Length of triangulated categories

We introduce the notion of composition series of triangulated categories. Their lengths yield invariants for these categories. We then focus on composition series of derived categories of certain projective varieties and finite dimensional algebras. We construct composition series of different lengths, for some smooth projective rational surfaces and for certain smooth threefolds. On the other hand, we prove that for derived categories of finite dimensional representations of Dynkin and extended Dynkin quivers, for derived categories of some singular varieties, all composition series have the same length. If this property would also hold for the blow-up of $\mathbb{P}^2$ in $10$ general points, then Krah's phantom subcategory would have infinite length. In particular, it would have infinite chain of thick subcategories.
Comment: 35 pages. Results on toric surfaces are generalized. There was a gap in the argument of the length on higher genus curve, and the section on curves is now removed, which will appear in future work