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|a In wireless sensor networks (WSNs), the network lifetime (NL) is a crucial metric since the sensor nodes usually rely on limited energy supply. In this paper, we consider the joint optimal design of the physical, medium access control (MAC), and network layers to maximize the NL of the energy-constrained WSN. The problem of NL maximization can be formulated as a nonlinear optimization problem encompassing the routing flow, link scheduling, transmission rate, and power allocation operations for all active time slots (TSs). The resultant nonconvex rate constraint is relaxed by employing an approximation of the signal-to-interference-plus-noise ratio (SINR), which transforms the problem to a convex one. Hence, the resultant dual problem may be solved to obtain the optimal solution to the relaxed problem with a zero duality gap. Therefore, the problem is formulated in its Lagrangian form, and the Karush-Kuhn-Tucker (KKT) optimality conditions are employed for deriving analytical expressions of the globally optimal transmission rate and power allocation variables for the network topology considered. The nonlinear Gauss-Seidel algorithm is adopted for iteratively updating the rate and power allocation variables using these expressions until convergence is attained. Furthermore, the gradient method is applied for updating the dual variables in each iteration. Using this approach, the maximum NL, the energy dissipation per node, the average transmission power per link, and the lifetime of all nodes in the network are evaluated for a given source rate and fixed link schedule under different channel conditions.
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