## 1 Introduction  
Wireless sensor networks (WSNs) have received lots of attention in recently years due to their wide applications like military and disaster surveillance, industrial product line monitoring, agricultural and wildlife observation, healthcare, smart homes, _etc._[1]. Cheap and tiny sensor nodes are usually randomly deployed in a physical environment to be monitored and they will transmit their collected data to certain remote sink node (or base station) in an autonomous and unattended manner. Energy efficiency and balancing is one of the primary challenges to the successful application of WSNs since the sensor nodes are powered with limited batteries and they cannot be easily recharged once deployed. Up to now, many energy efficient routing algorithms or protocols have been proposed with techniques like clustering, data aggregation, multi-path and location tracking, _etc._, as can be seen from related work. However, many of them aim to minimize parameters like total energy consumption or delay during the routing process, which might cause some hotspot nodes as well as a partitioned network due to the overuse of certain nodes. Since the network lifetime is usually defined as the time when the first node dies from lack of energy, huge amounts of energy will be wasted by the remaining sensor nodes. Hop number and hop distance have a very important impact on many network metrics like energy consumption, routing overhead, interference, latency, _etc._ Intuitively, if the hop number is too large, the energy consumption can be reduced at the cost of long end-to-end latency and large control overhead. If the hop number is too small (e.g., direct transmission), the latency will be very small while the energy consumption can be very large due to the long distance wireless communication nature. Therefore, an optimal hop number with suitable individual distance(s) needs to be deduced as a tradeoff in order to achieve energy reduction and energy balancing. Hotspot problems are caused by an unbalanced energy consumption among the sensors [2-5]. It is a big challenge to overcome under the random and dynamic topology of WSNs. Besides, the routing scheme and traffic pattern vary under different applications, which add to the difficulty of energy balancing and will usually lead to hotspot nodes and partitioned networks. For example, when all sensors use direct transmission, the nodes far away from sink node will die earlier since the energy consumption is proportional to the fourth order of the long distance. Meanwhile nodes close to the sink node will have much residual energy. On the other hand, when multi-hop transmission is used, nodes near a sink node will have more traffic to forward and will die quickly while nodes far from a sink node will have much remaining energy by using short distance multi-hop transmission. This hotspot issue also appears in many other energy efficient routing algorithms or protocols like those described in [6-8], _etc._  
To effectively alleviate the hotspot problem, we need to balance energy consumption among all sensors by considering factors like manner of transmission, traffic patterns, hop number and distance, _etc._ Based on our previous work in [5], we find that the final residual energy is still not well balanced when the first node dies, even though the energy consumption is largely reduced and the network lifetime is extended. The main reason is that in [5] we tried to minimize the total energy consumption on each route so that some nodes close to sink node are overused, which causes a hotspot problem. In order to alleviate the hotspot problem, we study the energy consumption under different energy and traffic models and aim to let all sensor nodes consume their energy at similar rate. In other words,we are not trying to \(Min(\sum\limits_{i=1}^{n}E_{i})\) for the \(n\)-hop route but rather to equalize \(E_{I}\approx E_{2}\approx\cdots\approx E_{n}\) for the involved intermediate \(n\) nodes with proper individual distance \(d_{i}\).