  We study the problem of simulating a two-user multiple-access channel (MAC)
over a multiple access network of noiseless links. Two encoders observe
independent and identically distributed (i.i.d.) copies of a source random
variable each, while a decoder observes i.i.d. copies of a side-information
random variable. There are rate-limited noiseless communication links between
each encoder and the decoder, and there is independent pairwise shared
randomness between all the three possible pairs of nodes. The decoder has to
output approximately i.i.d. copies of another random variable jointly
distributed with the two sources and the side information. We are interested in
the rate tuples which permit this simulation. This setting can be thought of as
a multi-terminal generalization of the point-to-point channel simulation
problem studied by Bennett et al. (2002) and Cuff (2013). When the pairwise
shared randomness between the encoders is absent, the setting reduces to a
special case of MAC simulation using another MAC studied by Haddadpour et
al.~(2013). We establish that the presence of encoder shared randomness can
strictly improve the communication rate requirements. We first show that the
inner bound derived from Haddadpour et al.~(2013) is tight when the sources at
the encoders are conditionally independent given the side-information at the
decoder. This result recovers the existing results on point-to-point channel
simulation and function computation over such multi-terminal networks. We then
explicitly compute the communication rate regions for an example both with and
without the encoder shared randomness and demonstrate that its presence
strictly reduces the communication rates. Inner and outer bounds for the
general case are also obtained.
