The single-source and single-sink (SSSS) topology, in wireless sensor networks, is defined as the network topology, where every node behaves as a relaying node except the single source and the single sink. We consider the problem of finding a joint optimal scheme with consideration of physical, medium access control (MAC), and network layers to maximize the network lifetime (NL) for the SSSS topology in energy-constrained wireless sensor networks (WSNs). The optimization problem, when the communication link are restricted to an interference-free time division multiple access (TDMA) scheme, can be formulated as a mixed integer-convex programming and relaxed to the convex programming. In particular, this paper focuses on utilizing the Karush-Kuhn-Tucker (KKT) optimization conditions to analytically solve this convex problem for the SSSS linear topology. An analytical expression of upper bound on NL for cross-layer optimization is derived as a function of the initial energy and quantities of data held by the sources, taking the influence of data rates, link access method and routing into account. The result agrees well with numerical result of optimization solver, and can be applied to algorithms in WSNs aiming at maximizing NL.
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