The ionization of residual gas by an electron beam in an accelerator generates ions that can resonantly couple to the beam through a wave propagating in the beam-ion system. The original theory of the fast ion instability was developed assuming both a constant external focusing and the beam size. The theory predicts an instability in which an initial perturbation grows as ~exp(avt). In the present paper we consider a more realistic model that takes into account variation of the beta function in the lattice and associated with it variation of the beam size. We find that, in combination with ion decoherence effect, the spatial inhomogeneity can result in (1) purely exponential growth, ~exp(Gt), and (2) typically smaller growth rates. Detailed calculations are performed for the lattice of the Advanced Light Source at the LBL. Show Partial Abstract