Presented here is the QDS method modified to employ an arbitrary governing velocity probability distribution. An algorithm is presented for the computation of QDS particle "blueprints". The method, which employs a known continuous velocity probability distribution function, uses a set of fixed QDS particle "weights", which can be arbitrarily selected. Provided the weights, particle "blueprint" velocities are computed by taking multiple moments around the governing velocity probability distribution function to provide the discrete representation employed by QDS. In particular, we focus on the results obtained when the governing distribution function is the Chapman-Enskog distribution function. Results are shown for several benchmark tests including a one dimensional standing shock wave and a two dimensional lid driven cavity problem. Finally, the performance of QDS when applied to General Purpose computing on Graphics Processing Units (GPGPU) is demonstrated.