2.3 Unified frameworks for TCP/AQM systems
2.3.2 Duality model of TCP
In this section, it will be shown that the above feedback-control system solves a utility maximization problem at its equilibrium.
Suppose that the equilibrium rates and prices are given byx∗,y∗,p∗, andq∗. Based on
2A more accurate formulation is given in [96] that includes the internal variables of AQM in the parameters ofGl.
(2.3) and (2.2), we have following equilibrium relationships
y∗ =Rx∗, q∗ =RTp∗. (2.6)
Assume that equilibrium rates satisfy
x∗i =fi(q∗i), (2.7)
wherefi(·)is implicitly defined byFi(x∗, q∗i) = 0or given by the source static law, e.g., [97]. fi(·) is usually a positive, strictly monotonic decreasing function, since the source decreases its rate with increasing congestion.
Let fi−1(xi) be the inverse function of (2.7), and let a utility function Ui(xi) be its integral
Ui(xi) :=
Z
fi−1(xi)dxi. (2.8)
This relation implies thatUi(xi)is a monotonic increasing and strictly concave function. It is easy to check that the equilibrium ratex∗i uniquely solves
maxxi≥0Ui(xi)−xiq∗i. (2.9)
We interpretUi(xi)as the benefit the source receives by transmitting at ratexiandqi∗as the price per unit. Then (2.9) is a maximization of the source’s profit. This interpretation makes few assumptions regarding TCP and AQM and can be used for various TCP schemes.
The global optimization problem to maximize aggregate utility with capacity con- straints is formulated by Kelly in [77, 80],
maxx≥0
X
i
Ui(xi) (2.10)
subject to Rx ≤ c. (2.11)
It has a unique solution, since it is maximizing a concave function over a convex set. Now
we interpret the equilibrium price as the dual variables (or as the Lagrange multipliers) for the problem (2.10-2.11). Then its Lagrangian is
L(x, p) = X
i
Ui(xi)−X
l
pl(yl−c) = X
i
(Ui(xi)−qixi) +X
l
plcl. (2.12)
The dual problem is
minp≥0
X
i
Bi(qi) +X
l
plcl, (2.13)
where
Bi(qi) = max
xi≥0 Ui(xi)−xiqi. (2.14) Convex duality implies that at the optimum p∗, the corresponding x∗, which maximizes individual optimality (2.9), is exactly the unique solution to the primal problem (2.10-2.11) since (2.14) is identical to (2.9). Therefore, provided the equilibrium pricesp∗can be made to align with the Lagrange multipliers, the equilibrium ratex∗solves the primal problem in a distributed way. It is proven in [96] that any link algorithm that satisfies
yl∗ ≤cl with equality ifp∗ >0for anyl (2.15)
will guarantee this alignment. In this case, x∗ is the unique primal optimal solution, and p∗ is a dual optimal solution. It has been argued [96] that the condition (2.15) is satisfied by any AQM that stabilizes the queue, e.g., RED, REM, and Droptail. Therefore, various TCP/AQM protocols can be interpreted as different distributed primal-dual algorithms to solve the global optimization problem (2.10-2.11) with different utility functions.
The equilibrium structures of different congestion control schemes are characterized by their corresponding utility functions. This model provides us with a rigorous framework in which to study various equilibrium properties such as fairness, efficiency, and effects of different network parameters. In Chapter 6, I will present the methods and results following this approach.
This optimization framework can also be extended to study the interaction of TCP at a fast timescale and IP routing at a slow timescale. See Chapter 5 for details.
Chapter 3
Local Dynamics of Reno/RED
3.1 Introduction
It is well known that TCP Reno/RED can oscillate wildly and it is extremely hard to reduce the oscillation by tuning RED parameters, e.g., [110, 25]. This oscillation could be the outcome of the AIMD bandwidth probing strategy employed by TCP Reno and noise-like traffic that are not effectively controlled by TCP (e.g., short lived TCP source). Recent models e.g., [36, 59], imply however that oscillation is an inevitable outcome of the pro- tocol itself. We present more evidence to support this view. We argue that Reno/RED oscillates not only because of the AIMD probing and noise traffic, but more fundamentally, it is due to instability. Therefore, even if there is no AIMD, and the congestion window is periodically adjusted by the average of AIMD based on loss probability, the oscillation per- sists. We illustrate using ns-2 simulations that, after smoothing out the AIMD component of the oscillation, the average behavior can either be steady with small random fluctuations (when the protocol is stable), or exhibit limit cycles of amplitude much larger than ran- dom fluctuations (when it is unstable). Moreover, this qualitative behavior persists even when a large amount of noise traffic is introduced, and even when sources have different delays. We conclude that it is the protocol stability that largely determines the dynamics of Reno/RED.
This motivates the stability characterization of Reno/RED. In Section 3.3 we develop a general nonlinear model of Reno/RED. The equilibrium structure of this system is analyzed using duality model, and a unique equilibrium exists because it is the unique solving of the
underling utility maximization problem, see [96] for details. Here, we study local stability by linearizing the model around this equilibrium. The linear model generalizes the single link identical source model of [59]. We validate our model with simulations and illustrate the stability region of Reno/RED. We derive a sufficient stability condition for the special case of a single link with heterogeneous sources. It shows that Reno/RED becomes unstable when delay increases, or more strikingly, when link capacity increases!
In the linearized model, the gain introduced by TCP Reno increases rapidly with delay and link capacity. This induces instability and makes compensation by RED extremely difficult. In particular, RED parameters can be tuned to improve stability, but only at the cost of a large queue, even when they are dynamically adjusted. Our results suggest that Reno/RED is ill suited for future high-speed networks, which motivates the design of new distributed algorithms for high speed long latency networks.