At first, the adsorbent (usually activated carbon) is fresh with all its adsorption sites. Essentially none of the material to be removed escapes from the column. As time passes, some of the adsorption sites are used up, and concentration in the effluent rises.
The shape of the graph may vary considerably for different situations. Usually there is a long time before the effluent concentration rises sharply and then levels off. If all the sites were occupied, we would expect the inlet concentration and the outlet concentrations to become the same.
Note the lines in blue. The breakthrough concentration is determined by the process specifications. This is the allowable concentration. If a pollutant is being removed, the breakthrough concentration might be the government regulation for what your plant can discharge. For a commercial product where the column is removing color, the breaktrhough concentration is determined by your specification for product quality. The point is that breakthrough concentration is not some fundamental number but depends on how you decide to operate your process.
The exhaustion concentration also depends on process considerations. If there are not enough workers on the night shift, the column may have to be removed in time for regeneration by the day shift so that it can ready in time the next day to become the t rail column. Note that the column could still adsorb some more material. However, you get little benefit by running for a longer time. You pay for labor, for electricity, for plant costs, etc. When the benefits are not worth the costs, the column is considered exhausted.