Given a set of jobs (or items), each of which being characterized by its resource demand and its lifespan, and a sufficiently large number of identical servers (or bins), the busy time minimization problem (BTMP) requires to find a feasible jobs-to-servers assignment having minimum overall power-on time. Although BTMP can be classified as a variant of temporal bin packing, it actually represents an independent branch of research since, for instance, the related solution sets are generally different. Typically, such computations (and generalizations of it) are very important in data center workload management to keep operational costs low. Hence, finding efficient solution techniques for BTMP is a relevant topic in discrete optimization, both from a theoretical and a practical point of view. In this work, we first give an overview of heuristic approaches for the problem under consideration. As a main contribution, a new best-cost based heuristic is presented as well as theoretically analyzed, and it is empirically shown to improve the quality of feasible solutions obtained for a set of challenging benchmark instances.