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A channel optimized vector quantizer (COVQ) scheme is studied and evaluated for a recently introduced discrete binary-input 2q-ary-output channel with Markovian ergodic noise based on a finite queue. This channel can effectively model binary-modulated correlated Rayleigh fading channels with output quantization of resolution q. It is shown that the system can successfully exploit the channel's memory and soft-decision information. Signal-to-distortion gains of up to 2.3 dB are obtained for only 2 bits of soft-decision quantization over COVQ schemes designed for a hard-decision (q = 1) demodulated channel. Furthermore, gains as high as 4.6 dB can be achieved for a highly correlated channel, in comparison with systems designed for the ideally interleaved (memoryless) channel. Finally, the queue-based noise model is validated as an effective approximation of correlated fading channels by testing a COVQ trained using this model over the Rayleigh fading channel.