In this paper, we extend previous works on task assignment problems for a set of linear-pipeline tasks over a set of processors. Different from previous works, we revisit the problems with two additional system factors: deadline and energy-consumption, which are key factors in real-time and power-aware computation. For a set of linear-pipelined real-time tasks (each with same deadline) and a set of processors (or hosts), we discuss how to develop an energy-efficient resource-allocation scheme to assign processors (or hosts) to handle the tasks and set the processor speed properly so that all tasks can be completed by a given deadline and with minimal energy consumption. We propose an O(m²logm) time complexity algorithm to determine optimal task-assignment and speed-setting schemes leading to minimal energy consumption, for a given m real-time tasks running on n identical processors (with or without DVS supports). Also, an O(min (n²m², nm²logm)) time complexity algorithm is proposed for systems with an array of n heterogeneous processors in fixed order.