In typical applications, a priority queue is used to execute a sequence of n insert, m decrease, and n delete-min operations, starting with an empty structure. We study the performance of different priority queues for this type of operation sequences both theoretically and experimentally. In particular, we focus on weak heaps, weak queues, and their relaxed variants. We prove that for relaxed weak heaps the execution of any such sequence requires at most 2m + 1.5n lg n element comparisons. This improves over the best bound, at most 2m + 2.89n lg n element comparisons, known for the existing variants of Fibonacci heaps. We programmed six members of the weak-heap family of priority queues. For random data sets, experimental results show that non-relaxed versions are performing best and that rank-relaxed versions are slightly faster than run-relaxed versions. Compared to weak-heap variants, the corresponding weak-queue variants are slightly better in time but not in the number of element comparisons.