  Recently, there has been much progress in understanding stationary measures
for colored (also called multi-species or multi-type) interacting particle
systems, motivated by asymptotic phenomena and rich underlying algebraic and
combinatorial structures (such as nonsymmetric Macdonald polynomials).
  In this paper, we present a unified approach to constructing stationary
measures for most of the known colored particle systems on the ring and the
line, including (1) the Asymmetric Simple Exclusion Process (multispecies ASEP,
or mASEP); (2) the q-deformed Totally Asymmetric Zero Range Process (TAZRP)
also known as the q-Boson particle system; (3) the q-deformed Pushing Totally
Asymmetric Simple Exclusion Process (q-PushTASEP). Our method is based on
integrable stochastic vertex models and the Yang-Baxter equation. We express
the stationary measures as partition functions of new "queue vertex models" on
the cylinder. The stationarity property is a direct consequence of the
Yang-Baxter equation.
  For the mASEP on the ring, a particular case of our vertex model is
equivalent to the multiline queues of Martin (arXiv:1810.10650). For the
colored q-Boson process and the q-PushTASEP on the ring, we recover and
generalize known stationary measures constructed using multiline queues or
other methods by Ayyer-Mandelshtam-Martin (arXiv:2011.06117, arXiv:2209.09859),
and Bukh-Cox (arXiv:1912.03510). Our proofs of stationarity use the Yang-Baxter
equation and bypass the Matrix Product Ansatz used for the mASEP by
Prolhac-Evans-Mallick (arXiv:0812.3293).
  On the line and in a quadrant, we use the Yang-Baxter equation to establish a
general colored Burke's theorem, which implies that suitable specializations of
our queue vertex models produce stationary measures for particle systems on the
line. We also compute the colored particle currents in stationarity.
