  For an eventually periodic module, we have the degree and the period of its
first periodic syzygy. This paper studies the former under the name \lq\lq
periodic dimension\rq\rq. We give a bound for the periodic dimension of an
eventually periodic module with finite Gorenstein projective dimension. We also
provide a method of computing the Gorenstein projective dimension of an
eventually periodic module under certain conditions. Besides, motivated by
recent results of Dotsenko, G\'elinas and Tamaroff and of the author, we
determine the bimodule periodic dimension of an eventually periodic Gorenstein
algebra. Another aim of this paper is to obtain some of the basic homological
properties of eventually periodic algebras. We show that a lot of homological
conjectures hold for this class of algebras. As an application, we characterize
eventually periodic Gorenstein algebras in terms of bimodules Gorenstein
projective dimensions.
