Estimates for the Syracuse problem via a probabilistic model

We employ a simple stochastic model for the Syracuse problem (also known as the $(3x+ 1)$ problem) to get estimates for the average behavior of the trajectories of the original deterministic dynamical system. The use of the model is supported not only by certain similarities between the governing rules in the systems, but also by a qualitative estimate of the rate of approximation. From the model, we derive explicit formulae for the asymptotic densities of some sets of interest for the original sequence. We also approximate the asymptotic distributions for the stopping times (times until absorption in the only known cycle $\{1,2\}$) of the original system and give numerical illustrations of our results.