Optimal management of MPLS networks

(1)

Optimal Management of MPLS

Networks

J o h a n n e s

M a r t h i n u s

d e K o c k

T h e s i s p r e s e n t e d i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r t h e d e g r e e o f M a s t e r o f S c i e n c e a t t h e U n i v e r s i t y o f S t e l l e n b o s c h S u p e r v i s o r : P r o f . A . E . K r z e s i n s k i M a r c h 2 0 0 2

(2)

(3)

Declaration

I the undersigned

hereby declare that the work contained in this thesis is my own original work

and has not previously in its entirety or in part been submitted at any university for a degree.

Signature:

...

. ..

III

(4)

(5)

Abstract

M u l t i p r o t o c o l L a b e l S w i t c h i n g ( M P L S ) i s a r o u t i n g t e c h n o l o g y w h i c h c a n m a n a g e Q u a l i t y o f S e r v i c e ( Q o S ) i n s c a l a b l e c o n n e c t i o n l e s s n e t w o r k s u s i n g r e l a t i v e l y s i m p l e p a c k e t f o r w a r d i n g m e c h -a n i s m s . T h i s t h e s i s c o n s i d e r s t h e o p t i m i s a t i o n o f t h e Q o S o f f e r e d b y a n M P L S n e t w o r k . T h e Q o S m e a s u r e u s e d i s t h e e x p e c t e d p a c k e t d e l a y w h i c h i s m i n i m i s e d b y s w i t c h i n g p a c k e t s a l o n g o p t i m a l l a b e l s w i t c h e d p a t h s ( L S P s ) . T w o m a t h e m a t i c a l m o d e l s o f M P L S n e t w o r k s a r e p r e s e n t e d t o g e t h e r w i t h a p p r o p r i a t e a l g o r i t h m s f o r o p t i m a l l y d i v i d i n g t h e n e t w o r k t r a f f i c i n t o f o r w a r d i n g e q u i v a l e n c e c l a s s e s ( F E C s ) , f i n d i n g o p t i m a l L S P s w h i c h m i n i m i s e t h e e x p e c t e d p a c k e t d e l a y a n d s w i t c h i n g t h e s e F E C s a l o n g t h e o p t i m a l L S P s . T h e s e a l g o r i t h m s a r e a p p l i e d t o c o m p u t e o p t i m a l L S P s f o r s e v e r a l t e s t n e t w o r k s . T h e m a t h e m a t i c s o n w h i c h t h e s e a l g o r i t h m s a r e b a s e d i s a l s o r e v i e w e d . T h i s t h e s i s p r o v i d e s t h e M P L S n e t w o r k o p e r a t o r w i t h e f f i c i e n t p a c k e t r o u t i n g a l g o r i t h m s f o r o p t i m i s i n g t h e n e t w o r k 's Q o S .

v

(6)

(7)

Opsomming

M u ltip ro to c o l L a b e l S w itc h in g (M P L S ) is 'n ro e te rin g s m e to d e o m d ie d ie n s v la k (Q o S ) v a n 'n s k a le e rb a re , v e rb in d in g lo s e n e tw e rk te b e s tu u r d e u r m id d e l v a n re la tie f e e n v o u d ig e v e rs e n d in g s m e g a n -is m e s . H ie rd ie te s is b e s k o u d ie o p tim e rin g v a n d ie Q o S v a n 'n M P L S -n e tw e rk . D ie Q o S -m a a ts ta f is d ie v e rw a g te v e rt ra g in g v a n 'n n e tw e rk -p a k k ie . D it w o rd g e m in im e e r d e u r p a k k ie s la n g s o p tim a le

" la b e l s w itc h e d p a th s " (L S P s ) te s tu u r.

T w e e w is k u n d ig e m o d e lle v a n M P L S -n e tw e rk e w o rd o n d e rs o e k . T o e p a s lik e a lg o ritm e s w o rd v e rs k a f v ir d ie o p tim a le v e rd e lin g v a n d ie n e tw e rk v e rk e e r in " fo rw a rd in g e q u iv a le n c e c la s s e s " (F E C s ), d ie s o e k to g n a o p tim a le L S P s (w a t d ie v e rw a g te p a k k ie -v e rtra g in g m in im e e r) e n d ie s tu u r v a n d ie F E C s la n g s d ie o p tim a le L S P s . H ie rd ie a lg o ritm e s w o rd in g e s p a n o m o p tim a le L S P s v ir v e rs k e ie to e ts n e tw e rk e o p te s te l. D ie w is k u n d ig e te o rie w a a ro p h ie rd ie a lg o ritm e s g e g ro n d is , w o rd o o k h e rs ie n .

H ie rd ie te s is v e rs k a f d o e ltre ffe n d e ro e te rin g s a lg o ritm e s w a a rm e e 'n M P L S -n e tw e rk b e s tu u rd e rj-e s d ie n e tw e rk s e Q o S k a n o p tim e e r.

(8)

(9)

Acknowledgements

This work was performed within the

Siemens- Telkom

Centre of

Excellence for ATM

& Broadband

Networks

and

their Applications

and is supported

by grants from the South African National

Research Foundation,

Telkom SA Limited and Siemens Telecommunications.

I also spent a sabbatical

at the

Teletraflic Research Centre

at the University of Adelaide.

My host

was Prof PG Taylor from the Department

of Applied Mathematics

at the University of Adelaide.

(10)

(11)

List of Publications

1 . S A B erezn er, A M In g g s, JM d e K o ck an d A E K rzesin sk i. A ltern ativ e R o u tin g an d R eco n -fig u ratio n in C o m m u n icatio n N etw o rk s. In P ro ceed in g s o f

TeleTraflic

'9 7 , G rah am sto w n , S o u th A frica, S ep tem b er 1 9 9 7 .

2 . JM d e K o ck an d A E K rzesin sk i. C o m p u tin g an O p tim al V irtu al P ath C o n n ectio n N etw o rk b y S im u lated A n n ealin g . In P ro ceed in g s o f th e

South African Telecom m unications,

N etw orks

an d

Applications

C onference (SATN AC )

'9 8 , C ap e T o w n , S o u th A frica, S ep tem b er 1 9 9 8 . 3 . JM d e K o ck an d A E K rzesin sk i. D ealin g w ith In stab ility in th e R ed u ced L o ad A p p ro x im

a-tio n . In P ro ceed in g s o f th e

South African Telecom m unications,

N etw orks and Applications

C onference (SATN AC )

'9 9 , D u rb an , S o u th A frica, S ep tem b er 1 9 9 9 .

4 .

A

A rv id sso n , JM d e K o ck , A E K rzesin sk i an d P G T ay lo r. C o st-E ffectiv e D ep lo y m en t o f B an d w id th P artitio n in g in B ro ad b an d N etw o rk s. In P ro ceed in g s o f th e

South African

Telecom m unications,

N etw orks and Applications

C onference (SATN AC )

'9 9 , D u rb an , S o u -th A frica, S ep tem b er 1 9 9 9 .

5 . S A B erezn er, JM d e K o ck , A E K rzesin sk i an d P G T ay lo r. L o cal R eco n fig u ratio n o f A T M V irtu al P ath C o n n ectio n N etw o rk s. In P ro ceed in g s o f

IFIP Broadband

C om m unications

'9 9 , H o n g K o n g , N o v em b er 1 9 9 9 .

6 . S A B erezn er, JM d e K o ck an d A E K rzesin sk i. T h e D esig n o f O p tim al V irtu al P ath C o n -n ectio -n N etw o rk s w ith S erv ice S ep aratio n . In P ro ceed in g s o f th e

International

C onference

o n

Inform ation,

C om m unications

and Signal Processing (IC IC S)

'9 9 , S in g ap o re, D ecem b er 1 9 9 9 .

7 . JM d e K o ck an d A E K rzesin sk i. Q u ality o f S erv ice O p tim isatio n in M P L S N etw o rk s. In P ro ceed in g s o f th e

IFIP W orkshop

o n

Perform ance M odelling and Evaluation of ATM

&

IP

N etw orks

(IFIP ATM & IP)

2000,

Ilk ley , U K , Ju ly 2 0 0 0 .

8 . JM d e K o ck an d A E K rzesin sk i. O p tim isin g th e Q u ality o f S erv ice in M P L S N etw o rk s. In P ro ceed in g s o f th e

South African Telecom m unications,

N etw orks

an d

Applications

C

onfer-ence (SATN AC )

2000,

S o m erset W est, S o u th A frica, S ep tem b er 2 0 0 0 .

(12)

xu

9. A

A rvidsson,

JM de K ock, A E K rzesinski and PG Taylor.

The D esign of A TM V irtual

Path

Connection

N etw orks w ith Service Separation.

In Proceedings

of the

South African

Telecom-munications, Networks

and

Applications Conference (SATNAC) 2000,

Som erset W est, South

A frica, Septem ber

2000.

10. JM de K ock and A E K rzesinski.

Finding

O ptim al

Paths

in M PLS N etw orks.

In Proceedings

of

Africom 2001,

Cape Tow n, South A frica, M ay 2001.

11. JE Burns, TJ O tt, JM de K ock and A E K rzesinski.

Path Selection and Bandw idth

A llocation

in M PLS

N etw orks:

a N on-linear

Program m ing

A pproach.

In Proceedings

of

IT Com 2001,

(13)

v

V ll ix xi 1

1

2

3

5

7

12

16

20

21

23

(14)

x iv C O N T E N T S

3.3 T he L S R 's Q ueueing D iscipline

3.4 F inding O ptim al L S P s ...

C onclusion .

3.5.2 A L arger N etw ork

25

26

30

33

34

35

36

T w o S m all N etw orks Initial F easible F low V ector

3.5.1 3.4.1 S om e A pplications . . . . .

3.6

3.5

4.1 T he M odel .

4.2 A nalysis of the L ink D elays

4.4 F inding O ptim al L abel S w itched P aths.

4 M in im isin g L in k D elay s in M P L S N etw o rk s 3 7

37

38

40

45 45

46

60

T he L S P S ets

R

T he C onvergence of the F low D eviation A lgorithm s 4.5.1

4.5.2

R esults. . . .

C onclusion .

A Q uantitative M easure of the L ink D elay .

4.6

4.5

4.3

5 C o n clu sio n

61

A C o n v ex an d C o n cav e F u n ctio n s

63

B T h e F ran k -W o lfe M eth o d

B .1 S om e R esults from L inear A lgebra

B .2 S om e R esults from O ptim isation T heory

B .3 T he F rank-W olfe M ethod .

6 7

67

70

79

C T h e F ib o n acci S earch M eth o d 8 7

(15)

D.1 The Newton-Raphson

Method

.

D.2 A Cauchy-Type

Steepest Descend Method

D.3 A Gradient Projection

Method

.

Bibliography

xv

91

93

94

97

(16)

(17)

List of Tables

3.1 The arrival and service rates for the 4 node network

3.2 The packet arrival rates for the SA network

3.3 The LSPs used by the SA network

4.1 LSP Correlation

.

34

35

36

47

4.2 LSP Statistics

of the Kleinrock Algorithm

for the 50 Node Network.

. . . ..

49

4.3 LSP Statistics

of the Bertsekas-Gallager

Algorithm

for the 50 Node Network

...

51

4.4 Normalised

LSP Statistics

of the Kleinrock Algorithm

for the 50 Node Network..

51

4.5 Normalised

LSP Statistics

of the Bertsekas-Gallager

Algorithm

for the

50 Node

Network.

. . . ..

52

4.6 LSP Multiplicity

of the Kleinrock Algorithm

for the 50 Node Network

52

4.7 LSP Multiplicity

of the Bertsekas-Gallager

Algorithm

for the 50 Node Network

..

52

4.8 LSP Statistics

of the Kleinrock Algorithm

for the 100 Node Network

. . . ..

55

4.9 LSP Statistics

of the Bertsekas-Gallager

Algorithm

for the 100 Node Network

. ..

58 4.10 Normalised

LSP Statistics

of the Kleinrock Algorithm

for the 100 Node Network.

59 4.11 Normalised

LSP Statistics

of the Bertsekas-Gallager

Algorithm

for the 100 Node

Network

. . . ..

59 4.12 LSP Multiplicity

of the Kleinrock Algorithm

for the 100 Node Network

59 4.13 LSP Multiplicity

of the Bertsekas-Gallager

Algorithm

for the 100 Node Network

.

60

(18)

(19)

List of Figures

4 .2 T h e T o p o lo g y o f th e 2 0 N o d e N e tw o rk 4 .1 T h e T o p o lo g y o f th e 1 0 N o d e N e tw o rk 3 .4 A 4 n o d e n e tw o rk . 3 .5 T h e S A M P L S n e tw o rk 2 .1 T h e R o u te S e ts 6

22

23

34

3 4 3 5 4 5 4 5 4 6

...

C o n v e rg e n c e fo r th e 1 0 N o d e N e tw o rk A 6 n o d e n e tw o rk . 4 .3 3 .3 3 .1 T h e S e t

R

j 3 .2 T h e S e t

B

n 4 .4 C o n v e rg e n c e o f th e B e rts e k a s -G a lla g e r A lg o rith m fo r th e 1 0 N o d e N e tw o rk .... 4 7 4 .5 C o n v e rg e n c e fo r th e 2 0 N o d e N e tw o rk . . . .. 4 8 4 .6 C o n v e rg e n c e o f th e B e rts e k a s -G a lla g e r A lg o rith m fo r th e 2 0 N o d e N e tw o rk .... 4 9 4 .7 L S P C o rre la tio n fo r th e 1 0 N o d e N e tw o rk 4 .1 3 P e rc e n ta g e u s e o f S h o rte s t P o s s ib le L S P s A .1 A C o n v e x F u n c tio n . . . . 4 .1 1 L S P C o rre la tio n fo r th e 5 0 N o d e N e tw o rk 4 .1 2 L S P C o rre la tio n fo r th e 1 0 0 N o d e N e tw o rk 4 .9 T h e T o p o lo g y o f th e 5 0 N o d e N e tw o rk . 4 .1 0 T h e T o p o lo g y o f th e 1 0 0 N o d e N e tw o rk 5 0 5 3 5 4

55

56

5 7 5 8

63

L S P C o rre la tio n fo r th e 2 0 N o d e N e tw o rk 4 .8 x ix

(20)

xx

LIST OF FIGURES

A .2 A Concave Function

...

64

B.l Convex Feasible Region

9

Form ed by Linear Constraints . . . 75

B.2 A Supporting Line

P

x

. . .

.

-

...

76

B.3 A Supporting H yperplane

P

x

...

77

B.4 A N on-O ptim al Point x

...

78

(21)

Chapter 1 Introduction

T he Q uality of S ervice (Q oS ) offered by a

connection-oriented

netw ork can be m anaged by opti-m ally utilising the transm ission capacity or bandw idth of the underlying physical netw ork. T here are several possible criteria of optim ality, for exam ple netw ork throughput, blocking probability or rate of earning revenue. S everal approaches to bandw idth m anagem ent (and thus Q oS m an-agem ent) have been discussed in the literature. T hese include V P (virtual path) distribution algorithm s [2]' V P C N (virtual path connection netw ork) optim isation m ethods [5, 7, 8] and the design of virtual subnetw orks [15].

T he success of the Internet is m aking

connectionless

netw orks based on the Internet P rotocol (IP ) increasingly popular. C urrent generation

IP

netw orks do not provide effective m echanism s (apart from priority queueing flags in packets) for m anaging Q oS . T his problem w as eventually addressed by several vendors including T oshiba, IB M , Ipsilon and C isco. E ach vendor cam e up w ith som e version of a technology now know n as

label switching.

B y allow ing explicit routing, label sw itching enables netw ork operators to m anage Q oS by m eans of optim al routing.

1.1 Label Switching

A label is a field in an

IP

packet that is used to determ ine the route follow ed by a packet. A label sw itching netw ork consists of a group of interconnected

label switching

routers

(L S R s). A n L S R perform s

label swapping

on incom ing packets. D uring label sw apping an incom ing packet's local (or global) label is exam ined and replaced by an appropriate global (or local) label. W hether a local label is replaced by a global label or vice versa depends on the type of

label binding

("dow nstream " or "upstream ") used. A packet is typically created by an application on a com puter in a subnetw ork connected to an L S R referred to as the

ingress LSR.

T he ingress L S R is also referred to as the originating node (O N ) (see A sh

et al.

[1]' for exam ple). U pon receiving the unlabeled packet, the ingress L S R assigns a label to the packet and forw ards the packet to the next L S R on the

(22)

2

Chapter 1. Introduction

packet's route. T he L SR uses the inform ation in the label and in the L SR routing table, or

label

information

base

(L IB ) to identify the next L SR on the packet's route. T he other L SR s traversed by the packet's route perform label sw apping on the incom ing packet and forw ard the packet to the next node on its route. Finally, the last L SR on the packet's route, know n as the

egress LSR,

rem oves the label and passes the packet to the apropriate application in the subnetw ork connected to it.

L abel sw ithing netw orks can use either

destination-based

or

explicit

routing. T he routing decision in destination-based is based only on the packet's destination address.

In

the case of explicit routing the route is specified by the packet's label. T he route w hich a packet follow s through the label sw itching netw ork is know n as a

label switched path

(L SP). W e do not allocate bandw idth to an L SP. T hus w e neither set up

trunks,

nor do w e construct

constraint-based

routing label switched

paths

(C R L SP) as m entioned in [1].

A label sw itching netw ork using explicit routing allow s

fine forwarding

granularity

-

the set of packets w hich an L SR can receive is partitioned into disjoint subsets know n as

forwarding

equiv-alence classes

(FE C s). T he set of packets belonging to a particular service class and travelling betw een a given O -D pair can be assigned to a distinct FE C or the FE C that a packet belongs to can be based on the com puter in the subnetw ork w here the packet originated and/or (correspond-ing to even finer forw ard(correspond-ing granularity) the application on the com puter w hich generated the packet. Fine forw arding granularity m akes the netw ork m ore flexible since it allow s routing based on service class [13]. A sh et

al.

[1] divides all path selection m ethods into four categories, nam ely

hierarchical

fixed routing

(FR ),

timedependent

'rOuting

(T D R ),

state-dependent

routing

(SD R ) and

event-dependent

routing

(E D R ). O ur m odel of explicit routing is such that FR , T D R and SD R are all m odelled. E D R is the only path selection category not covered since E D R routing tables are updated locally, w hereas ours is a centralised approach.

T his thesis exam ines how fine forw arding granularity can be used to optim ise the netw ork's Q oS. T he

expected packet

delay

is used as the netw ork's perform ance criterion. Fine forw arding gran-ularity enables the netw ork operator to m inim ise the expected packet delay. T his is done by allow ing several L SPs betw een the sam e O -D pair. T he total packet load offered to an O -D pair is sw itched am ong several L SPs in such a w ay that the expected packet delay is m inim ised. T hus the set of all packets offered to an O -D pair is partitioned into FE C s and the packets in each FE C are sw itched along a particular L SP.

1.2 Multiprotocol Label Switching

T he label sw itching protocol used in this thesis is

Multiprotocol

Label Switching

(M PL S) w hich w as introduced by the Internet E ngineering T ask Force (IE T F). A s the nam e indicates, it com bines the label sw itching approaches of the four vendors m entioned above. T he expected packet delay is m inim ised in M PL S netw orks w ith explicit routing.

(23)

1.3 Flow Deviation

3

T h e e x p e c te d p a c k e t d e la y d e p e n d s o n th e d e la y s in th e lin k s (d u e to b a n d w id th lim ita tio n s) a n d th e d e la y s in th e n o d e s L S R s (d u e to th e lim ite d sp e e d w ith w h ic h a n L S R c a n c o p y a p a c k e t o n a n o u tg o in g lin k ). T h is th e sis c o n sid e rs th e se sc e n a rio s se p a ra te ly .

In th e first sc e n a rio th e lin k b a n d w id th s a re a ssu m e d to b e in fin ite . It h a s b e e n p re d ic te d th a t th e d e c re a sin g c o st o f o p tic a l fib re a n d o th e r tra n sm issio n m e d ia w ill p ro v id e fu tu re n e tw o rk s w ith n e a rly u n lim ite d b a n d w id th . If th is "in fin ite b a n d w id th " m o d e l is a p p lie d to a n IP n e tw o rk , th e n e tw o rk 's Q o S is n o lo n g e r in flu e n c e d b y th e a v a ila b ility o f su ffic ie n t tra n sm issio n c a p a c ity . T h e

nodal delays

d u e to th e se rv ic e a n d q u e u e in g o f th e p a c k e ts in th e L S R s w ill p rim a rily d e te rm in e th e Q o S o ffe re d b y th e n e tw o rk . W e th e re fo re c o n sid e r th e e x p e c te d p a c k e t d e la y to b e c o m p o se d so le ly o f th e n o d a l d e la y s. C h a p te r 3 c o v e rs th is sc e n a rio .

In th e se c o n d (m o re re a listic sc e n a rio ) w e c o n sid e r th e lin k s to h a v e fin ite b a n d w id th a n d th e m a jo r so u rc e o f d e la y c o n sists o f th e q u e u e in g a n d se rv in g o f th e p a c k e ts a t th e lin k s. T h e

link

delays

d o m in a te to su c h a n e x te n t th a t th e n o d a l d e la y s a re c o n sid e re d n e g lig ib le . C h a p te r 4 c o n sid e rs th is sc e n a rio .

1.3 Flow Deviation

T h e o b je c t o f th e la b e l sw itc h in g a p p ro a c h is d iffe re n t fro m th e A T M n e tw o rk c a se . In a n A T M n e tw o rk th e p e rfo rm a n c e c rite rio n is th e e x p e c te d

rate of earning revenue

(o r p ro fit) w h e re a s th e e x p e c te d

packet delay

is th e p e rfo rm a n c e c rite rio n in a n M P L S n e tw o rk . A n A T M n e tw o rk o p e ra -to r c a n a c h ie v e th is b y c o n stru c tin g a fu lly -m e sh e d V P C N w h ic h m a x im ise s th e n e tw o rk 's e x p e c te d ra te o f e a rn in g re v e n u e . O n th e o th e r h a n d , th e o p e ra to r o f a n M P L S n e tw o rk sw itc h e s p a c k e t flo w s o n to o p tim a l L S P s in o rd e r to m in im ise th e e x p e c te d p a c k e t d e la y . In th e la b e l sw itc h in g sc e n a rio c o n sid e re d in th is th e sis

flow deviation

is p e rfo rm e d ra th e r th a n

capacity reservation

-n o C R L S P

virtual

network

(V N E T ) (th e M P L S e q u iv a le n t o f a V P C N , se e [1 ]) is c o n stru c te d b u t IP p a c k e ts a re sw itc h e d a lo n g a p p ro p ia te L S P s. C h a p te r 2 c o n sid e rs th e o rig in a n d d iffe re n t v e rsio n s o f th e flo w d e v ia tio n a lg o rith m . T h e m a th e m a tic a l m e th o d s o n w h ic h th e flo w d e v ia tio n a lg o rith m s a re b a se d a re re v ie w e d in th e a p p e n d ic e s.

(24)

(25)

Chapter 2 The Flow Deviation Algorithm

T h i s c h a p t e r d i s c u s s e s t h e

flow deviation

a l g o r i t h m i n a b s t r a c t t e r m s . T h e b a s i s o f t h e d i s c u s s i o n i s a g e n e r a l

label switching

network

w i t h a

cost function.

T h e p r e c i s e m e a n i n g o f a g e n e r a l l a b e l s w i t c h i n g n e t w o r k i s g i v e n i n s e c t i o n 2 . 1 . O n l y t h e p r o p e r t i e s o f t h e c o s t f u n c t i o n a r e s p e c i f i e d a n d n e i t h e r t h e f u n c t i o n n o r i t s p h y s i c a l i n t e r p r e t a t i o n ( e g d e l a y ) i s f i x e d . S e c t i o n 2 . 1 i n t r o d u c e s t h e n e t w o r k m o d e l a n d s e c t i o n 2 . 2 s t a t e s t h e o p t i m i s a t i o n p r o b l e m t o b e s o l v e d b y f l o w d e v i a t i o n . S e c t i o n s 2 . 3 a n d 2 . 4 d i s c u s s t w o v a r i a n t s o f t h e f l o w d e v i a t i o n a l g o r i t h m .

2.1 The Model

T h e n e t w o r k h a s

N

n o d e s w h e r e e a c h n o d e r e c e i v e s p a c k e t s o n i n c o m i n g l i n k s a n d f o r w a r d s t h e m o n a p p r o p r i a t e o u t g o i n g l i n k s . T h e n o d e s a r e n u m b e r e d f r o m o n e a n d i d e n t i f i e d b y t h e i r n u m b e r s . L e t

N

=

{ I , 2 , . . . ,

N}

d e n o t e t h e s e t o f n o d e s . E a c h O - D p a i r i s a s s i g n e d a u n i q u e i n t e g e r . L e t

J

b e t h e s e t o f a l l O - D p a i r s ( i d e n t i f i e d b y t h e i r n u m b e r s ) . T h e r e f o r e

J

=

IJI

=

N(N

-1).

E a c h p h y s i c a l l i n k c o r r e s p o n d s t o a n O - D p a i r , c o n s e q u e n t l y t h e l i n k s a r e i d e n t i f i e d b y t h e n u m b e r s o f t h e O - D p a i r s w h i c h t h e y c o n n e c t . L e t

£

d e n o t e t h e s e t o f l i n k s ( i d e n t i f i e d b y t h e i r l i n k n u m b e r s ) a n d l e t

L

=

1 £ 1

t h e n

£ ~

J

a n d L : S

J.

T h e f u n c t i o n . : L ':

N

x

N

f-+

J

i s d e f i n e d a s

.:L'(o,d)

=

j

i f 0 - D p a i r

(0, d)

i s n u m b e r e d

j.

L e t

Of.

d e n o t e t h e ( p h y s i c a l ) c a p a c i t y ( z e r o i f t h e p h y s i c a l l i n k d o e s n o t e x i s t ) o f t h e u n i - d i r e c t i o n a l l i n k 0 -

d

f o r e a c h O - D p a i r

(0, d)

s u c h t h a t . : L '(0 ,

d)

=

f.

T h u s t h e s e t £ c a n b e e x p r e s s e d a s £

=

{j

E

J

I

OJ

>

O } . L e t

Aj

d e n o t e t h e a r r i v a l r a t e o f p a c k e t s t o O - D p a i r

j

a n d l e t

A

b e t h e t o t a l p a c k e t a r r i v a l r a t e t o t h e n e t w o r k : 5

(26)

6

Chapter 2. The Flow Deviation Algorithm

J

A

=

L A j .

j=1

A is a ls o r e f e r r e d to a s th e to ta l e x t e r n a l t r a f f i c a tte m p tin g to e n te r th e n e tw o r k .

A r o u t e r

=

( 0 ,

d)

is a s e q u e n c e o f p h y s ic a l lin k s 0 - 0 1 , 0 1 - 0 2 , . . . ,Om -

d

c o n n e c tin g n o d e s 0

a n d

d.

r

=

(0,

d)

c a n a ls o c o n s is t o f th e s in g le lin k 0 -

d.

N o te th e n o ta tio n d is tin c tio n :

(0,

d)

d e n o te s th e 0 - D p a ir , 0 -

d

d e n o te s th e p h y s ic a l lin k a n d

(0, d)

d e n o te s a r o u te c o n n e c tin g n o d e s 0 a n d

d.

L e t

R

j d e n o te th e s e t o f r o u te s b e tw e e n 0 - D p a ir

j.

F ig u r e 2 .1 s h o w s th e s e t

R

j

=

{ r l ' r 2 } w h e r e 2 ( 0 1 ,

d

_{1 )}

=

j.

L e t

R

b e th e s e t o f a ll r o u te s in th e n e tw o r k : R =

U

Rj. j E J L e t A £ d e n o te th e s e t o f r o u te s th a t tr a v e r s e lin k £ f o r e a c h £ E L o T h e s e t A £

=

{ r 3 , r 4 , r 5 } is s h o w n in f ig u r e 2 0 1 . T h e r o u te s r 3 , r 4 a n d r 5 a ll tr a v e r s e lin k 0 2 -

d

2 w ith 2 ( 0 2 ,

d

2 )

=

£ 0 .• r5

..

• r3 ".

..•

.

." ...

. H. ~--~

•...•

7"4 • 0 2 d2 •

Tl.\

• d1 r 2 _...•/

....

0 1 /

\

....

(a) Rj (b) A i F ig u r e 2 .1 : T h e R o u te S e ts A ll p a c k e ts o f f e r e d to 0 - D p a ir

j

a r e s w itc h e d a lo n g r o u te s in R j. L e t Srj b e th e p o r tio n o f th e p a c k e ts o f f e r e d to O - D p a ir

j

w h ic h tr a v e l a lo n g r o u te r . I f l/r is th e a r r iv a l r a te o f p a c k e ts to r o u te r , th e n l/r

=

SrjAj f o r a ll

j

E

J

a n d r E R j. A ll th e p a c k e ts o f f e r e d to O - D p a ir

j

a r e s w itc h e d a lo n g r o u te s in R j, th e r e f o r e th e f o llo w in g tw o r e la te d e q u a tio n s h o ld

Aj =

L

l/r rEn;

L

Srj

=

1

rEn;

(1)

(2 ) f o r a ll

j

E J .

(27)

2.2 The Optimisation

Problem

7

T h e n e tw o rk m o d e l sp e c ifie d h e re is th a t o f a g e n e ra l la b e l sw itc h in g n e tw o rk . T h e sta n d a rd la b e l sw itc h in g n o tio n s su c h a s p a c k e ts, sw itc h in g , L S P s a n d F E C s a re in c o rp o ra te d in th is m o d e l. L S P s a re re fe rre d to a s ro u te s to k e e p th e d isc u ssio n g e n e ra l. F E C s a re n o t e x p lic itly m e n tio n e d b u t a re im p lie d b y th e p o rtio n s Sr j ' T h e m a in o b je c t o f th is th e sis is th e o p tim a l m a n a g e m e n t o f M P L S

n e tw o rk s. T h e m a n a g e m e n t g o a ls c a n b e sp e c ifie d in te rm s o f a n o n -lin e a r o p tim isa tio n p ro b le m . T h e n e x t se c tio n d isc u sse s a n o p tim isa tio n p ro b le m w h ic h sp e c ifie s th e o p tim a l m a n a g e m e n t o f th e g e n e ra l la b e l sw itc h in g n e tw o rk w h ic h w e a re c o n sid e rin g in th is c h a p te r.

2.2 The Optimisation Problem

T h is se c tio n in tro d u c e s th e g e n e ra l o p tim isa tio n p ro b le m w h ic h is a d a p te d to d e sc rib e sp e c ific M P L S n e tw o rk m o d e ls in c h a p te rs 3 a n d 4 .

L e t ' Y f d e n o te th e ra te a t w h ic h p a c k e ts a rriv e to lin k

£.

T h e re la tio n sh ip b e tw e e n ' Y f a n d lIr is

' Y f

=

L lIr fo r a ll

£

E

£.

rEA,

(3 )

A p a c k e t a rriv a l ra te w ill b e re fe rre d to a s af l o w in th e re m a in d e r o f th is c h a p te r. A c c o rd in g ly ,

(1 ) a n d (2 ) a re k n o w n a s p r e s e r v a t i o n o f f l o w e q u a tio n s sin c e th e y sp e c ify th a t th e to ta l flo w in

th e n e tw o rk is p re se rv e d .

L e t Df b e th e c o s t f u n c t i o n fo r lin k

£.

S o m e g e n e ra l p ro p e rtie s o f Df a re sp e c ifie d b u t n o t Df

itse lf. D f d e n o te s a c la ss o f fu n c tio n s o f w h ic h th e in d iv id u a l m e m b e rs d e sc rib e p a rtic u la r la b e l

sw itc h in g n e tw o rk s. T h e m o st im p o rta n t p ro p e rty is th a t Df is a fu n c tio n o f th e ra te ' Y f (th e l i n k f l o w) a t w h ic h p a c k e ts a re o ffe re d to lin k

£.

D f is a lso a fu n c tio n o f a fix e d se rv ic e ra te in

c h a p te r 3 a n d a fu n c tio n o f th e c a p a c ity G f in c h a p te r 4 . H o w e v e r, th e c a p a c itie s a n d se rv ic e

ra te s a re re g a rd e d a s c o n sta n ts fo r a g iv e n n e tw o rk . T h u s w e c o n sid e r D f a s a fu n c tio n o f a sin g le

v a ria b le ' Y f a n d th e d e riv a tiv e s o f Df w ith re sp e c t to ' Y f a re c o n sid e re d a s o rd in a ry d e riv a tiv e s a s

o p p o se d to p a rtia l d e riv a tiv e s. W e m a k e a n im p o rta n t a ssu m p tio n a b o u t D f .

Assumption

2.1 (Differentiability

of

Df ) . Df is a d i f f e r e n t i a b l e f u n c t i o n o f ' Y f a n d is d e f i n e d o n t h e i n t e r v a l

[0,

G f ) f o r e a c h £

E

£.

L e t D b e th e c o st fu n c tio n fo r th e n e tw o rk , L Db )

=

L Df ( " ( f ) , f =1 (4 )

w h e re I

=

(" (f)fE L is th e l i n k f l o w v e c t o r . D e fin e

v

=

(lIr)rER a s th e r o u t e f l o w v e c t o r . T h e c o st

(28)

8

Chapter 2. The Flow Deviation

Algorithm

D (v )

=

tD £

(2:

vr).

£ = 1 r E A e

T h e o b ject o f th is ch ap ter is to so lv e th e fo llo w in g n o n -lin ear o p tim isatio n p ro b lem .

Optimisation

Problem

2.1.

M in im is e : D (v )

=

tD £

(2:

vr)

£ = 1 r E A e S u b je c t to : (5 )

Aj =

2:

Vr rER; Vr ~ 0 fo r all

j

E

J

fo r all r E

R.

W e in tro d u ce an im p o rtan t assu m p tio n b efo re d iscu ssin g th e so lu tio n .

Assumption

2.2 (Flow Independence).

T h e 1 £'s a s s o c ia te d w ith d iffe r e n t lin k s a r e in d e p e n

-d e n t a n d th e V_r 's a s s o c ia te d w ith d iffe r e n t r o u te s a r e in d e p e n d e n t.

D efin e a c o s t r a te v e c to r c

=

( C 1 , C 2 , ... , C L ) w h ere

aD

Cf

=

0 1 £ .

Cf is th e rate at w h ich th e n etw o rk co st

D

in creases w ith an in fin itesim al in crease in th e lin k flo w 1 £ . T h u s, if th e flo w alo n g so m e ro u te r is in creased su ch th at th e flo w o n lin k

£

in creases b y an in fin itesim al am o u n t 0 " £ , th e in crease in D is (jD

=

O " £ C £ .T h e v ecto r c is th e g rad ien t v ecto r o f

D

at th e p o in t

I

w h ich is d en o ted b y V

Db).

N o te th at an im p o rtan t co n seq u en ce o f th e flo w in d ep en d en ce assu m p tio n is th at a D _ d D t

=

D ~ . C£

=

0 1 £ - d 1 £ L et r

E

Rj an d d efin e th e co st o f ro u te r as

aD

Cr

=

O V r

(29)

2.2 The Optimisation Problem

th e n

a

L cr

=

a ;;

L D e r e = l L

= ~

d D e a r y e L J d -£ = 1 r y e a Vr

= L D ~

e E r

=

L c e

e E r 9 (6 )

w h ic h is o b ta in e d b y d iffe re n tia tin g (4 ), a p p ly in g th e flo w in d e p e n d e n c e a s s u m p tio n a n d n o tin g th a t (3 ) im p lie s

{

I

a r y e _

aV

r

a

iff! E r o th e rw is e . T h e r o u t e c o s t Cr is a ls o k n o w n a s th e f i r s t d e r i v a t i v e l e n g t h o f a ro u te . C o n s id e r a p a rtic u la r O -D p a ir j . A ro u te r j

E

R j fo r w h ic h th e ro u te c o s t is m in im a l (ie cr ;

=

m in { cr

IrE

R j } ) is k n o w n a s a l e a s t - c o s t r o u t e c o n n e c tin g 0 - D p a ir j . A ro u te r E R j fo r w h ic h Cr is n o t m in im a l is re fe rre d to a s a n e x t r e m a l r o u t e . S im ila rly , th e flo w s o n th e le a s t-c o s t ro u te s a n d th e e x tre m a l ro u te s a re re fe rre d to a s l e a s t - c o s t f l o w s a n d e x t r e m a l f l o w s re s p e c tiv e ly .

L e t i /

=

( v r ) r E 'R . b e a n o p tim a l ro u te flo w v e c to r (ie a n o p tim a l s o lu tio n to O p tim is a tio n P ro b -le m 2 .1 ). If

v

r ;

>

a

fo r s o m e rj

E

Rj,

a n in fin ite s im a l a m o u n t o f flo w 8 vr ;

>

a

c a n b e s h ifte d fro m

r j to s o m e ro u te r E Rj w ith o u t d e c re a s in g th e c o s t D . T h e c h a n g e 8 D in c o s t is 8 D

=

8 vr ; a a D ( i / ) - 8 vr ; a a D ( i / ) Vr Vr ; a n d s in c e 8 D m u s t b e n o n -n e g a tiv e 8 vr ; f D ( i / ) ; : : :8 vr ; a a D ( i / ) Vr Vr ; a a D ( i / ) ; : : : a a D ( i / ) Vr Vr ; T h is im p lie s th a t if

V

r;

>

a

fo r rj

E

R

j, th e n

(30)

10 Chapter 2. The Flow Deviation Algorithm

a n d

f

D(i/) 2:

f) f)

D(i/)

Vr vrj

fo r a ll

r

E

R

j (7 )

f)~rj

D(i/) =

m in { f)~r

D(i/)

IrE

R

j}

o r a lte rn a tiv e ly c_rj

=

m in _{cr

IrE

R

_j} w h ic h id e n tifie s rj a s a le a s t-c o s t ro u te . T h u s a n o p tim a l ro u te flo w

v

rj is p o s itiv e o n ly o n a ro u te rj w ith a m in im a l firs t d e riv a tiv e le n g th . F u rth e rm o re , a t

a n o p tim a l ro u te flo w v e c to r, th e ro u te s a m o n g w h ic h A j is s p lit m u s t h a v e e q u a l firs t d e riv a tiv e le n g th s (ie th e y m u s t a ll b e le a s t-c o s t ro u te s ). T h u s (7 ) is a n e c e s s a ry c o n d itio n fo r o p tim a lity . If th e c o s t fu n c tio n

D

is c o n v e x , (7 ) is a ls o a s u ffic ie n t c o n d itio n fo r o p tim a lity (s e e B e rts e k a s

et

al. [9]).

W e d e fin e a ro u te flo w v e c to r a s

feasible

if th e c o n s tra in ts o f O p tim is a tio n P ro b le m 2 .1 a re s a tis fie d . T h u s w e h a v e th e fo llo w in g d e fin itio n .

Definition 2.1 (Feasible Route Flow Vector). A route flow vector v

is

feasible if

A j =

L

Vr rEnj Vr

2:

0 fo r a ll

j

E

:J

fo r a ll

r

E

R.

(8 )

(9)

G iv e n a fe a s ib le ro u te flo w v e c to r v, c o n s id e r c h a n g in g v a lo n g a d ire c tio n

_{D..v = (D..v}

r) rEn a n d th u s o b ta in in g th e ro u te flo w v e c to r

v

+

aD..v.

B e rts e k a s

et al.

[9 ] m e n tio n th re e re q u ire m e n ts w h ic h

_D..v

h a s to m e e t.

1 . T h e firs t re q u ire m e n t is th e fe a s ib ility o f th e re s u ltin g ro u te flo w v e c to r. F o rm a lly th is m e a n s th a t fo r s o m e a_{m a x}

>

0 a n d a n y

a

E [0 , am a x ] , th e flo w v e c to r

v + aD..v

m u s t b e fe a s ib le . T h u s b y (8 )

A j

=

L

(v

r

+

aD..v

r) rEnj a n d s in c e

v

is a ls o fe a s ib le ,

0=

L

aD..v

r rEnj a n d th e re fo re 0

=

L

D..v

r rEnj

(31)

2.2 The Optimisation

Problem

fo r a ll

j

E

:7 .

S im ila r to (1 ) a n d (2 ), th is a ls o im p lie s th e p re s e rv a tio n o f flo w .

11

2 . T h e s e c o n d re q u ire m e n t fo r fe a s ib ility (9 ) im p lie s th a t V

r

+

a!::lvr ~

0 fo r a ll

r

E

R

a n d th u s

!::lvr ~

0 fo r a ll

r

E

R

s u c h th a t V

r

=

O .

3 . T h e fin a l re q u ire m e n t is th a t

!::lv

is a d e s c e n t d ire c tio n a n d th e re fo re th e c o s t fu n c tio n

D

c a n b e d e c re a s e d b y m a k in g s m a ll m o v e s fro m

v

in th e d ire c tio n

!::lv.

L e t V

D (v)

d e n o te th e g ra d ie n t v e c to r o f

D

a t th e p o in t

v.

T h e re q u ire m e n t th a t

!::lv

s h o u ld b e a d e s c e n t d ire c tio n im p lie s th a t

VD(v).!::lv

< o.

T h e re q u ire m e n ts o n th e th e d ire c tio n

!::lv

a re s a tis fie d b y a fa m ily o f ite ra tiv e a lg o rith m s th a t d o th e fo llo w in g . L e t th e c u rre n t ro u te flo w v e c to r (a t ite ra tio n m ) b e

vm

=

(v;:')rER'

T h e ro u te flo w v e c to r

v

m

is n o w c h a n g e d to

V~1+l

=

(1- ar)v~

+

arbvr

(1 0 )

w h e re a_r E [0 ,1 ] fo r e a c h

r

E R a n d

bv

is a fe a s ib le ro u te flo w v e c to r. T h e d ire c tio n

!::lv

a t ite ra tio n m is n o w g iv e n b y

!::lv = bv - v

m.

E a c h o f th e s e a lg o rith m s e n s u re s th a t th e p re s e rv a tio n o f flo w c o n d itio n is m e t

L

!::lvr

=

L

(bvr - V~1)

rERj rERj

=

L

bV

r -

L

v;:'

rERj rERj

=

A j - A j

=0

(32)

12 Chapter 2. The Flow Deviation

Algorithm

T h e s e c o n d re q u ire m e n t o f B e rts e k a s e t a l . is a ls o s a tis fie d s in c e { y v s a tis fie s (9 ) a n d th e re fo re

6 . vr

=

{ y vr - v ; : '

=

( y vr

2':

0 fo r a ll r E R s u c h th a t V ~ 1

=

O .

T h e fin a l re q u ire m e n t is m e t b y c h o o s in g th e v e c to r a

=

( ar ) r E R s u c h th a t ar E [0 , 1 ] fo r e a c h

r E R a n d D ( v m + l )

<

D ( vm) .

W e c o n s id e r tw o a lg o rith m s th a t s o lv e O p tim is a tio n P ro b le m 2 .1 in th e n e x t tw o s e c tio n s . T h e firs t a lg o rith m b e lo n g s to th is c la s s . T h e s e c o n d a lg o rith m is s im ila r to th e a lg o rith m s in th is c la s s b u t th e flo w m o v e m e n t is n o t s p e c ifie d b y (1 0 ).

2.3 The Kleinrock Algorithm

T h e v e rs io n o f flo w d e v ia tio n p re s e n te d h e re m o v e s th e s a m e p o rtio n a o f flo w to a ll le a s t-c o s t ro u te s . T h u s

a

r

=

a fo r a ll

r E

R a n d fro m (1 0 ):

V ; 1 + l

=

(1 -

a ) v ~ 1

+

a { y vr fo r e a c h r E R

w h ic h c a n b e w ritte n a s a v e c to r e q u a tio n v m + l

=

(1 - a ) vm

+

a { y v . T h e fa c to r a is c h o s e n s u c h th a t D ( ( l - a ) v m

+

a { y v ) is m in im is e d . T h e re s u lt o f th is is th a t D ( vm+1) : : : ;D ( vm) . T h is c a n b e s e e n b y n o tin g th a t a

=

0 y ie ld s D ( v m + l )

=

D ( vm) a n d th e re fo re th e m in im is a tio n p ro c e s s w ill y ie ld a v a lu e o f a s u c h th a t D ( v m + l ) is a t m o s t e q u a l to D ( vm) . T h e flo w d e v ia tio n a lg o rith m te rm in a te s w h e n D ( vm+1) is n o lo n g e r s tric tly le s s th a n D ( vm) . T h u s , d u rin g th e e x e c u tio n o f th e a lg o rith m D ( v m + l )

<

D ( vm) w h ic h s a tis fie s th e th ird re q u ire m e n t m e n tio n e d a t th e e n d o f s e c tio n 2 .2 .

L in k a n d ro u te flo w s a re re la te d b y (3 ) a n d th e re fo re O p tim is a tio n P ro b le m 2 .1 c a n b e s o lv e d w ith th e lin k flo w s a s th e d e c is io n v a ria b le s . T h e m a in a d v a n ta g e is th a t th e n u m b e r o f d e c is io n v a ria b le s is s m a lle r s in c e (in g e n e ra l) R ~ N !

»

N ( N

-1)

2':

L . Im p lic itly th e p ro b le m s till w o rk s w ith ro u te flo w s . T h is w ill b e e x p la in e d s h o rtly .

Optimisation

Problem

2.2.

M i n i m i s e : L

Db)

=

L

Debe)

e = 1 S u b j e c t t o : A j

=

L

Vr r E R ;

,e

2':

0 fo r a ll

j

E

:J

fo r a ll e E L .

(33)

2.3 The Kleinrock Algorithm

13

T h e a l g o r i t h m w h i c h i s n o w d i s c u s s e d i s a s p e c i a l c a s e o f t h e

Frank- Wolfe method

( s e e a p p e n d i x B ) a n d i s d u e t o K l e i n r o c k [ 2 3 ] .

W e s t a r t w i t h a n i m p o r t a n t a s s u m p t i o n .

Assumption

2.3. For each

£ E

L,

Df

i s

a convex function

of

,f.

D

( g i v e n b y ( 4 ) ) i s t h e r e f o r e a l s o a c o n v e x f u n c t i o n o f

,f.

T h i s i m p l i e s t h a t i f a f e a s i b l e l i n k f l o w v e c t o r e x i s t s , a n y l i n k f l o w v e c t o r "I w h i c h m i n i m i s e s

D

y i e l d s t h e g l o b a l m i n i m u m o f

D

( s e e a p p e n d i x A f o r a p r o o f ) . T h e a l g o r i t h m i s i n i t i a l i s e d w i t h a f e a s i b l e l i n k f l o w v e c t o r c jJ 0 w h i c h i s r e p e a t e d l y m o d i f i e d u n t i l

D

i s m i n i m i s e d . K l e i n r o c k [ 2 3 ] p r o v i d e s a m e t h o d o f f i n d i n g a f e a s i b l e i n i t i a l l i n k f l o w v e c t o r w h i c h i s d i s c u s s e d i n c h a p t e r s 3 a n d 4 s i n c e i t d e p e n d s o n t h e l i n k c o s t f u n c t i o n

Df.

A t e a c h s t e p m o f t h e a l g o r i t h m , a f e a s i b l e l i n k f l o w v e c t o r c jJ m

= (cPi)fEC

i s c o n s t r u c t e d f r o m t h e p r e v i o u s f e a s i b l e l i n k f l o w v e c t o r c jJ m - 1 b y s h i f t i n g a p o r t i o n a o f t h e c u r r e n t f l o w f r o m e x t r e m a l r o u t e s o n t o l e a s t - c o s t r o u t e s a n d t h e n c a l c u l a t i n g t h e l i n k f l o w v e c t o r c jJ m b a s e d o n t h e r e s u l t i n g r o u t e f l o w s . T h i s i s e q u i v a l e n t t o s h i f t i n g f l o w t o a n d f r o m l i n k s ; t h u s t h e e a r l i e r c o m m e n t t h a t t h e a l g o r i t h m i m p l i c i t l y w o r k s w i t h r o u t e s . A n e w l i n k f l o w v e c t o r c jJ m i s c o n s t r u c t e d a t e a c h s t e p m o f t h e a l g o r i t h m i n t h e f o l l o w i n g w a y . C o m p u t e t h e c o s t r a t e v e c t o r c f o r t h e p r e v i o u s l i n k f l o w v e c t o r c jJ m - 1 , t h u s

aDI

Cf

= -

m - l

a,f

I F " ', f o r e a c h

£

E

L.

N o t e t h a t c

=

VD(cjJm-1).

C o m p u t e t h e i n c r e m e n t a l n e t w o r k c o s t

b..D

f o r t h e l i n k f l o w v e c t o r c jJ m - 1 w h e r e L

b..D

=

c . c jJ m - 1

=

L

CfcPr;-l.

f = l

(11)

b..D

i s t h e d e r i v a t i v e o f

D

a t t h e p o i n t c jJ m - 1 i n t h e d i r e c t i o n i n d i c a t e d b y c jJ m - 1 ( s e e F l e m i n g [ 1 7 ] f o r e x a m p l e ) . W e n e x t d e t e r m i n e w h e t h e r a l i n k f l o w v e c t o r

cjJm,

w h i c h y i e l d s a l o w e r v a l u e o f

D

t h a n c jJ m - 1 , c a n b e c o n s t r u c t e d . I f n o s u c h f l o w v e c t o r e x i s t s t h e n c jJ m - 1 i s t h e o p t i m a l l i n k f l o w v e c t o r a n d t h e a l g o r i t h m i s t e r m i n a t e d . D i j k s t r a 's a l g o r i t h m ( s e e M a n b e r [ 2 5 ] f o r e x a m p l e ) i s u s e d t o f i n d a r o u t e r b e t w e e n e a c h O - D p a i r

j

s u c h t h a t C_{r i s m i n i m a l .} L e t

R

c d e n o t e t h e s e t o f t h e s e r o u t e s s i n c e t h e s e l e a s t - c o s t r o u t e s a r e b a s e d o n t h e c u r r e n t c o s t r a t e v e c t o r c . L e t CT

=

(CJf)fEC

d e n o t e t h e l e a s t - c o s t l i n k f l o w v e c t o r

(34)

14

C h a p t e r 2 . T h e F l o w D e v i a t i o n A l g o r i t h m w h i c h r e s u l t s w h e n , f o r e a c h O - D p a i r j , a l l t h e f l o w

A j

i s s e n t a l o n g t h e l e a s t - c o s t r o u t e c o n n e c t i n g j . N o t e t h a t

u

m a y n o t b e a f e a s i b l e l i n k f l o w v e c t o r1. G i v e n t h a t

c/Jm -1

i s a f e a s i b l e l i n k f l o w v e c t o r , w e m u s t e n s u r e t h a t ( 1 -

a )c/Jm -1

+

a u

( f o r t h e o p t i m a l v a l u e o f

a)

i s a l s o f e a s i b l e . T h e v a l i d i t y o f t h e a l g o r i t h m d e p e n d s o n t h i s . T h e i n c r e m e n t a l n e t w o r k c o s t

8 D

f o r t h e l e a s t - c o s t l i n k f l o w v e c t o r

u

i s g i v e n b y L

8 D =

c .

u = L C £ (j£

£ = 1 w h i c h i s t h e d e r i v a t i v e o f

D

a t t h e p o i n t

c/Jm -1

i n t h e d i r e c t i o n i n d i c a t e d b y

u .

C o m b i n i n g t h i s w i t h e q u a t i o n ( 1 1 ) y i e l d s

8 D - 6 .D

=

c .

(u - c/Jm -1 )

= V

D (c/Jm -1 ) . (u - c/Jm -1 )

w h i c h i s t h e d e r i v a t i v e o f

D

a t t h e p o i n t

c/Jm -1

i n t h e d i r e c t i o n

u - c/Jm -1 .

I f

8 D

<

6 .D

t h e d e r i v a t i v e o f

D

i n t h e d i r e c t i o n

u - c/Jm -1

i s n e g a t i v e a n d

D

c a n b e d e c r e a s e d b y a s s i g n i n g a p o r t i o n a o f t h e f l o w i n t h i s d i r e c t i o n . T h i s i s d o n e b y m o v i n g f l o w t o t h e r o u t e s i n

R

c ' A l i n e s e a r c h i s u s e d t o f i n d t h e o p t i m a l v a l u e o f a

E

[ 0 , 1 ] s u c h t h a t , ( a )

=

( 1 -

a )c/Jm -1

+

a u

m i n i m i s e s

D

a n d c /J mi s s e t t o ,( a ) . S u i t a b l e s e a r c h m e t h o d s i n c l u d e t h e g o l d e n s e c t i o n s e a r c h a l g o r i t h m ( s e e P r e s s e t a l. [ 2 7 ] f o r e x a m p l e ) a n d t h e F i b o n a c c i s e a r c h m e t h o d ( s e e a p p e n d i x C ) . I f

8 D

>

6 .D

t h e n e t w o r k c o s t

D

w i l l i n c r e a s e i f f l o w i s m o v e d t o t h e r o u t e s i n

R

c .

c/Jm -1

r e p r e s e n t s a n o p t i m a l l i n k f l o w v e c t o r i n t h i s c a s e a n d t h e a l g o r i t h m i s t e r m i n a t e d .

A l g o r i t h m 2 . 1 ( K l e i n r o c k F l o w D e v i a t i o n ) . W e sta rt w ith a n in itia l fe a sib le lin k flo w v e c to r

c /J 0 a n d c h o o se a te rm in a tio n c rite rio n lO rd > O .

l.m + -1 2 . C o m p u t e t h e c u r r e n t c o s t r a t e v e c t o r c , w h e r e C£

=

D ~ (cP 'F -1 )

f o r e a c h £

E

L.

3 . C o m p u t e t h e i n c r e m e n t a l c o s t

6.D

f o r t h e c u r r e n t l i n k f l o w v e c t o r : L

6 .D

=

L

C £ cP 'F -

1 .

£ = 1 1N o t e t h a t t h e r o u t e f l o w v e c t o r i s r e q u i r e d t o b e f e a s i b l e ( i e s a t i s f y d e f i n i t i o n 2 . 1 ) . T h e r e q u i r e m e n t s f o r a f e a s i b l e l i n k f l o w v e c t o r d e p e n d o n t h e e x a c t f o r m o f t h e l i n k c o s t f u n c t i o n Df, w h i c h i n t u r n d e p e n d s o n t h e M P L S n e t w o r k b e i n g m o d e l l e d . O n e o b v i o u s r e q u i r e m e n t i s t h a t 'Yf :'::: Cf.

(35)

2 . 3 T h e K l e i n r o c k A l g o r i t h m

15

4 . U s e D ijk s tr a 's a lg o r ith m to f in d th e le a s t- c o s t r o u te s b a s e d o n th e c u r r e n t c o s t r a te v e c to r c . D e n o te th e le a s t- c o s t lin k f lo w v e c to r ( w h ic h r e s u lts w h e n a ll th e f lo w

Aj

is d e v ia te d a lo n g th e le a s t- c o s t r o u te s c o n n e c tin g j ) b y

u.

5 . C o m p u te th e in c r e m e n ta l c o s t

8D

f o r th e le a s t- c o s t lin k f lo w v e c to r : L

8D

=

Le£a£.

£ = 1 6 . i f

I~D - 8DI

<

C f d t h e n s t o p e l s e c o n t i n u e e n d i f

7 . F in d a ( w ith 0

:S

a

:S

1 ) s u c h th a t th e lin k f lo w v e c to r ( 1 -

a)qr-

1 +

au

m in im is e s

D

a n d s e t < jJm

=

( 1 -

a)<jJm-1

+

au.

8 . m

+-

m

+

1 a n d r e t u r n to s te p 2 .

A f te r th e a lg o r ith m h a s te r m in a te d ( a t ite r a tio n m

=

M)

w e s e t

,£

=

_1Jfl-1

_{f o r e a c h}

£

E

L.

N o te th a t th e te r m in a tio n c r ite r io n

I~D - 8DI

<

cfd c a n b e r e p la c e d b y in s e r tin g th e s im p le c o n v e r g e n c e te s t

ID(v

m) -

D(v

m-1)

I

<

cfd b e tw e e n s te p s 7 a n d 8 .

C o m p u ta tio n a lly , th e m o s t e x p e n s iv e s te p s a r e 4 a n d 7 . T h e s ta n d a r d im p le m e n ta tio n o f D ijk -s tr a '-s a lg o r ith m h a s c o m p le x ity

O(N

3). T h is c a n b e r e d u c e d to O ( N2lo g

N)

b y u s in g h e a p s ( s e e M a n b e r [ 2 5 ] f o r e x a m p le ) . T h e v e r s io n s o f D ijk s tr a 's a lg o r ith m u s e d in th is th e s is a ll m a k e u s e o f h e a p s . T h e lin e s e a r c h m e th o d s u s e d in th is th e s is a ll h a v e lo w e r c o m p le x itie s th a n D ijk s tr a 's a lg o r ith m . T h u s th e c o m p le x ity o f o n e ite r a tio n o f th e K le in r o c k a lg o r ith m is

O(N

2 lo g

N).

T h e n u m b e r o f ite r a tio n s w h ic h th e a lg o r ith m r e q u ir e s to c o n v e r g e c a n n o t b e e s tim a te d . C o n s e q u e n tly , a n u p p e r b o u n d o n th e c o m p le x ity o f th e e n tir e a lg o r ith m c a n n o t b e g iv e n .

T h e d is c u s s io n g iv e n h e r e is s im p ly a m o tiv a tio n f o r th e a lg o r ith m . T h e a lg o r ith m is f o r m a lly d e r iv e d f r o m th e F r a n k - W o lf e m e th o d in a p p e n d ix B . N o te th a t s te p 4 is e q u iv a le n t to s o lv in g O p tim is a tio n P r o b le m B .7 .

(36)

16 Chapter 2. The Flow Deviation

Algorithm

2.4 The Bertsekas-Gallager Algorithm

T h is s e c tio n p r e s e n ts a n o th e r v e r s io n o f th e f lo w d e v ia tio n a lg o r ith m , n a m e ly th e

Bertsekas-Gallager

algorithm

( s e e B e r ts e k a s

et al.

[ 9 ] ) . T h is a lg o r ith m s o lv e s O p tim is a tio n P r o b le m 2 .1 b y a p p ly in g a g r a d ie n t p r o je c tio n m e th o d to a C a u c h y - ty p e s te e p e s t d e s c e n t m e th o d ( b a s e d o n N e w to n 's m e th o d ) a s d e s c r ib e d in a p p e n d ix D .

T h e le a s t- c o s t r o u te s a r e c a lc u la te d in e a c h ite r a tio n o f th e a lg o r ith m a n d f lo w is th e n m o v e d f r o m th e e x tr e m a l r o u te s o n to th e le a s t- c o s t r o u te s . T h e a m o u n t o f f lo w m o v e d is O - D p a ir d e p e n d e n t a n d c a lc u la te d b y m e a n s o f a g r a d ie n t p r o je c tio n m e th o d . T h is is d if f e r e n t f r o m th e K le in r o c k a lg o r ith m ( d is c u s s e d in th e p r e v io u s s e c tio n ) in w h ic h th e s a m e p o r tio n o f f lo w is m o v e d f r o m a ll e x tr e m a l r o u te s c o n n e c tin g a ll O - D p a ir s . C o n s id e r ite r a tio n m o f th e B e r ts e k a s - G a lla g e r a lg o r ith m a n d le t

r

j b e th e c u r r e n t le a s t- c o s t r o u te b e tw e e n 0 - D p a ir j . T h e e q u a lity c o n s tr a in t

Aj

=

L

Vr r E R ; c a n b e e x p r e s s e d a s Vr;

= Aj -

L

Vr 'rER; ropr; (1 2 )

w h ic h c a n b e s u b s titu te d in to O p tim is a tio n P r o b le m 2 .1 to e lim in a te th e e q u a lity c o n s tr a in ts a n d th u s y ie ld th e f o llo w in g p r o b le m .

Optimisation

Problem 2.3.

Minimise:

D(i/)

Subject

to:

L

v

r:::;

Aj

r E R ; rop'r; Vr ::::0 f o r a ll

j

E

.J

f o r a ll

j

E

.J,

r

E

R

j a n d

r

i-

r

j . i/ c o n s is ts o f e a c h r o u te f lo w Vr s u c h th a t

r

is n o t a le a s t- c o s t r o u te . T h e g r a d ie n t p r o je c tio n m e th o d ( d e s c r ib e d in a p p e n d ix D ) c a n n o w b e a p p lie d to th is p r o b le m . T h e m a in ite r a tio n ( 5 4 ) o f th e g r a d ie n t p r o je c tio n m e th o d f o r th is p r o b le m is

(37)

2.4 The Bertsekas-Gallager

Algorithm

17 v

m

₌

_{m a x}

{o

_v

m -1 -T , T

[::; D(v

m -1)] -1

O~T

D(v

m-1)} , ( 1 3 ) w h e r e

r

i s n o t a l e a s t - c o s t r o u t e . T h e a b o v e c a l c u l a t i o n i s p e r f o r m e d f o r e a c h e x t r e m a l r o u t e b e t w e e n a n 0 - D p a i r j . F i n a l l y , t h e r o u t e f l o w o f t h e l e a s t - c o s t r o u t e

r

j i s c a l c u l a t e d a s

v:.';

=

Aj -

LV;!".

TERj TOPTj T h e f i r s t a n d s e c o n d p a r t i a l d e r i v a t i v e s o f

D(v)

w i t h r e s p e c t t o t h e r o u t e f l o w s a r e c a l c u l a t e d b y u s i n g ( 1 2 ) a n d b y a p p l y i n g t h e c h a i n r u l e f o r f u n c t i o n s o f s e v e r a l v a r i a b l e s t o y i e l d

O~_

0

0 -D(v)

= -D(v)

- -D(v)

oV

T

oV

T

oV

Tj f o r 0 - D p a i r

j

a n d

r

-I-

r

j' T h i s c a n b e d e v e l o p e d f u r t h e r b y u s i n g ( 6 ) :

o

0 aD(v)

=

L

aD(v)

=

L

Cf

v,.

fET rf f E " w h i c h l e a d s t o

0-_

,,0

-D(v)

= ~

-D(v)

- ~

-D(v)

OV,.

n arf n arf

< E T <ETj

=

LCf -

I::C f

fET fETj

=

CT -

c

Tj. D e f i n e

a

2

D(v)

2 - - 2 CT -

(av

T)

a n d t h e

second derivative

cost rate vector

c2

= (C I,

C~, ... ,

c i )

a s

2

a

2

D

Cf

=

a

(rd

f o r e a c h

£

E

'c.

( 1 4 )

(38)

1 8 C h a p t e r 2 . T h e F l o w D e v i a t i o n A l g o r i t h m ( 1 5 ) a n d t h e s a m e a r g u m e n t u s e d i n t h e d e r i v a t i o n o f ( 6 ) g i v e

C ;

= ~

[L

~ D (v )

-

L

~ D (V )]

8v

r n

8"(e

< E r

eE rj

82 82

=

L

- - 2

D

( V ) -

L

- - 2

D(v)

eE r

8

b e)

eE rj

8

b e)

8

2

=

L

- - 2

D

( V )

e U 'r 8 b e )

=

L C ~ '

e E £ r w h e r e

I:-

r i s t h e s e t o f a l l l i n k s i n r o r r j b u t n o t i n b o t h :

I:-

r

=

{ £ E

I:-

1 £ E r o r £ E

rj}

\

{ £ E

I:-

1 £ E r a n d £ E

rj}.

C o n s e q u e n t l y t h e i t e r a t i v e s t e p ( 1 3 ) o f t h e g r a d i e n t p r o j e c t i o n m e t h o d b e c o m e s

111_ { C - }

V

_r

-

m a x 0 ,

v;:,-l -

r

2

C

rj

.

c

r

A l g o r i t h m 2 . 2 ( B e r t s e k a s - G a l l a g e r F l o w D e v i a t i o n ) . W e s ta r t w ith a n in itia l fe a s ib le lin k

flo w v e c to r

cPo

a n d c h o o s e a te r m in a tio n c r ite r io n Efd >

0

a n d a s te p -s iz e 'T]>

O.

1.

m + - l 2 . C o m p u t e t h e c u r r e n t c o s t r a t e v e c t o r c , w h e r e

c e

=

D 'e (4 );'-l)

f o r e a c h £

E

1:-.

3 . C o m p u t e t h e i n c r e m e n t a l c o s t

D.D

f o r t h e c u r r e n t l i n k f l o w v e c t o r : L

D.D

=

L

c e 4 > ;'-l.

e= l

4 . U s e D i j k s t r a 's a l g o r i t h m t o f i n d t h e l e a s t - c o s t r o u t e s b a s e d o n t h e c u r r e n t c o s t r a t e v e c t o r c . D e n o t e t h e l e a s t - c o s t l i n k f l o w v e c t o r ( w h i c h r e s u l t s w h e n a l l t h e f l o w

Aj

i s d e v i a t e d a l o n g t h e l e a s t - c o s t r o u t e s c o n n e c t i n g j ) b y a . 5 . C o m p u t e t h e i n c r e m e n t a l c o s t J D f o r t h e l e a s t - c o s t l i n k f l o w v e c t o r : L J D

= L cerJe.

e= l

(39)

2.4 The

B e r ts e k a s - G a lla g e r

Algorithm

6. if

I~D - 8DI

<

Efd

then

s t o p e ls e

continue

endif

7 . S e t c /J '"

+-

c/J",-1

a n d p e rfo rm th e fo llo w in g p ro c e d u re fo r e a c h O -D p a ir j.

19

(a ) C o m p u te th e c o s t ra te v e c to r c , w h e re Cf

=

D~(</J'l')

fo r e a c h

I!

E L a n d th e s e c o n d d e riv a tiv e c o s t ra te v e c to r c2, w h e re

cE

=

D~' (</J'l')

fo r e a c h

I!

E

L.

(b ) L e t

r

j b e th e le a s t-c o s t ro u te b e tw e e n 0 - D p a ir j . T h e n fo r e a c h ro u te

r

E

Rj, r

=I=-

r

j s e t Cr -

c

rj Wr

=7)7

r a n d s e t

'"

{o

" ,-1

}

V_r

=

m a x ,Vr - Wr . (c ) C a lc u la te th e flo w a lo n g th e le a s t-c o s t ro u te c o n n e c tin g O -D p a ir j:

"'-'

"""'

'"

v

rj - A j - ~

v

r . r E " R j r=f.rj (d ) U p d a te th e lin k flo w v e c to r

c/J"',

w h e re

</J'l'

=

L

v~

rEAt D e c re m e n t th e s te p -s iz e 7 ) in a n a p p ro p ria te w a y . 8 . m

+-

m

+

1 a n d

return

to s te p 2 . fo r e a c h

I!

E

L.

A fte r th e a lg o rith m h a s te rm in a te d (in ite ra tio n m

=

M)

w e s e t

'"'if =

</J"7-

1 fo r e a c h

I!

E

£.

N o te th a t th e te rm in a tio n c rite rio n

I~D - 8DI

<

Efd c a n b e re p la c e d b y in s e rtin g th e s im p le c o n v e rg e n c e te s t

ID(v"')

- D(V"'-l)j

<

Efd b e tw e e n s te p s 7 a n d 8 .

T h e lin k flo w v e c to r c /J '" is u p d a te d (in s te p 7 ) a fte r th e L S P flo w s Vr E

Rj

h a v e b e e n c h a n g e d fo r e a c h 0 - D p a ir j . T h e re fo re c /J '" is o n ly fix e d a t th e c o m p le tio n o f ite ra tio n m . T h is im p lie s th a t th e c o s t ra te v e c to rs c a n d c2

(40)

20 Chapter 2. The Flow Deviation

Algorithm

A step-size 1] is introduced in the iterative step. B ertsekas et

al.

[9] suggest an initial step-size of 1]

=

1. T hey also m ention the possibility of keeping 1] fixed as opposed to several decrem ent procedures.

C om putationally, the m ost expensive step is the calculation of the shortest paths. T he version of D ijkstra's algorithm used in this thesis has com plexity O (N2log

N)

(see the explanation for the K leinrock algorithm ). T hus one iteration of the B ertsekas-G allager algorithm has com plexity

O(N

2

_logN).

2.5 _{Applying Flow Deviation to MPLS Networks}

T he next tw o chapters adapt the flow deviation algorithm s to optim ise the Q oS offered by specific M PL S netw orks. C hapter 3 considers an M PL S netw ork m odel in w hich the m ajor packet delays occur in the nodes (L SR s). T his m odel accom m odates different service classes. C hapter 4 applies flow deviation to M PL S netw ork m odels in w hich the lim ited link bandw idths cause the m ajor delays. T his m odel is only applicable to traffic belonging to a single service class.