  A real-time communication system with two encoders communicating with a
single receiver over separate noisy channels is considered. The two encoders
make distinct partial observations of a Markov source. Each encoder must encode
its observations into a sequence of discrete symbols. The symbols are
transmitted over noisy channels to a finite memory receiver that attempts to
reconstruct some function of the state of the Markov source. Encoding and
decoding must be done in real-time, that is, the distortion measure does not
tolerate delays. Under the assumption that the encoders' observations are
conditionally independent Markov chains given an unobserved time-invariant
random variable, results on the structure of optimal real-time encoders and the
receiver are obtained. It is shown that there exist finite-dimensional
sufficient statistics for the encoders. The problem with noiseless channels and
perfect memory at the receiver is then considered. A new methodology to find
the structure of optimal real-time encoders is employed. A sufficient statistic
with a time-invariant domain is found for this problem. This methodology
exploits the presence of common information between the encoders and the
receiver when communication is over noiseless channels.
