## 4 Performance Analysis  
### Definition of the Error Model  
Generally, the route of DD is established along the shortest path between source and sink. As the time passes, the energy of a node can be drained very quickly on the shortest path. The rapid dissipation of a node's energy brings node failure resulting in lossy situation. Consequently, a routing algorithm extends the shortest path including adjacent neighbor nodes instead of failed nodes to keep the connectivity of WSNs. We defined this extended path including the shortest path as a Main Communication Path (MCP). The MCP is an extended path constructed by a group of nodes enclosing the shortest path and adjacent neighbors of the path.  
DD is designed to transmit most data over the MCP consisting of the shortest distances between nodes. As referred from [12,23], the behavior of the data transmission through the MCP is the main communication characteristics of DD. Although we can select an error node irrespective of the MCP, if we select error nodes with this method, the selected node does not participate in the transmission as a forwarding node. For this reason, the selection of the error nodes located on the MCP can emulate lossy situation better than that of uniform random selection regardless of MCP to clarify the problems of lossy WSNs.  
For example, in the case of one source and one sink at each corner (left-bottom and right-top), the shortest path extended to adjacent neighbors forming a diagonal shape.  
Figure 4 shows MCP used for each topology in the simulation. In Figure 4, a gray area indicates MCP and red triangle represents error-prone node with a given error rate (i.e., 10%) for each topology. The error group stands for the potential error nodes in the MCP. The error rate indicates the number of selected nodes in all nodes of topology, but the error-prone node should be randomly selected from MCP. The error rate represents the instability of lossy WSNs at a given error rates (%).  
For example, the error group on the MCP is composed of 56 nodes for the grid topology and 112 nodes for the random topology. If the error rate is 5%, five nodes are selected as error-prone nodes from the nodes located on MCP at every selection. That is, in each selection, error nodes will be randomly selected from the sensor group on the MCP according to given error rate.  
To model lossy WSNs, we make the randomly selected nodes to be blinked for a random time. Blinking selected nodes emulate the error-prone nodes to generate lossy situation as a random failure. The random failure of the selected error nodes is to avoid influence on interest propagation period for setting up gradient in DD. The interest propagation period is important to recover the lossy situation in DD. However, the lossy situation may occur in lossy WSNs independent with the interest propagation period as random failure. The author of DD mentioned the interest propagation period is a protocol design factor to trade-off between overhead and robustness.  
The interest propagation period on time of original DD is a sum of fixed interval (30 seconds) and random time with large variance (range: a few \(\sim\) hundreds seconds). However, this randomness might affect the number of resulted packets and the small interest interval of DD made a lot of interests flooding which might be a burden to analyze the effect of lossy WSNs.  
Since the main purpose of lossy WSNs model is to generate unpredictable errors independent with the interest propagation period, we eliminates the randomness of DD to avoid difficulties in analyzing the result. Instead, in order to make the unexpected lossy situation, the fixed interval (60 seconds) is used for interest propagation and randomly selected nodes blinked for random time on MCP during the simulation.