## 4 Reliability Comparison and Evaluation  
Sensor networks for monitoring underwater pipelines are usually linear. Sensors can forward their sensed information either through their left side, their right side or both. In this section, we develop an analytical reliability model to analyze and compare the connectivity reliability of the different architectures discussed in this paper. Let us assume that the sensors are uniformly distributed along the pipeline. In addition, they are generating messages at the same rate. Let us assume that there are \(n\) sensors and \(n+l\) links in the network. The sensors are \(S_{l}\), \(S_{2}\), \(S_{3}\),..., and \(S_{n}\) from the left to the right of the network. In our model, we will assume that a sensor can send its sensed information to the NCC if the link from the left side of the sensor is healthy or if the link from the right side of the sensor is healthy. In other words, the availability of either connection will allow the sensor to send its information. This means that the NCC is linked to both sides of the network. We will develop a general reliability model and then we will customize that model to the different types of sensor network architectures. To start developing the model we will consider one sensor on the network. A sensor cannot send its information to the NCC if there is no connectivity from that sensor to the NCC from the left and right sides of the sensor. If we take one side of the network, the sensed information cannot be transferred if there is a broken link which cannot be used to forward the information to next link towards the NCC. Let \(L_{i}\) be the probability that the sensor S\({}_{i}\) cannot forward its sensed information to the NCC from the left side and \(R_{i}\) be the probability that the sensor \(S_{i}\) cannot forward its sensed information to the NCC from the right side. We have:  
Figure 7: Integrated wired/RF wireless sensor network. \[L_{i}=(1-w^{j}) \tag{1}\]  
\[R_{i}=(1-w^{n-i}) \tag{2}\]  
where \(w\) is the probability that a link or a hop can forward the information to the next link or hop. This is from the fact that a senor can send its sensed information to the NCC if and only if all links from this sensor to the NCC are operating; otherwise, when there is one or more faulty links in that direction, the sensed information will not be able to reach the NCC. Now, let \(Q_{i}\) be the probability that sensor \(S_{i}\) cannot send its information from both sides. We have:  
\[Q_{i}=L_{i}\cdot R_{i} \tag{3}\]  
\[Q_{i}=(1-w^{j})\cdot(1-w^{n-i}) \tag{4}\]  
Let \(P_{i}\) be the probability that sensor \(S_{i}\) can send its sensed information to the NCC. We have:  
\[P_{i}=1-Q_{i} \tag{5}\]  
\[P_{i}=1-(1-w^{j})\cdot(1-w^{n-i}) \tag{6}\]  
Now, the expectation for all sensors on the network to be able to send their sensed information to the NCC is:  
\[E_{n}=\frac{\sum_{l-1}^{n}P_{i}}{n} \tag{7}\]  
\[E_{n}=\frac{\sum_{l-1}^{n}(1-(1-w^{i})(1-w^{n-1}))}{n} \tag{8}\]  
The value of \(w\) can be calculated in different ways in the mentioned architectures. For wired sensor networks, a long network can be divided into multiple segments. A segment has a probability \(d\) of being healthy. In this network, we have  
\[w=d \tag{9}\]  
For the wireless sensor network:  
\[L_{i}=(1-w^{j}) \tag{1}\]0  
where \(x\) is the probability that a node cannot forward a message to a next hop and \(j\) is the maximum jump factor that a node can reach to other nodes in one direction. For example, if each sensor can jump over 3 sensors from each side, then \(j=3\). Here we have \(x^{j}\) is the probability that a sensor cannot forward its messages in one direction; while \(1-x^{i}\) is the probability that a sensor can forward its messages. This is calculated from the fact that a sensor will not be able to forward its messages in one direction if and only if all next \(j\) sensors are faulty.