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This paper is concerned with positive α-times resolvent families on an ordered Banach space E (with normal and generating cone), where 0<α≤2. We show that a closed and densely defined operator A on E generates a positive exponentially bounded α-times resolvent family for some 0<α<1 if and only if, for some ω∈ℝ, when λ>ω,λ∈ρA, Rλ,A≥0 and supλRλ,A: λ≥ω<∞. Moreover, we obtain that when 0<α<1, a positive exponentially bounded α-times resolvent family is always analytic. While A generates a positive α-times resolvent family for some 1<α≤2 if and only if the operator λα−1λα−A−1 is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.