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The placement of mesh routers (MRs) in building a wireless mesh network (WMN) is the first step to ensure the desired network performance. Given a network domain, the fundamental issue in placing MRs is to find the minimal configuration of MRs so as to satisfy the network coverage, connectivity, and Internet traffic demand. In this paper, the problem is addressed under a constraint network model in which the traffic demand is non-uniformly distributed and the candidate positions for MRs are pre-decided. After formulating the MR placement problem, we first provide the theoretical analysis to validate the traffic demand and determine the optimal position of Internet gateway (IGW). To reduce complexity of determining the locations of MRs while satisfying the traffic constraint, we propose an effective heuristic algorithm to obtain an close-to-optimal solution. Finally, our simulation results verify our analytical model and show the effectiveness of our proposed algorithm.