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

Perfect and $\varepsilon$-Perfect Simulation Methods for the One Dimensional Kac Equation