Proof of Lemma 5.1

5.8 Appendix

5.8.3 Proof of Lemma 5.1

We will prove the lemma by induction. Note that b0 < Ta implies that a1 = h⁻¹(b0) >

h⁻¹(T_a) =r_a. Sincea≥T(1)andh(1) <0,a₁ =h⁻¹(b₀)<1(see Figure 5.6). Hence

h(r)

a₂

T(r)

b₁

T(0)

b₀ r

a₁ a₀

Figure 5.6: Proof of Lemma 5.1.

0 =a₀ < r_a< a₁ <1.

This implies thatb₁ =T(a₁)satisfies

T(0) = b₀ < T_a< b₁ < T(1).

Sinceb1 < T(1) < h(0),a2 =h⁻¹(b1)> h⁻¹(h(0)) = 0, we have 0 =a₀ < a₂ < r_a< a₁ <1.

Let the induction hypothesis be

a₀ < . . . < a_2n < r_a < a_2n−1 < . . . < a₁ b₀ < . . . < b_2n−2 < T_a < b_2n−1 < . . . < b₁.

Thenb_2n =T(a_2n)> T(a_2n−2) = b_2n−2 andb_2n=T(a_2n)< T(r_a) =T_a. Hence, b2n−2 < b2n < Ta.

This implies thatr_a < a_2n+1 < a_2n−1, which in turn implies thatT_a < b_2n+1 < b_2n−1. This completes the induction.

Chapter 6

Throughput, Fairness, and Capacity

6.1 Introduction

Recent studies, e.g. [80, 97, 116, 164, 101, 88, 96], have shown that a bandwidth allocation policy can be formulated as a utility maximization problem where the bandwidth allocation x^∗ (source rates) solves [80]

maxx

X

i

U_i(x_i) subject toRx ≤c. (6.1)

It is remarkable that as long as traffic sources adapt their rates to the aggregate (sum of) congestion in their paths, they are implicitly maximizing some utility. The optimization problem (6.1) is a convenient characterization of the equilibrium properties of various TCP/AQM systems. We can derive the underlying utility functions of various TCP al- gorithms and use them to study the relations among network throughput, fairness, and ca- pacity. Our work reveals some counter-intuitive behaviors, which will be briefly presented in this chapter. See [144, 146] for more detailed results and proofs.

We refer to network throughput as the total traffic through the network, which measures the efficiency of the bandwidth allocation policy under which the network operates. There are many examples in the literature that point to an inevitable tradeoff between fairness and aggregate throughput (efficiency), yet there is no general theorem clarifying this folklore.

How do we balance fairness and efficiency in designing bandwidth allocation policies?

Will adding additional link capacities necessarily result in higher aggregate throughput?

In this chapter, we rigorously study these questions in general networks using an ana- lytical model . Here are our main results.

Suppose that the bandwidth allocation policies, represented by utility functions, are parameterized by a common scalarα ≥0. We derive explicit expressions for the changes in source rates and congestion prices when the parameter α or the capacities change for general utility functions.

We specialize to a particular class of utility functions [116] that characterize various TCP variants and include various fairness criterions as special cases. The parameterα in these utility functions can be interpreted as a quantitative measure of fairness [107, 16], and an allocation is fair ifαis large. All examples in the literature indicate that a fair allocation is necessarily inefficient. We quantitatively formulate the relations between fairness and efficiency in general networks. This characterization allows us both to produce the first counter-example (Theorem 6.3) and trivially explain all the previous supporting examples (Corollary 6.2). Surprisingly, the class of networks in our counter-example indicates that a fairer allocation could be always more efficient. In particular it implies that max-min fairness can achieve a higher aggregate throughput than proportional fairness.

Intuitively, we might expect that the aggregate throughput will always rise when some links increase their capacities. This turns out to be wrong, and we characterize exactly the condition under which this is true (Theorem 6.4). Not only can the aggregate throughput be reduced when some link increases its capacity, more strikingly, it can also be reduced even when all links increase their capacities by the same amount (Theorem 6.5). Moreover, this holds under all bandwidth allocation policies . This paradoxical result seems less surprising in retrospect: raising link capacities always increases the aggregate utility, but mathemat- ically there is no a priori reason that it should also increase the aggregate throughput. If all links increase their capacities proportionally, however, the aggregate throughput will indeed increase, under the class of utility functions proposed in [116] (Theorem 6.6).

It is well known that counter-intuitive behavior can arise in a distributed system where agents optimize their own objectives, e.g., the Braess paradox in transportation networks.

It was discovered theoretically in 1968 [18, 120, 44] and verified in real world years later [37]. It shows that adding a new road to a transportation network may cause longer travel

time for every car. Subsequent paradoxes have been discovered in mechanical and electrical networks [27], in queueing networks [28, 136, 83, 11, 84], and in computer systems [72, 73]. Even though our results have the same flavor, they differ in important ways from the Braess paradox.

First, in the Braess paradox, the performance degradation is due to misalignment of individual and social optimalities. In our case, it is due to misalignment of two social objectives (utility maximization versus throughput maximization). Second, in the Braess paradox, the addition of new road leads to degraded performance for all flows, and hence the new equilibrium point is not Pareto optimal. In our case, all equilibrium points are Pareto optimal, and hence some flows are worse off and some better off in the new equi- librium point. Finally, examples of the Braess paradox always involve the addition of new paths and flows that re-route to maximize their own objectives. In our case, only link capacities are changed, while network topology and routing are fixed.