# Value of Information Based Data Retrieval in UWSNs
## 5 Performance Measures, Simulation Setup and Results  
### Performance Measures and Experiments  
The basic measures of performance that we will use are Vol accumulated \(\textsc{Y}_{Acc}\)' and Vol lost \(\textsc{Y}_{L}\)'. To discuss what these measures are we refer to Figure 2b. \(\textsc{Y}_{Acc}(t)\) is the decaying Vol profile in the system and is represented in abstract terms by a straight line. It is a general statement on the depreciation of the valuation, whereas, in an actual situation the dynamics of this depreciation, i.e., the actual shape of the curve, will be governed by system variables and type of information recorded. This chart assumes that no measurements are recorded after \(t=0\), hence, the chart only shows a monotonic decay in the valuation after \(t=0\). \(\tau_{start}\) and \(\tau_{finish}\) are the start and end times for the complete AUV tour. In our case, a tour is a visitation sequence that is a permutation on the set of all sensor nodes, i.e., the tour is a'simple path' on a graph that has the sensor nodes as vertices. \(Y_{Ava}\) is the VoI available in the system at the start of the tour while the \(Y_{Acc}\) is the VoI accumulated by the AUV by the end of its tour. \(Y_{L}\) is determined as  
\[Y_{L}=Y_{Ava}-Y_{Acc} \tag{23}\]  
\[Y_{L}=\sum_{i=1}^{n}\sum_{j=1}^{k}\left[A_{ij}e^{-B_{ij}(\tau_{Start}-\tau_{ij })}-A_{ij}e^{-B_{ij}(\tau_{ij}-\tau_{ij})}\right] \tag{24}\]  
The loss in \(Y_{Acc}\) is a result of a combination of factors that can be attributed to physical system limitations such as the AUV speed, delay in starting time of the tour \(\tau_{Start}\), and inefficiencies resulting from the planned path. It is the path planning part of this problem that we want to explore in this paper. To filter out effects of a delay in \(\tau_{Start}\) we use \(Y_{L}\) as it is a measure of loss between the range \(t=[\tau_{Start}:\tau_{Finish}]\). Any loss other than \(Y_{L}\) is not a result of path planning inefficiencies. \(Y_{L}\) gives a more clear picture in terms of comparative performance as compared to \(Y_{Acc}\). Other than VoI based performance markers we can also use time as a metric to determine certain aspects of performance. Time taken by an AUV for a complete traversal of the set of sensor nodes can be employed for this purpose. The earlier an AUV completes the tour of a UWSN, the more readily it is available for a new tour of this or another neighboring UWSN. Time can be useful in understanding energy \(E_{AUV}\) consumed by the AUV to complete its tour. Energy consumed is proportional to the distance it travels \(D_{tour}\) which is proportional to the time taken by the AUV to complete the tour \(T_{tour}\). Shorter tours also result in less wear and tear of the AUVs. \[\Delta\tau=\tau_{finish}-\tau_{start}=T_{tour}\propto D_{tour}\propto E_{AUV} \tag{25}\]  
A measure of the efficiency \(\Omega\) of a planner can be determined by combining \(Y_{L}\) and \(T_{tour}\). \(\Omega\) is inversely proportional to both \(Y_{L}\) and \(T_{tour}\), therefore,  
\[\Omega\propto\frac{1}{Y_{L}}\cdot\frac{1}{T_{tour}}\implies\Omega=\frac{p}{qY _{L}\cdot rT_{tour}}=\frac{k}{Y_{L}\cdot T_{tour}} \tag{26}\]  
where, \(k=\frac{p}{q\cdot r}\) is a constant and is set to 1 in this experimental study. Other than the measures of \(Y_{Acc}\), \(Y_{L}\), \(T_{tour}\) & \(\Omega\), we also need to have some specific measures for'response to emergency situations'. Figure 2: Infotentials: (**a**) The figure on the left is an example of exponentially decaying infotentials that we use for modelling and analyzing high-priority and low-priority events; (**b**) The figure on the right shows various parameters and measures that we use for comparative analysis of path planning algorithms. These would be VoI accumulated from first hot-spot \(v_{HS_{Acc}}\), VoI lost from first hot-spot \(v_{HS_{L}}\), time taken to arrive at first hot-spot \(\tau_{HS}\) and a measure of urgency \(\Psi\).