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This paper studies the dynamic bin packing problem, in which items arrive and depart at arbitrary times. We want to pack a sequence of unit fractions items (i.e., items with sizes 1/w for some integer w≥1) into unit-size bins, such that the maximum number of bins ever used over all time is minimized. Tight and almost-tight performance bounds are found for the family of any-fit algorithms, including first-fit, best-fit, and worst-fit. In particular, we show that the competitive ratio of best-fit and worst-fit is 3, which is tight, and the competitive ratio of first-fit lies between 2.45 and 2.4942. We also show that no on-line algorithm is better than 2.428-competitive.