refer to this approach as “evenly spaced”. In this letter we propose a mathematical model which focuses on the maximization of the lifetime of the path of nodes involved in a data ﬂow. This model allows us to ﬁnd the best placement of the devices when they have different levels of residual energies. Previous works cited did not place an emphasis on the importance of different levels of residual energies among their working assumptions, for this reason we called our approach “energy spaced”. From the model we ﬁnd out that the optimal placement is on the straight Manuscript received February 1, 2008. The associate editor coordinating the review of this letter and approving it for publication was S. Buzzi. The authors are with DEIS - UNICAL, Via Pietro Bucci, cubo 42/d - 87036 Rende (CS), Italy (e-mail: vloscri@deis.unical.it). Digital Object Identiﬁer 10.1109/LCOMM.2008.080168. line between source and destination as in [4], but the nodes must be spaced according to their residual energies. When we compare our approach with the random and the evenly spaced deployments, the results show that the energy spaced solution achieves a much longer lifetime. In WSN, wasteful disconnections and inconvenient replacements of nodes are often caused by wide variations in the lifetime values of nodes. Our placement strategy avoids these problems by making all the nodes involved in the route last the same amount of time. To the best of our knowledge, no mathematical scheme based on the residual energy of nodes has been introduced for the placement of wireless devices to date. II. SYSTEM AND MODEL DESCRIPTION We consider a data ﬂow between a source and a destination in a sensor ﬁeld. The positions of the relay nodes have been chosen according to three different strategies of placement: • random; • evenly spaced along the straight line. This is according to [4] in order to minimize the energy consumption; • energy spaced along the straight line, taking into account the different levels of residual energy of the relay nodes. The last strategy is the focus of this letter, and it has been optimized as a result of the mathematical model which follows. The energy model we used to characterize the physical layer of our mathematical scheme is taken from [5]. By simplifying this model we obtain that the energy required to send one bit at the distance d is E = βd α , where α is the exponent of the path loss (2 ≤ α ≤ 5), β is a constant [J/(bit · m α )]. We set α equal to 2 and β equal to 10 pJ/(bit · m 2 ), which are typical values of a free space model. Next, we introduce the mathematical model of the system. Let v 1 and v n denote the known source and destination positions, respectively. Let {v i } n−1 i=2 be the positions of the n − 2 relay nodes. Let {T i } n−1 i=1 and {E i } n−1 i=1 be the life times and the residual energies of the nodes, respectively. Let P rec denote the minimum required power in order for a bit to be correctly received. We assume a power control system is in place so that the transmitter adjusts its power in order to deliver P rec at the receiver. This implies that each T i is a function of the positions of nodes i and i +1, i.e. T i = Ei Prec‖vi−vi+1‖ 2 . The distance between two successive nodes in the path is ‖v i − v i+1 ‖. Problem: Find {v i } n−1 i=2 such that min {T i } n−1 i=1 is maximized. This can be immediately solved by placing the nodes on the segment with the extremes v 1 and v n , the distance between adjacent nodes being chosen in order to have T 1 = T 2 = ··· = 1089-7798/08$25.00 c  2008 IEEE