Optical burst switching (OBS) paradigm has been proposed to achieve the packet switching level throughput in the all optical domains. To meet the recent OBS commercialization requirement, we study the edge/core node combined (ECNC) OBS networks where the core node performs the edge node function as well. In this type of networks, the fairness problem occurs because the remaining offset-time decides the priority under the JET-based OBS. To solve this problem, we propose a fairness guaranteeing scheme by grouping bursts based on the destination and the remaining offset-time and assigning them to each wavelength group. Also, we propose a new burst generating and scheduling scheme for the ingress node and analyze the networks throughput.