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We consider a heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of surveillance. An aircraft visits the area periodically and gathers data about the activity in the area from the sensor nodes. There are two types of nodes that are distributed over the area using two-dimensional homogeneous Poisson point processes; type 0 nodes with intensity (average number per unit area) {\lambda}_{0} and battery energy E_{0}; and type 1 nodes with intensity {\lambda}_{1} and battery energy E_{1}. Type 0 nodes do the sensing while type 1 nodes act as the cluster heads besides doing the sensing. Nodes use multihopping to communicate with their closest cluster heads. We determine the optimum node intensities ({\lambda}_{0}, {\lambda}_{1}) and node energies (E_{0}, E_{1}) that guarantee a lifetime of at least T units, while ensuring connectivity and coverage of the surveillance area with a high probability. We minimize the overall cost of the network under these constraints. Lifetime is defined as the number of successful data gathering trips (or cycles) that are possible until connectivity and/or coverage are lost. Conditions for a sharp cutoff are also taken into account, i.e., we ensure that almost all the nodes run out of energy at about the same time so that there is very little energy waste due to residual energy. We compare the results for random deployment with those of a grid deployment in which nodes are placed deterministically along grid points. We observe that in both cases \lambda_{1} scales approximately as \sqrt{\lambda_{0}}. Our results can be directly extended to take into account unreliable nodes.
Sensor networks, energy, lifetime, stochastic geometry, Voronoi cells.

C. Rosenberg, V. P. Mhatre, N. Shroff, D. Kofman and R. Mazumdar, "A Minimum Cost Heterogeneous Sensor Network with a Lifetime Constraint," in IEEE Transactions on Mobile Computing, vol. 4, no. , pp. 4-15, 2005.
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