  We introduce two new exactly solvable (stochastic) interacting particle
systems which are discrete time versions of q-TASEP. We call these geometric
and Bernoulli discrete time q-TASEP. We obtain concise formulas for
expectations of a large enough class of observables of the systems to
completely characterize their fixed time distributions when started from step
initial condition. We then extract Fredholm determinant formulas for the
marginal distribution of the location of any given particle.
  Underlying this work is the fact that these expectations solve closed systems
of difference equations which can be rewritten as free evolution equations with
k-1 two-body boundary conditions -- discrete q-deformed versions of the quantum
delta Bose gas. These can be solved via a nested contour integral ansatz. The
same solutions also arise in the study of Macdonald processes, and we show how
the systems of equations our expectations solve are equivalent to certain
commutation relations involving the Macdonald first difference operator.
