  We present a proof of an upper tail bound of the correct order (up to a
constant factor in the exponent) in two classes of stationary models in the KPZ
universality class.
  The proof is based on an exponential identity due to Rains in the case of
Last Passage Percolation with exponential weights, and recently re-derived by
Emrah-Jianjigian-Sepp\"ail\"ainen (EJS). Our proof follows very similar lines
for the two classes of models we consider, using only general monotonocity and
convexity properties, and can thus be expected to apply to many other
stationary models.
