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We present a computationally efficient algorithm that minimizes the transmission energy and latency associated with transmitting a file across a Gilbert Elliott fading channel. We formulate the optimal tradeoff between transmission energy and latency as a partially observed Markov decision process problem (POMDP). The channel state is not directly observed and hence transmission decisions must be based on ACK/NAK information provided over a feedback channel. The key idea is to reformulate the resulting POMDP as a Markovian search problem, with optimal transmission control policies that are threshold in nature. Threshold policies are computationally inexpensive to implement. Our analysis shows that for different parameter values of the Gilbert Elliott fading channel, the optimal transmission policy, while threshold in structure, exhibits vastly different behaviour - from persistent retransmission to back-off and wait. Numerical examples demonstrate the performance improvements that can be obtained using the optimal threshold policies as compared to existing heuristic algorithms.