Current intermittent demand inventory control models assume that the demand interval is memoryless: the probability of observing a positive demand does not depend on the time since the last demand occurred. Contrarily, several forecasting contributions suggest that demand intervals contain more distributional information. We find that the data of the M5 forecasting competition confirms this. Therefore, we propose an inventory control model that explicitly uses the full distributions of the demand sizes and intervals and thereby acknowledges that the probability of a demand occurrence may vary throughout the interval. To exploit this information, we also allow for time-varying order-up-to levels that flexibly adjust inventories according to the dynamic requirements. We derive the long-run average holding costs, non-stockout probability, order fill rate, and volume fill rate. Inspired by an analogy with multi-item inventory control models, we propose a greedy marginal-analysis heuristic to optimize the order-up-to levels, which we benchmark against the optimal solution on theoretical instances. In a simulation study on the M5 competition data we demonstrate this method's improved on-target service performance compared to that of traditional solutions. We furthermore show that target service levels can be achieved at significantly lower costs with time-varying than with fixed order-up-to levels.