T2002:05 ISSN:1100-3154
Objective Functions for Balance in
TraÆc Engineering
JuanAlonso,HenrikAbrahamsson,BengtAhlgren,
AndersAnderssonandPerKreuger
falonso,henrik,bengta,andersa,piakg@sics.se
May2002
Swedish InstituteofComputerScience
Box1263,SE-16429Kista,Sweden
Abstract. Weprovearesultconcerningobjectivefunctionsthatcanbe
usedto obtaineÆcient andbalanced solutionsto the multi-commodity
network owproblem.Thistypeofsolutionisofinterestwhenrouting
traÆc inthe Internet. A particular case of the resultprovedhere (see
Engineering ?
JuanAlonso,HenrikAbrahamsson,BengtAhlgren,AndersAndersson, and
PerKreuger
SICS{SwedishInstituteofComputerScience
E-mail:first.lastname@sics.se
Abstract. Weprovearesultconcerningobjectivefunctionsthatcanbe
usedto obtaineÆcient andbalanced solutionsto the multi-commodity
network owproblem.Thistypeofsolutionisofinterestwhenrouting
traÆc inthe Internet. A particular case of the resultprovedhere (see
Corollary2below)wasstatedwithoutproofinapreviouspaper.
1 Introduction
TraÆc engineeringencompassesperformanceevaluationandperformance
opti-misationofoperationalIPnetworks.An importantgoalwithtraÆcengineering
istousetheavailablenetworkresourcesmoreeÆcientlyfordi erenttypesofload
patternsandtoavoidcongestionbyhavingarelativelybalanceddistributionof
traÆcoverthenetwork.
Current routing protocols in the Internet calculate the shortest path to a
destinationinsomemetricwithoutknowinganythingaboutthetraÆcdemand.
Manualcon gurationbythenetworkoperatoristhereforenecessarytobalance
load between available paths to avoidcongestion. One way of simplifying the
task of the operator and improve use of the available network resources is to
maketheroutingprotocol sensitivetotraÆc demand.Routingthen becomesa
owoptimisationproblem.
Inanotherpaper[1]wediscussed anewroutingalgorithm basedon
multi-commodity ow optimisation. In this report we present and prove a theorem
thathas,asaspecialcase,aresultstatedwithoutproofinthementionedpaper.
Thetheoremconcernsoptimisationobjectivefunctionswhichallowthenetwork
operator to choose amaximum desired link utilisation level. Theoptimisation
will then nd themosteÆcientsolution, ifit exists, satisfying thisconstraint.
Theobjectivefunctionthusenablestheoperatortocontrolthetrade-o between
minimising thenetworkutilisationandbalancingloadovermultiplepaths.
Wehavetriedtomakethepaperself-containedbygivingenoughbackground
information.Theinterestedreaderisreferredtoapreviouspaper[1]andthe
ref-erencesquotedthereformoreinformationonIPnetworksandonoptimisation.
Therestofthisreportisorganisedasfollows.Section2recallstheformulation
ofthemulti-commodity owproblemasgiveninanearlierpaper[1].Section3
anditsproofarepresentedin Section4.
2 Formulation of the multi-commodity ow problem
Therouting problemin a network consists in ndinga pathor multiple paths
that send the requested traÆc throughthenetwork without exceedingthe
ca-pacityofthelinks.Inapreviouspaper[1] wemodelled theroutingproblem as
amulti-commoditynetwork owproblem(MCF)asfollows.
WerepresentthenetworkbyadirectedgraphG=(N;E),where N isaset
of nodes andE is aset of (directed)edges. Weassume thegraphis such that
itsedgescanbeuniquelyrepresentedbyanorderedpair(i;j)ofnodes,wherei
istheinitialpointoftheedgeandjits nalpoint.Everyedge(i;j)2E hasan
associatedcapacity k
ij
re ecting thebandwidth available to thecorresponding
link.Inaddition,weassumegivenademand matrix D=D(s;t)expressingthe
traÆcdemandfromnodestonodetinthenetwork.Theentriesofthedemand
matrix are non-negative and, to avoid trivialities, weassume that D(s;t) >0
for at least one pair of nodes. We model commodities as (only destination)
nodes, i.e., a commodity t is to be interpreted as \all traÆc to t". Then the
corresponding(MCF)problem canbeformulatedasfollows:
minff(y)jy2P 12 g (MCF 12 ) where y=(y t ij
); fort2N;(i;j)2E, andP
12
isthepolyhedronde nedbythe
equations: X fjj(i;j)2Eg y t ij X fjj(j;i)2Eg y t ji = d(i;t) 8i;t2N (1) X t2N y t ij k ij 8(i;j)2E (2) where d(i;t)= 8 > < > : X s2N D(s;t) ifi=t D(i;t) ifi6=t Thevariablesy t ij
denotetheamountoftraÆctotroutedthroughthelink(i;j).
The equation set (1) statethe conditionthat,at intermediate nodes i (i.e., at
nodesdi erentfromt),theoutgoingtraÆcequalstheincomingtraÆcplustraÆc
createdat i and destined to t,while at t theincoming traÆc equalsall traÆc
destinedtot.Theequationset(2)statetheconditionthatthetotaltraÆcrouted
overalinkcannotexceedthelink'scapacity.
It will also be of interest to consider the corresponding problem without
requiringthepresenceoftheequationset(2). Wedenote thisproblem:
1 everypoint y =(y t ij )inP 12 orP 1
representsapossiblesolutionto the routing
problem: itgivesawayto routetraÆcoverthenetwork sothat thedemand is
metand capacitylimitsarerespected(whenit belongs toP
12
), orthedemand
ismetbutcapacitylimitsarenotnecessarilyrespected(whenitbelongstoP
1 ).
A generallinearobjectivefunction foreither problemhas theform f(y)=
P t;(i;j) b t ij y t ij
.Wewill, however,consideronly thecasewhenallb t
ij
=1which
correspondstothecasewhereallcommoditieshavethesamecostonalllinks.
3 Preliminaries
This section contains the de nitions of eÆcient, (L;E)-balanced solutions and
somepreliminaryresults.Fortherestofthepaperwesimplifythenotationand
useeinsteadof(i;j)todenotedirected edges.Thefunctionconsideredabove,
f(y)= P t;e y t e
,willbeusedasameasure ofeÆciency. Wesaythaty
1 is more eÆcient thany 2 iff(y 1 )f(y 2 ),wherey 1 ;y 2 belongtoP 12 orP 1 .Tomotivate
thisde nition,notethatwhenevertraÆcbetweentwonodescanberoutedover
twodi erentpathsofunequallength,f willchoosetheshortestone.Incasethe
capacityoftheshortestpathisnotsuÆcienttosendtherequestedtraÆc,f will
utilise theshortestpathto 100%of itscapacityand sendtheremaining traÆc
overthelongerpath.
Givenapointy=(y t e )asabove,weletY e = P t2N y t e
denotethetotaltraÆc
sentthroughebyy.Everysuchy de nesautilisation ofedgesbytheformula
u(y;e)= 8 > < > : P t2N y t e k e = Y e k e ifk e >0 0 ifk e =0
RecallthatapartitionE=(E
1 ;:::;E
m
)(ofthesetofedgesE)isacollection
ofnon-empty,pair-wisedisjointsubsetsofEwhoseunionisE.Givenapartition
E andy2P
12 orP
1
, utilisationisde nedto bethem-dimensionalvector:
u(y;E)=( max e2E1 ;:::;max e2Em ) SupposethatL=(` 1 ;:::;` m
)isavectorofrealnumberssatisfying0<`
i <1
for i = 1;:::;m. We saythat y 2 P
12 orP
1
is (L;E)-balanced if u(y;E) L,
wheretheinequalityisto beunderstoodcomponent-wise.
GivenapartitionE,asequenceLasabove,andarealnumber>1,de ne
f L;E; (y)= m X i=0 X e2E i k e C `i; ( u(y;e))
wherethelinkcostfunction C `
i ;
(illustratedinFig.1)isde ned by
C ` i ; (U)= ( U ifU ` i U+(1 ) ` i ifU ` i L;E;
-
`i
Fig.1.ThelinkcostfunctionC `
i ;
.
Lemma1. Usingthe above notation,wehave:
1. Forall y2P 1 ,f(y)f L;E; (y). 2. Ify2P 1 is(L;E)-balanced, thenf L;E; (y)=f(y). Proof. 1)SinceC ` i ;
(U)U forallU 0,wehave:
f(y)= X e2E Y e = m X i=0 X e2Ei Y e = m X i=0 X e2Ei k e u(y;e) m X i=0 X e2E i k e C ` i ; (u(y;e))=f L;E; (y):
2) Suppose that y is(L;E)-balanced and e 2E
i . Then u(y;e)` i and hence C ` i ;
(u(y;e))=u(y;e).Thus
f L;E; (y)= m X i=0 X e2Ei k e C `i; (u(y;e))= m X i=0 X e2Ei k e u(y;e)=f(y):
Thiscompletestheproofofthelemma.
Corollary 1. Supposethaty
1 ;y
2 2P
1
are(L;E)-balancedandy
1 isoptimalfor f L;E; .Then f(y 1 )f(y 2 ).
Proof. Followsimmediatelyfromtheassumptionsandpart2)ofthelemma.
4 The result
Beforestatingthetheorem,weneedtode neafewconstants.Let
v=minff(y)jy2P
12
g and V =maxff(y)jy2P
12 g
Noticethat v>0sinceD(s;t)>0,andV <1sincethenetworkis nite and
weareenforcingthe( nite)capacityconditions.Thus, 0<vV <1.
GivenL=(` 1 ;:::;` m ),L+denotes(` 1 +;:::;` m +).Finally,letÆ>0
denotetheminimumcapacityoftheedgesofpositivecapacity.
Theorem1. LetE andLbeasabove,andletdenote arealnumbersatisfying
0 < < min 1im (1 ` i ). Suppose that y 2 P 1
is (L;E)-balanced, and let
1+ V
2
vÆ
. Then any solution x of MCF
1
with objective function f L;E;
is
(L+;E)-balanced. Moreover, x is more eÆcient than any other (L+;
Proof. Supposethatx2P
1
isasolutionofMCF
1
withobjectivefunctionf ,
forsome1+ V 2 vÆ ,andlety2P 1
be(L;E)-balanced.Weclaimthat
f L;E; (x)f L;E; (y)=f(y) V v v V v f(x) (3) sothat,in particular, f L;E; (x) V v f(x) (4)
Indeed, the rst (in)equalityin (3) is truebecausex is optimalfor f L;E;
, the
secondfollowsfrom2)ofthelemma,andthelasttwobythede nitionsofvand
V.
Wewillassume,forcontradiction,thatxisnot(L+;E)-balanced,i.e.that
for somei, 1i m, there isan edge e2E
i
such that u(x;e) >`
i +. Let E 0 i =fe2E i ju(x;e)>` i
+gand note that, by assumption, E 0
i
is notempty.
In(5)and(6)belowweusetheconvenientnotation:
X C = m X j=0 X e2E j X e2E 0 j SetX e = P t2N x t e
.In(6)belowweusethefactthatC ` i ; (U)U: f L;E; (x)= X C k e C ` i ; (u(y;e))+ X e2E 0 i k e C ` i ; (u(y;e)) (5) X C k e u(y;e)+ X e2E 0 i k e C `i; (u(y;e)) (6) = m X j=0 X e2Ej X e + X e2E 0 i k e C `i;
(u(y;e)) u(x;e) =f(x)+ X e2E 0 i k e (( 1)u(y;e)+(1 )` i ) =f(x)+( 1) X e2E 0 i (X e k e ` i )
Itfollowsfrom theinequalitywehavejust obtained,togetherwith(4), that
( 1) X e2E 0 i (X e k e ` i ) V v 1 f(x): (7)
But,takingintoaccountthefact that>1,weobtain
( 1) X e2E 0 i (X e k e ` i )=( 1) X e2E 0 i k e (u(x;e) ` i )>( 1) X e2E 0 i k e ( 1)Æ V 2 V f(x)
corollary.ThiscompletestheproofofTheorem1.
Corollary2belowisthespecial casewhen m=1,i.e.E =(E)andL=(`).
It was formulated without proof as Theorem 1 of [1]. In this special case we
simplify thenotationandsimplywritef `;
insteadoff L;E;
.
Corollary 2. Let `; be real numberssatisfying 0<` <1and 0<<1 `.
Suppose that y 2 P
1
is `-balanced, and let 1+ V
2
vÆ
. Then any solution x
of MCF
1
with objective function f `;
is (`+)-balanced. Moreover, x is more
eÆcientthanany other (`+)-balancedpointofP
1 .
Theattentivereadermayhavewonderedwhy,ifweareinterestedin nding
eÆcient(L;E)-balanced solutions,wehave not used the following moredirect
approach.Considertheproblem
minff(y)jy2P 1L g (MCF 1L ) whereP 1L
denotesthepolyhedronde nedbytheequationset(1)togetherwith
thefollowingequations
X t2N y t e ` i k e 8i(1im)8e2E i
When(L;E)-balancedsolutionsexist,solving(MCF
1L
)willproduceeÆcient
(L;E)-balancedsolutions,andthemethodofTheorem1willproduce(L+;
E)-balanced solutions. Since we can choose arbitrarily small, the two methods
areessentiallyequivalent.Whenno(L;E)-balancedsolutionsexist,however,the
methodsdi ermarkedly.Inthiscase(MCF
1L
)yieldsonlytheinformationthat
theproblemisinfeasible,whereasthemethodofTheorem1willproducea\best
e ort"solution(whichwill ofcoursenotbe(L;E)-balanced).Wecallthe
solu-tion\beste ort"becausef L;E;
,bypenalisingedgeswithhighutilisation,gives
preferencetosolutionsthatareas\balanced"aspossible.Giventhatthe
appli-cationwehavein mindisroutingtraÆcin theInternet(see[1]),andthattime
isimportant,itshouldbeclearthatthemethodproposedinTheorem1o ersa
considerablepracticaladvantageoverthealternativeprovidedby(MCF
1L ).
References
[1] H.Abrahamsson,B.Ahlgren,J.Alonso,A.Andersson,andP.Kreuger. AMulti
PathRoutingAlgorithmforIPNetworksBasedonFlowOptimisation
