Random variations in the well widths of a superlattice structure are shown theoretically to result in band structure features which mimic amorphous materials. In direct analogy to uniform superlattices in which the periodic potential results in minibands, a randomly varying potential within a superlattice, arising from random variations in well width, gives rise to the formation of states within the miniband gap resembling band tailing effects. Calculations are made of the electron transmissivity through the superlattice based on the effective mass solution of the Schrödinger equation. Superlattice structures consisting of 30 wells of both randomly varying and uniform width are studied. Random variations governed both by a Poisson and a uniform distribution are analyzed. A possible new switching device based on the Ovshinsky effect [Phys. Rev. Lett. 21, 1450 (1968)] is proposed.