Software rejuvenation is a proactive fault management technique that has been extensively studied in the recent literature. In this paper, we focus on an example for a telecommunication billing application considered in Huang et al. (1995) and develop the discrete-time stochastic models to estimate the optimal software rejuvenation schedule. More precisely, two software availability models with rejuvenation are formulated via the discrete semi-Markov processes, and the optimal software rejuvenation schedules which maximize the steady-state availabilities are derived analytically. Further, we develop statistically non-parametric algorithms to estimate the optimal software rejuvenation schedules, provided that the complete sample data of failure times are given. Then, a new statistical device, called discrete total time on test statistics, is introduced. Finally, we examine asymptotic properties for the statistical estimation algorithms proposed in this paper through a simulation experiment.