Rate of SHS (self-propagating high-tеmperature synthesis) reactions in solid nano-sized multilayers is controlled by the time and temperature dependent vacancy concentration. The increase of reaction temperature is typically faster than the rate of vacancy generation. Therefore, the finite relaxation rate of vacancies leads to drastic slowing down of SHS. On the other hand, as-prepared vacancy supersaturation due to fast deposition on the cold substrate may lead to a certain acceleration of SHS. Influence of (1) vacancy mean free path and (2) initial vacancy supersaturation on the SHS rate is investigated numerically. In wide region of parameters the front velocity appears to be inversely proportional to the square root of vacancy mean free path length.