  Both providers and consumers of distributed storage services benefit from the
quantification of the severity of consistency violations. However, existing
methods fail to capture a typical pattern of violation - the disorder among
operations with reference to the ideally strong atomic execution. Such disorder
are often seen in Internet applications based on distributed storage services,
such as instant messaging. To this end, we use inversions in a permutation with
reference to the totally-ordered permutation to measure the consistency of
data. The $i$-atomicity model is proposed and a (pseudo-)polynomial
verification algorithm for a restricted type of histories is proposed. The
basic idea of the verification algorithm is brute-force search of the
permutation with less than $i$ inversions, which requires exponential cost.
Reasonable assumptions abstracted from applications scenarios can be leveraged
to prune the search space, which yields an efficient polynomial verification
algorithm.
