Identifying almost sorted permutations from TCP buffer dynamics

Associate to each sequence A of integers (intending to represent packet IDs) a sequence of positive integers of the same length M(A). The i’th entry of M(A) is the size (at time i) of the smallest buffer needed to hold out-of-order packets, where space is accounted for unreceived packets as well. Call two sequences A, B equivalent (written A ≡FB B) if M(A) = M(B). We prove the following result: any two permutations A,B of the same length with SUS(A), SUS(B) ≤ 3 (where SUS is the shuffledup-sequences reordering measure [3]), and such that A ≡FB B are identical. The result (which is no longer valid if we replace the upper bound 3 by 4) was motivated by Restored, a receiver-oriented model of network traffic we introduced in [7].