Waldschmidt constants have been studied in different fields of mathematics, e.g., complex analysis, algebraic geometry, number theory and commutative algebra. After Nagata's work for the 14th Hilbert problem, these constants received great attention. In particular, in algebraic geometry, they recently have been rediscoverd by Bocci and Harbourne in the set-up of the containment relations between symbolic and ordinary powers of homogeneous ideals. In this paper, we study the Waldschmidt constant of a generalized fat point subscheme on the projective plane, which consists of essentially distinct points. Furthermore, we study various properties of the Waldschmidt constant of a generalized fat point subscheme which are related to complete ideal sheaves. Using these properties, we prove the lower semi-continuity of the Waldschmidt constants of generalized fat point subschemes which consists of less than or equal to 8 points. As an application, we also calculate the Waldschmidt constants of the generalized fat point subschemes which give rise to weak del Pezzo surfaces of degree 4.