## 3 Analysis of Energy Efficiency and Network Connectivity
### Analysis of the Energy Efficiency  
Here, we study the **one-hop energy-distance ratio**, which is defined as the ratio of the energy consumption for the one-hop transmission to the average **distance progress** of a packet during the one-hop transmission, where the **distance progress** refers to the difference between the before-hop distance (between the sender and the sink) and the after-hop distance (between the relay node and the sink) [15]. We consider that one-hop energy-distance ratio is able to represent the overall energy efficiency since any intermediate relay transmission can be viewed as a new one-hop transmission for the remaining route. The one-hop energy-distance ratio should be consistent with the overall energy-distance ratio for the entire route in a homogeneous environment, which will be later substantiated by simulations in Section 4. Note that the determination of the relationship between the transmission range and the energy efficiency is an extension of the works in [16, 17] with different definitions, setups and derivations. Let \(u\) be the distance between a sensor node \(S\) and the sink \(D\) as shown in Figure 2. The condition where \(u\leq r\) gives a direct transmission from \(S\) to \(D\), and the distance progress is \(u\). For \(u>r\), a  
Figure 2: An illustration of the forwarding progress. neighboring node, say \(M\), is required to relay packets between \(S\) and \(D\), and the distance progress is equal to (\(u-v\)) with \(v\) being the distance between the one-hop router \(M\) and the sink \(D\). In other words, a packet only travels (\(u-v\)) distance towards \(D\), and it has another \(v\) distance to travel. Denote by \(\mathbb{X}\) the random variable (r.v.) corresponding to the distance progress for a one-hop transmission, and denote \(\mathbb{U}\) and \(\mathbb{V}\) the r.v. for \(u\) and \(v\), respectively. Like [17], we also assume that each node knows the locations of all its neighbors and the location of the destination node. We now define the **transmission strategy**, which can be described as follows: (i) The source node \(S\) directly transmits a packet to the destination node \(D\) if \(D\) is located within distance \(r\) from \(S\); (ii) if the destination node \(D\) is outside the transmission range of the source node \(S\), the packet is forwarded to the neighbor that is closer in distance to the destination node \(D\) than the source node \(S\), and that is nearest to the source node \(S\) among all neighbors; and (iii) the source node S will not send out a packet when there does not exist any neighbor satisfying (ii), and will postpone the transmission until such a neighbor appears. It should be pointed out that, the transmission strategy adopted in our study is **Nearest with Forward Progress** (NFP), while the transmission strategy adopted in [17] is **Most Forward with Fixed Radius** (MFR). This is due to the fact that (i) as pointed out in [8], a transmission power from a sensor node just enough to reach its nearest neighbor in the direction towards the final destination gives the optimal use of energy, and (ii) as shown in [16], NFP yields the highest one-hop throughput. Appropriate adjustments to the derivation based on [17] are made for our study. For completeness, in the following we provide the full derivation based on [17] with highlight of our adjustments. Denote by \(\mathcal{G}\) the event that there exists at least one relay node that is closer to the destination node than the source node if the destination node is outside the transmission range of the source node, but nearest to the source node among all neighbors, and denote by \(\bar{\mathcal{G}}\) the complement of \(\mathcal{G}\). With the above denotations, \(\mathbb{X}\) can be expressed as:  
\[\mathbb{X}= \begin{cases}\mathbb{U},&\text{if }\mathbb{U}\leq r\\ \mathbb{U}-\mathbb{V},&\text{if }(\mathbb{U}>r)\cap\mathcal{G}\\ 0,&\text{if }(\mathbb{U}>r)\cap\bar{\mathcal{G}}\end{cases}