# End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks
## 4 Communication
### Frame Structure  
Since a single carrier is available for communication, here we recall the reuse concept that is implemented in cellular networks (see, e.g., [23]). Two or more motes (sensors or CHs) can transmit in the same time slot and with the same frequency if at a given receiver one of the transmissions is received with enough power level over the others, that is, the potential co-channel interference is below a given threshold. Then, from [23] we remember the reuse concept by which, for any two positive integers or _shift parameters_\(i\) and \(j\leq i\) the number of clusters operating at different frequency is given by \(N=i^{2}+ij+j^{2}\), see Table 2. Given that the work frequency in the WSN is unique, the assignment of this \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Sector** & **Condition** & **Axes** & **Condition** \\ \hline \(S_{0}\) & \(0<x_{p},0<y_{p}<x_{p}\) & \(A_{0}\) & \(0<x_{p},y_{p}=0\) \\ \(S_{1}\) & \(0<x_{p},x_{p}<y_{p}<2x_{p}\) & \(A_{1}\) & \(0<x_{p},y_{p}=x_{p}\) \\ \(S_{2}\) & \(0<x_{p},2x_{p}<y_{p}<3x_{p}\) & \(A_{2}\) & \(0<x_{p},y_{p}=2x_{p}\) \\ \(S_{3}\) & \(0<x_{p},3x_{p}<y_{p}<4x_{p}\) & \(A_{3}\) & \(0<x_{p},y_{p}=3x_{p}\) \\ \(S_{4}\) & \(0<x_{p},4x_{p}<y_{p}<5x_{p}\) & \(A_{4}\) & \(0<x_{p},y_{p}=4x_{p}\) \\ \(S_{5}\) & \(0<x_{p},5x_{p}<y_{p}<6x_{p}\) & \(A_{5}\) & \(0<x_{p},y_{p}=5x_{p}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of S\({}_{k}\) and Axes \(A_{k}\). operating frequency to the sensors and/or to the CHs must be in shifts and with non-overlapping time intervals. Then, for our two tier architecture were the intra-cluster and inter-cluster communication ranges are different (we provides details later), two different reuse factors are considered, \((i_{x},j_{x})\) so \(N_{x}=i_{x}^{2}+i_{x}j_{x}+j_{x}^{2}\) (\(x=Intra,x=Inter\)). \(N_{Intra}\) and \(N_{Inter}\) are the number of time intervals or time slots per frame that are assigned for, respectively, intra-cluster and inter-cluster communications. Clearly we expect that \(N_{Intra}<N_{Inter}\). The quantification of both parameters depends mainly on the maximum radio co-channel interference allowed and is out of the scope of this paper. Figure 3 shows the proposed _Combi-Frame_. It is basically composed of the contention sub-frame, hereinafter the CONT sub-frame, and the TDMA sub-frame. We must highlight the fact that each cluster uses exactly one contention slot per CONT sub-frame for intra-cluster communication and one TDMA slot per TDMA sub-frame for inter-cluster communication. While the first communication is between the sensors and its CH, the latter refers to the communication between CHs. Then, by choosing the two pairs \((i_{x},j_{x})\) (\(x=Intra,Inter\)), we can select cluster sizes \(N_{x}\) and consequently the length of the Combi-Frame. This allows us to balance the trade-off between interference and network throughput. For example, a high \(N_{intra}\) increases the geographical distance between CHs that use the same transmission slot, which implies a longer CONT sub-frame, and consequently longer delay. The same argument applies for \(N_{inter}\). Figure 3: Combi-Frame Structure. The duration of one contention slots is equal to \(T_{Intra}\) and the duration of one TDMA slot is equal to \(T_{Inter}\). Additionally, each contention slot and TDMA slot is divided, respectively into \(N_{msC}\) and \(N_{msT}\) mini-slots. Therefore, the duration of the Combi-Frame is,  
\[ \begin{split}& T_{CF}=T_{CONT}+T_{TDMA}=\\ &=N_{Intra}\cdot T_{Intra}+N_{Inter}\cdot T_{Inter}=N_{Intra}(N_{ msC}\cdot T_{msC})+N_{Inter}(N_{msT}\cdot T_{msT})=N_{msC}\cdot T_{ms}.\end{split}  \tag{1}\]  
In Equation (1) the duration of both sub-frames are identified as, \(T_{CONT}=N_{intra}\cdot T_{Intra}\) and \(T_{TDMA}=N_{inter}\cdot T_{Inter}\) respectively. Also, \(T_{Intra}=N_{msC}\cdot T_{msC}\), \(T_{Inter}=N_{msT}\cdot T_{msT}\) and \(N_{msC}=N_{Intra}\cdot N_{msC}+N_{Inter}\cdot N_{msT}\).