In this paper we use the Erlang theory to quantitatively analyse the trade offs between energy conservation and quality of service in an ad-hoc wireless sensor network. Nodes can be either sleeping, where no transmission or reception can occur, or awake where traffic is processed. Increasing the proportion of time spent in the sleeping state will decrease throughput and increase packet loss and delivery delay. However there is a complex relationship between sleeping time and energy consumption. Increasing the sleeping time does not always lead to an increase in the energy saved. We identify the energy consumption profile for various levels of sensor network activity and derive an optimum energy saving curve that provides a basis for the design of extended-life ad hoc wireless sensor networks.Keywords: Sensor networks, Ad hoc networks, Energy efficient design, QoS, Erlang formula IntroductionRecent advances in micro-electro-mechanical systems (MEMS) technology, wireless communications and digital electronics have enabled the development of low-cost, low-power, multifunctional smart sensor nodes [1]. Smart sensor nodes are autonomous devices equipped with heavily integrated sensing, processing, and wireless communication capabilities [2][3]. When these nodes are networked together in an ad-hoc fashion, they form a sensor network. The nodes gather data via their sensors, process it locally or coordinate amongst neighbors and forward the information to the user or, in general, a data sink. Due to the node's limited transmission range, this forwarding mostly involves using multi-hop paths through other nodes [3]. A node in the network has essentially two different tasks: (1) sensing its environment and processing the information for onward transmission, and (2) forwarding traffic from other sensors as an intermediate relay in the multi-hop path.The major design challenge for this type of network is to increase the operational lifetime of the sensors as much as possible [1][4]. Indeed, sensor nodes are miniature devices and operate on a tiny, non-replaceable battery. Energy efficiency is therefore the critical design constraint. Research can address two different perspectives of the energy problem: (1) an increase in battery capacity and (2) a decrease in the amount of energy consumed at the wireless terminal. The focus of battery technology research has been to increase battery power capacity while restricting the weight of the battery. However, unlike other areas of computer technology such as microchip design, battery technology has not experienced significant (compared to Moore's Law) advancement in the past 30 years. Therefore, unless a breakthrough occurs in battery technology, a goal of research should be to decrease the energy consumed in the wireless terminal [5].In terms of energy consumption, the wireless exchange of data between nodes strongly dominates other node functions such as sensing and processing [6] [7]. Moreover, actual radios consume power not only when sending and receiving data, but al...