Proving Optimality of DWS(Distance-Weighted Sampling) Probability Function for FMS IP Trace-Back Technique

Abstract

A convergence time is the time to construct the attack path. In FMS, each router sends its IP by fragments, and the victim should wait until the last router sends its last IP fragment. Therefore, the convergence time is determined by the slowest router. Kim et al.[1] proposed a new sampling theory, so called Distance-Weighted Sampling that did not penalize the furthest router. They showed that the sampling probability p= f(d ) where the f(d ) is a decreasing function of distance d traveled by the target packet. Since the convergence time will be determined by the slowest router, we have to maximize the minimum number of IP fragments incoming at each router station. The optimal choice was stated as $f(d)=\frac{1}{2(d+1)}$ and a small sample simulation study supported their claim. In this article we are going to prove rigorously that $\frac{1}{2(d+1)}$ is indeed the optimal sampling probability under mild assumptions.