We have investigated photoluminescence (PL) properties of GaAs (15 nm)/AlAs (15 nm) and GaAs (20 nm)/AlAs (20 nm) multiple quantum wells at 10 K under high density excitation conditions at excitation energies in the region of the fundamental excitons. It has been found that the PL due to exciton-exciton scattering, the so-called P emission, is observed with a threshold nature in addition to the appearance of the biexciton PL. The energy spacing between the P-PL band and the heavy-hole exciton depends on the layer thickness, which reflects the change of the exciton binding energy by the quantum size effect. The intensity of the biexciton-PL band is saturated by the appearance of the P-PL band. Both the exciton-exciton scattering process and the biexciton formation process require the collision of two excitons. Thus, the exciton-exciton scattering process prevents the formation of biexcitons, which leads to the saturation behavior of the biexciton-PL intensity. Furthermore, we have confirmed the existence of optical gain leading to stimulated emission due to the exciton-exciton scattering process with use of a variable-stripe-length method.