In the present paper new light is shed on the non-central extensions of the Dirichlet distribution. Due to several probabilistic and inferential properties and to the easiness of parameter interpretation, the Dirichlet distribution proves the most well-known and widespread model on the unitary simplex. However, despite its many good features, such distribution is inadequate for modeling the data portions next to the vertices of the support due to the strictness of the limiting values of its joint density. To replace this gap, a new class of distributions, called Conditional Non-central Dirichlet, is presented herein. This new model stands out for being a more easily tractable version of the existing Non-central Dirichlet distribution which maintains the ability of this latter to capture the tails of the data by allowing its own density to have arbitrary positive and finite limits.