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A new channel model for binary additive noise communication channel with memory, called weighted queue-based channel (WQBC), is introduced. The proposed WQBC generalizes the conventional queue-based channel (QBC) such that each queue cell has a different contribution to the noise process, i.e. the queue cells are selected with different probabilities. Suitably selecting the modeling function, the generalization introduced by the WQBC does not increase the number of modelling parameters required compared to the QBC. The statistical and information-theoretical properties of the new model are derived. The WQBC and the QBC are compared in terms of capacity and the accuracy in modeling a family of hard decision frequency-shift keying demodulated correlated Rayleigh and Rician fading channels. It is observed that the WQBC requires a much smaller Markovian memory than the QBC to achieve the same capacity, and provides a very good approximation of the fading channels as the QBC for a wide range of channel conditions.