  We consider the $q$-totally asymmetric simple exclusion process ($q$-TASEP)
in the stationary regime and study the fluctuation of the position of a
particle. We first observe that the problem can be studied as a limiting case
of an $N$-particle $q$-TASEP with a random initial condition and with particle
dependent hopping rate. Then we explain how this $N$-particle $q$-TASEP can be
encoded in a dynamics on a two-sided Gelfand-Tsetlin cone described by a
two-sided $q$-Whittaker process and present a Fredholm determinant formula for
the $q$-Laplace transform of the position of a particle. Two main ingredients
in its derivation is the Ramanujan's bilateral summation formula and the Cauchy
determinant identity for the theta function with an extra parameter. Based on
this we establish that the position of a particle obeys the universal
stationary KPZ distribution (the Baik-Rains distribution) in the long time
limit.
