  We provide a general framework for computing upper bounds on mixing times of
finite Markov chains when its minimal ideal is left zero. Our analysis is based
on combining results by Brown and Diaconis with our previous work on stationary
distributions of finite Markov chains. Stationary distributions can be computed
from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of
the finite semigroup underlying the Markov chain. Using loop graphs, which are
planar graphs consisting of a straight line with attached loops, there are
rational expressions for the stationary distribution in the probabilities. From
these we obtain bounds on the mixing time. In addition, we provide a new Markov
chain on linear extension of a poset with $n$ vertices, inspired by but
different from the promotion Markov chain of Ayyer, Klee and the last author.
The mixing time of this Markov chain is $O(n \log n)$.
