## 5 Performance Analysis
### Network Overheads and Computational Complexity  
One of the consequences of the \(\Delta T\)_Vector_ extension is a larger RREP packet as it must contain the individual \(\Delta T_{i}\) values of all intermediate nodes of a route, while the original TTHCA mechanism only requires the sum \(\Delta T_{TOT}\). The size of the \(\Delta T\)_Vector_ is dependent on the route HC, so if for example each \(\Delta T_{i}\) value is represented by 32 bits, then on a route from a source node S to a destination node D with intermediate nodes I\({}_{1}\) and I\({}_{2}\), RREP will comprise a \(\Delta T\)_Vector_ length of 32 bits, 64 bits and 96 bits when respectively received by I\({}_{2}\), I\({}_{1}\) and S. This contrasts with the corresponding RREP packet in the TTHCA algorithm which will have a 32 bits \(\Delta T_{TOT}\) value for each node. While a \(\Delta T\)_Vector_ with more than one element theoretically increases the transmission and reception time requirements for the routing packet, when cognisance is taken of the high bandwidths available in modern wireless technologies, the extended RREP packets will have negligible impact upon performance.  
A second ramification of the \(\Delta T\)_Vector_ extension is the increased FP detection rate. From the network performance perspective, this means that the shortest route in terms of HC is not always available, as highlighted in both Sections 5.1 and 5.2. This does not necessarily imply decreased performance in terms of route delay since FP detection can in many cases lead to a positive outcome as routes with intermediate nodes with very high traffic loads will be omitted.  
A formal complexity analysis for the new \(\Delta T\)_Vector_ extension reveals the only supplementary cost incurred compared with the original TTHCA algorithm is the outlier detection scheme performed by the source node. If the Dixon Q-test is used as the outlier method, the only extra computations needing to be performed relate to the ranking of \(\Delta T\)_Vector_ values. Since the \(\Delta T\)_Vector_ length equals the route HC, the time complexity for ranking is _O_(HC\({}^{2}\)). This ranking however, can be implemented as a linear search of 4 \(\Delta T\) values, since the Q-test only uses the three largest and the smallest \(\Delta T\) value. This results in a time complexity for the new \(\Delta T\)_Vector_ extension of _O_(HC), which is the same as TTCHA [17]. The corresponding FP performance of \(\Delta T\)_Vector_ extension also needs to be analysed because these are identified even when there are no errors in the measured node processing times. If the probability of a FP is \(p\), then the probability of \(i\) FP occurring before a healthy route is located will be (_1-p_):_pi_. The average number \(<\)\(i\)\(>\) of route discovered before a healthy route can therefore be expressed as _p_/(_1-p_). So for \(p<\)_0.5, on average up to one FP will be discovered before a healthy route is identified for the \(\Delta T\)_Vector_ extension. The worst case in a single wormhole MANET is thus, on average three algorithm executions when a wormhole infected route is found before a healthy route is located. In contrast, the impact of FP on the TTHCA algorithm is less problematic because a FP is only identified when there are time measuring errors [17].  
In summary, this formal analysis has shown the new \(\Delta T\)_Vector_ extension has the same linear time complexity as the original TTHCA algorithm, with the rider that because of FP occurrences, one additional execution cycle of the \(\Delta T\)_Vector_ extension may be necessitated, though this still affords a very effective lightweight protection mechanism against time tampering for TTHCA.