Estimation of Congestion Price Using Probabilistic Packet Marking

One key component of recent pricing-based congestion control schemes is an algorithm for probabilistically setting the Explicit Congestion Notification bit at routers so that a receiver can estimate the sum of link congestion prices along a path. We consider two such algorithms—a well-known algorithm called Random Exponential Marking (REM) and a novel algorithm called Random Additive Marking (RAM). We show that if link prices are unbounded, a class of REM-like algorithms are the only ones possible. Unfortunately, REM computes a biased estimate of total price and requires setting a parameter for which no uniformly good choice exists in a network setting. However, we show that if prices can be bounded and therefore normalized, then there is an alternate class of feasible algorithms, of which RAM is representative and furthermore, only the REM-like and RAM-like classes are possible. For properly normalized link prices, RAM returns an optimal price estimate (in terms of mean squared error), outperforming REM even if the REM parameter is chosen optimally. RAM does not require setting a parameter like REM, but does require a router to know its position along the path taken by a packet. We present an implementation of RAM for the Internet that exploits the existing semantics of the time-to-live field in IP to provide the necessary path position information.