Time-Scale Decomposition and Equivalent Rate-Based Marking

Differential equation models for Internet congestion control algorithms have been widely used to understand network dynamics and the design of router algorithms. These models use a fluid approximation for user data traffic and describe the dynamics of the router queue and user adaptation through coupled differential equations. The interaction between the routers and flows occurs through marking, where routers indicate congestion by appropriately marking packets during congestion. In this paper, we show that the randomness due to short and unresponsive flows in the Internet is sufficient to decouple the dynamics of the router queues from those of the end controllers. This implies that a time-scale decomposition naturally occurs such that the dynamics of the router manifest only through their statistical steady-state behavior. We show that this time-scale decomposition implies that a queue-length based marking function (e.g., RED-like and REM-like algorithms, which have no queue averaging, but depend only on the instantaneous queue length) has an equivalent form which depends only on the data arrival rate from the end-systems and does not depend on the queue dynamics. This leads to much simpler dynamics of the differential equation models (there is no queueing dynamics to consider), which enables easier analysis and could be potentially used for low-complexity fast simulation. Using packet-based simulations, we study queue-based marking schemes and their equivalent rate-based marking schemes for different types of controlled sources (i.e., proportional fair and TCP) and queue-based marking schemes. Our results indicate a good match in the rates observed at the intermediate router with the queue-based marking function and the corresponding rate-based approximation. Further, the window size distributions of a typical TCP flow with a queue-based marking function as well as the equivalent rate-based marking function match closely, indicating that replacing a queue-based marking function by its equivalent rate-based function does not statistically affect the end host's behavior