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The multi-source and single-sink (MSSS) topology of wireless sensor networks (WSNs) is defined as the network topology, where all of nodes can gather, receive and transmit data to the sink. We study the issue of finding the joint optimal scheme with consideration of physical, medium access control (MAC), and network layers to maximize the network lifetime (NL) for such topology in the energy-constrained WSNs. This paper presents a duality approach by utilizing the Karush-Kuhn- Tucker (KKT) optimality conditions to analytically provide a solution to this optimization problem. Analytical expression of the globally optimal NL is derived as a function of the amount of nodes, the initial energy and the data rate arrived at the nodes, taking the influence of data rates, link access and routing method into account. The analytical result agrees well with numerical result of the optimization solver and can be applied to algorithms in WSNs aiming at maximizing NL.