  In this paper, we present new Poisson-type deviation inequalities for
continuous-time Markov chains whose Wasserstein curvature or $\Gamma$-curvature
is bounded below. Although these two curvatures are equivalent for Brownian
motion on Riemannian manifolds, they are not comparable in discrete settings
and yield different deviation bounds. In the case of birth--death processes, we
provide some conditions on the transition rates of the associated generator for
such curvatures to be bounded below and we extend the deviation inequalities
established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for
continuous time random walks on graphs. Probab. Theory Related Fields 116
(2000) 573--602] for continuous-time random walks, seen as models in null
curvature. Some applications of these tail estimates are given for
Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues.
