The scattered Optical Network Units (ONUs) in an optical access network have different propagation delays. In an Ethernet passive optical network (EPON) adopting the traditional Interleaved Polling with Adaptive Cycle Time (IPACT) scheme, such ONU propagation delay difference will waste network resources and will degrade both network delay performance and network energy efficiency in delivering low-latency services. In this paper, to deal with the heterogeneous ONU propagation delays problem, two solutions are proposed and investigated. The first solution, namely the Upstream Postponing with ONU Dozing (UP-OD) scheme, is to properly postpone the upstream transmissions of those ONUs having relatively short propagation delays to improve channel utilization efficiency, and ONU doze mode is incorporated to enhance network energy efficiency. The second solution, namely the Identical Fiber Length with ONU Sleeping (IFL-OS) scheme, is to adopt an identical distribution fiber length for ONUs to enhance channel utilization, and ONU sleep mode is incorporated for energy consumption reduction. Simulation results show that both the UP-OD scheme and the IFL-OS scheme reduce network delay and improve network energy efficiency in delivering low-latency (<1 ms) data, and the IFL-OS scheme shows lower energy consumption in transmitting per bit of low-latency (<1 ms) data compared with the UP-OD scheme. Further practical value discussion shows that for the case of serving services requiring 1 ms delay, the UP-OD scheme is suitable for applying in the 1G-EPONs, whereas the IFL-OS scheme is considerable for the 10G-EPONs.