![2 Mathematics of Internet architecture
In this chapter, we will develop a mathematical formulation of the problem of resource allo-
cation in the Internet. A large communication network such as the Internet can be viewed
as a collection of communication links shared by many sources. Congestion control algo-
rithms are protocols that allocate the available network resources in a fair, distributed,
and stable manner among the sources. In this chapter, we will introduce the network util-
ity maximization formulation for resource allocation in the Internet, where each source
is associated with a utility function Ur(xr), and xr is the transmission rate allocated to
source r. The goal of fair resource allocation is to maximize the net utility
r Ur(xr) sub-
ject to resource constraints. We will derive distributed, congestion control algorithms that
solve the network utility maximization problem. In a later chapter, we will discuss the
relationship between the mathematical models developed in this chapter to transport layer
protocols used in the Internet. Optimality and stability of the congestion control algorithms
will be established using convex optimization and control theory. We will also introduce
a game-theoretical view of network utility maximization and study the impact of strategic
users on the efficiency of network utility maximization. Finally, routing and IP addressing
will be discussed. The following key questions will be answered in this chapter.
• What is fair resource allocation?
• How do we use convex optimization and duality to design distributed resource allocation
algorithms to achieve a fair and stable resource allocation?
• What are the game-theoretic implications of fair resource allocation?
Mathematical background: convex optimization
..................................................................................................................................................
2.1
In this section, we present some basic results from convex optimization which we will find
useful in the rest of the chapter. Often, the results will be presented without proofs, but
some concepts will be illustrated with figures to provide an intuitive feel for the results.
2.1.1 Convex sets and convex functions
We first introduce the basic concepts from optimization theory, including the definitions of
convex sets and convex functions.
Definition 2.1.1 (Convex set) A set S ⊆ Rn is convex if αx + (1 − α)y ∈ S whenever
x, y ∈ S and α ∈ [0, 1]. Since αx + (1 − α)y, for α ∈ [0, 1], describes the line segment](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-25-320.jpg)
![8 Mathematics of Internet architecture
between x and y, a convex set can be pictorially depicted as in Figure 2.1: given any two
points x, y ∈ S, the line segment between x and y lies entirely in S.
x
y
S ⊆ R2
Figure 2.1 A convex set, S ⊆ R2.
Definition 2.1.2 (Convex hull) The convex hull of set S, denoted by Co(S), is the smallest
convex set that contains S, and contains all convex combinations of points in S, i.e.,
Co(S) =
k
i=1
αixi
xi ∈ S, αi ≥ 0,
k
i=1
αi = 1
.
See Figure 2.2 for an example.
Figure 2.2 The solid line forms the boundary of the convex hull of the shaded set.
Definition 2.1.3 (Convex function) A function f(x) : S ⊆ Rn → R is a convex function
if S is a convex set and the following inequality holds for any x, y ∈ S and α ∈ [0, 1] :
f(αx + (1 − α)y) ≤ αf(x) + (1 − α)f(y);
f(x) is strictly convex if the above inequality is strict for all α ∈ (0, 1) and x = y.
Pictorially, f(x) looks like a bowl, as shown in Figure 2.3.
Definition 2.1.4 (Concave function) A function f(x) : S ⊆ Rn → R is a concave func-
tion (strictly concave) if −f is a convex (strictly convex) function. Pictorially, f(x) looks
like an inverted bowl, as shown in Figure 2.4.](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-26-320.jpg)
![covered with a coating of ice, as if they had been glazed with glass,
while the white waters streaming beneath us fell into a deep,
eddying pool. We managed, after some difficulty, to cross the stream
in the second cave, and to penetrate a considerable distance along
the treacherous rocks into the very centre, as it were, of the great
mountain; but, just as we were winding along a kind of
subterranean passage, which looked like a short cut into eternity,
our lights went out, owing to the water falling from above, and, as
we could hear nothing but rushing waters ahead, we, with some
difficulty, beat a retreat into the first cave, which looked like a fairy
palace in comparison with the dark cavern we had just left. These
caves were at an altitude of 7000 feet above the level of the sea,
and we were now at the true source of the remarkable river.
Wherever the water poured over the rocks it left a white deposit,
and when we tasted it, it produced a marked astringent feeling upon
the tongue, leaving a strong taste of alum, sulphur, and iron, with all
of which ingredients, especially the two former, it appeared to be
strongly impregnated.[51]
It is a remarkable and interesting geographical fact that the waters
which form the source of the Waikato River burst from the sides of
Ruapehu, within a short distance of the Whangaehu, and at almost
the same altitude. Both streams run almost parallel to each other for
a long distance from their source, and then, as they reach the
desert, they gradually diverge and divide the two great watersheds
of this portion of the country, the Waikato flowing to the north into
Lake Taupo, and the Whangaehu to join the sea in the south. There
is, I believe, no place in the world where two great rivers may be
seen rising at an altitude of over 7000 feet in the sides of a glacier-
clad mountain, and rolling for miles, side by side, down its rugged
slopes, the waters of the one of alabaster whiteness, and the waters
of the other as pure and as limpid as crystal, and each forming the
dividing waters of an area of country of nearly 100 miles in length.
It had taken us nine hours to reach the ice caves, and as it was now
late in the day we began to descend with all haste, in order, if](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-59-320.jpg)
![FOOTNOTES:
[51] Near to this point, on the summit of the mountain, there is a
lake formed by an extinct crater, filled by subterranean springs,
and it is likely that the Whangaehu may in some way be
connected with it. It is, however, clear that there must, of
necessity, be strong subterranean springs in this portion of the
mountain, to account for the large volume of water forming the
source of this river, as likewise extensive deposits of alum, of
some form or another, to cause the complete discoloration of the
waters by that mineral. I believe that this singular river will be
found to possess great medicinal properties for the cure of
rheumatic affections and cutaneous disorders.](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-61-320.jpg)
![country, our pleasant party at Karioi was composed of
representatives of many nations. A Mr. Rees, who had come up from
Whanganui, was a native of Australia, and had served in the armed
constabulary at Parihaka; Mr. Newman, our host, hailed from the
South of England; one of the hands was a New Zealander, another
an Austrian, a third came from the Alpine districts of the Tyrol, and
another from the Land o' Cakes, while the native race was here
represented by several hapus of one of the principal Whanganui
tribes. To listen to the spirited description given by Mr. Rees of the
Parihaka campaign, and to his delineation of Te Whiti[52] and other
notable chiefs, to participate in the varied conversation upon the
wonders of the surrounding country, to chat with the Tyrolese in his
native tongue, and to feel that a great vacuum had been filled in our
insides, was so great a change to what we had recently experienced,
that we now seemed to be partaking of the pleasures of the varied
society and seductive luxuries of a first-class antipodean
caravansary, where hospitality was boundless and good-fellowship
the order of the day.
In the evening we visited the native kainga, and spent some time
with the Maoris in the wharepuni. There were about twenty natives
present, men, women, and children, and in the centre of the
primitive apartment blazed a huge fire, which threw out a terrific
heat, and rendered the place almost unbearable. The natives were
mostly short of stature, with hard features, and I remarked that they
spoke with a much harsher accent than those further to the north,
and that they clipped many of their words in a remarkable way.
When Turner inquired for an explanation of this habit, they stated
that their great ancestor, Ngatoroirangi, when he came over in the
Arawa canoe was engaged in baling out that craft during a storm,
and that whilst so doing he caught a severe cold, which caused him
to speak in a sharp, halting kind of way, which has been imitated
ever since by many of the Whanganui tribes, who claim descent
from that celebrated chief, and who has been before alluded to in a
previous chapter as the first explorer of the country.](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-64-320.jpg)
![[52] Te Whiti and Tohu, the Maori prophets, were captured in
1882, at the instance of the Government, by the armed
constabulary at the native settlement of Parihaka, for inciting
their followers to commit acts of lawlessness against the
European settlers.](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-66-320.jpg)
![which time the natives displayed no token of friendship, the only
manifestations we received in this respect being from the dogs and
pigs, the latter even going so far as to scratch their backs against
our legs.
RUAKAKA.
At last an old woman, who had been watching the proceedings
keenly, and whose appearance reminded me of one of the witches in
Macbeth, suddenly rose, and stepping with an excited air into the
middle of the circle, waved her bare right arm round her head, and
shouted at the top of her voice, Haeremai! Haeremai! Haeremai!
[53] And then turning to the natives, in an equally excited way, said,
The pakehas have been following up the rivers of great names, and
have come to our homes; they are hungry, and we must give them
food. The words of this weird dame, whom we afterwards found
was the chieftainess Hinepareoterangi, and mother of the chief of
the hapu, acted like magic upon the natives, who at once took
charge of our horses, while the women hastened to prepare a meal,
old Hinepareoterangi opening the feast by presenting us with some
of the finest apples I had ever tasted.[54] In a short time we were](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-79-320.jpg)
![man, and then, looking us full in the face, she exclaimed in her wild,
weird way, We believe in nothing here, and get fat on pork and
potatoes. This brought down roars of laughter from the assembled
Hauhaus, and we dropped the religious question.
It was, in fact, very clear that these natives were as deeply wrapped
in the darkness of heathenism as were their forefathers centuries
ago, and beyond a superstitious species of Hauhauism, no germ of
religious teaching appeared to have found its way into their breasts.
They were, however, always ready to sing Hauhau chants to the
glorification of Te Whiti and Te Kooti, who appeared to be the
presiding deities of these wild tribes. At night, when the wind and
rain raged without, and the river rushed through its rock-bound
channel with a noise like thunder, both men and women would chant
these wild refrains in droning, melancholy notes, but in perfect
harmony, the airs in most cases being exceedingly pretty and
touching.
The two following chants were sung to us by Te Pareoterangi and
other natives in chorus, and were taken down in Maori verbatim by
Turner. I am indebted for their spirited translation to the able pen of
Mr. C.O. Davis.
TE KOOTI'S LAMENT.
I stood alone awhile, then moving round
I heard of Taranaki's doings. The rumours
Reached me here, and then I raised
My hand to Tamarura,[55] that deity
Above. Ah me! 'twas on the third
Of March that suffering came,
For then, alas! Waerangahika[56] fell;
And I was shipped on board a vessel,
And borne along upon the ocean.
We steer for Waikawa,[57] and then we bear
Away to Ahuriri,[58] to thee, McLean.[59]
Ah, now I'm seated on St. Kilda's[60] deck,](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-83-320.jpg)
![And looking back to gaze upon the scene
My tears like water freely flow; now
Whanganui's[61] shore is seen, now
Whangaroa,[62]
Where mountain waves are raising up their
crests
Near Wharekauri.[63] O, my people,
Rest ye at home; arise and look around,
nd northward look. The lightsome clouds
Are lingering in the sky, and wafted hither
Day by day, yes, from my distant home,
Turanga, from which I now am separated,
Separated now from those I love.
O, my people! respect the queen's authority,
That we may prosper even to the end.
Suffice the former things thrown in our path
As obstacles. Uphold the governor's laws
To mitigate the deeds of Rura, who brought
Upon us all our troubles.
HAUHAU HYMN.
Let us arise, O people!—the whole of us arise.
Lo, Tohu and Te Whiti now have reached
The pits of darkness—the house of Tangaroa,
[64]
And gateway of the spirit-world of Miru,[65]
Where men are bound all seasons of the year.
The offspring, too, of David they would bind.
The bright and morning star, Peace, at the end
Will come, and in the times of David
Feelings of vindictiveness will cease.
'Tis not from thee; it is from Moses](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-84-320.jpg)
![And the Prophets—from Jesus Christ
And His Apostles, that lines of demarcation
Were set up to shield thee from man's wrath.
The termination comes by thee, O Tohu!
And while it wears a pleasing aspect,
I am lighted into day.
FOOTNOTES:
[53] Haeremai is the usual cry of welcome with the Maoris.
[54] When afterwards we asked the natives how it was they
appeared to be mistrustful of us when we first arrived, they
replied that they had always been suspicious of half-castes and
pakehas, especially since the capture of Winiata by Barlow. That
Te Takaru, the murderer of Moffat, came there sometimes, and
they thought we were after him. They then related to us the
circumstances of Moffat's death. It would appear that the
murdered man, on his last journey, came to Ruakaka, and
induced several of the natives to accompany him to the Tuhua
country. Moffat, who had been driven from that district by the
natives, had been warned not to enter it again; but,
notwithstanding this caution, he determined to revisit it, in order
to prospect for gold. The party left by one of the bush tracks, and
when it had nearly reached its destination, Moffat was fired upon
by a native from behind a tree, and mortally wounded in the
back. At the same moment he fell from his horse, when another
native jumped forward, and split his skull open with a tomahawk.
[55] Tamarura—probably a supposed angel recognized by the
Hauhau parties.
[56] Waerangahika—one of the pas at Poverty Bay, which was
taken by our forces.
[57] Waikawa—now known as Open Bay.
[58] Ahuriri—the great Maori name of Hawke's Bay.
[59] The late Sir Donald McLean, the Superintendent of the
province of Hawke's Bay (Napier).
[60] St. Kilda was the name of the vessel in which Te Kooti was
transported to the Chatham Islands.](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-85-320.jpg)
![[61] Whanganui—name of a places on the Chatham Islands.
[62] Whangaroa—name of a place on the Chatham Islands.
[63] Wharekauri is the native name of the Chatham Islands.
[64] The god of the sea, and guardian of fishes.
[65] Supposed being armed with authority in Hades.](https://image.slidesharecdn.com/88566-250226061726-a4534078/85/Communication-Networks-An-Optimization-Control-and-Stochastic-Networks-Perspective-Srikant-R-86-320.jpg)
Communication Networks An Optimization Control and Stochastic Networks Perspective Srikant R.
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- 5. Communication Networks An Optimization Control and Stochastic Networks Perspective Srikant R. Digital Instant Download Author(s): Srikant R., Ying L. ISBN(s): 9781107036055, 1107036054 Edition: draft File Details: PDF, 2.49 MB Year: 2013 Language: english
- 7. Communication Networks Communication Networks blends control, optimization, and stochastic network theories with features that support student learning to provide graduate students with an accessible, modern approach to the design of communication networks. • Covers a broad range of performance analysis tools, including important advanced topics that are made accessible to graduate students for the first time. • Taking a top-down approach to network protocol design, the authors begin with the deterministic model and progress to more sophisticated models. • Network algorithms and protocols are tied closely to the theory, engaging students and helping them understand the practical engineering implications of what they have learnt. • The background behind the mathematical analyses is given before the formal proofs and is supported by worked examples, enabling students to understand the big picture before going into the detailed theory. • End-of-chapter exercises cover a range of difficulties; complex problems are broken down into several parts, many with hints to guide students. Full solutions are available to instructors. R. Srikant is the Fredric G. and Elizabeth H. Nearing Endowed Professor of Electrical and Computer Engineering, and a Professor in the Coordinated Science Laboratory, at the University of Illinois at Urbana-Champaign, and is frequently named in the university’s list of teachers ranked as excellent. His research interests include communication networks, stochas- tic processes, queueing theory, control theory, and game theory. He has been a Distinguished Lecturer of the IEEE Communications Society, is a Fellow of the IEEE, and is currently the Editor-in-Chief of the IEEE/ACM Transactions on Networking. Lei Ying is an Associate Professor in the School of Electrical, Computer and Energy Engineering at Arizona State University, and former Northrop Grumman Assistant Professor at Iowa State University. He is a winner of the NSF CAREER Award and the DTRA Young Inves- tigator Award. His research interests are broadly in the area of stochastic networks, including wireless networks, P2P networks, cloud computing, and social networks.
- 9. Communication Networks AN OPTIMIZATION, CONTROL, AND STOCHASTIC NETWORKS PERSPECTIVE R. SRIKANT University of Illinois at Urbana-Champaign LEI YING Arizona State University
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107036055 c Cambridge University Press 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Srikant, R. (Rayadurgam) Communication networks : an optimization, control, and stochastic networks perspective / R. Srikant, University of Illinois at Urbana-Champaign, Lei Ying, Arizona State University. pages cm Includes bibliographical references and index. ISBN 978-1-107-03605-5 (Hardback) 1. Telecommunication systems. I. Ying, Lei (Telecommunication engineer) II. Title. TK5101.S657 2013 384–dc23 2013028843 ISBN 978-1-107-03605-5 Hardback Additional resources for this publication at www.cambridge.org/srikant Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. v To Amma, Appa, Susie, Katie, and Jenny RS To my parents, Lingfang and Ethan LY
- 12. “This book by Srikant and Ying fills a major void – an analytical and authoritative study of communication networks that covers many of the major advances made in this area in an easy-to-understand and self-contained manner. It is a must read for any networking student, researcher, or engineer who wishes to have a fundamental understanding of the key operations of communication networks, from network dimensioning and design to congestion control, routing, and scheduling. Throughout the book, the authors have taken pains to explain highly mathematical material in a manner that is accessible to a beginning graduate student. This has often required providing new examples, results, and proofs that are simple and easy to follow, which makes the book attractive to academics and engineers alike. A must have networking book for one’s personal library!” Ness B. Shroff The Ohio State University “Communication Networks provides a deep, modern and broad yet accessible coverage of the analysis of networks. The authors, who made many original contributions to this field, guide the readers through the intuition behind the analysis and results. The text is ideal for self-study and as a basis for a graduate course on the mathematics of communication networks. Students in networking will benefit greatly from reading this book.” Jean Walrand University of California, Berkeley “Communication Networks, by Srikant and Ying, provides a mathematically rigorous treat- ment of modern communication networks. The book provides the essential mathematical preliminaries in queueing theory, optimization and control, followed by a rigorous treat- ment of network architectures, protocols and algorithms that are at the heart of modern-day communication networks and the Internet. It is the best textbook on communication net- works from a theoretical perspective in over 20 years, filling a much needed void in the field. It can be an excellent textbook for graduate and advanced undergraduate classes, and extremely useful to researchers in this rapidly evolving field.” Eytan Modiano Massachusetts Institute of Technology “This book presents a view of communication networks, their architecture and protocols, grounded in the theoretical constructs from optimization and queuing theory that underpin the modern approach to the design and analysis of networks. It is a superb introduction to this approach.” Frank Kelly University of Cambridge “This textbook provides a thoughtful treatment of network architecture and network proto- col design within a solid mathematical framework. Networks are required to provide good stable behavior in random environments. This textbook provides the tools needed to make this happen. It provides needed foundations in optimization, control, and probabilistic tech- niques. It then demonstrates their application to the understanding of current networks and the design of future network architectures and protocols. This is a ‘must’ addition to the library of graduate students performing research in networking, and engineers researching future network architectures and protocols.” Donald F. Towsley University of Massachusetts Amherst
- 13. CONTENTS Preface page xi 1 Introduction 1 I Network architecture and algorithms 5 2 Mathematics of Internet architecture 7 2.1 Mathematical background: convex optimization 7 2.1.1 Convex sets and convex functions 7 2.1.2 Convex optimization 11 2.2 Resource allocation as utility maximization 15 2.2.1 Utility functions and fairness 17 2.3 Mathematical background: stability of dynamical systems 19 2.4 Distributed algorithms: primal solution 21 2.4.1 Congestion feedback and distributed implementation 24 2.5 Distributed algorithms: dual solution 26 2.6 Feedback delay and stability 27 2.6.1 Linearization 29 2.7 Game-theoretic view of utility maximization 30 2.7.1 The Vickrey–Clarke–Groves mechanism 31 2.7.2 The price-taking assumption 34 2.7.3 Strategic or price-anticipating users 35 2.8 Summary 41 2.9 Exercises 42 2.10 Notes 47 3 Links: statistical multiplexing and queues 49 3.1 Mathematical background: the Chernoff bound 49 3.2 Statistical multiplexing and packet buffering 51 3.2.1 Queue overflow 52 3.3 Mathematical background: discrete-time Markov chains 55 3.4 Delay and packet loss analysis in queues 64 3.4.1 Little’s law 64 3.4.2 The Geo/Geo/1 queue 67 3.4.3 The Geo/Geo/1/B queue 69 3.4.4 The discrete-time G/G/1 queue 70
- 14. viii Contents 3.5 Providing priorities: fair queueing 72 3.5.1 Key properties 76 3.6 Summary 78 3.7 Exercises 79 3.8 Notes 85 4 Scheduling in packet switches 86 4.1 Switch architectures and crossbar switches 87 4.1.1 Head-of-line blocking and virtual output queues 88 4.2 Capacity region and MaxWeight scheduling 90 4.2.1 Intuition behind the MaxWeight algorithm 96 4.3 Low-complexity switch scheduling algorithms 96 4.3.1 Maximal matching scheduling 96 4.3.2 Pick-and-compare scheduling 102 4.3.3 Load-balanced switches 102 4.4 Summary 105 4.5 Exercises 106 4.6 Notes 109 5 Scheduling in wireless networks 110 5.1 Wireless communications 110 5.2 Channel-aware scheduling in cellular networks 114 5.3 The MaxWeight algorithm for the cellular downlink 116 5.4 MaxWeight scheduling for ad hoc P2P wireless networks 122 5.5 General MaxWeight algorithms 125 5.6 Q-CSMA: a distributed algorithm for ad hoc P2P networks 129 5.6.1 The idea behind Q-CSMA 129 5.6.2 Q-CSMA 130 5.7 Summary 134 5.8 Exercises 135 5.9 Notes 140 6 Back to network utility maximization 142 6.1 Joint formulation of the transport, network, and MAC problems 142 6.2 Stability and convergence: a cellular network example 151 6.3 Ad hoc P2P wireless networks 155 6.4 Internet versus wireless formulations: an example 157 6.5 Summary 159 6.6 Exercises 160 6.7 Notes 163 7 Network protocols 165 7.1 Adaptive window flow control and TCP protocols 166 7.1.1 TCP-Reno: a loss-based algorithm 167 7.1.2 TCP-Reno with feedback delay 170
- 15. ix Contents 7.1.3 TCP-Vegas: a delay-based algorithm 171 7.2 Routing algorithms: Dijkstra and Bellman–Ford algorithms 175 7.2.1 Dijkstra’s algorithm: link-state routing 176 7.2.2 Bellman–Ford algorithm: distance-vector routing 179 7.3 IP addressing and routing in the Internet 182 7.3.1 IP addressing 183 7.3.2 Hierarchical routing 184 7.4 MAC layer protocols in wireless networks 186 7.4.1 Proportionally fair scheduler in cellular downlink 187 7.4.2 MAC for WiFi and ad hoc networks 188 7.5 Summary 191 7.6 Exercises 192 7.7 Notes 194 8 Peer-to-peer networks 195 8.1 Distributed hash tables 195 8.1.1 Chord 196 8.1.2 Kademlia 202 8.2 P2P file sharing 207 8.2.1 The BitTorrent protocol 208 8.3 Structured P2P streaming 210 8.4 Unstructured P2P streaming 215 8.5 The gossip process 219 8.6 Summary 221 8.7 Exercises 222 8.8 Notes 225 II Performance analysis 227 9 Queueing theory in continuous time 229 9.1 Mathematical background: continuous-time Markov chains 229 9.2 Queueing systems: introduction and definitions 237 9.3 The M/M/1 queue 239 9.4 The M/M/s/s queue 241 9.4.1 The PASTA property and blocking probability 242 9.5 The M/M/s queue 242 9.6 The M/GI/1 Queue 243 9.6.1 Mean queue length and waiting time 246 9.6.2 Different approaches taken to derive the P-K formula 247 9.7 The GI/GI/1 queue 249 9.8 Reversibility 251 9.8.1 The M/M/1 queue 253 9.8.2 The tandem M/M/1 queue 254 9.9 Queueing systems with product-form steady-state distributions 254
- 16. x Contents 9.9.1 The Jackson network 255 9.9.2 The multi-class M/M/1 queue 256 9.10 Insensitivity to service-time distributions 258 9.10.1 The M/M/1-PS queue 259 9.10.2 The M/GI/1-PS queue 259 9.11 Connection-level arrivals and departures in the internet 263 9.12 Distributed admission control 267 9.13 Loss networks 269 9.13.1 Large-system limit 271 9.13.2 Computing the blocking probabilities 274 9.13.3 Alternative routing 275 9.14 Download time in BitTorrent 276 9.15 Summary 280 9.16 Exercises 282 9.17 Notes 289 10 Asymptotic analysis of queues 290 10.1 Heavy-traffic analysis of the discrete-time G/G/1 queue 291 10.2 Heavy-traffic optimality of JSQ 294 10.3 Large deviations of i.i.d. random variables: the Cramer–Chernoff theorem 302 10.4 Large-buffer large deviations 307 10.5 Many-sources large deviations 312 10.6 Summary 317 10.7 Exercises 318 10.8 Notes 321 11 Geometric random graph models of wireless networks 323 11.1 Mathematical background: the Hoeffding bound 323 11.2 Nodes arbitrarily distributed in a unit square 325 11.3 Random node placement 328 11.4 Summary 335 11.5 Exercises 336 11.6 Notes 339 References 340 Index 349
- 17. PREFACE Why we wrote this book Traditionally, analytical techniques for communication networks discussed in textbooks fall into two categories: (i) analysis of network protocols, primarily using queueing theo- retic tools, and (ii) algorithms for network provisioning which use tools from optimization theory. Since the mid 1990s, a new viewpoint of the architecture of a communication network has emerged. Network architecture and algorithms are now viewed as slow-time- scale, distributed solutions to a large-scale optimization problem. This approach illustrates that the layered architecture of a communication network is a natural by-product of the desire to design a fair and stable system. On the other hand, queueing theory, stochas- tic processes, and combinatorics play an important role in designing low-complexity and distributed algorithms that are viewed as operating at fast time scales. Our goal in writing this book is to present this modern point of view of network protocol design and analysis to a wide audience. The book provides readers with a comprehen- sive view of the design of communication networks using a combination of tools from optimization theory, control theory, and stochastic networks, and introduces mathematical tools needed to analyze the performance of communication network protocols. Organization of the book The book has been organized into two major parts. In the first part of the book, with a few exceptions, we present mathematical techniques only as tools to design algorithms implemented at various layers of a communication network. We start with the transport layer, and then consider algorithms at the link layer and the medium access layer, and finally present a unified view of all these layers along with the network layer. After we cover all the layers, we present a brief introduction to peer-to-peer applications which, by some estimates, form a significant portion of Internet traffic today. The second part of the book is devoted to advanced mathematical techniques which are used frequently by researchers in the area of communication networks. We often sacri- fice generality by making simplifying assumptions, but, as a result, we hope that we have made techniques that are typically found in specialized texts in mathematics more broadly accessible. The collection of mathematical techniques relevant to communication networks is vast, so we have perhaps made a personal choice in the selection of the topics. We have chosen to highlight topics in basic queueing theory, asymptotic analysis of queues, and scaling laws for wireless networks in the second part of the book. We note that two aspects of the book are perhaps unique compared to other textbooks in the field: (i) the presentation of the mathematical tools in parallel with a top-down view of communication networks, and (ii) the presentation of heavy-traffic analysis of queueing models using Lyapunov techniques.
- 18. xii Preface The background required to read the book Graduate students who have taken a graduate-level course in probability and who have some basic knowledge of optimization and control theory should find the book accessible. An industrious student willing to put in extra effort may find the book accessible even with just a strong undergraduate course in probability. Researchers working in the area of com- munication networks should be able to read most chapters in the book individually since we have tried to make each chapter as self contained as possible. However, occasionally we refer to results in earlier chapters when discussing the material in a particular chapter, but this overlap between chapters should be small. We have provided a brief introduction to the mathematical background required to understand the various topics in the book, as and when appropriate, to aid the reader. How to use the book as an instructor We have taught various graduate-level courses from the material in the book. Based on our experience, we believe that there are two different ways in which this book can be used: to teach either a single course or two courses on communication networks. Below we provide a list of chapters that can be covered for each of these options. • A two-course sequence on communication networks. • Course 1 (modeling and algorithms): Chapters 1–6 except Section 3.5, and Sec- tions 7.1, 7.2, 7.4.1, and Chapter 8. The mathematical background, Sections 2.1 and 2.3, can be taught as and when necessary when dealing with specific topics. • Course 2 (performance analysis): Chapters 9 (cover Section 8.2 before Section 9.14), 10, and 11. We recommend reviewing Chapter 3 (except Section 3.5), which would have been covered in Course 1 above, before teaching Chapter 10. • A single course on communication networks, covering modeling, algorithms, and per- formance analysis: Chapters 1–6 except Section 3.5, Sections 9.1–9.10 of Chapter 9, and Chapters 10 and 11. Acknowledgements This book grew out of courses offered by us at the University of Illinois at Urbana- Champaign, Iowa State University, and Arizona State University. The comments of the students in these courses over the years have been invaluable in shaping the material in the book. We would like to acknowledge Zhen Chen, Javad Ghaderi, Juan Jose Jaramillo, Xiaohan Kang, Joohwan Kim, Siva Theja Maguluri, Chandramani Singh, Weina Wang, Rui Wu, Zhengyu Zhang, and Kai Zhu in particular, who read various parts of the book carefully and provided valuable comments. We also gratefully acknowledge collaborations and/or discussions with Tamer Başar, Atilla Eryilmaz, Bruce Hajek, Frank Kelly, P. R. Kumar, Sean Meyn, Sanjay Shakkottai, Srinivas Shakkottai, Ness Shroff, and Don Towsley over the years, which helped shape the presentation of the material in this book.
- 19. 1 Introduction A communication network is an interconnection of devices designed to carry information from various sources to their respective destinations. To execute this task of carrying infor- mation, a number of protocols (algorithms) have to be developed to convert the information to bits and transport these bits reliably over the network. The first part of this book deals with the development of mathematical models which will be used to design the protocols used by communication networks. To understand the scope of the book, it is useful first to understand the architecture of a communication network. The sources (also called end hosts) that generate information (also called data) first convert the data into bits (0s and 1s) which are then collected into groups called packets. We will not discuss the process of converting data into packets in this book, but simply assume that the data are generated in the form of packets. Let us consider the problem of sending a stream of packets from a source S to destination D, and assume for the moment that there are no other entities (such as other sources or destinations or intermediate nodes) in the network. The source and destination must be connected by some communication medium, such as a coaxial cable, telephone wire, or optical fiber, or they have to communicate in a wireless fashion. In either case, we can imagine that S and D are connected by a communication link, although the link is virtual in the case of wireless communication. The protocols that ensure reliable transfer of data over such a single link are called the link layer protocols or simply the link layer. The link layer includes algorithms for converting groups of bits within a packet into waveforms that are appropriate for transmission over the communication medium, adding error correction to the bits to ensure that data are received reliably at the destination, and dividing the bits into groups called frames (which may be smaller or larger than packets) before converting them to waveforms for transmission. The process of converting groups of bits into waveforms is called modulation, and the process of recovering the original bits from the waveform is called demodulation. The protocols used for modulation, demodulation, and error correction are often grouped together and called the physical layer set of protocols. In this book, we assume that the physical layer and link layer protocols are given, and that they transfer data over a single link reliably. Once the link layer has been designed, the next task is one of interconnecting links to form a network. To transfer data over a network, the entities in the network must be given addresses, and protocols must be designed to route packets from each source to their des- tination via intermediate nodes using the addresses of the destination and the intermediate nodes. This task is performed by a set of protocols called the network layer. In the Inter- net, the network layer is called the Internet Protocol (IP) layer. Note that the network layer protocols can be designed independently of the link layer, once we make the assumption that the link layer protocols have been designed to ensure reliable data transfer over each link. This concept of independence among the design of protocols at each layer is called
- 20. 2 Introduction layering and is fundamental to the design of large communication networks. This allows engineers who develop protocols at one layer to abstract the functionalities of the protocols at other layers and concentrate on designing efficient protocols at just one layer. Next, we assume that the network layer has been well designed and that it somehow generates routes for packets from each possible source to each possible destination in the network. Recall that the network is just an interconnection of links. Each link in the net- work has a limited capacity, i.e., the rate at which it can transfer data as measured in bits per second (bps). Since the communication network is composed of links, the sources produc- ing data cannot send packets at arbitrarily high rates since the end-to-end data transfer rate between a source and its destination is limited by the capacities of the links on the route between the source and the destination. Further, when multiple source-destination (S-D) pairs transfer data over a network, the network capacity has to be shared by these S-D pairs. Thus, a set of protocols has to be designed to ensure fair sharing of resources between the various S-D pairs. The set of protocols that ensures such fair sharing of resources is called the transport layer. Transport layer protocols ensure that, most of the time, the total rate at which packets enter a link is less than or equal to the link capacity. However, occasion- ally the packet arrival rate at a link may exceed the link capacity since perfectly efficient transport layer protocol design is impossible in a large communication network. During such instances, packets may be dropped by a link and such packet losses will be detected by the destinations. The destinations then inform the sources of these packet losses, and the transport layer protocols may retransmit packets if necessary. Thus, in addition to fair resource sharing and congestion control functionalities, transport layer protocols may also have end-to-end (source-destination) error recovery functionalities as well. The final set of protocols used to communicate information over a network is called the application layer. Application layer protocols are specific to applications that use the net- work. Examples of applications include file transfer, real-time video transmission, video or voice calls, stored-video transmission, fetching and displaying web pages, etc. The application layer calls upon transport protocols that are appropriate for their respective applications. For example, for interactive communication, occasional packet losses may be tolerated, whereas a file transfer requires that all packets reach the destination. Thus, the former may use a transport protocol that does not use retransmissions to guarantee reliable delivery of every packet to the destination, while the latter will use a transport protocol that ensures end-to-end reliable transmission of every packet. In addition to the protocol layers mentioned above, in the case of wireless communi- cations, signal propagation over one link may cause interference at another link. Thus, a special set of protocols called Medium Access Control (MAC) protocols are designed to arbitrate the contention between the links for access to the wireless medium. The MAC layer can be viewed as a sublayer of the link layer that further ensures reliable operation of the wireless “links so that the network layer continues to see the links as reliable carriers of data. A schematic of the layered architecture of a communication network is provided in Figure 1.1. To ensure proper operation of a communication network, a packet generated by an application will not only contain data, but also contain other information called the header. The header may contain information such as the transport protocol to be used and the address of the destination for routing purposes. The above description of the layered architecture of a communication network is an abstraction. In real communication networks, layering may not be as strict as defined
- 21. 3 Introduction Physical layer: bits over wire/wireless channels. Link layer: reliable transmission of frames (collections of bits). MAC sublayer: multiple links over a shared medium. Shared medium Network layer: data transmitted in the form of packets. Each packet has source and destination addresses, and data. Each node in the network contains routing information to route the packets. Transport layer: reliable end-to-end data transmission. Sources may use feedback from destinations to retransmit lost packets. Sources may also use the feedback information to adjust data transmission rates. Application layer: applications. Protocols such as HTTP, FTP, and SSH transmit data over the network. destination 2 destination 1 source 1 source 2 source 1 destination 1 data packets feedback HTTP, FTP, SSH Figure 1.1 Schematic of the layered architecture of a communication network. above. Some protocols may have functionalities that cut across more than one layer. Such cross-layer protocols may be designed for ease of implementation or to improve the efficiency of the communication network. Nevertheless, the abstraction of a layered architecture is useful conceptually, and in practice, for the design of communication networks. Having described the layers of a communication network, we now discuss the scope of this book. In Part I, we are interested in the design of protocols for the transport, net- work, and MAC sublayers. We first develop a mathematical formulation of the problem of resource sharing in a large communication network accessed by many sources. We show how transport layer algorithms can be designed to solve this problem. We then drill deeper
- 22. 4 Introduction into the communication network, and understand the operation of a single link and how temporary overload is handled at a link. Next, we discuss the problem of interconnecting links through a router in the Internet and the problem of contention resolution between multiple links in a wireless network. The algorithms that resolve contention in wireless links form the MAC sublayer. As we will see, the algorithms that are used to interconnect links within a wireline router share a lot of similarities with wireless MAC algorithms. We devote a separate chapter to network protocols, where we discuss the actual protocols used in the Internet and wireless networks, and relate them to the theory and algorithms developed in the earlier chapters. Part I concludes with an introduction to a particular set of application layer protocols called peer-to-peer networks. Traditional applications deliver data from a single source to a destination or a group of destinations. They simply use the lower layer protocols in a straightforward manner to perform their tasks. In Peer-to-Peer (P2P) networks, many users of the network (called peers) are interested in the same data, but do not necessarily download these data from a single destination. Instead, peers down- load different pieces of the data and share these pieces among themselves. This type of sharing of information make P2P systems interesting to study in their own right. Therefore, we devote a separate chapter to the design of these types of applications in Part I. Part II is a collection of mathematical tools that can be used for performance analysis once a protocol or a set of protocols have been designed. The chapters in this part are not organized by functionalities within a communication network, but are organized by the commonality of the mathematical tools used. We will introduce the reader to tools from queueing theory, heavy-traffic methods, large deviations, and models of wireless networks where nodes are viewed as random points on a plane. Throughout, we will apply these mathematical tools to analyze the performance of various components of a communication network.
- 23. Part I Network architecture and algorithms
- 25. 2 Mathematics of Internet architecture In this chapter, we will develop a mathematical formulation of the problem of resource allo- cation in the Internet. A large communication network such as the Internet can be viewed as a collection of communication links shared by many sources. Congestion control algo- rithms are protocols that allocate the available network resources in a fair, distributed, and stable manner among the sources. In this chapter, we will introduce the network util- ity maximization formulation for resource allocation in the Internet, where each source is associated with a utility function Ur(xr), and xr is the transmission rate allocated to source r. The goal of fair resource allocation is to maximize the net utility r Ur(xr) sub- ject to resource constraints. We will derive distributed, congestion control algorithms that solve the network utility maximization problem. In a later chapter, we will discuss the relationship between the mathematical models developed in this chapter to transport layer protocols used in the Internet. Optimality and stability of the congestion control algorithms will be established using convex optimization and control theory. We will also introduce a game-theoretical view of network utility maximization and study the impact of strategic users on the efficiency of network utility maximization. Finally, routing and IP addressing will be discussed. The following key questions will be answered in this chapter. • What is fair resource allocation? • How do we use convex optimization and duality to design distributed resource allocation algorithms to achieve a fair and stable resource allocation? • What are the game-theoretic implications of fair resource allocation? Mathematical background: convex optimization .................................................................................................................................................. 2.1 In this section, we present some basic results from convex optimization which we will find useful in the rest of the chapter. Often, the results will be presented without proofs, but some concepts will be illustrated with figures to provide an intuitive feel for the results. 2.1.1 Convex sets and convex functions We first introduce the basic concepts from optimization theory, including the definitions of convex sets and convex functions. Definition 2.1.1 (Convex set) A set S ⊆ Rn is convex if αx + (1 − α)y ∈ S whenever x, y ∈ S and α ∈ [0, 1]. Since αx + (1 − α)y, for α ∈ [0, 1], describes the line segment
- 26. 8 Mathematics of Internet architecture between x and y, a convex set can be pictorially depicted as in Figure 2.1: given any two points x, y ∈ S, the line segment between x and y lies entirely in S. x y S ⊆ R2 Figure 2.1 A convex set, S ⊆ R2. Definition 2.1.2 (Convex hull) The convex hull of set S, denoted by Co(S), is the smallest convex set that contains S, and contains all convex combinations of points in S, i.e., Co(S) = k i=1 αixi xi ∈ S, αi ≥ 0, k i=1 αi = 1 . See Figure 2.2 for an example. Figure 2.2 The solid line forms the boundary of the convex hull of the shaded set. Definition 2.1.3 (Convex function) A function f(x) : S ⊆ Rn → R is a convex function if S is a convex set and the following inequality holds for any x, y ∈ S and α ∈ [0, 1] : f(αx + (1 − α)y) ≤ αf(x) + (1 − α)f(y); f(x) is strictly convex if the above inequality is strict for all α ∈ (0, 1) and x = y. Pictorially, f(x) looks like a bowl, as shown in Figure 2.3. Definition 2.1.4 (Concave function) A function f(x) : S ⊆ Rn → R is a concave func- tion (strictly concave) if −f is a convex (strictly convex) function. Pictorially, f(x) looks like an inverted bowl, as shown in Figure 2.4.
- 27. 9 2.1 Mathematical background: convex optimization The line segment connecting the two points (x, f(x)) and (y, f(y)) lies “above” the plot of f(x). y x Figure 2.3 Pictorial description of a convex function in R2. The line segment connecting the two points (x, f(x)) and (y, f(y)) lies “below” the plot of f(x). y x Figure 2.4 Pictorial description of a concave function in R2. Definition 2.1.5 (Affine function) A function f(x) : Rn → Rm is an affine function if it is a sum of a linear function and a constant, i.e., there exist α ∈ Rm×n and a ∈ Rm such that f(x) = αx + a. The convexity of a function may be hard to verify from the definition given above. Therefore, next we present several conditions that can be used to verify the convexity of a function. The proofs are omitted here, and can be found in most textbooks on convex analysis or convex optimization. Result 2.1.1 (First-order condition I) Let f : S ⊂ R → R be a function defined over a convex set S. If f is differentiable and the derivative f (x) is non-decreasing (increasing) in S, f(x) is convex (strictly convex) over S.
- 28. 10 Mathematics of Internet architecture Result 2.1.2 (First-order condition II) Let f : S ⊂ Rn → R be a differentiable function defined over a convex set S. Then f is a convex function if and only if f(y) ≥ f(x) + f(x)(y − x), ∀x, y ∈ S, (2.1) where f(x) = ∂f ∂x1 (x), ∂f ∂x2 (x), . . . , ∂f ∂xn (x) and xi is the ith component of vector x. Pictorially, if x is one-dimensional, this condition implies that the tangent of the function at any point lies below the function, as shown in Figure 2.5. Note that f(x) is strictly convex if the inequality above is strict for any x = y. f(y ) f(x ) + f⬘(x ) (y − x) f(x ) x y Figure 2.5 Pictorial description of inequality (2.1) in one-dimensional space. Result 2.1.3 (Second-order condition) Let f : S ⊂ Rn → R be a twice differentiable function defined over the convex set S. Then, f is a convex (strictly convex) function if the Hessian matrix H with Hij = ∂2f ∂xi∂xj (x) is positive semidefinite (positive definite) over S. Result 2.1.4 (Strict separation theorem) Let S ⊂ Rn be a convex set and x be a point that is not contained in S. Then there exists a vector β ∈ Rn, β = 0, and constant δ 0 such that n i=1 βiyi ≤ n i=1 βixi − δ holds for any y ∈ S.
- 29. 11 2.1 Mathematical background: convex optimization 2.1.2 Convex optimization We first consider the following unconstrained optimization problem: max x∈S f(x), (2.2) and present some important results without proofs. Definition 2.1.6 (Local maximizer and global maximizer) For any function f(x) over S ⊆ Rn, x∗ is said to be a local maximizer or local optimal point if there exists an 0 such that f(x∗ + δx) ≤ f(x∗ ) for δx such that δx ≤ and x + δx ∈ S, where · can be any norm; x∗ is said to be a global maximizer or global optimal point if f(x) ≤ f(x∗ ) for any x ∈ S. When not specified, maximizer refers to global maximizer in this book. Result 2.1.5 If f(x) is a continuous function over a compact set S (i.e., S is closed and bounded if S ⊆ Rn), then f(x) achieves its maximum over this set, i.e., maxx∈S f(x) exists. Result 2.1.6 If f(x) is differentiable, then any local maximizer x∗ in the interior of S ⊆ Rn satisfies f(x∗ ) = 0. (2.3) If f(x) is a concave function over S, condition (2.3) is also sufficient for x∗ to be a local maximizer. Result 2.1.7 If f(x) is concave, then a local maximizer is also a global maximizer. In gen- eral, multiple global maximizers may exist. If f(x) is strictly concave, the global maximizer x∗ is unique. Result 2.1.8 Results 2.1.6 and 2.1.7 hold for convex functions if the max in the optimiza- tion problem (2.2) is replaced by min, and maximizer is replaced by minimizer in Results 2.1.6 and 2.1.7. Result 2.1.9 If f(x) is a differentiable function over set S and x∗ is a maximizer of the function, then f(x∗ )dx ≤ 0 for any feasible direction dx, where a non-zero vector dx is called a feasible direction if there exists α such that x + adx ∈ S for any 0 ≤ a ≤ α.
- 30. 12 Mathematics of Internet architecture Further, if f(x) is a concave function, then x∗ is a maximizer if and only if f(x∗ )δx ≤ 0 for any δx such that x∗ + δx ∈ S. Next, we consider an optimization problem with equality and inequality constraints as follows: max x∈S f(x), (2.4) subject to hi(x) ≤ 0, i = 1, 2, ..., I, (2.5) gj(x) = 0, j = 1, 2, ..., J. (2.6) A vector x is said to be feasible if x ∈ S, hi(x) ≤ 0 for all i, and gj(x) = 0 for all j. While (2.5) and (2.6) are inequality and equality constraints, respectively, the set S in the above problem captures any other constraints that are not in equality or inequality form. A key concept that we will exploit later in the chapter is called Lagrangian duality. Duality refers to the fact that the above maximization problem, also called the primal problem, is closely related to an associated problem called the dual problem. Given the constrained optimization problem in (2.4)–(2.6), the Lagrangian of this optimization problem is defined to be L(x, λ, μ) = f(x) − I i=1 λihi(x) + J j=1 μjgj(x), λi ≥ 0 ∀i. The constants λi ≥ 0 and μj are called Lagrange multipliers. The Lagrangian dual function is defined to be D(λ, μ) = sup x∈S L(x, λ, μ). Let f∗ be the maximum of the optimization problem (2.4), i.e., f∗ = maxx∈S f(x). Then, we have the following theorem. Theorem 2.1.1 D(λ, μ) is a convex function and D(λ, μ) ≥ f∗. Proof The convexity comes from a known fact that the pointwise supremum of affine functions is convex (see Figure 2.6). To prove the bound, note that hi(x) ≤ 0 and gj(x) = 0 for any feasible x, so the following inequality holds for any feasible x: L(x, λ, μ) ≥ f(x). This inequality further implies that sup x∈S h(x)≤0 g(x)=0 L(x, λ, μ) ≥ sup x∈S h(x)≤0 g(x)=0 f(x) = f∗ .
- 31. 13 2.1 Mathematical background: convex optimization Figure 2.6 The solid line is the pointwise supremum of the four dashed lines, and is convex. Since removing some constraints of a maximization problem can only result in a larger maximum value, we obtain sup x∈S L(x, λ, μ) ≥ sup x∈S h(x)≤0 g(x)=0 L(x, λ, μ). Therefore, we conclude that D(λ, μ) = sup x∈S L(x, λ, μ) ≥ f∗ . Theorem 2.1.1 states that the dual function is an upper bound on the maximum of the optimization problem (2.4)–(2.6). We can optimize over λ and μ to obtain the best upper bound, which yields the following minimization problem, called the Lagrange dual problem: inf λ≥0,μ D(λ, μ). (2.7) Let d∗ be the minimum of the dual problem, i.e., d∗ = infλ≥0,μ D(λ, μ). The difference between d∗ and f∗ is called the duality gap. For some problems, the duality gap is zero. We say strong duality holds if d∗ = f∗. If strong duality holds, then one can solve either the primal problem or the dual problem to obtain f∗. This is often helpful since sometimes one of the problems is easier to solve than the other. A simple yet frequently used condition to check strong duality is Slater’s condition, which is given below. Theorem 2.1.2 (Slater’s condition) Consider the constrained optimization problem defined by (2.4)–(2.6). Strong duality holds if the following conditions are true:
- 32. 14 Mathematics of Internet architecture • f(x) is a concave function and hi(x) are convex functions; • gj(x) are affine functions; • there exists an x that belongs to the relative interior1 of S such that hi(x) 0 for all i and gj(x) = 0 for all j. As mentioned earlier, when strong duality holds, we have a choice of solving the original optimization in one of two ways: either solve the primal problem directly or solve the dual problem. Later in this chapter, we will see that resource allocation problems in communi- cation networks can be posed as convex optimization problems, and we can use either the primal or the dual formulations to solve the resource allocation problem. We now present a result which can be used to solve convex optimization problems. Theorem 2.1.3 (Karush–Kuhn–Tucker (KKT) conditions) Consider the constrained opti- mization problem defined in (2.4)–(2.6). Assume that f and hi (i = 1, 2, . . . , I) are differentiable functions and that Slater’s conditions are satisfied. Let x∗ be a feasible point, i.e., a point that satisfies all the constraints. Such an x∗ is a global maximizer for the optimization problem (2.4)–(2.6) if and only if there exist constants λ∗ i ≥ 0 and μ∗ j such that ∂f ∂xk (x∗ ) − i λ∗ i ∂hi ∂xk (x∗ ) + j μ∗ j ∂gj ∂xk (x∗ ) = 0, ∀k, (2.8) λ∗ i hi(x∗ ) = 0, ∀i. (2.9) Further, (2.8) and (2.9) are also necessary and sufficient conditions for (λ∗, μ∗) to be a global minimizer of the Lagrange dual problem given in (2.7). If f is strictly concave, then x∗ is also the unique global maximizer. The KKT conditions (2.8) and (2.9) can be interpreted as follows. Consider the Lagrangian L(x, λ, μ) = f(x) − i λihi(x) + j μjgj(x). Condition (2.8) is the first-order necessary condition for the maximization problem maxx∈S L(x, λ∗, μ∗). When strong duality holds, we have f(x∗ ) = f(x∗ ) − i λ∗ i hi(x∗ ) + j μ∗ j gj(x∗ ), which results in condition (2.9) since gj(x∗) = 0 ∀j, and λ∗ i ≥ 0 and hi(x∗) ≤ 0 ∀i. We remark that condition (2.9) is called complementary slackness. 1 For convex set S, any point in the relative interior is a point x such that for any y ∈ S there exist z ∈ S and 0 λ 1 such that x = λy + (1 − λ)z.
- 33. 15 2.2 Resource allocation as utility maximization Resource allocation as utility maximization .................................................................................................................................................. 2.2 The Internet is a shared resource, shared by many millions of users, who are connected by a huge network consisting of many, many routers and links. The capacity of the links must be split in some fair manner among the users. To appreciate the difficulty in defining what fairness means, let us consider an everyday example. Suppose that one has a loaf of bread which has to be divided among three people. Almost everyone will agree that the fair allocation is to divide the loaf into three equal parts and give one piece to each person. While this seems obvious, consider a slight variant of the situation, where one of the people is a two-year-old child and the other two are football players. Then, an equal division does not seem appropriate: the child cannot possibly consume the third allocated to her, so a different division based on their needs may appear to be more appropriate. The situation becomes more complicated when there is more than one resource to be divided among the three people. Suppose that there are two loaves of bread, one wheat and one rye, so a fair division has to take into account the preferences of the individuals for the different types of bread. Economists solve such problems by associating a so-called utility function with each individual, and then finding an allocation that maximizes the net utility of the individuals. We now formally describe and model the resource allocation problem in the Internet. Consider a network consisting of a set of links L accessed by a set of sources S. We will use the terms source and user interchangeably. Each source is associated with a route, where a route is simply a collection of links. Thus, we assume that the route used by a source to convey packets to their destination is fixed. Since the route is fixed for a source, we use the same index (typically r or s) to denote both a source and its route. We allow multiple sources to share exactly the same route. Thus, two routes can consist of exactly the same set of links. Each user derives a certain utility Ur(xr) when transmitting at rate xr. The utility can be interpreted as the level of satisfaction that a user derives when its transmission rate is xr. We assume that Ur(xr) is an increasing, continuously differentiable function. It is also usually the case that the rate at which the utility increases is larger at smaller rates than at larger rates. For example, a user’s level of satisfaction will increase by a larger amount when the rate allocated to him or her increases from 0 Mbps to 1 Mbps than when the rate increases from 1 Mbps to 2 Mbps. Thus, we also assume that Ur(xr) is a strictly concave function. The goal of resource allocation is to solve the following optimization problem, called Network Utility Maximization (NUM): max xr r∈S Ur(xr) (2.10) subject to the link capacity constraints r:l∈r xr ≤ cl, ∀l ∈ L, (2.11) xr ≥ 0, ∀r ∈ S. (2.12) Next, we present an example of such a resource allocation problem and its solution.
- 34. 16 Mathematics of Internet architecture Example 2.2.1 Consider a network with L links numbered 1 through L, and L + 1 sources numbered 0 through L. All link capacities are assume to be equal to 1. Source 0’s route includes all links, while source r uses only link r, as shown in Figure 2.7. x1 x2 x3 xL x0 Figure 2.7 An L-link line network with L + 1 sources. Assuming log utility functions, the resource allocation problem is given by max x L r=0 log xr (2.13) with constraints x0 + xl ≤ 1, ∀l = 0, 1, . . . , L x ≥ 0, where x is the vector consisting of x0, x1 through xL. Now, since log x → −∞ as x → 0, the optimal solution will assign a strictly positive rate to each user, and so the last constraint can be ignored. Let pl be the Lagrange multiplier associated with the capacity constraint at link l and let p denote the vector of Lagrange multipliers. Then, the Lagrangian is given by L(x, p) = L r=0 log xr − L l=1 pl(x0 + xl − 1). Setting ∂L/∂xr = 0 for each r gives x0 = 1 L l=1 pl , xl = 1 pl , ∀l ≥ 1. (2.14) Further, the KKT conditions require that pl(x0 + xl − 1) = 0 and pl ≥ 0, ∀l ≥ 1. Substituting for the pl from (2.14), we obtain pl = L + 1 L , ∀l ≥ 1.
- 35. 17 2.2 Resource allocation as utility maximization Thus, the optimal data rates for the sources are given by x0 = 1 L + 1 , xr = L L + 1 , ∀r ≥ 1. We note an important feature of the solution. The optimal rate of each source in (2.14) explicitly depends on the sum of the Lagrange multipliers on its route. Thus, if a simple algorithm exists to compute the Lagrange multipliers on each link and feed back the sum of the Lagrange multipliers on its route to each source, then the source rates can also be computed easily. This feature of the optimal solution will be exploited later to derive a distributed algorithm to solve the resource allocation problem. .............................................................................................................................. A closed-form solution, as in Example 2.21, will not be easy to obtain for a network such as the Internet. The number of sources is typically in the millions, the number of links is in the thousands, and there is no central entity that even knows the topology of the network in its entirety. Therefore, we will develop distributed algorithms to solve the resource allocation problem later. 2.2.1 Utility functions and fairness In our network utility maximization framework, we have associated a utility function with each user. The utility function can be interpreted in one of two different ways: one is that there is an inherent utility function associated with each user, and the other interpretation is that a utility function is assigned to each user by the network. In the latter case, the choice of utility function determines the resource allocation to the users. Thus, the utility function can be viewed as imposing different notions of fair resource allocation. Of course, there is no notion of fair allocation that is universally accepted. Here, we will discuss some commonly used notions of fairness. A popular notion of fairness is proportional fairness. Proportionally fair resource allo- cation is achieved by associating a log utility function with each user, i.e., U(xr) = log xr for all users r. If f(x) is a concave function over a domain D, then it is well known that f(x∗ )(x − x∗ ) ≤ 0, ∀x ∈ D, (2.15) where x∗ is the maximizer of f(x). Thus, the optimal rates {x∗ r }, when U(xr) = log xr, satisfy r xr − x∗ r x∗ r ≤ 0, where {xr} is any other set of feasible rates. An allocation with such a property is called proportionally fair. The reason for this terminology is as follows: if one of the source rates is increased by a certain amount, the sum of the fractions (also called proportions) by which the different users’ rates change is non-positive. A consequence of this observation is that, if the proportion by which one user’s rate changes is positive, there will be at least one other user whose proportional change will be negative. If the utility functions are of the form wr log xr for some weight wr 0 for user r, the resulting allocation is called weighted proportionally fair.
- 36. 18 Mathematics of Internet architecture Another widely used notion of fairness in communication networks is called max-min fairness. An allocation {x∗ r } is called max-min fair if it satisfies the following property: if there is any other allocation {xr} such that a user s’s rate increases, i.e., xs x∗ s , there has to be another user u with the property xu x∗ u and x∗ u ≤ x∗ s . In other words, if we attempt to increase the rate for one user, the rate for a less-fortunate user will suffer. The definition of max-min fairness implies that min r x∗ r ≥ min r xr, for any other allocation {xr}. To see why this is true, suppose that there exists an allocation such that min r x∗ r min r xr. (2.16) This implies that, for any s such that minr x∗ r = x∗ s , the following holds: x∗ s xs. Oth- erwise, our assumption (2.16) cannot hold. However, this implies that if we switch the allocation from {x∗ r } to {xr}, we have increased the allocation for s without affecting a less- fortunate user (since there is no less-fortunate user than s under {x∗ r }). Thus, the max-min fair resource allocation attempts first to satisfy the needs of the user who gets the least amount of resources from the network. Yet another form of fairness that has been discussed in the literature is called minimum potential delay fairness. Suppose that user r is associated with the utility function −1/xr. The goal of maximizing the sum of the user utilities is equivalent to minimizing r 1/xr. The term 1/xr can be interpreted as follows: suppose user r needs to transfer a file of unit size. Then, 1/xr is the delay associated with completing this file transfer since the delay is simply the file size divided by the rate allocated to user r. Hence, the name minimum potential delay fairness. All of the different notions of fairness discussed above can be unified by considering utility functions of the form Ur(xr) = x1−α r 1 − α , (2.17) for some α 0. Resource allocation using the above utility function is called α-fair. Dif- ferent values of α yield different ideas of fairness. First consider α = 2. This immediately yields minimum potential delay fairness. Next, consider the case α = 1. The utility func- tion is not well defined at this point, but note that maximizing the sum of x1−α r /(1 − α) yields the same optimum as maximizing the sum of x1−α r − 1 1 − α . Now, by applying l’Hospital’s rule, we obtain lim α→1 x1−α r − 1 1 − α = log xr, thus yielding proportional fairness in the limit as α → 1.
- 37. 19 2.3 Mathematical background: stability of dynamical systems Next, we argue that the limit α → ∞ gives max-min fairness. Let x∗ r (α) be the α-fair allocation. Assume that x∗ r (α) → x∗ r as α → ∞ and x∗ 1 x∗ 2 · · · x∗ n. Let be the minimum difference of {x∗ r }, i.e., = minr(x∗ r+1 − x∗ r ). Then, when α is sufficiently large, we have |x∗ r (α) − x∗ r | ≤ /4, which implies that x∗ 1(α) x∗ 2(α) · · · x∗ n(α). Now, by the property of concave functions mentioned earlier (inequality (2.15)), r xr − x∗ r (α) x∗α r (α) ≤ 0. Considering an arbitrary flow s, the above expression can be rewritten as s−1 r=1 (xr − x∗ r (α)) x∗α s (α) x∗α r (α) + (xs − x∗ s (α)) + n i=s+1 (xi − x∗ i (α)) x∗α s (α) x∗α i (α) ≤ 0. Since |x∗ r (α) − x∗ r | ≤ /4, we further have s−1 r=1 (xr − x∗ r (α)) x∗α s (α) x∗α r (α) + (xs − x∗ s (α)) − n i=s+1 |xi − x∗ i (α)| x∗ s + /4 α x∗ i − /4 α ≤ 0. Note that x∗ i − /4 − (x∗ s + /4) ≥ /2 for any i s, so, by increasing α, the third term in the above expression will become negligible. Thus, if xs x∗ s (α), the allocation for at least one user whose rate satisfies x∗ r (α) x∗ s (α) will decrease. The argument can be made rigorous and extended to the case x∗ r = x∗ s for some r and s. Therefore, as α → ∞, the α-fair allocation approaches max-min fairness. Mathematical background: stability of dynamical systems .................................................................................................................................................. 2.3 Consider a dynamical system defined by the following differential equation: ẋ = f(x), f : Rn → Rn , (2.18) where ẋ is the derivative of x with respect to the time t. The time variable t has been omitted when no confusion is caused. Assume that x(0) is given. Throughout, we will assume that f is a continuous function and that it also satisfies other appropriate conditions to ensure that the differential equation has a unique solution x(t), for t ≥ 0. A point xe ∈ Rn is said to be the equilibrium point of the dynamical system if f(xe) = 0. We assume that xe = 0 is the unique equilibrium point of this dynamical system. Definition 2.3.1 (Globally, asymptotically stable) xe = 0 is said to be a globally asymp- totically stable equilibrium point if lim t→∞ x(t) = 0 for any x(0) ∈ Rn. We first introduce the Lyapunov boundedness theorem.
- 38. 20 Mathematics of Internet architecture Theorem 2.3.1 (Lyapunov boundedness theorem) Let V : Rn → R be a differentiable function with the following property: V(x) → ∞ as x → ∞. (2.19) Denote by V̇(x) the derivative of V(x) with respect to t, i.e., V̇(x) = ∇V(x)ẋ = ∇V(x)f(x). If V̇(x) ≤ 0 for all x, there exists a constant B 0 such that x(t) ≤ B for all t. Proof At any time T, we have V(x(T)) = V(x(0)) + T 0 V̇(x(t)) dt ≤ V(x(0)). Note that condition (2.19) implies that {x : V(x) ≤ c} is a bounded set for any c. Letting c = V(x(0)), the theorem follows. Theorem 2.3.2 (Lyapunov global asymptotic stability theorem) If, in addition to the conditions in the previous theorem, we assume that V(x) is continuously differentiable and also satisfies the following conditions: (1) V(x) ≥ 0 ∀x and V(x) = 0 if and only if x = 0, (2) V̇(x) 0 for any x = 0 and V̇(0) = 0, the equilibrium point xe = 0 is globally, asymptotically stable. Proof We prove this theorem by contradiction. Suppose x(t) does not converge to the equilibrium point 0 as t → ∞. Note that V(x(t)) is non-increasing because its derivative with respect to t is non-positive (V̇(x) ≤ 0) for any x. Since V(x(t)) decreases as a function of t and is lower bounded (since V(x) ≥ 0, ∀x), it converges as t → ∞. Suppose that V(x(t)) converges to, say, 0. Define the set C {x : ≤ V(x) ≤ V(x(0))}. The set C is bounded since V(x) → ∞ as ||x|| → ∞ and it is closed since V(x) is a continuous function of x. Thus, C is a compact set. Let −a = sup x∈C V̇(x), where a 0 is finite because V̇(x) is continuous in x and C is a compact set. Now we write V(x(t)) as V(x(t)) = V(x(0)) + t 0 V̇(x(s)) ds ≤ V(x(0)) − at,
- 39. 21 2.4 Distributed algorithms: primal solution which implies that V(x(t)) = 0, ∀ t ≥ V(x(0)) a , and x(t) = 0, ∀ t ≥ V(x(0)) a . This contradicts the assumption that x(t) does not converge to 0. The Lyapunov global asymptotic stability theorem requires that V̇(x) = 0 for any x = 0. In the case V̇(x) = 0 for some x = 0, global asymptotic stability can be studied using Lasalle’s invariance principle. The proof of the theorem is omitted in this book. Theorem 2.3.3 (Lasalle’s invariance principle) Replace condition (2) of Theorem 2.32 by V̇(x) ≤ 0, ∀x, and suppose that the only trajectory x(t) that satisfies ẋ(t) = f(x(t)) and V̇(x(t)) = 0, ∀t, is x(t) = 0, ∀t. Then x = 0 is globally, asymptotically stable. Distributed algorithms: primal solution .................................................................................................................................................. 2.4 In Section 2.2, we formulated the resource allocation problem as a convex optimization problem. However, the technique used to solve the optimization problem in Example 2.2.1 assumed that we had complete knowledge of the topology and routes. Clearly this is infea- sible in a giant network such as the Internet. In this section and the next, we will study distributed algorithms which only require limited information exchange among the sources and the network for implementation. The approach in this section is called the primal solution. Instead of imposing a strict capacity constraint on each link, we append a cost to the sum network utility: W(x) = r∈S Ur(xr) − l∈L Bl s:l∈s xs . (2.20) Here, x is the vector of rates of all sources and Bl(·) is the cost or price of sending data on link l. Thus, W(x) represents a tradeoff: increasing the data rates x results in increased utility, but there is a price to be paid for the increased data rates at the links. If Bl is interpreted as a “barrier” function associated with link l, it should be chosen so that it increases to infinity when the arrival rate on link l approaches the link capacity cl. Thus, it will ensure that the arrival rate is smaller than the capacity of a link. If Bl is interpreted as a “penalty” function which penalizes the arrival rate for exceeding the link capacity, rates slightly larger than the link capacity may be allowable, but this will result in packet losses over the link. One can also interpret cl as a virtual capacity of the link which is smaller
- 40. 22 Mathematics of Internet architecture than the real capacity, in which case, even if the arrival rate exceeds cl, one may still be operating within the link capacity. While it is not apparent in the deterministic formulation here, later in the book we will see that, even when the arrival rate on a link is less than its capacity, due to randomness in the arrival process, packets in the network will experience delay or packet loss. The function Bl(·) may thus be used to represent average delay, packet loss rate, etc. We assume that Bl is a continuously differentiable convex function so that it can be written equivalently as Bl s:l∈s xs = s:l∈s xs 0 fl(y)dy, (2.21) where fl(·) is an increasing, continuous function. We call fl(y) the congestion price func- tion, or simply the price function at link l, since it associates a price with the level of congestion on the link. It is straightforward to see that Bl defined in the above fash- ion is convex, since integrating an increasing function results in a convex function (see Result 2.1.1). Note also that the convexity of Bl ensures that the function (2.20) is a strictly concave function. We will assume that Ur and fl are such that the maximization of (2.20) results in a solution with xr 0, ∀r ∈ S. Then, the first-order condition for optimality states that the maximizer of (2.20) must satisfy Ur(xr) − l:l∈r fl s:l∈s xs = 0, r ∈ S. (2.22) Clearly, it is not practically feasible to solve (2.22) offline and implement the resulting data rates in the network since, as mentioned earlier, the topology of the network is unknown. Therefore, we will develop a decentralized algorithm under which each user can collect limited information from the network and solve for its own optimal data rate. A natural candidate for such an algorithm is the so-called gradient ascent algorithm from optimiza- tion theory. The basic idea behind the gradient ascent algorithm is intuitive, especially if the concave function is a function of one variable: since a concave function has a deriva- tive which is a decreasing function and the optimal solution is obtained at the point where the derivative is zero, it makes sense to seek a solution by moving in the direction of the derivative. More generally, for a function of many variables, the gradient ascent algorithm suggests moving in the direction of the gradient. Consider the algorithm ẋr = kr(xr) Ur(xr) − l:l∈r fl s:l∈s xs . (2.23) The right-hand side of the above differential equation is simply the derivative of (2.20) with respect to xr, while kr(·) is simply a step-size parameter which determines how far one moves in the direction of the gradient. The scaling function kr(·) must be chosen such that the equilibrium of the differential equation is the same as the optimal solution to the resource allocation problem. For example, if kr(xr) 0, setting ẋr = 0 for all r yields the same set of equations as (2.22). Algorithm (2.23) is called a primal algorithm since it arises from the primal formulation of the utility maximization problem. Note that the primal
- 41. 23 2.4 Distributed algorithms: primal solution algorithm is a congestion control algorithm for the following reasons: when the route price qr = l:l∈r fl( s:l∈s xs) is large, the congestion controller decreases its transmission rate. Further, if xr is large, U (xr) is small (since Ur(xr) is concave) and thus the rate of increase is small. Thus, the network can be viewed as a control system with the network providing the feedback to allow the sources to adjust their rates. Control systems are typically viewed as block diagrams, and to visualize our congestion control algorithm as a block diagram, we introduce some notation. Let R be the routing matrix of the network, i.e., the (l, r) element of this matrix is given by Rlr = 1, if route r uses link l, 0, otherwise. Let yl = s:l∈s xs (2.24) be the load on link l. Thus, yl = s Rlsxs. Letting y be the vector of all yl (l ∈ L), we have y = Rx. (2.25) Let pl(t) denote the price of link l at time t, i.e., pl(t) = fl s:l∈s xs(t) = fl(yl(t)). (2.26) Then the price of a route is just the sum of link prices pl of all the links in the route. So we define the price of route r to be qr = l:l∈r pl. (2.27) Also let p be the vector of all link prices and q be the vector of all route prices. We thus have q = RT p. (2.28) This notation allows us to depict the primal algorithm as a block diagram, as shown in Figure 2.8. We now establish the global asymptotic stability of (2.23) using the Lyapunov technique described in Section 2.3. Since W(x) is a strictly concave function, it has a unique mini- mizer x̂. Further, W(x̂) − W(x) is non-negative and is equal to zero only at x = x̂. Thus, W(x̂) − W(x) is a natural candidate Lyapunov function for the system (2.23). We use this Lyapunov function to prove the following theorem.
- 42. 24 Mathematics of Internet architecture source control link control x sources q R p RT y links Figure 2.8 A block diagram view of the congestion control algorithm. The controller at the source uses congestion feedback from the links to perform its action. Theorem 2.4.1 Consider a network in which all sources adjust their data rates according to the primal control algorithm (2.23). Define V(x) = W(x̂) − W(x), where W(x) is given by (2.20). Assume that the functions Ur(·), kr(·) and fl(·) are such that V(x) → ∞ as ||x|| → ∞, x̂i 0 for all i, and the equilibrium point of (2.23) is the maximizer of (2.20). Then, the controller in (2.23) is globally asymptotically stable. Proof Differentiating V(·), we get V̇ = − r∈S ∂V ∂xr ẋr = − r∈S kr(xr) Ur(xr) − qr 2 0, ∀x = x̂, (2.29) and V̇ = 0 if x = x̂. Thus, all the conditions of the Lyapunov theorem are satisfied, and so the system state will converge to x̂, starting from any initial condition. In the proof of Theorem 2.4.1, we have assumed that the utility, price, and scaling functions are such that W(x) satisfies the conditions required to apply the Lyapunov sta- bility theorem. It is easy to find functions that satisfy these properties. For example, if Ur(xr) = wr log(xr) and kr(xr) = xr, the primal congestion control algorithm for source r becomes ẋr = wr − xr l:l∈r fl(yl), and thus the unique equilibrium point can be obtained by solving wr/xr = l:l∈r fl(yl). Further, if fl(·) is such that Bl(·) is a polynomial function, W(x) goes to −∞, as ||x|| → ∞, and thus V(x) → ∞ as ||x|| → ∞. 2.4.1 Congestion feedback and distributed implementation For a congestion control algorithm to be useful in practice, it should be amenable to decentralized implementation. We now present one possible manner in which the primal algorithm could be implemented in a distributed fashion. We first note that each source
- 43. 25 2.4 Distributed algorithms: primal solution simply needs to know the sum of the link prices on its route to adjust its data rate as sug- gested by the algorithm. Suppose that every packet has a field (a certain number of bits) set aside in its header to collect the price of its route. When the source releases a packet into the network, the price field can be set to zero. Then, each link on the route can add its price to the price field so that, by the time the packet reaches its destination, the price field will contain the route price. This information can then be fed back to the source to implement the congestion control algorithm. A noteworthy feature of the congestion con- trol algorithm is that the link prices depend only on the total arrival rate to the link, and not on the individual arrival rates of each source using the link. Thus, each link has only to keep track of the total arrival rate to compute the link price. If the algorithm required each link to keep track of individual source arrival rates, it would be infeasible to imple- ment since the number of sources using high-capacity links could be prohibitively large. Thus, the primal algorithm is both amenable to a distributed implementation and has low overhead requirements. Packet headers in the Internet are already crowded with a lot of other information, such as source/destination addresses to facilitate routing, so Internet practitioners do not like to add another field in the packet header to collect congestion information. In view of this, the overhead required to collect congestion information can be further reduced to accommodate practical realities. Consider the extreme case where there is only one bit available in the packet header to collect congestion information. Suppose that each packet is marked with probability 1−e−pl when the packet passes through link l. Marking simply means that a bit in the packet header is flipped from a 0 to a 1 to indicate congestion. Then, along a route r, a packet is marked with probability 1 − e l:l∈r pl . If the acknowledgement for each packet contains one bit of information to indicate whether a packet is marked or not, then, by computing the fraction of marked packets, the source can compute the route price l:l∈r pl. The assumption here is that the xr’s change slowly so that each pl remains roughly constant over many packets. Thus, one can estimate pl approximately. While marking, as mentioned above, has been widely studied in the literature, the pre- dominant mechanism used for congestion feedback in the Internet today is through packet drops. Buffers used to store packets at a link have finite capacity, and therefore a packet that arrives at a link when its buffer is full is dropped immediately. If a packet is dropped, the destination of the packet will not receive it. So, if the destination then provides feed- back to the source that a packet was not received, this provides an indication of congestion. Clearly, such a scheme does not require even a single bit in the packet header to collect congestion information. However, strictly speaking, this type of congestion feedback can- not be modeled using our framework since we assume that a source’s data rate is seen by all links on the route, whereas, if packet dropping is allowed, some packets will not reach all links on a source’s route. However, if we assume that the packet loss rate at each link is small, we can approximate the rate at which a link receives a source’s packet by the rate at which the source is transmitting packets. Further, the end-to-end drop probability on a route can be approximated by the sum of the drop probabilities on the links along the route if the drop probability at each link is small. Thus, the optimization formulation approx- imates reality under these assumptions. To complete the connection to the optimization
- 44. 26 Mathematics of Internet architecture framework, we have to specify the price function at each link. A crude approximation to the drop probability (also known as packet loss rate) at link l is ((yl − cl)/yl)+ , which is non-zero only if yl = r:l∈ r xr is larger than cl. This approximate formula for the packet loss rate can serve as the price function for each link. Distributed algorithms: dual solution .................................................................................................................................................. 2.5 In this section we consider another distributed algorithm based on the dual formulation of the utility maximization problem. Consider the resource allocation problem that we would like to solve, max xr r∈S Ur(xr), (2.30) subject to the constraints r:l∈r xr ≤ cl, ∀l ∈ L, (2.31) xr ≥ 0, ∀r ∈ S. (2.32) The Lagrange dual of the above problem is obtained by incorporating the constraints into the maximization by means of Lagrange multipliers as follows: D(p) = max {xr≥0} r Ur(xr) − l pl s:l∈s xs − cl . (2.33) Here the pl’s are the Lagrange multipliers that we saw in Section 2.1. The dual problem may then be stated as min p≥0 D(p). As in the case of the primal problem, we would like to design an algorithm that ensures that all the source rates converge to the optimal solution. Note that, in this case, we are looking for a gradient descent (rather than the gradient ascent we saw in the primal for- mulation), since we would like to minimize D(p). To find the direction of the gradient, we need to know ∂D/∂pl. We first observe that, in order to achieve the maximum in (2.33), xr must satisfy Ur(xr) = qr, (2.34) where, as usual, qr = l:l∈r pl is the price of a particular route r. Note that we have assumed that xr 0 in writing down (2.34). This would be true, for example, if the utility function is an α-utility function with α 0. Now, ∂D ∂pl = r Ur(xr) ∂xr ∂pl − (yl − cl) − k pk ∂yk ∂pl = r Ur(xr) ∂xr ∂pl − (yl − cl) − k pk r:k∈r ∂xr ∂pl
- 45. 27 2.6 Feedback delay and stability = r Ur(xr) ∂xr ∂pl − (yl − cl) − r ∂xr ∂pl k:k∈r pk = r Ur(xr) ∂xr ∂pl − (yl − cl) − r ∂xr ∂pl qr. Thus, using (2.34), we have ∂D ∂pl = −(yl − cl). (2.35) Recalling that to minimize D(p) we have to descend in the direction of the gradient, from (2.34) and (2.35) we have the following dual control algorithm: xr = Ur −1 (qr) (2.36) and ṗl = hl(yl − cl)+ pl , (2.37) where hl 0 is a constant and (g(x))+ y denotes (g(x))+ y = g(x), y 0, max(g(x), 0), y = 0. Note that pl is not allowed to become negative because the KKT conditions impose such a condition on the Lagrange multipliers. Also, we remark that pl has the same dynamics of the queue length at the link when hl ≡ 1. It increases at rate yl, which is the arrival rate, and decreases at rate cl, the capacity of the link. Thus, the links do not have to compute their dual variables explicitly when hl ≡ 1; the dual variables are simply the queue lengths. The stability of this algorithm follows in a manner similar to that of the primal algorithm by considering D(p) as the Lyapunov function, since the dual algorithm is simply a gradient algorithm for finding the minimum of D(p). In Chapter 7, we will discuss practical TCP protocols based on the primal and dual formulations. When we discuss these protocols, we will see that the price functions and congestion control mechanisms obtained from the two formulations have different interpretations. Feedback delay and stability .................................................................................................................................................. 2.6 We have seen in Sections 2.4 and 2.5 that the primal and dual algorithms are globally, asymptotically stable if link prices were fed back instantaneously to sources and rate adjustments at sources were reflected instantaneously at links. Both assumptions are not true in reality due to delays. In this section, to illustrate how to analyze congestion control algorithms in the presence of delays, we will consider a simple one-link network and study the impact of feedback delay on the stability of a primal congestion controller.
- 46. 28 Mathematics of Internet architecture Consider a simple system with one link of capacity c and one source with utility func- tion log x. In this case, the proportionally fair congestion controller derived in Section 2.4 becomes ẋ = k(x) 1 x − f(x) . Choosing k(x) = κx for some κ 0, so that when x is close to zero the rate of increase is bounded, this controller can be written as ẋ = κ(1 − xf(x)). (2.38) Suppose that f(x) is the loss probability or marking probability when the arrival rate at the link is x. Then, the congestion control algorithm can be interpreted as follows: increase x at rate κ and decrease it proportional to the rate at which packets are marked, with κ being the proportionality constant. In reality, there is a delay from the time at which a packet is released at the source to the time at which it reaches the link, called the forward delay, and a delay for any feedback to reach the source from the link, called the backward delay. So, one cannot implement the above congestion controller in the form (2.38). Let us denote the forward delay by Tf and the backward delay by Tb. Taking both delays into consideration, we have y(t) = x(t − Tf ) and q(t) = p(t − Tb) = f(y(t − Tb)) = f(x(t − Tf − Tb)) = f(x(t − T)), where T = Tf + Tb is called the Round-Trip Time (RTT). The congestion controller becomes the following delay differential equation: ẋ = κ(1 − x(t − T)f(x(t − T))). (2.39) Let x̂ be the unique equilibrium point of this delay differential equation, i.e., 1 − x̂f(x̂) = 0. To analyze this system, we will assume that x(t) is close to the equilibrium and derive conditions under which the system is asymptotically stable, i.e., z(t) x(t) − x̂ converges to zero as t → ∞. To this end, we will linearize the system around the equi- librium point and derive conditions under which the resulting linear delay differential equation is asymptotically stable. Clearly, we would like to study the global, asymptotic stability of the system instead of just the asymptotic stability assuming that the system is close to its equilibrium, but typically studying the global, asymptotic stability is a much harder problem when there are many links and many sources.
- 47. 29 2.6 Feedback delay and stability 2.6.1 Linearization To study the asymptotic stability of (2.39), we substitute x(t) = x̂ + z(t) and linearize the delay differential equation as follows: ż = κ 1 − (x̂ + z(t − T))f(x̂ + z(t − T)) ≈ κ 1 − (x̂ + z(t − T))( f(x̂) + f (x̂)z(t − T)) , where in the second line we only retain terms linear in z in Taylor’s series. The rationale is that, since x is close to x̂, z is close to zero and z = x − x̂ dominates z2, z3, etc. Now, using the equilibrium condition 1 − x̂f(x̂) = 0, we obtain ż = −κ( f(x̂) + x̂f (x̂))z(t − T), (2.40) where we have again dropped z2 terms for the same reasons as before. To understand the stability of the linear delay differential equation (2.40), we introduce the following theorem, which characterizes the necessary and sufficient condition for a general linear delay differential equation to be asymptotically stable. Theorem 2.6.1 Consider a system governed by the following linear delay differential equation: ẋ(t) = −ax(t) − bx(t − T), (2.41) where the initial condition x(t), −T ≤ t ≤ 0, is specified. For any choice of the initial condition, x(t) converges to zero (the unique equilibrium of the system) as t → ∞ if and only if there exists a χ such that 0 χ π T , a = −c cos χT, c = χ sin χT , and − a ≤ b ≤ c. (2.42) If a = 0, the condition simplifies to 0 ≤ b ≤ π 2T . The shaded area in Figure 2.9 is the set of a and b that satisfies condition (2.42). Theorem 2.6.1 is a well-known result in the field of delay differential equations, and we will not prove it here. The following proposition is an immediate result of the theorem. Proposition 2.6.2 The linearized version of the proportionally fair controller, given in (2.40), is asymptotically stable if and only if κT( f(x̂) + x̂f (x̂)) ≤ π 2 . (2.43) Equation (2.43) suggests that the parameter κ should be chosen inversely proportional to T. This means that the congestion control algorithm should react more slowly when the feedback delay is large. This is very intuitive since, when T is large, the algorithm is
- 48. 30 Mathematics of Internet architecture b 0 a (−1/T, 1/T ) (0, p/(2T )) Figure 2.9 The set of a and b that satisfies condition (2.42). reacting to events that occurred a long time ago and thus should change the source rate x very slowly in response to such old information. In this section, we considered the problem of deriving delay differential equation models for congestion control protocols operating on a single link accessed by a single source. The modeling part can be easily extended to general networks with an arbitrary number of links and sources. However, the stability analysis of the resulting equations is considerably more challenging than in the case of the simple single-source, single-link model. We will not address this problem in this book, but we provide references in the Notes section at the end of the chapter. Game-theoretic view of utility maximization .................................................................................................................................................. 2.7 In earlier sections, we used utility maximization as a tool to understand network archi- tecture and algorithms. In doing so, it was assumed that all users will act as a team to maximize total network utility. It is interesting to relax this assumption and understand what happens if each user attempts selfishly to maximize its own utility minus any price incurred for transmitting at a certain data rate. In this section, we will first introduce a pric- ing scheme called the VCG mechanism, under which no user has an incentive to lie about their utility function. Therefore, the network can learn the true utility functions and thus solve the network utility maximization problem. However, the VCG mechanism requires each user to convey its utility function to the network. We will show that this communica- tion burden can be lessened under a reasonable assumption. Specifically, it will be shown that, in a large network such as the Internet, the selfish goals of the users coincide with the system-wide goal if users are price-takers, i.e., they take the price given by the network without attempting to infer the impact of their actions on the network. This is a reason- able assumption for the Internet since no one user can impact the network significantly and hence cannot use any reasonable inference algorithm to determine the impact of its own actions on the network price. On the other hand, it is also of theoretical interest to
- 49. 31 2.7 Game-theoretic view of utility maximization understand how far away we are from the team-optimal solution if users are strategic, i.e., the users are not price-takers. We will address this issue in this section and present a lower bound on the total network utility if the users are strategic. 2.7.1 The Vickrey–Clarke–Groves mechanism Recall that the goal of the utility maximization problem introduced in Section 2.2 was to solve the following optimization problem: max x≥0 r Ur(xr) subject to r:l∈r xr ≤ cl, ∀l, where xr is the rate allocated to user r, Ur is the user’s utility function, and cl is the capacity of link l. In a large network such as the Internet, it is not feasible to solve the utility maximization problem in a centralized fashion. However, to understand the game-theoretic implica- tions of the solution, let us suppose that the network is small and that a central network planner wants to solve the utility maximization problem. To solve the problem, the net- work planner first has to ask each user to reveal its utility function. However, there may be an incentive for a user to lie about its utility function; for example, by incorrectly reporting its utility function, a user may receive a larger data rate. Suppose that user r reveals its utility function as Ũr(xr), which may or may not be the same as Ur(xr). Then, the network will solve the maximization problem max x≥0 r Ũr(xr) subject to r:l∈r xr ≤ cl, ∀l, and allocate the optimal rate x̃r to user r. Suppose that the user is charged a price qr by the network for receiving this rate allocation. The goal is to select the price such that there is no incentive for each user to lie about its utility function. Surprisingly, the following simple idea works: user r is charged an amount equal to the reduction in the sum utility of the other users in the network due to the presence of user r. Specifically, the network first obtains the optimal solution {x̄s} to the following problem, which assumes that user r is not present in the network: max x≥0 s=r Ũs(xs) subject to s=r:l∈s xs ≤ cl, ∀l.
- 50. 32 Mathematics of Internet architecture The price qr is then computed as qr = s=r Ũ(x̄s) − s=r Ũ(x̃s), which represents the increase in the sum utility of other users due to the absence of user r. This pricing mechanism is called the Vickrey–Clarke–Groves (VCG) mechanism. Let us now consider how users will report their utilities knowing that the network plans to implement the VCG mechanism. Clearly, each user r will announce a utility function such that its payoff, i.e., the utility minus the price, is maximized: Ur(x̃r) − qr. We will now see that the payoff obtained by user r when lying about its utility function is always less than or equal to the payoff obtained when it truthfully reports its utility function. In other words, we will show that there is no incentive to lie. If user r reveals its utility function truthfully, its payoff is given by Ut = Ur(x̃t r) − ⎛ ⎝ s=r Ũs(x̄t s) − s=r Ũs(x̃t s) ⎞ ⎠ , where {x̃t s} is the allocation given to the users by the network and {x̄t s} is the solution of the network utility maximization problem when user r is excluded from the network, both being computed using the submitted utility functions. The superscript t indicates that user r has revealed its utility function truthfully. Next, suppose that user r lies about its utility function and denote the network planner’s allocation by x̃l, where superscript l indicates that user r has lied. Now, the payoff for user r is given by Ul = Ur(x̃l r) − ⎛ ⎝ s=r Ũs(x̄t s) − s=r Ũs(x̃l s) ⎞ ⎠ . Since {x̄t s} is the allocation excluding user r, it is independent of the utility function submitted by user r. If truth-telling were not optimal, Ul Ut for some incorrect utility function of user r, which would imply Ur(x̃l r) + s=r Ũs(x̃l s) Ur(x̃t r) + s=r Ũs(x̃t s). The above expression contradicts the fact that x̃t is the optimal solution to max x≥0 Ur(xr) + s=r Ũs(xs) subject to the capacity constraints. It is worth noting that we have not made any assump- tions about the strategies of other users in showing that user r has no incentive to lie. A strategy that is optimal for a user, independent of the strategies of others, is called a dominant strategy in game theory parlance. Thus, truth-telling is a dominant strategy under the VCG mechanism.
- 51. Another Random Scribd Document with Unrelated Content
- 52. mountain. Dark, bright, and shining with a metallic lustre, it looked like a solid wall of bronze built by Cyclopean hands, the stupendous jagged ridge which crowned it resembling the rampart of an embattled fortress. This appeared to be one of the grandest specimens of a trachytic lava bed to be found in any part of the world, and it formed one of the most interesting geological phenomena I had ever beheld. Looking at this stupendous mass, one could fairly realize how widespread and how tremendous in its proportions must have been the volcanic action of Ruapehu. The stream of lava which had formed this great deposit had evidently come from one of the many central craters of the mountain, and had rolled down in a molten stream for a distance of several miles, until it had gradually cooled into its present form. When gazing up at this singular monument, it could be seen that there was not a single flaw in its whole surface to mar the general outline of its colossal proportions. Here and there from the hard metallic surface, which shone like bronze by some powerful agency difficult to comprehend, blocks of the adamantine rock had fallen into the ravine below, but even every line of their surface was as sharp and as angular as if they had been just wrought into form under our eyes.
- 53. GREAT TRACHYTIC LAVA BED. When we had travelled a considerable distance up to the head of this wild gorge, we found it impossible to get out of it except by the way we had come, so we headed back again, and climbed, with great difficulty and at considerable risk, up the enormous bluff forming the entrance to the gorge, the sharp edges of the lava being particularly rough on our hands. Once at the summit of the bluff, we gained a long spur which formed the top of the great bed of lava we had examined in the ravine below, and which was here about 600 yards in width, as evidenced by the rugged outcrops of black rock that rose above the surface of the ground on every side. Travelling for a short distance up this steep ridge, we descended a rocky precipice to the right into another weird gorge, where the milky waters of the Whangaehu came bounding in a rapid descent over boulders and rocky precipices. We crossed the river at this point, and we kept the stream on our left for a considerable distance up the mountain. When we had followed up this ravine for a long distance we came to another scoria spur, mounting upwards towards the mountain. About two miles up this the ravine widened out, with high lava walls on either side, while right in the centre rose a high ridge of lava, which ended in steep, sloping ridges of fine scoria. The great snow peaks beyond now came into full view, and at a height of 5300 feet the ravine opened out on our left, and over the flat terrace above a large waterfall fell from a height of 150 feet over a semicircular precipice into a deep, rocky basin, and, as the vast volume of water poured on to the great rocks beneath, it resounded through the ravine like the echo of distant thunder. We named this the Horseshoe Fall from the shape of the precipice over which the water fell. From the Horseshoe Fall we mounted still higher up a very steep ascent on to a flat-topped scoria spur, which immediately to the right descended into a rugged ravine over a sheer precipice of 400 feet, while to the left of the ridge, which we followed up, rolled the Whangaehu, at a depth of about 300 feet in the gorge below, and
- 54. beyond which the giant form of one of the principal spurs of the mountain, built up of scoria and layers of lava, rose to a height of about 1000 feet above us. We were now high up in the mountain, and the cold wind from the snow-crowned glacier above swept over us with a chilly blast, while the colossal walls of rock, towering above on every side, cast their weird shadows around, and blocked out every ray of sunlight. We climbed for about three miles further up the dreary scoria spurs, the monotonous appearance of which was only relieved by the fantastic outcrops of lava rock, which jutted up above the surface in every direction, as if still hot and quaking with subterranean heat. One of the most remarkable features about these fantastic outcrops of lava was that time and the devastating effects of the elements to which they must have been subjected for hundreds, nay, thousands, of years, appeared to have left no traces upon them, the hard, metallic-looking surface of the rock being as sharp in outline as if it had but just got cool from the terrific heat of the stupendous fires, which had left their impress in every direction over the face of the mountain. Not a sign of vegetation was to be seen anywhere. We could not even get a glimpse of the country around, as the windings of the enormous gorge had led us, as it were, into the very heart of the mountain, and had surrounded us with its high, rugged walls. As we climbed still further to the glacier- crowned heights above us, the appearance of this wild ravine became still more desolate; rugged, craggy boulders of black rock were scattered about the slopes in every direction, and we had to climb over huge masses of rock that barred our pathway. Thick icicles now covered the ground, hung in festoons from the rocks, and bedecked the high precipices in the form of a glittering fringe, while the snow was not only on the heights above, but in the deep ravines beneath us. In the distance we could hear the loud roaring of a cataract, and, as we pressed on, the sound of the falling water resounded louder and louder, and at an altitude of 6250 feet another waterfall, far larger and more beautiful than the one we had previously discovered, burst into view. We had hoped that this would prove the source of the river, as it was now late in the day, and it was clear that we would not have much more time for climbing if we
- 55. wished to gain our camp before nightfall. We soon found, however, that the great gorge still wound into the mountain for 1000 feet above, and that the true source of the river was yet further ahead. We took our first rest at this stage, and gazed in admiration at the leaping volume of water in front of us. Here, on our right, rose a gigantic bluff of lava and conglomerated rock, while round this frowning point and coursing down the steep incline of the gorge, up which we were ascending, swept the white waters of the Whangaehu, until the whole volume, concentrated into a narrow rocky channel, burst over a precipice with a fall of 300 feet into the rocky gorge below. This was one of the most beautiful and unique cascades I had ever seen. All around the craggy rocks were white with a deposit of alum from the spray of the fall, while the water, of a milky hue, poured over the precipice in a continuous frothy stream, which appeared by its whiteness like folds of delicate lace. This beautiful cascade had not the sparkle and glitter of ordinary waterfalls, but a soft, milky appearance different to anything I had ever beheld before. The big, circular, rock-bound basin, into which the water fell, was decorated around its sides with fantastic clusters of icicles, all of the same milky whiteness, and mingling as they did with the still whiter snow, they served to complete one of the most singular and attractive features of this weird ravine. We named this the Bridal Veil Fall on account of its peculiar lace-like appearance.
- 56. THE BRIDAL VEIL FALL. Leaving the Bridal Veil Fall to dart over its echoing rocks, we struck up the steep, precipitous ridge ahead, where we could still see the white waters of the river coming down, as it were, from the very summit of the mountain. Here the whole surroundings had a most wild and romantic appearance, and we seemed to have entered a dismal solitude where there was no sound but the rushing of waters as they dashed over the rocky precipices, or rolled among the stupendous boulders which lay scattered about the winding channels of the deep ravines. We pushed on as fast as we could over an
- 57. enormous outcrop of lava, and when we had reached 6750 feet fresh wonders still seemed to call us onward. At this elevation we discovered two cascades falling over a steep, bluff-like precipice, and only at a short distance apart from each other. These two shoots of water, which appeared to be of the same proportions, fell from a height of about 100 feet into the ravine below, and then dashed onward to leap over the precipice of the Bridal Veil. All around the rocks were resplendent with icicles, and with the white coating of alum appeared like alabaster. We named these the Twin Waterfalls on account of their singular resemblance to each other. From this point of the great ravine we again mounted up precipitous rocks and lava ridges, one of which we had to climb hand over hand for a height of fifty feet. The river now, as far as we could discern, appeared to pour out of the snow as it came down in a rapid torrent through a precipitous ravine, along the side of which we crawled with difficulty. As we mounted higher the stupendous rocks, over which we had to make our way, were piled about in the most intricate confusion, and in one place we had to pass over an outcrop of trachytic rock which was broken into angular pieces, as sharp as flint, and fractured in every direction, as if it had been subjected at some period to the force of a terrific explosion. It required great care to get over this difficult point, as there was only room enough to crawl along between the wall of rock on one side, and a precipice of 200 feet on the other, which fell with a sheer descent into a big, circular, ice-bound pool, into which the milky waters of the Whangaehu poured in the form of foaming cascades. Here, around on every side, rose steep precipices, great buttresses of black lava mounted up in the form of stupendous bluffs that supported, as it were, the rampart-like heights above, while right in front of us, and towering to an altitude of over 1000 feet, was a glacier slope crowned with craggy peaks, which stood out in bold relief against the sky. This rugged locality was one of the most singular of the whole mountain. No region could be wilder or more desolate in appearance. There was nothing but the blue heavens above to relieve the frigid glare of the ice, the cold glitter of the snows, and
- 58. the dreary tints of the frowning fire-scorched rocks. We now seemed to be in a new world, where solitude reigned supreme, and where Nature, casting aside her most radiant charms, looked stern and awe-inspiring in her mantle of ice and snow. Right under the snowy glacier above us were wide, yawning apertures, arched at the top, and framed, as it were, with ice, in the form of rude portals, through which the white waters of the river burst in a continuous stream. These were ice caves. Climbing over the rough boulders, and then descending into a rocky channel, where the water mounted over our knees, we entered the largest of these singular structures, when a wonderful sight met the gaze. We found ourselves in a cave of some 200 feet in circumference, whose sides of black volcanic rock were sheeted with ice, and festooned with icicles, all grandly and marvellously designed. At the further end from where we entered was a wide, cavernous opening, so dark that the waters of the river, as they burst out of it in a foaming, eddying stream down the centre of the cave in which we stood, looked doubly white, in comparison with the black void out of which they came. We were now right under the enormous glacier that covered the summit of the mountain, and the roof of the cave was formed of a mass of frozen snow, which had been fashioned by some singular law of Nature into oval-shaped depressions of about two feet in height, and a foot and a half broad, all of one uniform size, and so beautifully, and so mathematically precise in outline, as to resemble the quaint designs of a Moorish temple; while, from all the central points to which the edges of these singular designs converged, a long single icicle hung down several inches in diameter at its base, perfectly round and smooth and clear, tapering off towards its end with a point as sharp as a needle. High up on our left, in the walls of the cave, were two apertures like the slanting windows of a dungeon, through which the light streamed, giving a soft, mysterious halo to the whole scene, which looked weird and indescribably curious. We had brought candles with us, and lighting them, we pressed forward to explore the deep cavern beyond, but to do so we had to climb over sharp, slippery rocks, which were
- 59. covered with a coating of ice, as if they had been glazed with glass, while the white waters streaming beneath us fell into a deep, eddying pool. We managed, after some difficulty, to cross the stream in the second cave, and to penetrate a considerable distance along the treacherous rocks into the very centre, as it were, of the great mountain; but, just as we were winding along a kind of subterranean passage, which looked like a short cut into eternity, our lights went out, owing to the water falling from above, and, as we could hear nothing but rushing waters ahead, we, with some difficulty, beat a retreat into the first cave, which looked like a fairy palace in comparison with the dark cavern we had just left. These caves were at an altitude of 7000 feet above the level of the sea, and we were now at the true source of the remarkable river. Wherever the water poured over the rocks it left a white deposit, and when we tasted it, it produced a marked astringent feeling upon the tongue, leaving a strong taste of alum, sulphur, and iron, with all of which ingredients, especially the two former, it appeared to be strongly impregnated.[51] It is a remarkable and interesting geographical fact that the waters which form the source of the Waikato River burst from the sides of Ruapehu, within a short distance of the Whangaehu, and at almost the same altitude. Both streams run almost parallel to each other for a long distance from their source, and then, as they reach the desert, they gradually diverge and divide the two great watersheds of this portion of the country, the Waikato flowing to the north into Lake Taupo, and the Whangaehu to join the sea in the south. There is, I believe, no place in the world where two great rivers may be seen rising at an altitude of over 7000 feet in the sides of a glacier- clad mountain, and rolling for miles, side by side, down its rugged slopes, the waters of the one of alabaster whiteness, and the waters of the other as pure and as limpid as crystal, and each forming the dividing waters of an area of country of nearly 100 miles in length. It had taken us nine hours to reach the ice caves, and as it was now late in the day we began to descend with all haste, in order, if
- 60. possible, to reach the point where we had left our horses before nightfall. As the sun went down the wind blew with a freezing blast, and as we descended precipice after precipice, and ridge after ridge, and the tints of evening crept gradually over the dismal sides of the mountains, our course appeared long beyond measure. When we got near to the immense mass of lava we had beheld in wonder in the morning, the shades of night overtook us, and it was with great difficulty we could pick our way over the rough boulders of the dark, weird gorge, which now looked like Dante's Inferno with the fires put out. We again struck the waters of the Whangaehu, and shining as they did like a white streak in the darkness, we were enabled to follow them up until we came to our camp. We soon had our tent erected under the lee of a cluster of scrub, which served to protect us from the fury of the wind, which now swept in strong blasts across the scoria plains. Our camping-place was as near as possible in the centre of the desert, and at a point which indicated an elevation of 3000 feet above the level of the sea. It might, in fact, be considered as the highest point of the great central table-land, for it was here that the watershed divided, and flowed on the one hand to the north, and on the other to the south, as previously described. A drink of tea and a biscuit formed our only meal, and then we lay down to pass one of the roughest and most uncomfortable nights we had ever experienced. About midnight a great storm of wind swept over the plains, and dark clouds gathered over the heavens, and the rain continued to descend in torrents throughout the night. Fortunately for us, the few straggling bushes around served to break the force of the blast, otherwise everything would have been blown away.
- 61. FOOTNOTES: [51] Near to this point, on the summit of the mountain, there is a lake formed by an extinct crater, filled by subterranean springs, and it is likely that the Whangaehu may in some way be connected with it. It is, however, clear that there must, of necessity, be strong subterranean springs in this portion of the mountain, to account for the large volume of water forming the source of this river, as likewise extensive deposits of alum, of some form or another, to cause the complete discoloration of the waters by that mineral. I believe that this singular river will be found to possess great medicinal properties for the cure of rheumatic affections and cutaneous disorders.
- 62. CHAPTER XXI. KARIOI. Our commissariat gives out—The Murimotu Plains—The settlement—The homestead—The welcome—Society at Karioi—The natives—The Napier Mail. WHEN morning broke over our camp on the Onetapu Desert the rain poured down without intermission, the flood waters of the great mountain swept over the plains in every direction, and the whole country, obscured for the most part by heavy mists, looked indescribably desolate. To remain camped where we were was simply to court starvation. We were now nearly 100 miles from where we had started, and, while our horses were so weak as to be hardly able to walk, through exposure and want of proper food, our own commissariat was reduced to its lowest. Yet, up to this point, we had not accomplished one-half of our intended journey. It is true we had ascended the great mountains, and had seen their wonders, but there were still dense forests and unknown regions to be traversed. We had been told before setting out from Tapuwaeharuru that a sheep-station known as Karioi could be reached by travelling in the direction of Whanganui. This was out of our course, but there was no alternative but to make for it, in order to recruit our horses and replenish our commissariat. We therefore looked towards this place as a kind of Land of Promise, flowing with the proverbial milk and honey. Once clear of the sterile desert, we took a southerly course along the Whangaehu River, until we reached the magnificent tract of open country known as the Murimotu Plains. This wide district, which forms, as it were, the southern slope of the great central table-land, stretches in the west to the borders of the forest country which
- 63. extends to the valley of the Whanganui, while to the eastward it is bounded by the lower hills which branch out in the form of extensive ridges from the southern end of the Kaimanawa Mountains. These plains, which resemble in general features those to the north of the desert forming the Rangipo plateau, are covered with a network of streams and rivers, and, for the most part, with a luxuriant growth of native grasses, the ridges and lower hills which dot them towards the east being carpeted with low fern. We travelled across the plains principally by compass bearing, and we had to cross many swollen streams in our course, the waters of one pouring in the form of a cascade into a deep circular basin. Beyond this point we again struck the Whangaehu, which had now become a wide stream, but its waters were still quite white. After a journey of nine hours, during which time the rain and wind never ceased, we sighted a three-rail fence, which we joyously hailed as the first sign of civilization we had seen for some time. The fence proved to be the horse-paddock of the station, and following it along, we soon came to our destination. We found the various whares and rustic huts composing the settlement of Karioi scattered promiscuously about the banks of the Tokiahuru River, a tributary of the Whangaehu, which wound through the station in its course to the south. The site of the settlement was most delightfully chosen, and the views from every part of it were most attractive. Upon arrival at the homestead all hands came to greet us, although nobody knew who we were, nor where we had come from; nor were we asked whether we were hungry. With true bush etiquette, that was taken as a matter of course, and we were soon invited to partake of what was to us a magnificent repast. We found the good people of Karioi true cosmopolites, ready to enter into conversation and to furnish all the news in their power in exchange for what we could tell them of the country we had passed through. Strange as it may appear, in this small settlement of whites and natives, which formed the last link in the chain of European settlement stretching from the East Coast into this portion of the
- 64. country, our pleasant party at Karioi was composed of representatives of many nations. A Mr. Rees, who had come up from Whanganui, was a native of Australia, and had served in the armed constabulary at Parihaka; Mr. Newman, our host, hailed from the South of England; one of the hands was a New Zealander, another an Austrian, a third came from the Alpine districts of the Tyrol, and another from the Land o' Cakes, while the native race was here represented by several hapus of one of the principal Whanganui tribes. To listen to the spirited description given by Mr. Rees of the Parihaka campaign, and to his delineation of Te Whiti[52] and other notable chiefs, to participate in the varied conversation upon the wonders of the surrounding country, to chat with the Tyrolese in his native tongue, and to feel that a great vacuum had been filled in our insides, was so great a change to what we had recently experienced, that we now seemed to be partaking of the pleasures of the varied society and seductive luxuries of a first-class antipodean caravansary, where hospitality was boundless and good-fellowship the order of the day. In the evening we visited the native kainga, and spent some time with the Maoris in the wharepuni. There were about twenty natives present, men, women, and children, and in the centre of the primitive apartment blazed a huge fire, which threw out a terrific heat, and rendered the place almost unbearable. The natives were mostly short of stature, with hard features, and I remarked that they spoke with a much harsher accent than those further to the north, and that they clipped many of their words in a remarkable way. When Turner inquired for an explanation of this habit, they stated that their great ancestor, Ngatoroirangi, when he came over in the Arawa canoe was engaged in baling out that craft during a storm, and that whilst so doing he caught a severe cold, which caused him to speak in a sharp, halting kind of way, which has been imitated ever since by many of the Whanganui tribes, who claim descent from that celebrated chief, and who has been before alluded to in a previous chapter as the first explorer of the country.
- 65. On the second evening after our arrival at Karioi, and when all hands were assembled in the homely whare watching the big pots boiling for supper, in fact, when everything looked couleur de rose, a horseman rode up bespattered with mud from head to foot, bringing a packet of papers and a handful of letters. This was the Napier mail, and we hailed it with delight, as it was the first tidings of civilization we had obtained since we left Tapuwaeharuru, over twenty-four days past. We anxiously scanned the telegrams, to see what had arisen with regard to the Mahuki difficulty, when we learned that the native minister was about to leave Alexandra to travel by way of the Mokau River to Taranaki, in company with a body-guard of armed natives, under the chief Hone Te Wetere, that Mahuki's tribe was going to oppose his journey through that portion of the country, and that a gallows had been erected at Te Kumi, to hang the native minister and all other whites that might be caught across the aukati line. This news, which was about the most exciting item of intelligence the papers contained, was discussed with much gusto. The mere idea of war in the King Country—Alexandra in flames and a minister hanged—seemed to act like magic upon the heroic hearts of the cosmopolitan community at Karioi. This new phase of the native difficulty Turner and myself treated with apparent indifference, but in reality, coming as it did at that moment, we secretly deemed it of no small concern, as we had determined to leave Karioi on the following day, re-enter the King Country at its southern end, and come out somehow across the northern frontier. In the suggestive words of the schoolboy, we never let on; but, as a matter of fact, from the time we left Karioi until we crossed the aukati line at Alexandra, five weeks afterwards, this significant item of intelligence was our bête noire, as during our progress northward we could never tell from day to day what difficulties we might run into with the natives by reason of the Hursthouse-Mahuki episode. FOOTNOTES:
- 66. [52] Te Whiti and Tohu, the Maori prophets, were captured in 1882, at the instance of the Government, by the armed constabulary at the native settlement of Parihaka, for inciting their followers to commit acts of lawlessness against the European settlers.
- 67. CHAPTER XXII. FOREST COUNTRY. The start from Karioi—On the track—Te Wheu maps the country —The primeval solitude—Terangakaika Forest—The flora— Difficulties of travel—The lakes—Birds—Pakihi— Mangawhero River—Gigantic vines—Fallen trees—Dead forest giants—Mangatotara and Mangatuku Rivers—A Slough of Despond—Dismal Swamp. WE were invited to stay as long as we liked at Karioi, but as we were anxious, as the weather was breaking, to push forward as soon as possible, we had to content ourselves with two days' rest, and on the morning of the 24th of April we again set out. Having examined all the principal natural features of the country for over 200 miles northward of this point, I determined to traverse the plains to the southward of Ruapehu, and then pass through the great forest to the westward of that mountain, in order to reach the Manganui-a-te- Ao River near to its junction with the Whanganui, and afterwards proceed northward through the King Country, by the best route we could find. We had heard from the Maoris that there was an unfrequented native track, leading somewhere in the direction of the Manganui-a- te-Ao River, through the region we were going to explore, but it was at all times difficult to travel, and still more difficult to find, unless by those well acquainted with the country. We were told that it led over high mountains and steep hills, and across rivers and boggy creeks innumerable. With these difficulties ahead, we endeavoured to secure the services of a native guide to accompany us as far as Ruakaka, the Maori settlement on the Manganui-a-te-Ao, but no one among the many natives we treated with was willing to make the
- 68. journey; all excusing themselves upon the plea that they did not like to undertake the responsibility of introducing Europeans into the country. At last, after considerable parleying, a native, named Te Wheu, agreed to put us on to the track for a consideration, so we set out without delay. As it was clear that we should have to traverse the great forest on foot, and have much difficult travelling, we abandoned our sumpter-horse at Karioi, together with our gun, which, up to this time, had been of little service, and reducing our camp equipage to the lowest, packed our horses with the tent and blankets, and carried just sufficient provisions to last us for three days, by which time we hoped to reach Ruakaka. We picked up our guide Te Wheu at the Whakahi kainga, and took a westerly course across the Murimotu Plains, which extended, in the form of a well-grassed tract of country, as far as the southern base of Ruapehu, and beyond which a thick, and apparently impenetrable, forest rose, in the form of a barrier of varied and beautiful vegetation. Near to the southern end of the great mountain we passed the Maori settlement of Ohinepu, situated on a slope, with low mounds on its western side, on which were several tombs. We crossed the Waitaki Creek, flowing southerly from the mountain, and near to a native kainga, situated on a rock-bound hill, beneath which the Mangaehu stream flowed like a moat. From this point, after passing a swamp, we soon hit the so-called track, which would have been impossible to find without native assistance, hidden as the entrance to it was away in the winding of the dense forest. Here the colossal trees rose up on every side, a thick undergrowth of the most varied shrubs hedged us in wherever we turned, and coiling roots of trees, and black, swampy mud, with here and there a blazed tree, was the only indication of our course. To ride through this was impossible, and we therefore had to dismount and lead our horses. Te Wheu accompanied us to the summit of a densely-wooded hill, which rose 500 feet above the plain we had recently left. Before leaving us, however, we induced him to sketch out roughly, on the
- 69. ground, the lay of the country we were about to traverse, when he gave us the names and directions of the principal rivers and creeks we should have to cross. He then told us that as he was known at Ruakaka we might mention his name to the natives, but that he could not guarantee our safety, as the Maoris of that part were true Hauhaus, and objected to pakehas going into their territory. As soon as Te Wheu had disappeared on his homeward track we bent on our way through the great primeval solitude. We had been so much out in the open country hitherto, that the scenery of the forest seemed at first like a pleasant change, but this idea was completely altered after a journey through it of seventy miles. The Terangakaika Forest, which extends from the western slope of Ruapehu, forms part of the wide expanse of bush country which stretches into the valley of the Whanganui, and thence, westerly, to Taranaki. It grows to within 1000 feet or so of the snow-line of the great mountain, and covers nearly the whole of its western side, as well as the wide plateau near this portion of its base. When we had got well on our way, we found this enormous wilderness spreading itself out over a perfect network of broken, rugged ranges, which in many places appeared to have been hurled about by the terrific throes of an earthquake. The soil was everywhere of the richest description, and many of the colossal trees averaged from thirty to forty feet in circumference at the base, and towered above us to a height of considerably over 100 feet, forming a grand canopy of foliage, above and beyond which nothing could be seen but the blue of the sky and the golden rays of sunlight as they lit up the bright- green tints of the splendid vegetation. Among the largest trees was the towai, which here attained to a larger growth than any we had previously seen, its enormous branches supporting a canopy of small, shining, green leaves, giving it a very beautiful appearance. Next to the towai in size was the rimu, its pendulous branches making it everywhere a conspicuous and attractive feature, but it is worthy of remark that where on the volcanic soil, formed by the decomposition of rocks of that kind, the
- 70. towai attained to its largest size, we found that the rimu grew to larger proportions on the marly soil we afterwards met with as we approached the valley of the Whanganui. It was also in the latter locality that the rata likewise attained to its most colossal proportions; many of these parasitical giants clasping the enormous rimus in a death-like struggle for existence. Besides these grand representatives of the vegetable world, which formed by far the greater part of the forest growth, we also found many noble specimens of the hinau, the tawa, the miro, and matai, the berries of the three former trees being scattered over many parts of our track in enormous quantities. In fact, almost all the principal trees peculiar to the forests of the North Island here flourished in wonderful luxuriance, together with an extensive variety of shrubs and ferns, while mosses, lichens, and trailing vines clothed the tall trees to the topmost branches in gay festoons of vegetation, which presented the brightest and most variegated hues. With all these marvellous creations of the vegetable world around us, we soon, however, found that travelling through the great forest wilderness was both fatiguing and difficult. There was not 100 yards of level ground, and the native track, what little there was of it, led over steep precipitous ridges, from 200 to 400 feet in height, which were constantly ascending and descending in a way which rendered our progress not only slow, but difficult and tedious. The steep ascents, up which we had to drag our animals at every turn, were as slippery as glass with the dank humidity of the surrounding vegetation, and were encumbered with the gnarled roots of trees in every direction, while the descents were in many places so precipitous that it was impossible for us to lead our horses without the risk of them rolling over on us, so we were compelled to let them go their own way down, when they would, owing to the slippery nature of the soil, slide down on their haunches and never stop until they were pulled up by a boggy creek below. These creeks, filled with thick, black mud, impeded our progress at every descent, and struck terror into our animals, so that we would often have to flog them across, when their struggles to climb the slippery
- 71. ascents on the opposite side fatigued them fearfully. It was not as if we had only to encounter these difficulties now and again, but they presented themselves in the most aggravated forms at every few hundred yards of our journey, from morning until night, and for day after day. Thus, amid solitude and shade, we pursued our onward way, now plunging into the deep and gloomy chasms of the mountains, and anon rising to the opposite ascent, till the distant openings in the forest, restoring the welcome sunlight, revealed mountain and valley yet to be traversed. Our first day's journey brought us to two lakes, which Te Wheu told us we would find somewhere along our track, and which would serve as our first camping-place. A little before dusk we came suddenly out of the forest into a small, circular, open flat, fringed with toetoe, and covered with a luxuriant growth of native grass. On our left, a grassy ridge rose in a semicircle, and all around the open space the trees rose one above the other in the most attractive way, while a variety of shrubs dispersed about in the most picturesque order, made the place appear like a perfect garden. Right in the very centre of the natural parterre was Rangitauaiti, a beautiful lake of a complete circular form, and the water of which, looking like a polished mirror, was of the deepest blue. Beyond this flat, the native name of which was Rangitanua, and separated only by a low ridge crowned with a luxuriant growth of vegetation, was another open space, in the centre of which was Rangitauanui, an oval-shaped lake larger than the former, but in which the water was of the same limpid blue. The trees on the further side rose in a dense forest growth, and as they came close down to the water, they were reflected in the depths below with grand and beautiful effect. In fact, the whole surroundings of these lakes appeared so attractive after our long journey through the forest, that we seemed to have got into a quiet corner of paradise. We remained here the following day, as much to rest ourselves as our horses, and we enjoyed the quiet romance of the place immensely. The primeval region was a perfect elysium for birds of all kinds, and at daylight the forest was alive with their warblings, and
- 72. with the soft note of the tui came the harsh screech of the kaka; flocks of pigeons circled about the tree-tops, and gaily-plumed parrots winged in a rapid flight through the air. One of the latter birds, which we found dead, had a green body and a light green breast, with a dark crimson patch on the head, and a small patch under the eye of the same colour. This was the first bird of the kind I had seen in New Zealand, and it resembled very much one of the green mountain-parrots of Australia. When we left our camp at Rangitanua it was in the hope that we should be able to reach the Manganui-a-te-Ao by nightfall, but in this calculation we were greatly out. We passed round the western end of Lake Rangitauanui and entered a boggy, densely-wooded country, where the trees, especially the rimu, were larger and more gigantic in proportions than any we had yet seen. The dense forest here literally rained with moisture, and, as we had to lead our horses, we were at places compelled to plunge through swamps where the big roots of trees threatened to break our legs and those of our struggling animals. We crossed a branch of the Mangawhero, and towards sundown came to a small open flat called Pakihi, surrounded entirely by the forest, and where we found excellent feed for our horses. It had taken us seven hours of hard travelling to reach this spot, and during that time we had to cross no less than ten boggy creeks, besides other streams. The Mangawhero River ran round the western side of this small oasis, the towai-trees forming a conspicuous feature along the banks of the stream. We camped at Pakihi for the night, the stillness of the place being only broken now and again by the shrill note of the whistling duck. We struck camp at Pakihi early on the following day, but had some difficulty in crossing the Mangawhero, which we found to be a broad, rapid, boulder-strewn stream. The banks were very steep and slippery, and when we had our horses down on one side we had great difficulty in getting them up the other. As we got again into the thick of the forest the vegetation became denser, and the rimu-trees, seeming to increase in size, shot up for
- 73. over a hundred feet as straight as gun-barrels. Where some of these giants of the forest had fallen across our track, we had often to cut a way round them for our horses, through the thick shrub and tangled vines, the latter of which impeded our progress at every turn, by tripping us up, and winding round the legs and necks of our animals like treacherous snares. The enormous rata-vines had been very troublesome up to this point, but now we had to do battle not only against them, but against the supple-jacks, which we found growing everywhere in a perfect network of snakelike coils on the soft, marly soil of the country we were now in. It was nothing to have a supple- jack round the neck and a rata-vine round the legs at the same time, while our horses would often get so entangled that they would refuse to move until we had cut them a clear passage out of their difficulties. In many instances, owing to surrounding obstacles, there was no alternative but to make them leap over the fallen trees in our way, and when not able to do this, the animals would jump on to them and leap down like dogs. Indeed, the tricks that they had to go through to get over these and other impediments rendered them almost as clever as circus-horses. Another frequent feature we noticed was that where the great trees had apparently been lying for some time, the seeds of other trees had fallen upon them, and, germinating into life, had sent their roots down into the very heart of these decaying vegetable monsters. In this way it was no uncommon sight to see three or four different species of large trees living and flourishing upon the dead trunks of these forest giants. We crossed the Mangatotara River twice, and after passing through a very rough and broken portion of the great wilderness, we fell in with another river, called the Mangatuku, and which we had to cross three times in its winding course. Both of these streams appeared to drain a large area of country, and so dense was the vegetation along their banks that it was only here and there that a ray of sunlight shot through the thick canopy of green upon them.
- 74. During this portion of our journey we came across a complete network of tracks made by herds of wild cattle, and which led us about to all points of the compass, until we found it impossible to make out in what direction we should shape our course. We climbed a tree on the summit of a high ridge, but we could see nothing but the snowy summit of Ruapehu in the distance, while all around us, in every direction, was an apparently endless expanse of forest. From this point the country began to fall rapidly, and it was evident that we were descending into the valley of the Whanganui. After nine hours of incessant travelling, from the time we left our camp in the morning, we had crossed no less than thirty boggy creeks, besides other streams, and now that dusk had overtaken us, we found it impossible to proceed any further. We were now in the midst of a swampy portion of the forest, which seemed like a veritable Slough of Despond, and which, judging from the way the ground had been rooted up in every direction, appeared to be a kind of wild-pig elysium. Throughout the whole distance we had come, the country had been grubbed up by these animals, many of which we saw of great size, and apparently of true wild-boar ferocity. We were compelled to pitch camp in this uninviting spot, our horses faring badly, as there was little or no food for them beyond what they could get from the trees and shrubs. This was one of the most dreary places in which we had camped during our journey. The night was dark and wet, the colossal trees rose like spectres around us, the enormous vines that twisted and twirled about them like coils of vipers, were covered with grey moss, which hung in dank festoons often over two feet in length, like enormous spider-webs, and as the rain poured down from the branches above, the whole place looked as if it had been saturated with moisture for centuries. We cut down branches of the nikau, and made a tolerably good bed for ourselves after smoothing down the ground where the pigs had been rooting; and we named the place Dismal Swamp on account of the swampy nature of the country and the truly dismal character of the whole surroundings. This camp was situated at an altitude of 1700 feet
- 75. above the level of the sea, or just 560 feet lower than our camp at the lakes.
- 76. CHAPTER XXIII. RUAKAKA. The wharangi plant—Enormous ravines—Ruakaka—Reception by the Hauhaus—The chief Pareoterangi—The parley— Hinepareoterangi—A repast—Rapid fall of country—The Manganui-a-te-Ao—Shooting the rapids—The natives— Religion—Hauhauism—Te Kooti's lament—A Hauhau hymn. WE struck camp at Dismal Swamp at daybreak, and travelled on for many miles through the same character of country we had been traversing for the past five days. Before leaving us, at the entrance to the forest, Te Wheu had warned us not to allow our horses to eat a certain shrub, called by the natives wharangi, which we found growing for many miles along our course, with broad, oval-shaped, light-green leaves. This plant, when eaten by horses or cattle, is said to produce stupefaction, followed by convulsions and death, the only known cure being instant bleeding from the ears. Our own animals were now ready to eat anything, and made desperate efforts to devour the foliage of the trees, and, as we went along, we had great difficulty in keeping them away from this poisonous shrub, which they would devour greedily. During this journey the boggy creeks and fallen trees became more troublesome than before, and the hills steeper and more difficult to climb. We passed along one ridge, with enormous ravines below, some of which were of circular shape, and in appearance not unlike extinct craters, while deep down in their depths, all around their sides, and up to their very topmost ridges, nothing was to be seen but a luxuriant growth of the most varied and beautiful vegetation. Here, too, the geological character of the country changed, the trachytic rocks giving place to a sandstone
- 77. formation, covered with a stratum of thick, marly earth, which was so slippery in places that we could hardly manage to get along. During the greater part of the morning the rain had been pouring down in torrents, and what with the swollen condition of the creeks, the slippery nature of the soil, and the starved condition of our horses, our prospects of ever reaching Ruakaka seemed to be hopeless. At last, about two o'clock in the afternoon, we hailed with delight a break in the forest, and we came suddenly into a hilly region, where the tall fern grew higher than our horses' heads. After travelling a considerable distance through this country, we mounted to the top of a high hill, when we beheld, 200 feet beneath us, a fine, open valley, sunk like a pit, as it were, in the heart of a mountainous region, where enormous forests stretched away as far as the eye could reach on every side. Right down the centre of the valley, as far as we could see, we could trace the winding course of the Manganui-a-te-Ao, marked by precipitous cliffs of grey rock, which rose perpendicularly from the waters of the river to a height of 300 feet, while above these, again, on the further side of the stream, were terraces of rounded hills, backed by conical mountains, which mounted, one above the other, to a height of 3000 or 4000 feet, covered from base to summits with a thick mantle of luxuriant vegetation. On the side where we had emerged from the forest the valley was bounded by round-topped, fern-clad hills and flat, terrace-like formations that descended, in the form of gigantic steps, into the plain below, where the whares and cultivations of the natives, stretching for miles along the course of the stream, appeared dotted about in the most picturesque way. Taken altogether, the whole place had a singularly wild appearance as we gazed upon it, and now that we could see everything from our point of vantage without being seen, we wondered what kind of a welcome we should meet with from the natives. We led our horses down the steep, slippery track into the valley, and as we were now seen by some of the Maoris, there were loud shouts that pakehas had arrived, and the natives came out of the whares and awaited our approach in front of the wharepuni. We could see at
- 78. a glance that the words of Te Wheu were correct, and that the natives, so far as we could discern by outward signs, were veritable Hauhaus, alike in dress and bearing, while both men and women had a singularly wild and even savage appearance when compared with all other tribes I had seen in different parts of the country. It was likewise clear that they did not welcome us at first with any demonstrations of cordiality, and upon Turner inquiring for the chief, they replied that he was away at a wild-pig hunt, and that we must wait till he came. The natives then squatted around us, and scanned us narrowly, while we looked on with an air of apparent indifference. In the meanwhile a messenger had been despatched for the chief, whose name, we now learned, was Te Pareoterangi, and after a short delay he appeared before us, with half a dozen wild-looking natives, carrying a double-barrelled gun over his shoulder. He was a man below medium height, but of singularly massive build, broad- chested and broad-shouldered, with a well-formed head, and singularly well-moulded features. Indeed, his heavily-knit frame, intelligent air, and almost oriental cast of countenance made him stand out in marked contrast to the other natives, who were, for the most part, unlike the generality of their race, remarkable for their diminutive stature and ungainly appearance. When Te Pareoterangi came up, he squatted down with a sullen air, without going through any form of salutation, and then, after a pause, asked us what we had come for, and upon Turner telling him that he had brought the pakeha, who was travelling for pleasure, a titter ran round the circle, for, if we did not look it, we felt half- starved, we were drenched to the skin, and covered from head to foot with mud, and the chief, evidently realizing all the unpleasant features of our position, naively remarked, How can the pakeha travel for pleasure through such a forest as you have come? At which an old tattooed savage observed, Their horses are only rats; how did they get here? These pakehas have singular ways. This was said with a sinister smile from the old man, and in anything but a complimentary tone. Many other questions were put to us, and the parleying kept on, by fits and starts, for a good half-hour, during
- 79. which time the natives displayed no token of friendship, the only manifestations we received in this respect being from the dogs and pigs, the latter even going so far as to scratch their backs against our legs. RUAKAKA. At last an old woman, who had been watching the proceedings keenly, and whose appearance reminded me of one of the witches in Macbeth, suddenly rose, and stepping with an excited air into the middle of the circle, waved her bare right arm round her head, and shouted at the top of her voice, Haeremai! Haeremai! Haeremai! [53] And then turning to the natives, in an equally excited way, said, The pakehas have been following up the rivers of great names, and have come to our homes; they are hungry, and we must give them food. The words of this weird dame, whom we afterwards found was the chieftainess Hinepareoterangi, and mother of the chief of the hapu, acted like magic upon the natives, who at once took charge of our horses, while the women hastened to prepare a meal, old Hinepareoterangi opening the feast by presenting us with some of the finest apples I had ever tasted.[54] In a short time we were
- 80. invited into the wharepuni, and a big tin dish of potatoes and pork was set before us, the old chieftainess remarking, You are now in a 'Tongariro country,' and must not look for such delicacies as bread. As we had only had two meals for the past two days, and those of the most visionary description, we found this repast most acceptable. The pork, which had been preserved by being rendered down in its own fat, was delicious, while the potatoes were of the finest kind. Owing to the heavy rain and the flooded state of the Manganui-a-te- Ao, we were compelled to wait at Ruakaka for two days, during which time we visited many parts of the district. I found that the altitude of Ruakaka was 800 feet above the level of the sea, and it is worthy of remark, as showing the rapid fall of the country in this direction, that, in order to reach this place from the great central table-land where we had at first entered the forest, we had descended by the circuitous way we had come no less than 1600 feet in about forty miles. These figures will give some idea of the swift current of the Manganui-a-te-Ao, which, taking its rise near the north-western side of Ruapehu, cuts its way through a mountainous country in a deep, rock-bound channel, and receives the waters of innumerable tributaries along its entire course. The volume of water poured down by this impetuous stream, especially in the rainy season, and during the melting of the snows of Ruapehu, is something prodigious, while I believe the rapidity of its current is unequalled by any other river in New Zealand. Along its entire length its rocky bed is strewn with large boulders and masses of rock of colossal size, while its precipitous cliffs, crowned with towering, forest-clad mountains, impart to it a singularly grand and wild appearance. Besides its rapid course, it is remarkable for its windings and dangerous rapids. We found that the river was known by three native names—viz. Manganui-a-te-Ao, or great river of light; Te Waitahupara, and Te Wairoahakamanamana-a-Rongowaitahanui, or the river of ever- dancing waters and steep, echoing cliffs—while the Whanganui, into
- 81. which it fell, was not only known by the latter name, but likewise as Te Wainui-a-Tarawera, or the great waters of Tarawera. The two rivers form the principal means of communication for the natives of Ruakaka with the outer world. From the Manganui-a-te-Ao they travel in canoes to the Whanganui, and thence southward to the coast. The distance is accomplished in a few days, owing to the rapid current, but the journey up stream often takes over a month. The natives are experienced canoemen, as they must be in order to navigate their frail canoes over the many rapids and winding turns that mark the whole course of the river, as well as that of the Whanganui. At most of the rapids the water shoots over enormous boulders and between narrow channels, and the canoes, guided by poles, are carried over the treacherous places with wonderful dexterity. As may well be imagined, the frail craft often gets upset, but the natives, who are expert swimmers, right them again with little difficulty. During our stay at Ruakaka we were guests of Pareoterangi and his family, which consisted of the old chieftainess, Hinepareoterangi, or the woman of the heavenly crest, as her name implied; Ani, wife of Pareoterangi, a tall, gaunt woman with blunt features, and who wore her hair in short, thick ringlets about her head; Te Ahi, her daughter; and Toma, the tattooed savage who had called our horses rats. We took up our quarters in the wharepuni with these people, but the dismal, and, I may say, dirty, tenement was constantly filled with the natives, who kept continually dropping in to chat or to have a look at us. In this way we had a good opportunity of studying the manners and customs of the Hauhaus of Ruakaka, and, all things considered, they seemed to be following about the same mode of life as they must have done before the arrival of Cook, their manners still presenting that mixture of rude freedom and simplicity suggestive of the infancy of society, before art had taught men to restrain the sentiments of their nature, or to disguise the original features of their character. Shut up in the midst of their forest wilderness, and having little or no connection with the outer world, they seemed to know nothing or to care for nothing beyond their own day-to-day
- 82. existence. We learned that since time immemorial this wild and secluded valley had been a place of settlement for different hapus of the tribes inhabiting the region of the Whanganui River, and that those at present dwelling there were the Ngatihau, Ngatiapa, Ngatimaringi, Ngatitamakana, Ngatiatamira, Ngatiruakopiri, Ngatiikewaia, and Ngatitara. We were informed that their common ancestor was Uenuku, and that their forefathers came from Hawaiki in the Tainui, Arawa, and Aotea canoes. In former times the whole valley of the Manganui-a-te-Ao was fortified with formidable pas, so that it was impossible for an enemy to get up the river. During the troubled times of the great war with the Europeans Ruakaka was always considered as a safe meeting-place for the Hauhau tribes of this part of the country, since the pakehas did not know of its existence; and even if they had, as the natives reasonably remarked, they would never have attempted to penetrate into its fastnesses with any prospect of returning alive. I was anxious to test the religious principles of our Hauhau friends, just to see whether a ray of Christianity was to be found in this wild valley, and during an evening sitting, when the wharepuni was heated like a furnace, and all the motley crowd were assembled together, I got Turner to sound the old tattooed man, who had been a noted fighting-chief during the war, upon the present and upon the hereafter. This grim, antiquated warrior would sit and listen for hours to everything that was said, but he would never venture a remark. Now and again a diabolically sinister smile would pass over his blue- lined countenance, and he would mutter a word with a puff of smoke, but beyond this he was silent. When, however, the question as to his religious scruples was put straight to him, he spoke out frankly, and said, with an air of singular naïveté, At one time I thought there were two saints in the island—Tawhiao and Te Whiti— and I waited a long time to see if they would be taken up to heaven in a chariot of fire, but I have waited so long that I am tired, and now I think that there are no saints in heaven or on earth. Old Hinepareoterangi, who was always a good talker, and displayed at all times a facetious spirit, laughed heartily at the admission of the old
- 83. man, and then, looking us full in the face, she exclaimed in her wild, weird way, We believe in nothing here, and get fat on pork and potatoes. This brought down roars of laughter from the assembled Hauhaus, and we dropped the religious question. It was, in fact, very clear that these natives were as deeply wrapped in the darkness of heathenism as were their forefathers centuries ago, and beyond a superstitious species of Hauhauism, no germ of religious teaching appeared to have found its way into their breasts. They were, however, always ready to sing Hauhau chants to the glorification of Te Whiti and Te Kooti, who appeared to be the presiding deities of these wild tribes. At night, when the wind and rain raged without, and the river rushed through its rock-bound channel with a noise like thunder, both men and women would chant these wild refrains in droning, melancholy notes, but in perfect harmony, the airs in most cases being exceedingly pretty and touching. The two following chants were sung to us by Te Pareoterangi and other natives in chorus, and were taken down in Maori verbatim by Turner. I am indebted for their spirited translation to the able pen of Mr. C.O. Davis. TE KOOTI'S LAMENT. I stood alone awhile, then moving round I heard of Taranaki's doings. The rumours Reached me here, and then I raised My hand to Tamarura,[55] that deity Above. Ah me! 'twas on the third Of March that suffering came, For then, alas! Waerangahika[56] fell; And I was shipped on board a vessel, And borne along upon the ocean. We steer for Waikawa,[57] and then we bear Away to Ahuriri,[58] to thee, McLean.[59] Ah, now I'm seated on St. Kilda's[60] deck,
- 84. And looking back to gaze upon the scene My tears like water freely flow; now Whanganui's[61] shore is seen, now Whangaroa,[62] Where mountain waves are raising up their crests Near Wharekauri.[63] O, my people, Rest ye at home; arise and look around, nd northward look. The lightsome clouds Are lingering in the sky, and wafted hither Day by day, yes, from my distant home, Turanga, from which I now am separated, Separated now from those I love. O, my people! respect the queen's authority, That we may prosper even to the end. Suffice the former things thrown in our path As obstacles. Uphold the governor's laws To mitigate the deeds of Rura, who brought Upon us all our troubles. HAUHAU HYMN. Let us arise, O people!—the whole of us arise. Lo, Tohu and Te Whiti now have reached The pits of darkness—the house of Tangaroa, [64] And gateway of the spirit-world of Miru,[65] Where men are bound all seasons of the year. The offspring, too, of David they would bind. The bright and morning star, Peace, at the end Will come, and in the times of David Feelings of vindictiveness will cease. 'Tis not from thee; it is from Moses
- 85. And the Prophets—from Jesus Christ And His Apostles, that lines of demarcation Were set up to shield thee from man's wrath. The termination comes by thee, O Tohu! And while it wears a pleasing aspect, I am lighted into day. FOOTNOTES: [53] Haeremai is the usual cry of welcome with the Maoris. [54] When afterwards we asked the natives how it was they appeared to be mistrustful of us when we first arrived, they replied that they had always been suspicious of half-castes and pakehas, especially since the capture of Winiata by Barlow. That Te Takaru, the murderer of Moffat, came there sometimes, and they thought we were after him. They then related to us the circumstances of Moffat's death. It would appear that the murdered man, on his last journey, came to Ruakaka, and induced several of the natives to accompany him to the Tuhua country. Moffat, who had been driven from that district by the natives, had been warned not to enter it again; but, notwithstanding this caution, he determined to revisit it, in order to prospect for gold. The party left by one of the bush tracks, and when it had nearly reached its destination, Moffat was fired upon by a native from behind a tree, and mortally wounded in the back. At the same moment he fell from his horse, when another native jumped forward, and split his skull open with a tomahawk. [55] Tamarura—probably a supposed angel recognized by the Hauhau parties. [56] Waerangahika—one of the pas at Poverty Bay, which was taken by our forces. [57] Waikawa—now known as Open Bay. [58] Ahuriri—the great Maori name of Hawke's Bay. [59] The late Sir Donald McLean, the Superintendent of the province of Hawke's Bay (Napier). [60] St. Kilda was the name of the vessel in which Te Kooti was transported to the Chatham Islands.
- 86. [61] Whanganui—name of a places on the Chatham Islands. [62] Whangaroa—name of a place on the Chatham Islands. [63] Wharekauri is the native name of the Chatham Islands. [64] The god of the sea, and guardian of fishes. [65] Supposed being armed with authority in Hades.
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