Ao Tang

Equilibrium of heterogeneous congestion control: existence and uniqueness

When heterogeneous congestion control protocols that react to different pricing signals share the same network, the resulting equilibrium may no longer be interpreted as a solution to the standard utility maximization problem. We prove the existence of equilibrium in general multiprotocol networks under mild assumptions. For almost all networks, the equilibria are locally unique, finite, and odd in number. They cannot all be locally stable unless there is a globally unique equilibrium. Finally, we show that if the price mapping functions, which map link prices to effective prices observed by the sources, are sufficiently similar, then global uniqueness is guaranteed.

Is fair allocation always inefficient

This paper describes the tradeoff between fairness criteria of different allocation policies and throughput in a general network. A class of utility functions parameterized by a scalar defines a bandwidth allocation policy. An allocation is fair if the scalar is large and efficient if the aggregate source rate is large. The conjecture depends on the network topology in terms of routing and link capacities. A fairer allocation is always more efficient.

Counter-intuitive throughput behaviors in networks under end-to-end control

It has been shown that as long as traffic sources adapt their rates to aggregate congestion measure in their paths, they implicitly maximize certain utility. In this paper we study some counter-intuitive throughput behaviors in such networks, pertaining to whether a fair allocation is always inefficient and whether increasing capacity always raises aggregate throughput. A bandwidth allocation policy can be defined in terms of a class of utility functions parameterized by a scalar α that can be interpreted as a quantitative measure of fairness. An allocation is fair if α is large and efficient if aggregate throughput is large. All examples in the literature suggest that a fair allocation is necessarily inefficient. We characterize exactly the tradeoff between fairness and throughput in general networks. The characterization allows us both to produce the first counter-example and trivially explain all the previous supporting examples. Surprisingly, our counter-example has the property that a fairer allocation is always more efficient. In particular it implies that maxmin fairness can achieve a higher throughput than proportional fairness. Intuitively, we might expect that increasing link capacities always raises aggregate throughput. We show that not only can throughput be reduced when some link increases its capacity, more strikingly, it can also be reduced when all links increase their capacities by the same amount. If all links increase their capacities proportionally, however, throughput will indeed increase. These examples demonstrate the intricate interactions among sources in a network setting that are missing in a single-link topology.

Local stability of FAST TCP

The stability properties of FAST TCP is studied based on a discrete-time model. We prove that FAST TCP is locally asymptotically stable for an arbitrary network in the absence of feedback delay. When feedback delay is present, we derive a sufficient condition for local asymptotic stability for the case of a single link. The condition suggests that local stability depends on delays only through their heterogeneity.

Understanding CHOKe: throughput and spatial characteristics

A recently proposed active queue management, CHOKe, is stateless, simple to implement, yet surprisingly effective in protecting TCP from UDP flows. We present an equilibrium model of TCP/CHOKe. We prove that, provided the number of TCP flows is large, the UDP bandwidth share peaks at (e+1)/sup -1/=0.269 when UDP input rate is slightly larger than link capacity, and drops to zero as UDP input rate tends to infinity. We clarify the spatial characteristics of the leaky buffer under CHOKe that produce this throughput behavior. Specifically, we prove that, as UDP input rate increases, even though the total number of UDP packets in the queue increases, their spatial distribution becomes more and more concentrated near the tail of the queue, and drops rapidly to zero toward the head of the queue. In stark contrast to a nonleaky FIFO buffer where UDP bandwidth shares would approach 1 as its input rate increases without bound, under CHOKe, UDP simultaneously maintains a large number of packets in the queue and receives a vanishingly small bandwidth share, the mechanism through which CHOKe protects TCP flows.

An Accurate Link Model and Its Application to Stability Analysis of FAST TCP

This paper presents a link model which captures the queue dynamics when congestion windows of TCP sources change. By considering both the self-clocking and the link integrator effects, the model is a generalization of existing models and is shown to be more accurate by both open loop and closed loop packet level simulations. It reduces to the known static link model when flows round trip delays are similar, and approximates the standard integrator link model when the heterogeneity of round trip delays is significant. We then apply this model to the stability analysis of FAST TCP. It is shown that FAST TCP flows over a single link are always linearly stable regardless of delay distribution. This result resolves the notable discrepancy between empirical observations and previous theoretical predictions. The analysis highlights the critical role of self-clocking in TCP stability and the scalability of FAST TCP with respect to delay. The proof technique is new and less conservative than the existing ones.

Equilibrium and Fairness of Networks Shared by TCP Reno and Vegas/FAST

It has been proved theoretically that a network with heterogeneous congestion control algorithms that react to different congestion signals can have multiple equilibrium points. In this paper, we demonstrate this experimentally using TCP Reno and Vegas/FAST. We also show that any desired inter-protocol fairness is in principle achievable by an appropriate choice of Vegas/FAST parameter, and that intra-protocol fairness among flows within each protocol is unaffected by the presence of the other protocol except for a reduction in effective link capacities. Dummynet experiments and ns-2 simulations are presented to verify these results.

Network equilibrium of heterogeneous congestion control protocols

When heterogeneous congestion control protocols that react to different pricing signals share the same network, the resulting equilibrium may no longer be interpreted as a solution to the standard utility maximization problem. We prove the existence of equilibrium under mild assumptions. Then we show that multi-protocol networks whose equilibria are locally non-unique or infinite in number can only form a set of measure zero. Multiple locally unique equilibria can arise in two ways. First, unlike in the single-protocol case, the set of bottleneck links can be non-unique with heterogeneous protocols even when the routing matrix has full row rank. The equilibria associated with different sets of bottleneck links are necessarily distinct. Second, even when there is a unique set of bottleneck links, network equilibrium can still be non-unique, but is always finite and odd in number. They cannot all be locally stable unless it is globally unique. Finally, we provide various sufficient conditions for global uniqueness. Numerical examples are used throughout the paper to illustrate these results.

Reverse Engineering MAC

This paper reverse engineers backoff-based random-access MAC protocols in ad-hoc networks. We show that the contention resolution algorithm in such protocols is implicitly participating in a non-cooperative game. Each link attempts to maximize a selfish local utility function, whose exact shape is reverse engineered from the protocol description, through a stochastic subgradient method in which the link updates its persistence probability based on its transmission success or failure. We prove that existence of a Nash equilibrium is guaranteed in general. The minimum amount of backoff aggressiveness needed for uniqueness of Nash equilibrium and convergence of the best response strategy are established as a function of user density. Convergence properties and connection with the best response strategy are also proved for variants of the stochastic-subgradient-based dynamics of the game. Together with known results in reverse engineering TCP and BGP, this paper completes the recent efforts in reverse engineering the main protocols in layers 2-4.

An empirical study on the connectivity of ad hoc networks

This paper discusses the probability of connectivity of ad hoc networks. An empirical formula is proposed to fit the simulation results. The parameters of the formula are determined for different cases and the asymptotic behavior is discussed. Finally, a new metric is proposed to quantify the connectivity of an ad hoc network.