## 4 Reliability Model and Evaluation  
In this section, we model and compare the reliability of an end-to-end WSN communication under traditional and hotline-assisted routing. Our network has two different homogenous Poisson distributions. The normal sensor nodes are deployed according to a dense homogeneous Poisson distribution and gateway nodes are deployed according to a sparse homogenous Poisson distribution. Throughout this section, we consider a one-dimensional deployment scenario for simplicity. We assume that there are \(\lambda_{{}_{1}}\) arrivals of sensor nodes per unit area. In other words there are, on the average, \(\lambda_{{}_{1}}\)occurrences of sensor nodes per unit area. As shown in Figure 6, we can conclude that the average distance between two neighboring nodes in a unit distance will be \(\lambda_{{}_{1}}\). Similarly on the average there are \(\lambda_{{}_{2}}\) occurrences of sensor gateway nodes per unit distance. The average distance between two neighboring sensor gateway node is therefore \(\lambda_{{}_{2}}\). The gateway nodes are connected to each other via a backbone wired link and the occurrences of gateway nodes per unit distance are significantly lesser than that of sensor nodes, leading to the following relation: \(\lambda_{{}_{2}}<<\lambda_{{}_{1}}\). Alternatively, we can write the above equation as:  
\[\lambda_{{}_{2}}=\alpha\lambda_{{}_{1}} \tag{1}\]  
where \(0<\alpha<1\) is the ratio of the gateway node occurrence with respect to the number of sensor nodes per unit distance. The value of \(\alpha\) would vary depending upon specific scenario or the density of sensor nodes and sensor gateway nodes in a given network. If \(\alpha\) is closer to 1 then the network is expected to be a well managed network with a high number of gateway nodes. We want to compute the reliability and resilience of our proposed topology against the traditional WSN routing mechanism. We take the worst case scenario where we consider a border node that lies exactly at the center of two gateway nodes as shown in Figure 7. As we already know that the average distance between two gateway nodes is \(\sfrac{1}{\lambda_{2}}\). The node that lies right in the middle of two gateway nodes would have \(\sfrac{1}{2}\lambda_{2}\) distance to either of the gateway nodes. But we know that the average distance between two neighboring sensor nodes is \(\sfrac{1}{\lambda_{1}}\). Using (1), we obtain: \(\sfrac{1}{2}\lambda_{2}=\sfrac{1}{2}\alpha\sfrac{1}{\lambda_{1}}\). In the above equation \(\sfrac{1}{2}\alpha\) is the factor that provides us required number of \(\sfrac{1}{\lambda_{1}}\)'s to cover the whole distance of \(\sfrac{1}{2}\lambda_{2}\). We are now interested in quantifying the reliability of the traditional and hotline-assisted scenarios. The reliability of communication between a sender and a receiver is the end-to-end probability of  
Figure 6: Network model showing distribution scenario of sensor nodes and sensor gateway nodes. Figure 7: Worst case scenario with a sensor node lying exactly in the middle of two gateway nodes. successful transmission. As shown in Figure 8(a), there is no gateway available so the reliability would depend on the number of hops, \(n\), between the sender and the destination. Each of the hops would have the same probability of successful packet transmission; we call this probability \(P_{{}_{1}}\). The reliability without gateway nodes is then: \((P_{{}_{1}})^{{}_{n}}\)  
In order to find out the reliability of the scenario when gateway nodes are assisting the routing, the overall reliability would be dependent on three different phases. One is the probability of packet reaching from source to the associated source gateway, say \(P_{{}_{1}}\), second is the probability of packet reaching from the source gateway to the destination gateway, say \(P_{{}_{2}}\), and lastly the probability of packet reaching from the associated destination gateway to the destination, which in worst-case is \(P_{{}_{1}}\). This scenario is shown in the Figure 8(b).