  We review the derivation of the Kac master equation model for random
collisions of particles, its relationship to the Poisson process, and existing
algorithms for simulating values from the marginal distribution of velocity for
a single particle at any given time. We describe and implement a new algorithm
that efficiently and more fully leverages properties of the Poisson process,
show that it performs at least as well as existing methods, and give empirical
evidence that it may perform better at capturing the tails of the single
particle velocity distribution. Finally, we derive and implement a novel
"$\varepsilon$-perfect sampling" algorithm for the limiting marginal
distribution as time goes to infinity. In this case the importance is a proof
of concept that has the potential to be expanded to more interesting (DSMC)
direct simulation Monte Carlo applications.
