## 5 Simulations  
In this section, we present the simulation environment and examine the results of the simulation study conducted on the four categories of data gathering trees with respect to five performance metrics (see Section 5.1 for the list of metrics), including the aggregation delay-the primary metric of interest in this paper. We evaluate results of statistical tests conducted on the values obtained for the performance metrics and arrive at significant conclusions. We explore the contributions of the other four performance metrics for minimizing the aggregation delay as well as analyze the performance tradeoffs. We also calculate the margin of error values for the minimum aggregation delay (confidence interval: 95%). Finally, we discuss the impact of the algorithm design choices on the structure of the two categories of data gathering trees (bottleneck node-weight based and bottleneck link-weight based trees) and their performance.  
We conduct simulations in a discrete-event simulator implemented in Java. We implemented the algorithms to determine the four types of data gathering trees (MaxBNW-DG, MinBNW-DG, MaxBLW-DG and MinBLW-DG trees) described in Section 3 and the algorithms to determine the diameter and minimum aggregation delay for data gathering trees described in Section 4. We conducted the simulations with networks of 50 nodes and 100 nodes, with the transmission ranges of the nodes varied from 25 m to 50 m, in increments of 5 m. The network dimensions are  
Figure 1: Examples to Determine the Minimum Aggregation Delay for a Data Gathering Tree.  
100 m \(\times\) 100 m. These are the usual parameters considered for simulation in sensor networks (e.g., refer [1, 3, 5, 6, 8]). The average number of neighbors per node is given by \(\pi R^{2}\)/\(A\), where \(R\) is the transmission range and \(A\) is the network area. We consider the 50-node network to be of moderate to high node density (20 to 40 neighbors per node on average) and the 100-node network to be of high to very high node density (40 to 80 neighbors per node on average).  
We assume an ideal medium access control (MAC) channel protocol [28] for the underlying wireless channel (_i.e._, there are no collisions during channel access). We also assume the number of unique CDMA codes to be sufficient enough for the intermediate nodes at the same level or different levels to simultaneously aggregate data from their respective child nodes, if the latter are ready with the data. Such idealistic assumptions have been successfully applied in the literature for wireless ad hoc networks and sensor networks (e.g., [29, 30, 31, 32, 33]) to extract optimal performance from the benchmarking algorithms run in the higher layers and to delineate the effects of the design choices.  
Figure 2: Performance of the Max/Min Bottleneck Node Weight and Link Weight-based DG Trees.  
The weights for the edges are uniform-randomly assigned in the range [0...1] and the weights for the nodes are also uniform-randomly assigned in the range [0...1]; the node weights are independent of the link weights. We generate 200 snapshots of the network graphs for each combination of the number of nodes and the transmission range per node and run the algorithms to determine the four DG trees on each of these graph snapshots and average the results (shown in Figure 2). Section 5.4 discusses about the margin of error values for the minimum aggregation delay (confidence interval is 95%).