A Note on cluster size of WSN

A large portion of energy-aware routing protocol for wireless sensor networks are cluster-based. The problem without clustering was that, battery of sensor nodes near the sink drained fast, which in turn induced successive layers of nodes from sink, to exhaust their battery quickly. In cluster based approaches too, power of nodes nearer to the cluster head (CH) are drained more rapidly compared to those at the periphery, as they are more often used as hopping nodes for nodes lying further from the CH. This is more so when the cluster is big. Uniform dissipation is achieved by reconfiguring the clusters at intervals. But, reconfiguring to a new set of clusters is a big overhead. In this work, we first emphasize that there is an optimum size of a cluster, for which the power dissipation at every node could be made uniform, by transmitting packets at different energy levels. If nodes in a cluster follow that protocol, we can do away with reconfiguring clusters, or at least do it at much longer intervals. I first define the goal formally as a constrained optimization problem, for linear array of sensor nodes. It turns out to be a non-linear optimization problem, which is translated to a linear optimization problem and solved. I show that the problem has a solution only when the cluster diameter is 6 (in terms of hop count) or less. Cluster of bigger size has no solution. We also formulate the problem, when nodes are uniformly distributed over a plane.