## 4 Reliability Model and Evaluation
Please note that in order to simplify the evaluation and comparison we have assumed that the probability of any packet traversing through wireless media (no matter from sensor node to sensor node or sensor node to gateway node) is P\({}_{1}\) and that of the wired media is P\({}_{2}\) we have not considered some factors like sleep state, transmission power, queue length, buffer overflow etc. We know that in our particular example the probability of successful transmission of packets on wireless domain, where they also have more number of hops, would be much less than that of the hotline-assisted gateway-to-gateway transmission. So we now have the relation: \(P_{{}_{1}}\ll P_{{}_{2}}\). The above equation can be written as: \(P_{{}_{1}}=\beta P_{{}_{2}}\), where \(0<\beta<1\). The probability of successful transmission in hotline assisted scenario is:  
\[(P_{{}_{1}})^{{}_{\sf Z\alpha}}(P_{{}_{2}})^{{}_{h}}(P_{{}_{1}})^{{}_{\sf Z \alpha}}=(P_{{}_{1}})^{{}_{\sf Z\alpha}}(P_{{}_{2}})^{{}_{h}}, \tag{2}\]  
Figure 8: (a) Probability of reliable packet transmission without gateway nodes. (b) Probability of reliable packet transmission with gateway nodes. where \(h\) is the number of wired hops between source and destination gateway nodes. The minimum value of \(h\) is 1. In case of traditional all wireless routing, the distance covered by the wired path in a hotline-assisted routing scenario will be replaced by a wireless path. We already know that the average distance between two neighboring gateway nodes is \(\sfrac{1}{\!\!\!\!\!/}_{\!\!\!\!\!\!/\lambda_{{}_{2}}}\) as shown in the Figure 4. From equation (3) we know that \(\sfrac{1}{\!\!\!\!\!/}_{\!\!\!\!\!\!/\alpha}\) represent the number of wireless hops in the given scenario. To cover the whole distance of \(h\) wired hops, we need the following number of wireless hops:  
\[\text{Average Number of Replaced Hops}=\,h\times\sfrac{1}{\!\!\!\!\!/}_{\!\!\!\!\!/ \alpha}\quad.\]  
In order to get probability for all wireless successful packet transmission we would get the following equation:  
\[\begin{array}{c}\left(P_{{}_{1}}\right)^{\!\!\!\!\!\!/}_{\!\!\!\!\!/\!\! \  
9(a) due to the fact that reliability in our proposed approach is directly proportional to the probability of successful packet transmission, provided value of \(h\) is constant. With very low value of \(\alpha\) the value of \(\mathbf{R}_{\iota}\) is observed to be very low and is much lesser than that of \(\mathbf{R}_{\mathbf{h}}\), therefore traditional routing provides much less reliability as compared to the proposed hotline assisted gateway routing. In the Figure 9(c) we have increased the number of wired hops (_i.e.,_\(h\)) from 1 to 3. The increase in wired hops has triggered a dramatic decrease in the overall reliability of traditional all wireless routing (\(\mathbf{R}_{\iota}\)). This is a direct consequence of the fact that in the traditional (all wireless) scenario many new wireless hops are needed to cover the distance that would be covered by the hotlines in our proposed topology. These new wireless hops are inherently unreliable and result in significantly increased packet losses. A slight decrease in the reliability of hotline assisted routing is also observed in Figure 9(c). This reduction in reliability is, nevertheless, negligible in comparison with the reliability penalties incurred by the all-wireless traditional routing scheme. Figure 9: Comparison of hotline based reliability with traditional wireless reliability.