In Duration-aware Pinwheel Scheduling (DAPS), each job \(J_i\) has a duration \(p_i\) (in days) and the objective is to find a schedule on a single machine such that each job \(J_i\) is completed at least once in every consecutive \(f_i\) days. This problem generalizes the well-known problem, (Ordinary) Pinwheel Scheduling, which assumes all jobs are unit-length (i.e., \(p_i = 1\) ). Previous studies have shown that in order to schedule jobs \(J_1, \dots , J_n\) feasibly, the density of the instance, defined as \(p_1 / f_1 + \dots + p_n/f_n\) , must be \( \le 1\) . Additionally, in the case of Ordinary Pinwheel Scheduling, every instance with density \(\le 5/6\) is known to be always schedulable. However, it remains open whether a similar density threshold would lead to an efficient algorithm for deciding schedulability in DAPS. In this paper, we present a negative solution to this open problem under the assumption that \(\textsf{P} \ne \textsf{NP}\) , namely, we show that no constant density threshold leads to a polynomial-time algorithm for deciding schedulability in DAPS unless \(\textsf{P} = \textsf{NP}\) . Furthermore, we demonstrate that another heuristic from Ordinary Pinwheel Scheduling fails to extend to DAPS. After that, we present an online scheduler which always schedule jobs feasibly for some DAPS instance class, by inventing a new measurement of schedulability of instances. This scheduler can be regarded as a 3-approximation algorithm for the optimization version of DAPS – known as Continuous Bamboo Garden Trimming on star graphs – which improves upon the previous best approximation ratio \(3 + 2\sqrt{2}\) .