# End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks
## WSN Scenario  
As can be observed in Figure 1, sensor are randomly distributed in a two-dimensional (2D) area. They are represented by small white circles. All sensors are homogeneous, equipped with identical hardware and software configuration. Also we assume omni-directional antennas with their transmission ranges as circles. Although it is not realistic, in the first instance we do not suppose any limitation in the range of coverage. Except the attenuation due to distance, we assume a perfect radio channel, that is, there are no fast or slow fading effects, nor bit errors. There is only a single transceiver unit per node. Hence, the WSN operates on a single frequency and nodes can be in one of the following three state: transmitting, receiving or sleeping. For cluster configuration purposes, we assume that all nodes know their actual physical location in the 2D-WSN area by applying some known location technique (for instance, see [16, 17, 18]). To cluster WSNs we could apply some well know algorithm such as in [19]; however the development of clustering algorithms is outside the scope of this works.  
Cells, Sensors and Cell Head Selection_  
Several strategies of coverage for a WSN could be considered, such as the force-based, the grid-based and the computational geometry-based approach [20]. However, since this is not the goal of our contribution, here we have adopted the hexagonal grid-based-approach as a start-up solution or initial analysis. As Figure 1 shows, the monitored area is divided into _cells_ with hexagonal perimeter. All hexagons have the same size, defined by radius \(R_{h}\) (in meters). The set of hexagons are placed in rings, forming a mosaic \(T\)[21], and each hexagon represent the coverage area of a cluster. Then, the number of clusters in the WSN area is determined by \(r_{\max}\), the number of rings that conforms the mosaic \(T\). Clearly, for a given \(r_{\max}\) the number of clusters is given by \(N_{c}(r_{\max})=1+3\cdot r_{\max}(r_{\max}+1)\). Figure 1 is an example with \(r_{\max}=3\) then \(N_{c}(3)=37\).  
All the sensors that are located within a given hexagon form a cluster. The number of motes (sensors) within a cluster is a 2D random variable that can be Poisson, uniform, etc. For the sake of brevity and conciseness, here we only assume that \(M_{s}\), a constant value, is the number of sensors spread throughout the service area. Also, for simplicity, here we assume that the number of sensors for each hexagon or group is given by \(M_{c}(r_{\max})=M_{s}/N_{c}(r_{\max})\).  
For each cluster, a single node among all members, called _cell head node_ (CH), acts as the receiver of the information sensed by the other CH members. We indicate that the number of techniques to elect one node as CH of the group is rather large [22]; but due to space limitations they are not discussed in this paper. Hence, as a first approach used in our study we have assumed that the CH is approximately located at the center of the hexagon.  
To identify each hexagon or its own CH, we use _hexagonal coordinates_, \(\langle x_{h},y_{h}\rangle=\langle x,y\rangle_{h}\). Also, for convenience, we introduce coordinates in polar-like (or ring) form \(\langle x_{p},y_{p}\rangle=\langle x,y\rangle_{p}\); which we call _polar coordinates_. Both are illustrated in Figure 2 where the sink is located at the center of the sensed area, i.e., \(\langle x,y\rangle_{h}=\langle x,y\rangle_{p}=\langle 0,0\rangle_{*=h,p}\). Finally, we divide the hexagonal grid into twelve disjoint zones: six sectors and six axes. Each sector is located between two axes. Sectors and axes are defined according to Table 1 where polar coordinates are used.  
Figure 1: Wireless Sensor Network Scenario with \(r_{\max}=3\).