In this paper we report novel experimental results of the erbium-doped-fiber ring resonator behaviors operating near threshold. The reaction of threshold on linewidth of ring's output light is pointed out. This provides a new auxiliary method to determine the threshold. The limits of finesse are analyzed. We also propose a new subsidiary method for finesse measurement based on self-homodyne by using of resonating of the intra-cavity amplified spontaneous emission (ASE) light, without external light source.