In this present paper we consider one generalization of the well-known problem of random filling a segment of large length with unit intervals. On the segment [0, x], if x ≥ 1, in accordance with the distribution law Fx we will place an open interval of unit length. Fx is the distribution of the left end of the unit interval, which is concentrated on the segment [0, x - 1]. Let the first allocated interval take the place of (t, t + 1) and divide the segment [0, x] into two parts [0, t] and [t + 1, x], which are further filled independently of each other according to the following rules. On the segment [0, t] a point t<sub>1</sub> is selected randomly in accordance with the law Ft, and the interval (t<sub>1</sub>, t<sub>1</sub> + 1) is placed. A point t<sub>2</sub> is selected randomly in the segment [t + 1, x] such that u = t2 - t - 1 is a random variable distributed according to the law F<sub>x-t-1</sub>, and we place the interval (t2, t2 +1). In the same way then the newly formed segments are filled. If x <
