On bounds for the pointwise availability of a repairable component

An alternating renewal process is considered with d.f. $A(t)$ and $B(t)$ of its up-phases and down-phases, respectively. It is assumed that an up-phase starts at the point $t=0$. Let $P(t)$ denote the up-state probability at time $t$. Assume that $A(+0)=0$, the mean duration of an up-phase equals 1 whereas that for a down-phase equals $\rho$. Introduce the function $\Delta(t)$ by the relation
$$ (1+\rho)P_0(t)=1+\rho\Delta(t). $$
Let then $B(t)=B_{\rho}(t)$, $\rho\to0$. It is proved that under a mild assumption for any non-exponential distribution $A(t)$ the equality
$$\sup\limits_{\delta<t<T}|\Delta(t)|\to0 as \rho\to0 $$
cannot hold for every positive $\delta$ and $T$. For the exponential distribution $A(t)$ see Kovalenko $\&$ Birolini [3].