  Let $Q$ be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and $\Rep(Q)$ the
category of finite dimensional representations over an algebraically closed
field. A representation is simply called a module. It will be shown that a
regular string module has, up to isomorphism, at most two Gabriel-Roiter
submodules. The quivers $Q$ with sink-source orientations will be characterized
as those, whose central parts do not contain preinjective modules. It will also
be shown that there are only finitely many (central) Gabriel-Roiter measures
admitting no direct predecessors. This fact will be generalized for all tame
quivers.
