We will formulate and discuss a theorem that under mild assumptions the Hausdorff dimension of limit sets of an analytic family of graph directed Markov systems varies in a real-analytic fashion. To a given tame meromorphic function f a conformal iterated function system will be ascribed by means of nice sets. The Hausdorff dimension of the limit set of every such iterated function system will be shown to be equal to h, the Hausdorff dimension of the radial Julia set of f. The h-dimensional Hausdorff measure of this radial set will be proved to be σ-finite. Based on the result for graph directed function systems, the real analyticity of the Hausdorff dimension of radial Julia sets will be proved for a large family of tame transcendental meromorphic functions.