networks
Fran ois Blan hy a
, Laurent Melon ay
and Guy Ledu a
a
Resear h Unit inNetworking, EECS department, University of Liege, Belgium
fblan hy,melon,ledu grun.monte ore.ulg.a .be
Multi-Proto olLabel Swit hing (MPLS) provides ways to ontrolthe Label Swit hed
Paths (LSPs) followed by traÆ trunks in a network and thereby to better traÆ
engi-neer it. In this ontext, we look at the problem of organizing the mapping of LSPs in
an optimalway throughout the network on the basis of a given obje tive fun tion. This
problemis highly ombinatorialand makes dynami and real-timefeatures a diÆ ult
is-sue for any LSProuting s heme. For this reason,we propose a omputationallyeÆ ient,
though approximate, on-line s heme adapted to an in remental optimization of the
net-work state. It is then applied to a seldom mentioned traÆ engineering problem: the
ompromise between load-balan ing and traÆ minimization. It is expe ted that lever
routingstrategies to balan ethe networkload willsometimes favor longerpaths inorder
toavoid ongestion,leadingtoanin rease oftheoverall networkutilization. This
reason-ingis on rmed byour study,andweshowthatanimprovementinnetworkmanagement
an be made by appropriately tuning this ompromise.
1. Introdu tion
RoutinginanIP network is on eptually rigid and straightforward froma traÆ
engi-neering point of view: any pa ket roughly follows the shortest path at its disposal. For
this reason, attempts have been made to allowInternet Servi e Providers (ISPs) to gain
a measure of ontrol and engineer their traÆ somewhat. For instan e, it is now made
possible inOSPFby tuning theweights assignedtothe links(asillustrated in[7℄). With
MPLS though, the notion of pa ket an be abstra ted to that of traÆ trunk arried in
LSPs, giving us the possibility toa hieve amu h moree e tive traÆ engineering when
setting up a route for su h a traÆ trunk. Indeed, it is a ommon belief that one of the
mostsigni antinterestofMPLSistraÆ engineeringofanetwork,allowingabetteruse
of its resour es and making provisioning abiteasier.
The existen eof expli itrouting me hanismsin MPLS makespossible anoverall
opti-mizationof the routes taken by all ows throughout the network. However, this requires
that a knowledge is available on the amount of resour es used by ea h LSP. This
knowl-edge an ome froma tive monitoringof the network orfrom apre eding request asking
This work has been arried out in the IST-1999-20675 ATRIUM proje t funded by the European
ommission.
kindof datais madeavailableone way oranotherand on entrate onthepath
optimiza-tion pro ess. Quite some work has already been done on this topi . There are two main
ategories of s hemes: o -line and on-line. The di eren e is based on the type of data
usedinthebest paths omputation: the rstmethodrequiresa ompleteknowledgeofall
LSPsgoingthroughthe network,whereasthese ondneedsonlyasyntheti imageofea h
link state in the topology. Any s heme an be implemented following two approa hes:
entralized (the omputation is done by a server) and de entralized (the omputation is
done atan ingress router).
An o -line pro edure is generally meant for a entralized implementation. A entral
route server periodi ally re omputes the best organization of the LSP map, thanks to
a generally ostly optimization pro edure that may take a lot of time. This way, the
response time of the network may fall down a long way in omparison with lassi al
IP routing (unless new LSPs are established using a lassi al shortest path, waiting for
reoptimization). Some papers however propose qui ker spe i s hemes thanks to
ap-proximations orgiven traÆ properties ([12℄ among others). Two persistent problems of
this approa h are that the network depends onvery few points of failure (two if there is
only one ba kup server), and the potentialrerouting of lotsof LSPs on e anew optimal
mappinghasbeen omputed. Indeed, theoptimalsolutionhasthe property ofbeing very
sensitive.
The on-line approa h, ompatible with both a entralized and a de entralized
imple-mentation (provided the omputational ost is not too high for a router), is favored by
a number of papers ([3℄,[10℄, [9℄, [1℄, the last being learly entralized). Generally,
ap-proximatealgorithmsareused to omputethe pathforany newrequested LSP. On ethe
omputationis done, the LSPis establishedimmediately,using RSVP-TE [2℄. This kind
ofs hemeo erstheadvantageofbeingsimpler,morerobust,andwithagenerallyshorter
response time.
Our work is learly part of the on-line de entralized approa h. We propose a very
e e tive approximation (O(NM) with N the number of nodes, M the number of links)
able to resist a very qui k hange of the network where plenty of LSPs are set up and
destroyed in a short span of time. This is a rst ontribution of this paper. Indeed, we
feel that most other s hemes are not qui k enough, would not s ale very welland might
put aheavy burden onarouter interms of omputationalpower(ex ept perhaps [9℄but
it fo uses mainly on restoration). Our approa h also o ers the advantage of being very
simple and generi , leaving aside, in a rst time, the question of the obje tives and the
onstraints of the optimization pro edure. While most mentioned methods use spe i
and elaborate te hniques, we think our s heme is potentially adequate (and generally
omputationallymoreeÆ ient)tosolvealarge eldof traÆ engineeringproblems. This
approximationpro edurewillbepresentedinse tion2. Inse tion3,wewillillustratehow
this s heme an be used to ta kle a basi traÆ engineering problem: the ompromise
between balan ing the available resour es throughout the network and minimizing the
overall use of the resour es. Indeed, it is obvious that load balan ing will favor paths a
The goal of this se tion is to introdu e the problem of optimizing the LSP mapping
throughout a given topology. We wish to onfront an optimal o -line s heme, using a
bran h-and-bound algorithm,with asimpleon-lineapproximation obtainedthanks toan
adapted ConstrainedShortest Path First(CSPF) pro edure. Of ourse, wealreadyknow
that the latter is by far the most e e tive (qui kest) method, but we an wonder how
goodanapproximationwe an rea h with su ha s heme.
2.1. General presentation of the problem addressed
We are given
Aset ofLSPs L,rangedoverby l,ea h hara terizedby asetof generi parameters
(l), in luding the sour e and destination nodes, bandwidth reservation, whatever
else is needed for spe i traÆ engineering onstraints.
Atopology,undertheformof a nite onne ted graphG =(X;U)whereX isaset
of nodes (verti es) and U a set of links (dire ted ar s) between these nodes. Ea h
link is hara terized by aset of generi parameters !(i;j)with (i;j)2U in luding
for instan e its apa ity as well as any feature needed for traÆ engineering. It is
assumed no linkof the kind(i;i)with i2X isin luded in U.
Asetof onstraintstoberespe tedbytheLSPmappingonea hlinkindependently.
That is,a onjun tion of propositions,
^
(i;j)2U
P(state L
(i;j)
;!(i;j)) w ith state L
(i;j)
=f(l)jl2L^(i;j)2path(l)g
that must be veri ed. It an spe ify, among others, that the apa ity threshold
must not beex eeded.
Two parti ular onstraintsallowed forea hLSPl2L. The rstisa ow onstraint
su h that path(l) must be a path in G from sour e node to destination node as
spe i edbythegraphtheoryvo abulary(apathinadire tedgraphisanysequen e
ofar swherethe nalvertexofoneisthe initialvertexofthenextone). These ond
isano-loop onstraintsu hthatpath(l)mustbeanelementarypathinG(ea hnode
is used atmost on e). No other onstraintsare allowed on aLSP, whi h meanswe
annotensurepath-widepropertiesforthesolution(forexample,we annotimpose
a maximalthreshold on the number of hops for aLSP).
An obje tive fun tion F(path(L);(L);!(U))tominimize.
We look for amapping fromL to G under the form of afun tion
path(:):L !(2 U
;order 3
)
spe ifying the path taken by a LSPfrom sour e todestination through G, minimizingF
and respe ting the given onstraints.
on the obje tive fun tion F) mixed integer problem with non-linear onstraints (again,
atworst, dependingonthe per-link onstraints). However, even the simplestformulation
(linear mixed integer problem with linear onstraints) leads to an at least NP- omplete
problem [8℄ ommonly solved with a omputationally (very) expensive algorithm alled
bran h-and-bound. That is why an o -line approa h (i.e. a powerful server omputes
all routes) an be seen as the most dire t way to ta kle this problem, though it is very
slow and highly stati . In a pra ti al use (day-to-day management of a network), an
additional onstraint omes from the fa t that the requests for LSPs o ur at di erent
points in time, requiring a onstant update of the mapping. An o -line approa h has
diÆ ulties to ope with this parti ular feature. It is therefore appropriate to look for
polynomial approximate s hemes following an in rementalapproa h, that is, adding the
LSPs one by one, intheorder ofo urren eof the requests. This isthe s opeof the next
se tion.
2.2. Double approximation using a CSPF
Withan in remental s heme, the same as with any other approximation, we an only
hope to rea h a lo aloptimum among the available LSP mappings, whereas the
bran h-and-bound nds the global optimum. But even if we lookonly at the problemof adding
a single LSP to a given topology (with orwithout any other LSPs running), omputing
the optimum (global for this redu ed problem) is again at least NP- omplete. Looking
for a polynomial s heme, we have to settle for a se ond approximation, yielding a lo al
optimum inour sear h forthe best route for a new LSP.
Basi ally, we limitthe explorationof the routes availableto those examined by a
on-strained shortest path algorithm, a CSPF, the osts on the links being the variation on
the obje tive fun tion aused by bringing the link intothe path for the LSP. A lassi al
Dijkstraalgorithm ould almostdoitex ept that those osts mightbenegative,
depend-ing on the obje tive fun tion F : nothing ensures that adding an LSP always makes
the obje tive in rease. We will then use the well-known Bellman-Kalaba s heme ([4℄)
(sometimes alled Bellman-Ford) and adaptits formulation toour needs.
Onthebasisofthehypothesisofse tion2.1,wehaveasetLofLSPsalreadyestablished
through the path(:) fun tion. We want to add a new LSP l new with parameters (l new ). LetL new =L[l new . We de ne
a predi ate that tellsusif (i,j) with i;j 2X is a validlink for l new
,
p(i;j)=(i;j)2U ^ P(state L
(i;j)
[f(l new
)g;!(i;j))
a no-loop lause applied to a potential partial route fi
0 ;i 1 ;::;i n 1 g for l new and a node i n , su h that nl(i n ;fi 0 ;i 1 ;::;i n 1 g)=:(i n 2fi 0 ;i 1 ;::;i n 1 g) i k 2X;k 2[0;n℄
as ore fun tion appliedto apotentialpartial routefi
0 ;i 1 ;::;i n g for l new , su h that s(fi 0 ;i 1 ;::;i n g)= 8 < : F(path(L new );(L new );!(U)) if (i k ;i k+1 )2U 8k2[0;n 1℄
with path(l )=fi 0 ;i 1 ;::;i n g i k 2X;k 2[0;n℄ a link- ostfun tion v(i;j)= 8 < : 1 if :p(i;j)_:nl(j; (i))
(i;j)=s( (i)[fjg) s( (i))
otherwise 8i;j 2X with (:):X !(2 U ;order 4
), yielding the urrent partial path fromthe sour e to
the spe i ed node already omputed in the Bellman-Kalaba pro edure. Of ourse,
it is preferable that the evaluation of (i;j) an be done without re omputing
the wholeroute s ores. A suitablepropertyof the s ore fun tion(i.e. the obje tive
fun tionF)isthatit anbeupdatede e tively,inanin rementalway,whenadding
a linkto the path.
Using these notations as well as those of se tion 2.1, the formulation of the
Bellman-Kalabaalgorithmisalmostun hangedfromthe original. Letd
k
(i)bethe ostof thebest
omputed route from sour e node 1 to node i at step k and note that in graph theory,
the (:) fun tionreturns the "destination set" of a node, following link orientation.
The algorithmfollows these steps:
1) d k (1)=0;8k 2N d 1 (i)=v(1;i);8i2X k+1) d k+1 (i)=min j2 1 (i) fd k (j)+v(j;i)g 8i2X
Note that v(j;i) above dynami ally hanges with the path used to rea h j. It stops
if d k (i)= d k+1 (i) 8i2 X, d k
(i) is then the minimal ost for a path from 1 to i and the
approximation to the optimal route for the new LSP is given by (destination). The
onvergen e isensured inatworst jXj steps, unlessthere is some negative ir uit(whi h
is made impossible by the parti ular no-loop onstraint presented in se tion 2.1). Note
that this algorithmis of omplexityO(NM)with Nthe numberof nodes, Mthe number
of links, whi h makes it quite eÆ ient. The ow onstraint is of ourse ensured by the
pro edure.
Whyisthesolutionapproximate? Thepro edureprogressesfromthesour etowardsthe
destinationalways usingthe linkwhi hminimizesthe variationof theobje tivefun tion.
However, thesevariationsdependonthepartialpathusedtorea hthelink. Whi hmeans
that, to nd the true optimalroute, wemay have to hoose alink whose variationis not
minimal, but su h that further variations will be smaller, and the nal obje tive value
lower. This an be ompared to the well known problem of ontinuous optimization of
a fun tion whose slope is unknown. In that situation, we an rea h a lo alminimum by
followingdowntheslopeofthe urve,butnothingtellsusthatthereisnobetterminimum
in another "valley", unrea hable ex ept by limbing a bit. But how do you know when
to limb? This is what makes our problemNP- omplete. There isno way to ir umvent
the ontinuous optimization problemjust mentioned, a onvex obje tive fun tion makes
sure any minimumwe nd isglobal. In general,a areful hoi eof the obje tivefun tion
an be suÆ ient to make sure the solution is not that mu hsuboptimal as itwill be the
ase with the load-balan ing obje tive of se tion 3.
In order to assess the onsisten y of this approximation, we have onfronted it to an
exa t algorithm. In [6℄ we presented a mixed-integer formulation for a basi instan e
of the problem introdu ed in se tion 2.1, and used it to draw a lower bound (a relaxed
solution) as a point of omparison. The on lusion was that the approximation is very
lose to the optimal solution provided the mean LSP size is small enough ompared to
the mean link apa ity.
3. TraÆ engineering point of view
Thepurposeof thisse tionisnot thepresentation ofelaboratedte hniquestoengineer
the traÆ of a network. Rather, we simply want to validate the approximate approa h
of se tion 2.2 by illustrating how it an be used to ontrol the ows passing through a
network, tominimizethe resour es used, the blo king probability,et . This willbedone
simply by an appropriate hoi e of the obje tive fun tion. Remember that, aside from
the sour eand destination,a LSPl2L is only hara terizedby itsrequested bandwidth
bw
(l). Ea h link of (i;j) 2 U has a spe i bandwidth apa ity ! ap
(i;j). The only
obligationwe have is torespe t the apa ity ona link whilerouting the LSPs.
Wewill rstpresent theobje tivefun tionsinuse,beforelayingdownsomeresultsand
observations obtained by simulations.
3.1. Obje tive fun tions for traÆ engineering
In the following, we denoteL
(i;j)
the load onlink (i;j)2U.
3.1.1. Load balan ing obje tive fun tion
X (i;j)2U L (i;j) ! ap (i;j) L ! ap 2 w ith L ! ap = 1 jUj X (i;j)2U L (i;j) ! ap (i;j)
the mean of the relative link load throughout the network. This fun tion is then the
varian e onthe relativelinkload and, assu h, represents the deviation fromthe optimal
load balan ing situation. In a perfe tly balan ed network, this deviation would be zero
for alllinks sothat allwould be o upied inexa tly the same proportions. Note that an
in rementalupdateoftheobje tivewhenaddingalinkispossibleaftersomedevelopment
(more details an befound in[6℄). This is a quadrati fun tion for two reasons:
Obviously, the deviation shouldbea positive notion.
Asthe deviationin reases,it ispenalizedwithas orethat risesmore thanlinearly.
fun tion prevents this tosome extent.
3.1.2. Load balan ing limitation (traÆ minimization)
The main problem with the load-balan ing fun tion presented above is that the only
thingittriestodoisto attentherelativeloadthroughoutthenetwork. Itwillnotmatter
ifsomeofthepathsgoalongwayaroundinordertoa hieveabetterload-balan ing. This
means thatLSPs maybe hara terized by very long delays,and that the network loadis
sus eptible torise very qui kly. Wemust thentry tolimitthelength ofthe paths hosen
for the LSPs by adding a kind of "shortest path length"term to the obje tive fun tion.
Butof ourse,this limitationmust be arefully hosensoastolettheload-balan ingterm
operate when needed, both terms must s ale in asimilar manner.
After a areful study of the variation of the load balan ing fun tion, we de ided to
use as our "traÆ minimization" term the sum of the square relative link loads. As a
onsequen e, the ompromise between load balan ing and traÆ minimization an be
expressed asfollows X (i;j)2U L (i;j) ! ap (i;j) L ! ap 2 + X (i;j)2U L (i;j) ! ap (i;j) 2
Why is this interesting? The (weighted) ombination of both terms will give more
importan etotheload-balan ingtermifthedeviationishighenoughtojustifythedetour,
elseitwillletthe "shortestpath"termminimizetheresour esused. Theweightingfa tor
allows to give more importan etoone aspe t orthe other.
3.1.3. Variant - Average delay obje tive fun tion
X (i;j)2U 1 ! ap (i;j) L (i;j)
Sin e, for a pa ket size P,
P
! ap
(i;j) L
(i;j)
approximates the queuing and transmission
delay on link (i;j), optimizing this fun tion will strive to minimize the average delay
throughoutthe network. Thiswillnaturallylead toload-balan ethe network. Moreover,
thisobje tivehasabuilt-intraÆ minimizationfeatureinthesensethatlongpathswould
degrade the averagedelay and are thereforedis ouraged.
3.2. "Load balan ing versus traÆ minimization" ompromise at work
Before presenting some simulations, let us onsider what we ould expe t. Load
bal-an ing the network should ideally produ e a network with an homogeneous blo king
probability by sour e-destination pair. Hopefully, it ould also lower the overall
blo k-ingprobability in omparison with a lassi al shortest path (innumberof hops). On the
otherhand,usingsu htraÆ engineeringte hniquessometimesfavordetoursanditmight
logi ally in rease further the load inside the network (again ompared with a minimum
hop routing). Clearly there must be a ompromise between load-balan ing and traÆ
drawn in the eld of network management.
The results presented orrespond to a generated topology with 20 nodes. 10 ingress
nodes have been sele ted and a sour e-destination probability distribution as well as a
LSP-bandwidth requested probability density have been randomly reated. These
prob-abilitiesare usedand respe tedbya randomLSPrequest(removal)generator. The ratio
between the mean link apa ity and the mean LSP size has been set to 50.In all
simula-tions, LSPs are added up toa ertain numberor up to a ertainnetwork load (a kindof
rise inpower). Then LSPs areaddedand removed insu hawayastokeep astationnary
state (in terms of the number of LSPs or the network load). The idea is to simulate a
ontinuous use of the topology, lots of paths being set up and destroyed for some time
while we tra k the blo kingprobability, the numberof failed establishment, the network
load, the numberof LSPs established,and the relativeload deviation(image ofthe
load-balan ing). Notethat the blo king probability is an exa t value, not an estimation, and
was omputed with asimple s heme des ribed in[6℄.
The topologyused has been "perfe tlyengineered" thanks to a Generalized maximum
on urrent ow algorithm. By "perfe tly engineered", we mean that a load of 100% is
rea hable throughout the network if the sour e-destination probabilities are respe ted
and the LSP-bandwidth are in nitesimal. This has been done be ause we realizeditwas
useless to try to engineer the traÆ on a network with engineering in onsisten ies su h
as huge links following very small ones (they an only rea h a very small relative load).
And indeed, the Generalized maximum on urrent ow approximation we used is lose
to the behaviour used by some network engineers we have met (e.g. "double up the link
apa ity when it rea hes 50% of load"). Note that we have appliedthe same simulations
to a ompletely random topology, the results were presented in [6℄ and are very similar,
thoughin a less marked way.
The results are expressed in terms of the mean overall blo king probability, the mean
numberof LSPsand themeannetworkload observed throughoutthe simulations. Figure
1 illustrates our ompromise in the ase of a network load frozen at 80%, while gure 2
shows the results with a xed number of LSPs (550). The shortest path in terms of hop
ounts is used as areferen e.
For a onstant network load, we an see on gure 1 that as the importan e of
load-balan ing in reases (lower ), the blo king probability de reases. However, the number
of LSPs established goesdown as welland falls verylowfor =0. In this ase,it would
seem that = 0:25 is an optimal "load-balan ing - traÆ minimization" ompromise,
sin e it has a low blo king probabilityfor a relatively high number of LSPs (3 times less
han e of LSP blo king than the shortest path, for an equivalent number of LSPs). We
an see that the ompromisea hieved by the delay obje tive is not quite asgood,but it
has the advantage of being self- ontained, without any need for tuning, besides it is still
mu h better than the shortest path. It is interesting to noti e that the number of LSPs
established by a onstrained shortest path in the hop sense is slightly lower than what
an be obtained with the superior values of , though the network load is the same. By
always hoosing the shortest possible path nomatter the onsequen es, we end up doing
more detoursto get to the destination assome links be ome saturated.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 250 500 750 1000 = 0 = 0 : 25 = 0 : 5 = 0 : 75 = 1 dela y hop Num b er of LSPs (gra y) Blo k proba (bla k)
Figure1. xed network load (80%)
0 0.1 0.2 0 24 48 72 96 = 0 = 0 : 25 = 0 : 5 = 0 : 75 = 1 dela y hop Net w ork load (gra y) Blo k proba (bla k)
Figure2. xed number of LSPs (550)
same. The network load logi ally in reases for a de reasing favoring load balan ing.
Theblo kingprobability,in uen edbythe networkload,shows aminimumfor =0:25.
The right ompromiseis a hieved for =0:25or =0:5. Theuntunabledelay obje tive
givesgoodresults,buttheshortestpathison eagainleftbehindwithaveryhighblo king
probability for anetwork load not that mu h lower.
By playing on this "load-balan ing versus traÆ minimization" ompromise in real
topologies,we anhopetogain mu h. Someproviderswehavemetuse todoublethelink
apa ity when the load rea hes 50%, and indeed a shortest path routing strategyis then
lose to its limits on erning blo king probabilities. With a more sophisti ated s heme
su h as this one, we an hope to make a better use of the resour es by allowing higher
network load innormal use.
4. Con lusion
We believe the kind of s heme presented in se tion 2.2 is well-adapted to solve traÆ
engineering problems of the ategory des ribed in se tion 2.1 and some others as well.
Though alternatives exist, whether o -line or on-line, entralized or de entralized, our
s heme seems an ex ellent ompromise between omputationaleÆ ien y and optimality.
It appeared in [6℄ that for a small enough LSP mean size ompared to link apa ities,
the a ura yof the solutionishigh. Thisapproa h isalsoeasilyupgradable,inthe sense
thatnotionssu haspreemption[5℄,restoration[11℄ anbeintegratedinastraightforward
manner. Moreover, the distin t underlying three-fold organization of the s heme
(ob-je tive or s ore fun tion, predi ate, omputation pro edure) leaves the door wide open
for re nement: we an de ne new obje tives, add new lauses to the predi ate and
im-prove the approximate s heme independently. Last, due to the omputational eÆ ien y
of the approximation, this methodis highlydynami ,almost ompatible with some kind
of "peer-to-peersession" LSPestablishment,and has averyshort responsetime. In fa t,
the time needed to establish anLSP is pra ti ally redu ed to that required to announ e
the path through the network.
Thoughse tion3wasmeantasabasi illustrationofwhat weintendwith thiss heme,
wethinkithastheinterestofshowingpra ti allya ompromisethatisseldommentioned:
load-balan ing against traÆ minimization. It is obvious that lever traÆ engineering
Finally, our s heme an be implemented quiteeasily. Weonly need anextension of an
existing link state proto ol (OSPF-TE [13℄ is an interesting possibility). We also need
a way to establish the omputed LSP path with the ne essary bandwidth reservations
throughoutthe network. This ouldbedone usingtheextended RSVP-TEde ned in[2℄.
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