One of the most important problems in communication network design is the stability of network after any disruption of stations or links. Since a network can be modeled by a graph, this concept is examined under the view of vulnerability of graphs. There are many vulnerability measures that were defined in this sense. In recent years, measures have been defined over some vertices or edges having specific properties. These measures can be considered to be a second type of measures. Here we define a new measure of the second type called the total accessibility. This measure is based on accessible sets of a graph. In our study we give the total accessibility number of well known graph models such as Pn, Cn, Km,n, W1,n, K1,n. We also examine this new measure under operations on graphs. A simple algorithm, which calculates the total accessibility number of graphs, is given. We observe that when any two graphs of the same size are compared in stability, it is inferred that the graph of higher total accessibility number is more stable than the other one. All the graphs considered in this paper are undirected, loopless and connected.