Identifying almost sorted permutations from TCP buffer dynamics

TL;DR

This paper characterizes almost sorted permutations in network buffer dynamics, showing that sequences with small shuffled-up-sequences and identical buffer profiles are essentially the same, advancing understanding of packet reordering.

Contribution

It proves that permutations with up to three shuffled-up-sequences and identical buffer profiles are identical, revealing a new structural property of nearly sorted sequences in network models.

Findings

Sequences with SUS ≤ 3 and same buffer profile are identical.

Buffer profile equivalence characterizes almost sorted permutations.

The result does not hold for SUS ≥ 4.

Abstract

Associate to each sequence $A$ of integers (intending to represent packet IDs) a sequence of positive integers of the same length ${\mathcal M}(A)$. The $i$'th entry of ${\mathcal M}(A)$ 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 $A$, $B$ {\em equivalent} (written $A\equiv_{FB} B$) if ${\mathcal M}(A)={\mathcal M}(B)$. We prove the following result: any two permutations $A,B$ of the same length with $SUS(A)$, $SUS(B)\leq 3$ (where SUS is the {\em shuffled-up-sequences} reordering measure), and such that $A\equiv_{FB} B$ 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 The result (which is no longer valid if we replace the upper bound 3 by 4) was motivated by RESTORED, a receiver-oriented model of network traffic we have previously introduced.