  We study a linear $q-$difference-differential Cauchy problem, under the
action of a perturbation parameter $\epsilon$. This work deals with a
$q-$analog of the research made in a previoues work, giving rise to a
generalization of a recent work by the second author. This generalization is
related to the nature of the forcing term which suggests the use of a
$q-$analog of an acceleration procedure.
  The proof leans on a $q-$analog of the so-called Ramis-Sibuya theorem which
entails two distinct $q-$Gevrey orders. The work concludes with an application
of the main result when the forcing term solves a related problem.
