## 1 Introduction  
Due to the technological developments of microelectromechanical systems, wireless communication and digital electronics, it becomes possible to produce low-cost wireless sensors. A wireless sensor network (WSN), which consists of a number of sensor nodes (several tens to thousands working together), is applied for information monitoring in the fields such as health monitoring [2010], military target tracking and surveillance [2011], natural disaster relief, hazardous environment exploration [2011], intelligent transportation [2010], and social networking and gaming [2011].  
Because the sensors have limited energy resource, the lifetime of a WSN is significantly affected by its energy consumption [2011]. Thus, improving the energy efficiency becomes a hot topic in this area [2011]. [PERSON] _et al._ [2011] state that the energy consumed by a WSN is mainly used for communication and data processing. Evidently, the energy consumed by communication is sensitive to the transmission distance. Thus, sensor deployment optimization is one of the most important methods used to reduce the energy consumption [2011]. In many applications, the data gathered via WSN are essential since the transmission errors may lead to system failures that may cause economic losses, environmental damage or casualties [2011]. As WSN is usually deployed in a harsh environment, according to the reports, the packet loss ratio can be as high as 70% in a typical WSN [2011]. Retransmission is now widely applied to improve the success rate of data transmission [2011]. It is obvious that data retransmission consumes extra energy. However, previous work usually assumes that the data transmission is always successful and neglects the possible retransmission.  
In this paper, the lifetime of a WSN is defined as the period starting from the initial working time until the WSN fails to satisfy its requirements (including coverage, connectivity and success transmission rate). Our goal is to maximize the lifetime for a homogeneous WSN that is used to monitor a circular target area with a base station in the center. We propose a new sensor deployment optimization model based on the energy consumption calculated under retransmission. The remainder of this paper is organized as follows. In Section 2011, we briefly review the related work of sensor deployment optimization. Section 2011 describes the static topology and dynamic behaviors of the WSN in our problem. The lifetime optimization models are proposed for both uniform and non-uniform deployment problems in Section 2011. We analyze the energy consumption in Section 2011, while the path probability and the success rate of data transmission, with and without retransmission, are quantified in Section 2011, respectively. In Section 2011, two case studies are presented to verify the effectiveness of considering retransmission in the lifetime optimization, and the advantage of non-uniform deployment scheme compared with a uniform one. Simulation further demonstrates that our optimal deployment scheme can provide much longer lifetime and higher successful transmission rate than a random scheme. Finally, Section 2011 concludes our results.