This paper discusses the important aspects of the reliability of systems with an imprecise general model of the structure function. It is assumed that the information about reliability behavior of components is restricted by the mean levels of component performance. In this case the classical reliability theory cannot provide a way for analyzing the reliability of systems. The theory of imprecise probabilities may be a basis in developing a general reliability theory which allows us to solve such problems. The basic tool for computing new reliability measures is the natural extension which can be regarded as a linear optimization problem. However, the linear programming computations will become impracticable when the number of components in the system is large. Therefore, the main aim of this paper is to obtain explicit expressions for computing the system reliability measures. We analyze the reliability of general structures and typical systems. The numerical examples illustrate the usefulness of the presented approach to reliability analyzing.