## 3 A Quality Aware Routing Metric for WOBAN
### Probability of Success  
We now evaluate the probability of successful reception of a transmitted packet on a test link. When the RTS/CTS option is enabled, a data packet is only transmitted when the RTS/CTS exchange is successful, i.e., the channel is found to be clear both at the transmitting and receiving nodes. However, the transmitted data packet can still be lost due to interference caused to the data packet or the ACK packet. Here, we analyze the possible events that can cause these transmission failures, which are explained with the help of Figure 2. In the figure, \(S\)\(\rightarrow\)\(D\) represents the test link, \(\Delta_{RTS}\) and \(\Delta_{CTS}\) denote the regions where the RTS and CTS packets for the test link can be received, and \(\Delta_{CS}\) denotes the area around \(S\) where nodes can sense the transmission from \(S\). For any node \(i\), \(I(i)\) denotes the area from where a transmission from any node \(j\in I(i)\) can interfere with a packet being received at \(i\). We assume that the nodes generate data packets based on a Poisson process with an arrival rate if \(\lambda\). The data packet length and the channel bandwidth are assumed to be \(DLEN\) bits and \(B\) bits/s, respectively. Our approach is to explore various cases where events can lead to unsuccessful data transmission. For each of these cases, we evaluate the factors that affect the POS, as outlined below:  
**Case-1:** The transmitted data packet is unsuccessful due to interference from nodes that are within the interference range of the \(D\) but outside its transmissions range. This is due to the fact that these nodes will not receive the CTS sent by \(D\). In Figure 2, we mark these nodes as \(PC_{i}\), of which we assume \(p\) nodes are sending and \(r\) nodes are receiving. As derived in [37], the probability of success of the DATA packet in the presence of these nodes are \(e^{\frac{-\lambda\times DLEN\times p}{B}}\) and \(e^{\frac{-\lambda\times DLEN\times r}{B}}\), respectively. **Case-2:** The test data packet can be unsuccessful due to interference from nodes that are within the transmission range of \(D\) but fail to receive the CTS packet. A node located in the \(\Delta_{CTS}\) that is outside the \(\Delta_{CS}\) of \(S\) (marked as \(NC_{i}\) in Figure 2) may not receive the CTS from \(D\) correctly due to an overlapping transmission from \(MC_{j}^{i}\) (refer to Figure 2). In general, if there are \(m\) such interferers among \(MC_{i}^{j}\), then the probability that the CTS is not received by \(NC_{i}\) is \(1-\prod_{i=1}^{m}e^{\frac{-\lambda\times DLEN}{B}}=1-e^{\frac{-\lambda\times DLEN \times m}{B}}\). The probability that \(NC_{i}\), having failed to receive the CTS from \(D\), interferes with the reception of the DATA packet at \(D\) is then given by \((1-e^{\frac{-\lambda\times DLEN}{B}})\), which is the probability that an RTS transmission from \(NC_{i}\) overlaps with the test DATA packet at \(D\). If there are \(n\) such nodes (\(NC_{1},NC_{2},\cdots,NC_{n}\)), then the DATA transmission will be successful with a probability of \(\prod_{i=1}^{n}1-(1-e^{\frac{-\lambda\times DLEN}{B}})(1-e^{\frac{-\lambda \times DLEN\times m}{B}})\). By taking into account all the factors described above, the probability of success of a transmitted data packet using the RTS/CTS handshake is given by  
\[P_{\mathrm{S}} = \left\{\prod_{i=1}^{n}\!1-\left(1-e^{\frac{-\lambda\times DIEN}{B} }\right)\left(1-e^{\frac{-\lambda\times DIEN\times m}{B}}\right)\right\}\times e ^{\frac{-\lambda\times DIEN\times q}{B}}\times e^{\frac{-\lambda\times DIEN \times q}{B}}. \tag{2}\]  
It must be noted that some additional factors affect the POS and hence the accuracy of Equation (2). Firstly, Equation (2) is based on the assumption that all interferers transmit independently, which is not exactly true as there are dependencies among these interferers. Secondly, in Equation (2), we assume that the arrival rate of all the interferers is identical.