Let $\{ \xi (t),0 \leqq t < \infty \} $ be a probability process such that the variable $\xi (t)$ takes on the two values 0 and 1. Let $\xi _1 ,\eta _1 ,\xi _2 ,\eta _2 , \cdots $ be the intervals of time, passed successively in states 0 and 1. We assume that $\xi _1 ,\eta _1 ,\xi _2 ,\eta _2 , \cdots $ are independent random variables with distribution functions ${\bf P}\{ \xi _n \leqq x\} = G(x)$ and ${\bf P}\{ \eta _n \leqq x\} = H(x)$. Let \[ \beta (t) = \int_0^t {\xi (u)du} \] give the time passed by the system in state 1. In this work the asymptotic distribution of the random variable $\beta (t)$ as $t \to \infty $ is investigated under different assumptions on the asymptotic behavior of the sums $\zeta _n = \xi _1 + \xi _2 + \cdots + \xi _n $ and $\chi _n = \eta _1 + \eta _2 + \cdots + \eta _n $ as $n \to \infty $. Here essentially a special method is used, which allows the study of the variable $\beta (t)$ to be reduced to the study of sums of a random number of independent random variables.