  Let $X$ be a smooth compact manifold with corners which has two embedded
boundary hypersurfaces $\partial_0 X , \partial_1 X$, and a fiber bundle
$\phi:\partial_0 X \to Y$ is given. By using the method of blowing up, we
define a pseudodifferential culculus $\Psi ^* _{\Phi,b} (X)$ generalizing the
$\Phi$-calculus of Mazzeo and Melrose and the (small) $b$-calculus of Melrose.
We discuss the Fredholm condition of such operators and prove the relative
index theorem. And as its application, the index theorem of "non-closed"
$\mathbb{Z}/k$ - manifolds is proved.
