I would like to express my deepest gratitude and appreciation to my advisors John Doyle and Steven Low. I would also like to thank my parents, to whom this thesis is dedicated.
Challenges of developing theories for the Internet
Internet innovation and development is the result of an engineering design cycle that relies heavily on intuition, heuristics, simulations, and experiments. However, great strides have been made in recent years to build a rigorous mathematical foundation of the Internet in several areas, such as Internet topology [92, 93], routing congestion control, etc.
Related work in congestion control
This approach is called the dual algorithm, where links adjust their prices dynamically and users' source prices are determined by a static function. Lav [96] provided a duality model that leads to a unified framework for understanding and designing TCP/AQM algorithms.
Summary of main results
Dynamics and stability
- Local stability of TCP/RED
- Modelling and dynamics of FAST TCP
FAST TCP [69] is one such algorithm designed based on this theoretical framework. Based on the existing continuous-time flow model, we prove that FAST TCP is globally stable for arbitrary networks when there is no round-trip delay.
Equilibrium and performance
- Relations among throughput, fairness, and capacity
- Joint utility optimization over TCP/IP
We show that the primal problem (1.2) is NP-hard and in general cannot be solved by minimal cost routing. We also show that this gap can be described as the penalty for not splitting the traffic across multiple paths in single-path routing.
Other related results
- Network equilibrium with heterogeneous protocols
- Control unresponsive flow–CHOKe
We elucidate the spatial characteristics of the buffer flowing under CHOKe that produce this throughput behavior. Specifically, we prove that, as the UDP input rate increases, even though the total number of UDP packets in the queue increases, their spatial distribution becomes increasingly concentrated near the tail of the queue and rapidly decreases to zero towards the next head. .
Organization of this dissertation
TCP Reno
When the congestion window is exhausted, the source must wait for an acknowledgment before sending a new one. Since there is approximately one window of packets transmitted for each CAT, the source rate is controlled by the window size divided by CAT.
TCP Vegas
TCP Vegas is proven to achieve weighted proportional fairness in equilibrium when there is sufficient buffer. Choe and Low [24] studied the dynamics of the TCP Vegas algorithm and showed that it can become unstable in the presence of network delay and provided modifications for greater stability.
FAST TCP
The scalable TCP [81], proposed by Kelly, uses multiplicative rise and multiplicative death instead of TCP Reno's AIMD. BIC TCP [163], proposed by Xu et al., uses binary search augmentation and additive augmentation.
Active Queue Management (AQM)
Droptail
The explicit congestion control protocol (XCP) [76], proposed by Katabi et al., is designed based on control theory and requires explicit feedback from the routers to achieve stability and fairness. The High Speed TCP (HSTCP) [39], proposed by Floyd, is a modification of current TCP to increase more aggressively and decrease more cautiously in large congestion window situations.
Random Early Detection (RED)
The dynamics of FAST TCP will be studied in Chapter 4 using droptail routers with sufficient buffering. When in between, packets will be dropped with a probability that varies linearly from 0 to maxp.
CHOKe
RED can also mark the incoming packets instead of dropping them with the implementation of Explicit Congestion Notification (ECN) [131] to prevent packet loss and improve throughput. If they both belong to the same flow, then they are both dropped; otherwise, the randomly selected packet is left intact and the arriving packet is allowed into the buffer with a probability that depends on the level of congestion (this probability is calculated exactly as in RED).
Unified frameworks for TCP/AQM systems
General dynamic model of TCP/AQM
In this framework, a complete feedback control system is specified by providing two additional blocks: the source rates are changed according to aggregate prices in the TCP algorithm, and the link prices are updated based on link utilization. The equilibrium properties will be investigated with the duality model introduced in the next subsection.
Duality model of TCP
We conclude that it is protocol stability that largely determines the dynamics of Reno/RED. The equilibrium structure of this system is analyzed using duality model, and a unique equilibrium exists because it is the unique solution of .
Motivation
It shows that Reno/RED becomes unstable as the delay increases, or more strikingly, as the link capacity increases. What is the effect of noise-like mouse traffic that is not effectively managed by Reno/RED.
Dynamic model
Nonlinear model of Reno/RED
Denote τi(t) as the round-trip time of source i at time t; it is the sum of the round-trip propagation delay and queuing delay P. The round-trip delays are related to the round-trip time through τi(t) = τlif(t) +τlib(t).
Linear model of Reno/RED
Including instantaneous queuing delay in the first place produces a qualitatively different model than when queuing delay is ignored or assumed constant. Time-varying delay in the second place makes linearization difficult, and we replace it by its (constant) equilibrium value (including equilibrium queue delay).
Validation and stability region
Also shown in Figure 3.4(b) are critical frequencies predicted from this static-switching model (with fl0(yl∗) = ρ this does not affect the critical frequency), using the same Nyquist plotting method described above. Intuitively, a larger delay or capacity, or a smaller load, leads to a larger equilibrium window; this confirms the folklore that TCP behaves poorly at large window sizes.
Local stability analysis
The left side of the (sufficient) stability condition depends on the network parameters (c and τi) and the RED parameters (α and ρ). The dependence of the stability condition on c, τ and N is most clearly shown in the case of equal sources, with τ =τi =τ =τ =Nˆτ.
RED parameter setting
More importantly, RED adds another cτ to the gain K, requiring a small αρ for stability and leading to a slow response and a large equilibrium queue. The high gain Ktcp in (3.10) is mainly responsible for instability at high delay, high capacity or low load.
Conclusion
Local stability of FAST TCP in the absence of feedback delay is proved in [69] for the case of a single link. Then we limit ourselves to a single link without feedback delay and prove the global stability of FAST TCP.
Model
Notation
Discrete and continuous-time models
We assume that the disruption in the queues due to changes in the congestion window decreases rapidly compared to the update period of the discrete-time model; see [160] for detailed justification and validation experiments for these arguments. A consequence of this assumption is that the link queue delay vector, p(t) = (pl(t), for all l), is implicitly determined by the congestion windows of sources in a static manner.
Validation
While the continuous-time model does not fully account for self-clocking, the discrete-time model ignores the fast dynamics on the connections. The sets of congestion window sizes are then used in and (4.6) to calculate the queuing delay predicted by the continuous-time model.
Stability analysis with the continuous-time model
Global stability without feedback delay
Therefore, the system is in its unique equilibrium under our assumption that it is full range. From the above argument, V(w(t), p(t)) is a system Lyapunov function and the system is globally asymptotically stable.
Local stability with feedback delay
From the proof, it is clear that V˙(w(t), p(t)) = 0 only implies that the source rates are in equilibrium. For example, this condition suggests that the equilibrium queue delay must be large to guarantee stability.
Numerical simulation and experiment
This suggests that the discrepancy is not in the stability theorem, but rather in the continuous-time model. It also provides justification for the discrete-time models in (4.4) and (4.7) based on the self-clock function introduced in the last section.
Stability analysis with the discrete-time model
Local stability with feedback delay
Since the open-loop system is stable, if we can show that the eigenvalue loci of L(ejw) do not include −1 for ω ∈ [0,2π, the closed-loop system is stable. If the spectral radius of L(ejw) is therefore strictly less than 1 for ω ∈[0,2π), the system will be stable.
Global stability for one link without feedback delay
We will show that the window update for sourcei is proportional toηi(t), and the system is in equilibrium if and only if allηi(t) are zero. The proof for ηmin(t) is similar. 4.28) The following lemma implies that the difference between differentηi(t) goes to zero exponentially fast.
Conclusion
Since both ηmax(t) andηmin(t) converge exponentially to zero from any initial value, the system converges to the equilibrium defined by ηi(t) = 0globally. Furthermore, we also found scenarios where predictions of the discrete-time model do not agree with experiment.
Appendix
- Proof of Lemma 4.1
- Proof of Theorem 4.3
- Proof of Lemma 4.3
- Proof of Lemma 4.4
- Proof of Lemma 4.8
The TCP/IP system equilibrium exists if and only if this problem has no duality gap. In particular, we show that the TCP/IP system over the special ring network is indeed unstable when the connection costs are pure prices.
Related work
First, our single-route routing problem is NP-hard (see Section 5.4) and generally has a duality gap, while the network flow problem is generally a linear program that is in P and has no duality gap. The instability of single-path routing is not surprising since it is well known that stability generally requires that the relative weight on the dynamic (traffic-sensitive) component of the link cost be small.
Model
The difference between single-path routing and multi-path routing is the integer constraint onW andR. We assume that a single path is chosen for each source-destination pair that minimizes the sum of the link costs in the path, for an appropriate definition of link cost.
Equilibrium of TCP/IP
It is easy to show that optimal solutions to both the primal problem (5.12) and its dual (5.13) exist, so the issue is whether there is a duality gap. Then, given a solution(R∗, x∗, p∗) of the primal and the dual problems, we will show that this is an equilibrium of TCP/IP.
Dynamics of TCP/IP
- Simple ring network
- Utility and stability of pure dynamic routing
- Maximum utility of minimum-cost routing
- Stability of minimum-cost routing
- General network
In the next period, each node i will choose the counterclockwise or clockwise direction accordingly since D−(i;r) or D+(i;r) is smaller. As expected, when a is small, routing is stable and the total utility increases with a, as in the ring network analyzed in Section 5.5.3 (Theorem 5.5).
Resource provisioning
Although the time-averaged service continues to increase after the introduction of path instability, it eventually peaks and drops to a level less than the service achievable by necessarily stable static routing. Hence there exists an optimal W∗ ∈ Ws, i.e., saving a minimum-cost path using αl as the link cost is optimal.
Conclusion
We specialize in a special ring network and show that routing is indeed unstable when switching costs are congestion prices. However, we show that if link capacity is optimally provisioned, purely static (and hence stable) routing is sufficient to maximize usability even for general networks, and link costs are proportional to provisioning costs.
Appendix
Proof of duality gap
It can be stabilized by adding a static component to the link cost definition, but the static component reduces the achievable utility. Thus, there appears to be an inevitable trade-off between routing stability and maximizing utility for a given set of link capabilities.
Proof of primal-dual optimality
Proof of Lemma 5.1
Intuitively, we can expect that the total throughput will always increase when some links increase their capacity. However, if all links increase their capacity proportionally, the total throughput will actually increase under the class of utility functions proposed in [116] (Theorem 6.6).
Model
Finally, examples of the Braess paradox always involve the addition of new paths and flows that redirect to maximize their own goals. Thus, in the remainder of the paper, we will focus on utility maximization with equality constraints representing only those constraints active at optimality.
Basic results
Since the optimalx always satisfies the constraints Rx =c, for a fixed c, the change in x should be in the null space of Rasαvarier.
Is fair allocation always inefficient?
- Conjecture
- Special cases
- Necessary and sufficient conditions
- Counter-example
Proportional justice (α = 1) is considered fairer, and max-min justice (α = . ∞) the fairest because it generalizes equal sharing of a single resource to a network of resources in a Pareto-maintaining way -optimality [45, 15] . Note that the condition is a function of α, although this is not explicit in the notation.
Does increasing capacity always raise throughput?
In fact, an easy and cheap upper bound for the total throughput increase iscS/2. To illustrate, we calculate the change in total throughput for the network in Figure 6.8 under the max–min policy α=∞.
Conclusion
- Introduction
- Model
- Existence of equilibrium
- Examples of multiple equilibria
- Local uniqueness of equilibrium
- Global uniqueness of equilibrium
- Conclusion
In contrast, the active constraint set in a multi-protocol network may be non-unique even if R has full rank as shown in Example 1. Note that the equilibrium in the center is a saddle point, and is therefore unstable.
Control unresponsive flow–CHOKe
Introduction
The number of equilibria is almost always finite and must be odd when they are associated with the same set of active constraints. Second, the number of equilibria associated with each set of tight links can be more than one, although always odd.
Model
A packet may be dropped, either on arrival due to CHOKe or congestion (eg according to RED), or after it has been queued when a future arrival from the same flow triggers a comparison. During this time period, on averageτ xi packets from flow arrive at the queue.
Throughput analysis
As the UDP sending rate grows indefinitely, even though UDP packets occupy up to half of the queue, its throughput drops to zero. The second result of the theorem can also be proved without using the three approximations.
Spatial characteristics
When the UDP input rate increases, although 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 quickly drops to zero towards the head of the queue. This means that most of the UDP packets are dropped before reaching the header.
Simulations
When x0 = 0.1c (Figure 7.10(a)), UDP packets are more or less uniformly distributed in the queue, with probability close to 0.08 at each position. Also marked in Figure 7.9(b) are the UDP bandwidth shares corresponding to UDP rates in Figure 7.10.
Conclusions
We have studied the equilibrium and dynamics of Internet congestion control using recently developed tools from feedback control theory and optimization theory. As we mentioned in Section 7.1, during the incremental deployment of congestion control schemes, there is an inevitable phase of heterogeneous protocols operating in the same network.
Congestion window of TCP Reno
RED dropping function
General congestion control structure
Window and queue traces without noise traffic
Queue traces with noise traffic
Queue traces with heterogeneous delays
Linear model validation
Stability region
Model validation–closed loop
Model validation–open loop
Numerical simulations of FAST TCP
Dummynet experiments of FAST TCP
Illustration of Lemma 4.3
Network to which integer partition problem can be reduced
A ring network
The routing r(a)
A random network
Aggregate utility as a function of a for random network
Proof of Lemma 5.1
Linear network
Linear network with two long flows
Fairness-efficiency tradeoff
Network for counter-example in Theorem 6.3
Throughput versus efficiency α in the counter-example
Source rates versus α in the counter-example
Counter-example for Theorem 6.5(1)
Counter-example for Theorem 6.5(2)
Example 2: two active constraint sets
Shifts between the two equilibria with different active constraint sets
Example 2: uncountably many equilibria
Example 3: construction of multiple isolated equilibria
Corollary : linear network
Illustration of Theorems 7.9 and 7.10
Experiment 1: effect of UDP rate x 0 on queue size and UDP share
Experiment 2: spatial distribution ρ(y)
Experiment 3: effect of UDP rate x 0 on queue size and UDP share with TCP
