The distributed program reliability (DPR) is a useful measure for reliability evaluation of distributed systems. In previous methods, a two-mode failure model (working or failed) is assumed for each computing node. However, this assumption is not realistic because data transfer may be possible by way of a computing node even when this node can neither execute programs nor handle its data files. In this paper, we define a new three-mode failure model for representing such a degraded operational state of computing nodes, and present a simple and efficient analysis method based on graph theory. In order to represent the degraded operational state, a given graph expressing a distributed system is augmented by adding new edges and vertices. By traversing this augmented graph, the reliability measure can be computed. Examples show the clear difference between the results of our proposed method and those of the previous ones.