The ASTRI Mini-Array is an INAF project aimed at observing astronomical objects that emit photons in the TeV and multi-TeV spectral bands. It consists of an array of nine innovative imaging atmospheric Cherenkov telescopes under deployment at the Observatorio del Teide (Tenerife, Spain). Detailed simulations of atmospheric showers of Cherenkov events using Monte Carlo methods are needed to estimate the expected performance of the ASTRI Mini-Array under different observing conditions, to validate the Monte Carlo simulation chain, for calibration purposes and for the development of ancillary instruments, and finally to allow the reconstruction of the real data collected by the array. The production of events detected by Cherenkov telescopes comprises the simulation of the development of particle cascade in the atmosphere with the associated emission of Cherenkov light and the simulation of the response of telescopes to the impinging light; the first task is carried out with the CoRSiKa software (which also simulates the emission of Cherenkov light), while the second is achieved with the sim_telarray package by simulating the transmission of Cherenkov light from the emission point to the telescope and the response of the telescope itself. In this contribution, we present the ASTRI Mini-Array simulation system, describing in detail the software pipeline adopted for the generation of Monte Carlo events, the hardware facilities dedicated to run the simulations at the offsite ASTRI Data Center, and the final products delivered for the Cherenkov data analysis.