Study of a small accelerator-driven subcritical research reactor in the Vinca Institute of Nuclear Sciences was initiated in 1999. The idea was to extract a beam of medium-energy protons or deuterons from the TESLA accelerator installation, and to transport and inject it into the reactor. The reactor core was to be composed of the highly enriched uranium fuel elements. The reactor was designated as ADSRR-H. Since the use of this type of fuel elements was not recommended any more, the study of a small accelerator-driven subcritical research reactor employing the low-enriched uranium fuel elements began in 2004. The reactor was designated as ADSRR-L. We compare here the results of the initial computer simulations of ADSRR-H and ADSRR-L. The results have confirmed that our concept could be the basis for designing and construction of a low neutron flux model of the proposed accelerator-driven subcritical power reactor to be moderated and cooled by lead. Our objective is to study the physics and technologies necessary to design and construct ADSRR-L. The reactor would be used for development of nuclear techniques and technologies, and for basic and applied research in neutron physics, metrology, radiation protection and radiobiology.