  We show that the joint probability generating function of the stationary
measure of a finite state asymmetric exclusion process with open boundaries can
be expressed in terms of joint moments of Markov processes called quadratic
harnesses. We use our representation to prove the large deviations principle
for the total number of particles in the system. We use the generator of the
Markov process to show how explicit formulas for the average occupancy of a
site arise for special choices of parameters. We also give similar
representations for limits of stationary measures as the number of sites tends
to infinity.
