A one dimensional random space-filling problem

Consider an interval of the real line (0, x ), x > 0; and place in it a random subinterval S ( x ) defined by the random variables X x and Y x , the position of the center of S ( x ) and the length of S ( x ). The set (0, x ) – S ( x ) consists of two intervals of length δ and η . Let a > 0 be a fixed constant. If δ ≦ a , then a random interval S ( δ ) defined by X δ , Y δ is placed in the interval of length δ . If δ < a , the placement of the second interval is not made. The same is done for the interval of length η . Continue to place non-intersecting random subintervals in (0, x ), and require that the lengths of all the random subintervals be ≦ a . The process terminates after a finite number of steps when all the segments of (0, x ) uncovered by random subintervals are of length < a . At this stage, we say that (0, x ) is saturated. Define N ( a, x ) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N ( a, x ), for large x .