We develop and evaluate SCIATA, a simplified combined inversion-and-thinning algorithm for simulating a nonstationary non-Poisson process (NNPP) over a finite time horizon, with the target arrival process having a given “rate” function and associated mean-value function together with a given variance-to-mean (dispersion) ratio. Designed for routine use when the dispersion ratio is at most two, SCIATA encompasses the following steps: (i) computing a piecewise-constant majorizing rate function that closely approximates the given rate function; (ii) computing the associated piecewise-linear majorizing mean-value function; (iii) generating an equilibrium renewal process (ERP) whose noninitial interrenewal times are Weibull distributed with mean one and variance equal to the given dispersion ratio; (iv) inverting the majorizing mean-value function at the ERP's renewal epochs to generate the associated majorizing NNPP; and (v) thinning the resulting arrival epochs to obtain an NNPP with the given rate function and dispersion ratio.