Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities of the system in an interval, i.e. the probabilities that the system performs above the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities exactly, but if the component performance processes are independent, it is possible to construct lower bounds based on the component availabilities to the different levels over the interval. In this paper we show that by treating the component availabilities over the interval as if they were availabilities at a single time point, we obtain an improved lower bound. Unlike previously given bounds, the new bound does not require the identification of all minimal path or cut vectors.