Tech Report: Throughput Guarantees for TCP Flows Using Adaptive Two Color Marking and Multi-Level AQM

This report provides technical details missing from the INFOCOMM ‘02 paper [1]. The details are related to the design and analysis of the control system. 1 Model Our starting point is the fluid-flow model developed in [2] for modeling TCP flows and AQM routers. In this section we will extend this model to account for two-color marking at the network edge and multi-level active queue management (AQM) running at the core. To begin, we assume a single edge router serving m sets of aggregate flows with each having Ni identical TCP flows. Each aggregate has a token bucket with rate Ai and size bi >> 1. The aggregation of these TCP flows feed a core router with link capacity C. At time t > 0, this router has queue length q(t). At time t > 0, each TCP flow is characterized by its window size Wi(t) and average round-trip time Ri(t) 4 = Ti + q(t) C where Ti is the propagation delay. The sending rate ri of an edge is ri = NiWi(t) Ri(t) . The fluid flow model is described by m + 1 coupled differential equations; one equation for each of the m TCP window dynamics and one for the AQM router. The differential equation for the AQM router is given by dq(t) dt = −C + m ∑ i=1 ri (1) while each TCP window satisfies dWi(t) dt = 1 Ri(t) − Wi(t)Wi(t−Ri(t)) 2Ri(t−Ri(t)) pi(t−Ri(t)) (2) where pi(t) denotes the probability that a mark is generated for this aggregate flow. To finish up, we model the color-marking process at the i-th edge and the multi-AQM action at the core. To model coloring, we let f i (t) be the fraction of fluid marked green; i.e., f i (t) = min { 1, Ai(t) ri(t) } and 1 − f i (t) the red fraction. At the core, we let pg(t) and pr(t) denote the probabilities that marks are generated for the green and red fluids, respectively.1 Consistent with Diffserv, we assume that 0 ≤ pg(t) < pr(t) ≤ 1. Probability pi(t) is then related to the green and red marks by pi(t) = f g i (t)pg(t) + (1− f i (t))pr(t). Let r̃i denote the minimum guaranteed sending rate (MGR) for the i-th edge (aggregate). We say that the router is over-provisioned if ∑m i=1 r̃i ≤ C and under-provisioned if ∑m i=1 r̃i > C. Last, we say that it is exactly-provisioned if ∑m i=1 r̃i = C. The objective of this paper is to develop control strategies at both the core and edges to ensure that the edge sending rates ri (1 ≤ i ≤ m) meet or exceed their respective MGRs when the system is over-provisioned. In the next section we address the steady-state feasibility problem; namely, determine whether values exist for {f i } and [pg, pr] such that the sending rates meet the MGRs More precisely, marks are embedded in the fluid as a time varying Poisson process, and the product of pg and pr with the green and red fluid throughputs respectively determine the intensity of this Poisson process