The Boolean Solution to the Congested IP Link Location Problem: Theory and Practice

Abstract

Abstract-Like other problems in network tomography or traffic matrix estimation, the location of congested IP links from end-to-end measurements requires solving a system of equations that relate the measurement outcomes with the variables representing the status of the IP links. In most networks, this system of equations does not have a unique solution. To overcome this critical problem, current methods use the unrealistic assumption that all IP links have the same prior probability of being congested. We find that this assumption is not needed, because these probabilities can be uniquely identified from a small set of measurements by using properties of Boolean algebra. We can then use the learnt probabilities as priors to find rapidly the congested links at any time, with an order of magnitude gain in accuracy over existing algorithms. We validate our results both by simulation and real implementation in the PlanetLab network.

Network neutrality inference

“…In our experimental evaluation, we use the performance metric introduced in [22], which is defined as follows:…”

“…Lemmas 1 and 2 will remain the same. Theorem 1 will change to refer to distinguishable sets of links (as opposed to distinguishable links); the proof will rely on the theory developed in [14] (as opposed to [22]). …”

“…According to step 1 and 2, (6) holds for |⃗ y| = |P |. In [22], the authors have proved that "P ⃗ x ( ⃗ Y = ⃗ y) = P ⃗ x ′ ( ⃗ Y = ⃗ y) for any snapshot ⃗ y always implies that ⃗ x = ⃗ x ′ ". Therefore, ∀⃗ x and ⃗ x ′ , equation (5) always implies that ⃗ x = ⃗ x ′ , i.e., ⃗ x = ⃗ x ′ .…”

“…It then estimates ⃗ x, either by solving this system of equations (when it has a unique solution [7,8,6,22,21,14]) or by picking a solution that has some desirable property, e.g., assumes the smallest number of problematic links [23,13,26,10] or occurs with the highest probability [22]. This approach fundamentally relies on the assumption that the network is neutral (each link treats traffic from all paths the same), otherwise it would be impossible to express path measurements as a function of link properties and form a solvable system of equations.…”

Shifting network tomography toward a practical goal

“…At first, this scenario sounded like a good match for Boolean Inference algorithms [12,8,6,11], which monitor paths during a particular time interval (on the order of a few minutes) and infer which particular links on these paths were congested during that interval. In principle, the source ISP could use one of these algorithms to infer which particular links of each peer were congested during each time interval, which would help answer all of the above questions.…”