  In a previous paper by the second author,two Markov chain Monte Carlo perfect
sampling algorithms -- one called coupling from the past (CFTP) and the other
(FMMR) based on rejection sampling -- are compared using as a case study the
move-to-front (MTF) self-organizing list chain. Here we revisit that case study
and, in particular, exploit the dependence of FMMR on the user-chosen initial
state. We give a stochastic monotonicity result for the running time of FMMR
applied to MTF and thus identify the initial state that gives the
stochastically smallest running time; by contrast, the initial state used in
the previous study gives the stochastically largest running time. By changing
from worst choice to best choice of initial state we achieve remarkable speedup
of FMMR for MTF; for example, we reduce the running time (as measured in Markov
chain steps) from exponential in the length n of the list nearly down to n when
the items in the list are requested according to a geometric distribution. For
this same example, the running time for CFTP grows exponentially in n.
