Abstract: The work is devoted to the study of a new model of random filling of a segment of large length with intervals of smaller length. Two new formulations of the problem are considered. In the first case, we consider a model in which unit intervals are placed on the segment in such a way that with each subsequent placement of an interval, there should be a free space (from the left and from the right) of length no smaller than a fixed value. In the second model, intervals with a length of 2 are located randomly and no two intervals should neighbor each other. In both cases, the behavior of the average number of located intervals depending on the length of the filled segment is investigated.