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WirelessHART is an open wireless sensor-actuator network standard for industrial process monitoring and control that requires real-time data communication between sensor and actuator devices. Salient features of a WirelessHART network include a centralized network management architecture, multi-channel TDMA transmission, redundant routes, and avoidance of spatial reuse of channels for enhanced reliability and real-time performance. This paper makes several key contributions to real-time transmission scheduling in WirelessHART networks: (1) formulation of the end-to-end real-time transmission scheduling problem based on the characteristics of WirelessHART, (2) proof of NP-hardness of the problem, (3) an optimal branch-and-bound scheduling algorithm based on a necessary condition for schedulability, and (4) an efficient and practical heuristic-based scheduling algorithm called Conflict-aware Least Laxity First (C-LLF). Extensive simulations based on both random topologies and real network topologies of a physical testbed demonstrate that C-LLF is highly effective in meeting end-to-end deadlines in WirelessHART networks, and significantly outperforms common real-time scheduling policies.