Derivations of the Expected Energy Costs of Search and Replication in Wireless Sensor Networks

We develop closed-form expressions of the expected minimum search energy cost and replication energy cost for both unstructured sensor networks (which use blind sequential search for querying) and structured sensor networks (which use efficient hash-based querying). We use both the square grid and random topology to derive each cost modeling. We find that the search cost of unstructured networks is proportional to the number of nodes N and inversely proportional to (r + 1) (where r denotes the number of copies of the target event). The search cost of structured networks is proportional to √ N/ √ r while the replication cost of both structured and unstructured networks is proportional to √ N(r − 1). Furthermore, the proportionality of those costs is independent of whether the topology is grid or random, which implies that the two topologies have common structural characteristics in terms of search and replication costs.