Response-Time-Optimized Distributed Cloud Resource Allocation

1.3 MB·

by  Matthias Keller

1
Response-Time-Optimized Distributed
Cloud Resource Allocation
Matthias Keller and Holger Karl
Abstract—Acurrenttrendinnetworkingandcloudcomputingistoprovidecomputeresourceswidelydistributedexemplifiedbyinitiatives
likeNetworkFunctionVirtualization.Thispavesthewayforawidespreadservicedeploymentandcanimproveservicequality;anearby
servercanreducetheuser-perceivedresponsetimes.Butalwaysusingthenearestserverisabaddecisionifthatserverisalready
highlyutilized.
Thispaperinvestigatestheoptimalassignmentofuserstodistributedresources–aconvexcapacitatedfacilitylocationproblemwith
integratedqueuingsystems.Wedeterminetheresponsetimesdependingonthenumberofusedresources.Thisenablesservice
providerstobalancebetweenresourcecostsandthecorrespondingservicequality.Wealsopresentalinearproblemreformulation
showingsmalloptimalitygapsandfastersolvingtimes;thisspeed-upenablesaswiftreactiontodemandchanges.Finally,wecompare
6 solutionsbyeitherconsideringorignoringqueuingsystemsanddiscusstheresponsetimereductionbyusingthemorecomplexmodel.
1 Ourinvestigationsarebackedbylarge-scalenumericalevaluations.
0
2 IndexTerms—cloudcomputing;virtualnetworkfunction;networkfunctionvirtualization;resourcemanagement;placement;facility
y location;queueingmodel;linearisation;optimization
a (cid:70)
M

9
2
1 INTRODUCTION thecrucialqualitymetricistheuser-perceivedresponsetime
]
I 1.1 ChallengesinDistributedClouds toarequestastheapplicationneedtoquicklyreactonuser
N interactions.Largeresponsetimesimpedeusability,increase
A current trend in networking and cloud computing is to
. userfrustration[16],[37],orpreventcommercialsuccess.
s providecomputeresourceswidelydistributed.Computation
c An obvious solution to provide small response times
will not only take place on desktops or large data centres,
[ wouldbetodeployanapplicationatmanysitessothateach
but also at smaller centres or within the network itself,
userfindsonesitenearby.This,however,isinfeasibleaseach
2 e.g.,insideindividualin-networkserverrackslocatednear
utilizedsiteincursadditionalcosts.Wearehencefacedwith
v backbonerouters.Thistrendisknownunderdifferentlabels,
2 thetasktodecidewhereauser’srequestshallbeprocessed,
forexample,CarrierClouds[6],[13],[51],DistributedCloud
6 usingasfewsitesaspossibleatabestpossibleresponsetime.
Computing [3], [15], [21], [46], or In-Network Clouds [28],
2 Werefertothistaskastheassignmentproblem.Thisproblem’s
6 [48], [50]. These In-Network Clouds tend to be less cost-
trade-offbetweencostandqualityisintuitiveyetdifficultto
0 efficientthanconventionalCloudsduetoaworseeconomy
captureinaconcreteproblemstatementandsolution.
. of scale; they are hence often geared towards specific
1 Thisdifficultyliesinthenatureoftheresponsetime.Itis
0 network services (e.g. firewalls, load balancers). Easing a asumofthreeparts:
6 moreflexibledeploymentoftheseservicesbecamepopular
• Thenetworklatencytakentosendtherequestfromthe
1 as Network Function Virtualization [22] not only inside a
usertothecloudresourceandsendingtheanswerback
: data centre but also beyond, in wide area networks [17],
v –theroundtriptime(RTT);
i [51],[52].Weconsidernotonlyexecutingnetworkfunctions
X • theactualprocessingtime(PT)oftherequest;
but more generically executing applications at those In-
• the queuing delay (QD) a request incurs at the cloud
r NetworkCloudsyieldinganimportantadvantage[4],[43]:
a resourcewhileotherrequestsarecurrentlyprocessedat
The resources of these Clouds are closer to end users
thatresource(??).
than those of conventional Clouds, have smaller latency
Inmanyapplications,wecanconsidertheprocessingtimeto
betweenuserandcloudresource,andarethereforesuitable
besignificantlysmallerthantheroundtriptime.Theround
for running highly interactive applications. Examples for
trip time depends on the choice where a user’s request is
such applications are latency-critical applications [7], [54],
processed and, to a much smaller degree, on the network
user-customizedstreamingservices[8],[18],[30],orCloud
loadalongtheway.Thequeuingdelay,however,depends
Gaming [37]; the computing tasks range from processing
onthesharingofaresourceamongmanyusersandisnot
the request, aggregating incoming data streams, up to
aneffectimmediatelyinfluencedbythedecisionforasingle
renderingandencodingvideostreams.Insuchapplications,
user;itdependsonthejointdecisionforallusers.
Fromqueuingtheoryweknowthat,forafixedutilization,
• ManuscriptreceivedJanuaryxx,yy;revisedJanuaryxx,yy.Thisworkwas
thequeuingdelayisshorterforhigherservicerates.??shows
partiallysupportedbytheGermanResearchFoundation(DFG)withinthe
CollaborativeResearchCentre“On-The-FlyComputing”(SFB901). queuing delays for different levels of system utilization
• TheauthorsarewiththeUniversityofPaderborn,WarburgerStr.100, ρ=λ/µforthreedifferentserviceratesµ.Forinstance,web
33098Paderborn,Germany.E-mail:[email protected],[email protected]
servers answering simple requests have high service rates
2
User Server
50
y
a ¹=0:10
el 40 ¹=1:00 180
g d 30 ¹=10:00 ms] 160 S1 S
queue euin 2100 me [ 140 2
u
cpu q 0 ti S
120 3
0 1 1 3 1 p.
4 2 4 s
e 100 S
utilization ½=¸¹ . r 4
Figure 1: Response time p 80
x
as the sum of round trip Figure 2: Queuing delays e S5
60
time RTT, queuing de- QDs for different service
10 30 50 70 90 110 130 150 170
layQD,processingtimePT. rates.
arrival rate ¸ [req/s]
andoftennegligiblequeuingdelays(say,below10ms).This Figure3:Thelowerplotshowsresponsetimesofdifferent
mayexplainswhytheyarecommonlyignoredinliterature arrivalratesλc forthreedifferentstrategiesforthetopology.
(??). We focus in this paper on computation-intensive ap- Theupperplotshowstheusedtopology.
plications, such as data processing, intrusion detection, or
game render application (see next paragraph) oppose to
light-weight application with nearly no computation like bothcaseswithdifferentproportionsofroundtriptimesand
web servers just returning static content. The processing processingtimes(??).
time of computation-intensive applications is longer (e.g.
up to 1s) than of light-weight applications, which implies 1.2 QueuingDelayEffects
a lower service rate and a longer queuing delay. A long
How does the queuing delay affect the response time? As
queuingdelaybecomesalargeportionoftheresponsetime,
a toy example, let us consider the network from ?? with
significant large enough to necessitates considering them
three locations of interest: One client c and two possible
whendecidingtheassignments.
facility locations fa, fb with compute resources to run the
Shah etal. [49] survey intrusion detection systems and
application.Theseresourcesareequallyfastandcanserve
citedifferentmeasurementsofpacketprocessingtimesofup
requests at rate µ=100req./s. Assume the round trip times
to10ms.Barkeretal.[7]studygameservermaploadslasting
20–110ms in their experiments. Ishii etal. [30] conduct between c and fa as 60ms and between c and fb as 70ms.
Requestsenterthenetworkatcwitharrivalrateλ.Withthis
experiments on AWS [1] using a parallel Data Processing
Applicationandobserveprocessingtimebetween400and setup,therequestscanbeservedatonlyfa,onlyfb,orsplit
1800ms. Lee etal. [37] and Claypool etal. [16] observe amongfa andfb.??shows,asafunctionofthearrivalrate,
theresultingresponsetimesRT=RTT+PT+QDforafew
drop in user experience when playing computer games
with artificially increased latency to larger than 200ms. In simplestrategiesSi2.
ThefirststrategyS1 minimizesonlytherequests’round
summary, we focus on applications with long processing
times,10ms–1s,andonaverageinternetroundtriptimes, trip time: Requests are assigned to the nearest facility fa
andifitscapacityµisexceededtheremainingrequestsare
60–600ms[36].
assignedtofb.Thedramaticresponsetimegrowthforλ>80
Differentproportionsbetweenroundtriptimesandpro-
istheresultoftoomanyrequestsassignedtofacilityfa.Let
cessingtimesarepossible.Decidingtheassignmentinsuch
λa betheassignedrequeststofa,thenfa’sutilizationρa is
scenarios can be simplified by ignoring the less dominant
λa/µ.Toavoidtoolargeutilizations,strategiesS2andS3limit
part: Very short round trip times inside data centres, say
lessthan20ms,leavesqueuingdelaysasthedominantpart themtoamaximumvalueρˆ,ρa, ρb ≤ ρˆ<1;S2 usesρˆ=0.9
andS3 usesρˆ=0.8.OntheonehandS3 withalowerlimit
of the response time. Similarly, long processing times, say
more than 1min, renders queuing delays as the dominant has a shorter RT (??) than S2 as the second facility is used
earlier.ButontheotherhandS2 canhandleahigherarrival
part.Inbothcases,roundtriptimescanbeignored;doing
rates than S3, 160<λ<180, because a higher limit enables
so, the assignment problem becomes a simpler mapping
handlingmorerequestsintotal;systemcapacityisλ<2µρˆ.To
problem.Ontheoppositesite,veryshortprocessingtimes,
say 0.1ms1, result in very low queuing delays rendering relaxsuchapredefinedupperbound,S4dynamicallyadjusts
thelimittothecurrentsystemutilization,ρˆ=λ/2µ.Withthe
theroundtriptimebeingthedominantpart.Whenignoring
same resources at both locations S4 boils down to evenly
queuingdelaysinthiscase,theassignmentproblembecomes
splitting the load between the two facilities. Compared to
asimpler,non-convexFacilityLocationProblem.Insummary,
S1..3 theresultingRTsareontheonehandsmallforλ>40
only if round trip times and processing times are not of
butontheotherhandlargerforλ<40.
thesamemagnitude,droppingthelessdominantpartisa
So far, all assignment strategies ignore the resulting
vitaloption.Ifbothtimesareofthesamemagnitude,also
queuingdelays.Incontrast,ourlaststrategyS5 additionally
queuing delays have to be considered when deciding the
assignment.Ignoringqueuingdelaysinsuchcasesworsen 1.Correspondingservicerateof10,000req./sofahighperformance
theassignmentresultinginhighresponsetimes;wecompare webserver,e.g.Apache,Nginx,deliveringstaticcontent.
3
1.0 ¹ of f magnitude). To proof this claim, we present an extended
¸ a FacilityLocationProblemwithintegratedqueuingsystems
of 0.8 (??),showitsconvexity,and,forthefirsttime,obtainoptimal3
o
ti solutionsforlargernetworksusingaconvexsolver(??). Due
c
a 0.6 toproblem’scomplexity,solvingtimeswerealreadylargefor
r
t f medium-sizednetworks(e.g.??ta2topology)hinderingthe
en 0.4 largescaledevaluationwehadenvisioned.Wewereableto
m
n l¢ =0 l¢ =20 shortenthesolvingtimeswithhighaccuracybynon-trivially
sig 0.2 l¢ =5 l¢ =40 linearisingtheconvexproblem(??).Thislinearisedproblem
as l¢ =10 l¢ =80 hasalargersearchspaceastheconvexproblem;despitethis,
itsolvestheoriginalproblemsignificantfaster(empirically
0.0
1 26 51 76 101 126 151 176 shown,??).Havingnowanadequateandfastsubstitution
arrival rate ¸ [req/s] athandenablesustocomparenumeroussolutionsobtained
byconsideringandignoringqueuingdelayssupportingour
Figure4:Requestassignmentstofa withdifferentdistances claimbyshowingsignificantresponsetimereductionwhen
l∆ betweenfa andfb. consideringqueuingdelays(??).Inaddition,weshowhow
theresponsetimeandqueuingdelaysincreaseswhenusing
lessercomputeresources(e.g.fewerlocations,??).
reducesthequeuingdelaysofbothresources.Thestrategy’s Thisevaluationin??extendsourownpreviouswork[33].
request assignment depends on the round trip times (lcf) We compare four factors influencing queuing delays and
betweenbothresources.TheresultingRTsarethelowestfor in addition vary input randomly in order to verify the
allstrategiesS1..5.Inconclusion,wewereabletoimprove statisticalrelevanceofourfindings.Insummary,wesolved
assignmentsbyconsideringqueuingdelays. andanalysed52,500configurations.
Hereafter,welisttheequationsoftheexpectedresponse
timesin??:S1 toS3 resultsin(1),S4 in(2),andS5 in(3).
2 RELATED WORK
60+ 1 , ifλ≤ρˆµ Assignment problems of the form described above have
 (cid:16) µ−λ (cid:17) been investigated before. We structure their comparison
fµ,ρˆ(λ):= ρˆλµ 60+ µ−1ρˆµ + (1) along four dimensions relevant to this work: Their model
 λ−ρˆµ(cid:16)70+ 1 (cid:17), else complexity, simplifications reducing the problem’s search
λ µ−λ+ρˆµ
space,optimizationgoals,andsolutionapproaches.Finally,
1(cid:18) 1 (cid:19) 1(cid:18) 1 (cid:19)
gµ(λ):= 2 60+µ−λ/2 + 2 70+µ−λ/2 (2) relatedsystemsofgeographicalloadbalancingarecompared.
 (cid:16) (cid:17) 
 λ1 60+ 1 +  2.1 ModelComplexity
hµ(λ):= λ1m∈[i0n,λ]λ−λλλ1 (cid:16)70+µµ−−λ1λ1+λ1(cid:17) (3) Thesimplestmodelconsidersonlytheroundtriptime(RTT)
when assigning users to cloud resources. They equate
?? shows strategy S5’s request assignments to resource response time with RTT. Clearly, this is a simplification of
fa as a fraction of λ on the vertical axis; the remaining reality,yetminimizingthisaverageRTTisequivalenttothe
requests are assigned to fb. The horizontal axis shows an well-knowncapacitatedFacilityLocationProblem(FLP).If
increasing arrival rate λ. The different lines correspond to theproblemisfurtherrestrictedtoonlyusepresources,it
distancesl∆ betweenfa andfb –howmuchlongerrequest becomesap-medianFLP,whichisNP-hard[31].
transportation takes to send to fb instead to fa. This way A step closer to reality is modelling also the processing
the original toy example is line l∆=10 and the other lines time (PT) in addition to the RTT. But as long as PT is
vary the round trip time to fb. If the resources have the constant, this still stays a Facility Location Problem of the
sameroundtriptime,theassignmentresultsinanevensplit. typedescribedabove.Thiscanbeeasilyseenbyextending
However, if one resource is farer away, l∆>0, at first the theoriginalnetworktopologybypseudo-links(attheserver
nearerresourceispreferredandonlywithincreasingarrivals oruserside)thatrepresenttheseprocessingtimesviatheir
do the assignments converge to an even split. Then, the latencies; this is a common rewriting technique for graph-
queuingdelayportionoftheresponsetimeissignificantly basedproblems(includingFacilityLocationProblems).
largerthanl∆. The real challenge occurs when we also consider the
queuingdelay(QD).Inthiscase,theadditionaltimecannot
1.3 Contribution beexpressedbyrewritingthenetworktopologyastheQD
dependsontheassignmentdecisions:Ahigherutilization
This paper discusses finding the optimal assignment be-
resultsinalongerwait,possiblytradingoffagainstashorter
tweenrequestingusersandcomputeresourceshostingthe
RTT.
answeringapplicationatdifferentlocationsinthenetwork.
Sofar,thismoregeneralmodelhasbeenconsideredonly
The assignment minimizes the expected average response
by few works discussed in the remaining of this section,
time for all users. We claim the necessity of considering
mostusesimplerassumptionsthanours(??)renderingthe
request’s queuing delays at used compute resources to
avoidsuboptimalassignmentstoe.g.over-utilizedresources
3.Numericallyobtainingsolutionsbysolverwithagapthresholdof
(whenroundtriptimesandqueuingdelaysareofthesame 10−6.
4
problemeasiertosolve.Vidyarthietal.[53]allowthesame 2.3 OptimizationGoal
degrees of freedom as we do. They approximate, similar
Existing literature uses queuing delays in FLPs with three
to us, the non-linear part of the objective function with a
optimizationgoals:classicFLP,min/maxFLP,andcoverage
piece-wiselinearfunction.However,incontrasttoourwork,
FLP.
theyusedacuttingplanetechniquewhichiterativelyrefines
Classic FLPs are problems that minimize the average
thepiece-wisefunctionasnecessary;itremainsunclearhow
responsetime,likeourproblem(??)orothers[10],[19],[47],
largetheirlinearisationerroris.Incontrast,ourevaluation
[53],[55],[59],allowingRTvariationsforindividualusers.
shows small linearisation errors; and this is achieved by
Aboolian etal. [2]’s min/max problem minimizes the
usingasimplertechnique.
maximumresponsetime.Intuitively,suchproblemsimprove
especiallytheusers’RTwithhighRTTstocloudresources.
However,ifonlyonesuchuserexistswithresourcesbeing
far away, assigning this user will negatively affect the
2.2 Simplifications
assignmentsofotherusers:Theirassignmentsarenowless-
restrictedlyconstrainedbyarelaxedupperboundandare
Otherauthorsinvestigateslightlydifferentscenarios,sothat
likelyworsethanwithoutthefirstuser.Incontrast,classical
theirproblemformulationsaresimilar,yetsimplerthanours.
FLPsdonotsufferthiswayfromaworsecaseuser.
Some authors [55], [59] replace the non-linear QD part
Anothertypeofproblemiscoverageproblems;theuser
withaconstantupperboundand,consequently,theresulting
assignment’s response times is upper bounded [40], [41],
problems become simpler to solve. But this also hides QD
[42]. Structurally, a coverage problem is a special, simpler
changesasaresultofassignmentchanges.Forinstance,in
case of a min/max problem; the first has a predefined
a situation where load balancing would reduce the QD,
bound, which is additionally minimized in the second.
this reduction is not visible as the QD part is constant.
Intuitively,suchproblemscanbeappliedinscenarioswhere
Consequently, the resulting solution has further potential
serviceguaranteesforacertainmaximalresponsetimewill
foroptimization–weexploitthispotential.
be provided and paid. In contrast, classical FLPs allow
Inanothersimplification,theassignmentsarepredefined minimizing the average response time below the lowest
byarule.Someauthors[2],[55],[59]alwaysassignrequests possibleresponsetimebound.
to the nearest cloud resource. In such a case, the problem
reduces to just finding the best resource location and is
easiertosolve.Theassignmentsarethenpredeterminedby 2.4 SolutionApproaches
therule.However,balancingtheassignmentscouldfurther
Acoupleofheuristicswereproposedsolvingrelatedprob-
reduce the QD but is not considered. We do not use any
lemswhicharevariantsoftheNP-hardcapacitatedFLP[27].
predefined assignment rule, so we have the freedom to
No work so far used solvers to obtain solutions (for non-
changeassignmentsinordertofurtherreducetheresponse
relaxedproblems)andfullenumerationsareknownforsmall
times.
instanceslimitedtoopenfivefacilities[2].Agreedydropping
Anothergroupofauthors[10],[19]usesaparametrized heuristicsuccessivelyremovesfromthesetofcandidatesthat
assignmentrulecalledthegravityrule:Weightsdetermine resourcewhichincreasestheresponsetimebythesmallest
how users are assigned to cloud resources. These config- amount [55]. Greedy adding heuristics successively add
urable weights are used to continuously solve the same resources,whichdecreasestheresponsetimebythelargest
problemwithnewweightsreflectingtheresourceutilizations amount [2], [10], [19]. Another heuristic probabilistically
oftheprevioussolution.Thisapproachdoesnotguaranteeto selects set changes of used resources [19] or performs a
converge,sotheauthorsproposeaheuristicthatattenuates breath-first-searchthrough“neighbouringsolutions”where
the changes in each iteration, enforcing convergence with twosolutionsareneighboursiftheirsetsofusedresources
an unknown linearisation error. In contrast, we solve the differinoneelement[2],[19].Suchheuristicscanbestocked
problem in one step by using all information to find the inlocaloptimaandtomitigatethisdrawbackmeta-heuristics
globaloptimum. areusedasasuperstructure[2],[10],[19],[47],[55].These
Liu etal. [39], Lin etal. [38], and Goudarzi etal. [25] meta-heuristicstypicallyrefinepreviouslygeneratedinitial
present a similar Facility Location Problem with convex solutions, which are obtained randomly or by combining
costs such as queuing delays or resource’s energy costs. existingsolutions.Thehopeisthatamongthefoundlocal
In contrast to our work, they relax the integer allocation optima, one solution is very close to the global optimum
decision variable simplifying the problem to the cost of – but without any guarantee. In contrast, we obtain global
a less accurate solution when rounding up the obtained optima.Thisisanimportantstepforheuristicdevelopment
continuous allocations. Our goal,in contrast, is to prevent asonlythisenablesaclearjudgementofheuristics’accuracy;
unexpected expenses by introducing an upper bound to theirsolution’sgaptotheglobaloptimum.
the number of used resources. Continuously relaxing our Others[53],[55]mayachievenearoptimalsolutionsby
problem can cause any location to be allocated a bit and, usingoptimizationtechniqueslikebranch-and-boundand
consequently, any site is used and paid. While the papers cuttingplanesbuttheirsolutionshaveunknownoptimality
[38], [39] only consider queuing delays as a cost function, gabs.Insummary,eitheroptimaforsmallinputorsolutions
this paper discusses a holistic queuing system integration withunknownoptimalitygapareobtained.Thismotivated
andadditionallyconsiderssplittingandjoining(assigning) ourworkonfindingnear-optimalsolutionswithanumeri-
ofthearrivalprocess. callyverysmalloptimalitygap.
5
Liuetal.[39]andWendelletal.[56]presentdistributed a)
algorithms for their global Geographical Load Balancing
problembydecomposingitintoseparatesubproblemssolved
by all clients. These subproblems converge to the optimal
solution only if they are executed in several synchronized
roundsinwhichassignmentandutilizationinformationare
exchangedamongallclients.Bothpapersstatethatthisdis-
tributedalgorithmswouldobtainoptimalsolutionfasterthan
gathering everything to a centralised solver. However, we
believethateachroundacommunicationdelayisintroduced
whensendingupdateinformationamongallclients;theyhad b) c)
ignored this delay in their evaluations. The resulting total
delay over all rounds is likely larger than communicating
Figure5:BipartitegraphofaFacilityLocationProblem(a);
with a centralised coordinator. In addition, our p-median
time-in-systemfunctionsateachfacility(b)and,alternatively,
Facility Location Problem has a global constraint on the
piece-wiselinearisedfunctions(c).
maximalusedresourcespreventingittobeeasilyseparated
intosubproblems.
Table1:Modelvariables
We observed that problem instances were solved only
Inputconstants:
exemplarysofar[2],[10],[19],[38],[39],[41],[47],[47],[56].
G=(V,E) Bipartite graph with V = C ∪F,
Consequently, the average performance of these solution C∩F =∅withclientnodesc∈C
approachesishardtopredict.Wegobeyondthisbyunder-
andfacilitynodesf ∈F
taking a statistical performance evaluation. We randomly lcf ∈R>0 Roundtriptimebetweencandf
varyourinputdataandverifythestatisticalrelevanceofour µf ∈R>0 Servicerateascapacityatf
findings. λc ∈R>0 Arrivalrateasdemandatc
Tµ ∈R>0 Timeinqueuingsystem(TiS)
2.5 GeographicalLoadBalancing αµs,βµs s-thbasepointTµ(αs)=βs ofT˜µ
AsystemforGeographicalLoadBalancing(GLB)comprises Decisionvariables:
twoparts:Thedecisionpartselectsappropriateserver,sites, xcf ∈R>0 Assignmentindemandunits
or Virtual Machines for requests of a certain origin – the yf ∈{0,1} Indicatoriff isopened(=1)
previous sections considers them. This section focuses on zfs ∈[0,1] Weightofs-thbasepointatf
therealisationpart,whichgathersmonitoringinformation
Helpervariables:
andimplementsselections.Differentmiddlewareshadbeen (cid:80) (cid:80)
proposed [23], [56], [57], [58] which are shared between Λ;Λf totalΛ= cλc;atf:Λf= cxcf
applications. In this way, each application benefits from τ ∈R>0 Sufficientsmallvalue
instancesoftheotherapplicationrunningatdiversesitesby ρ∈R>0=λ/µ Systemutilization
sharing monitoring information such as latency to servers
or to customers. They realise request assignments, e.g., to
correspondtolocationswhereuserrequestflowsenterthe
close-by or low utilised server by either configuring the
network.Facilitiesrepresentcandidatelocationstoexecute
DomainNameSystem(DNS)orareexplicitlyqueriedahead
theapplication,e.g.,datacentres.Moreprecisely,a(compute)
arequestsend.Slightlydifferent,Cardellinietal.[14]propose
resourcerefertoahostatsuchadatacentreexecutingthe
redirecting requests to different sites to balance the load.
application. ??a shows such a graph. The geographically4
Policies ranges from redirecting all, only largest, or only
distributeddemandismodelledbytherequestarrivalrate
grouprequeststoselectingsitesbasedonround-robin,site
utilization,orconnectionproperties.Ourpaperfocuseson λc foreachclientc. Computingcapacityismodelledasthe
solvingtheproblemandinvestigateswhetherthecomplexer request serving rate µf for each facility f. The round trip
problemwithqueuingsystemsisworththeadditionalefforts timelcf isthetimetosenddatafromctof andback.??lists
allvariables.
andourresultscanbeappliedtoimprovegeographicalload
Ourfirstproblemformulationrecapitulatestheknown
balancingsystems.
p-medianproblemP(G, λ, µ, p):
1 (cid:88)(cid:88)
T3hisPsRecOtiBonLEfiMrst formalises our scenario model and then mxin (cid:80)cλc c f xcflcf (objective) (4)
(cid:88)
details on practical realisations. Afterwards this section s.t. xcf =λc, ∀c (demand) (5)
discusses problem’s convexity and proposes a problem f
linearisationminimizingthemaximallinearisationerror. (cid:88)
xcf ≤yfµf, ∀f (capacity) (6)
c
3.1 Model
Ourscenarioisformalizedasacapacitatedp-medianFacility 4.Moreprecisely,therequestarrivalandservicepointsaretopologi-
callydistributed;theroundtriptimeofapathbetweentwopointsonly
LocationProblem[20].AbipartitegraphG=(C∪F, E)has
roughlymatchesitsgeographicallydistance.Weuse“geographically”
twotypesofnodes:clients(c∈C)andfacilities(f∈F).Clients foraconvenientexplanation.
6
(cid:88)
yf =p (limit) (7) Geographical Load Balancing systems (??) to our own
f ApplicationDeploymentToolkit[34].Theymonitortraffic,
decide assignments, and reconfigure the dispatching sub-
The formulation contains two decision variables:
xcf∈R≥0 describes which part of c’s request rate λc is sayvsetreamgedinntuimmbeepreorfioindcso.mTihnegarevqeuraegstesaartrriovuatlerractefoλrcthiesltahset
assignedtowhichf;yf∈{0, 1}describesiflocationf isused period. By solving problem QP once a period, the request
or not. The objective is to minimize the average response
assignments5 aredecidedforthenextperiod.Thedecisionis
time;butwithoutmodellingthequeuingdelayandservice
realisedbyconfiguringthedispatchingsubsystem,e.g.,DNS,
timeatfacilities,theresponsetimeonlyconsistsoftheround
and allocating cloud resources accordingly. The system is
triptime.TheRTTisminimizedwhilealldemandisserved
not meant to allow a fine grained assignment decision for
(5)andthecapacityisnotexceeded(6).
eachincomingrequest,e.g.atlinespeed.Onlongerterms,it
In addition, exactly p locations are used (7). This con-
decideswhichsitesarestrategicallyusedandhowincoming
straintservestwoproposes.First,bylimitingthenumberof
requestsareroughlydistributed.
locationwheretheapplicationisdevelopedto,theexpenses
fortheapplicationproviderwhenleasingCloudresourcesis
bound.InFacilityLocationvariantswherefacilityopening 3.3 ConvexOptimization
costs are directly integrated the resulting total costs are Previouswork(??)alsoconsideredourobjectivefunction(8)
unsure.Second,statingtheproblemwiththisboundallowus butdidnotsolvethecorrespondingproblemoptimally,ex-
toinvestigatetheresponsetimetrendwhileallowingmore ceptforsmallgraphsviafullenumeration.Thisisbecauseof
and more resources (??). Since 1979 the problem without thenon-linearityoftheobjectivefunctionwhichnecessitates
capacity is known to be NP-hard [27]. This problem is a non-linearsolvers.Thereexistacoupleofnon-linearsolvers
generalizationand,thus,alsoNP-hard. with different specializations: quadratic, convex, or non-
Untilnow,theresponsetimehasonlybeentheroundtrip convexobjectivefunctions.Bydeterminingthecomplexity
time. To predict the queuing times, the model is extended class of our objective function, we can choose a suitable
byqueuingsystemsateachfacility(??b).There,theservice solver,toefficientlyobtainaglobaloptimum.
timesareexponentialdistributed.Theinter-arrivaltimesat Thissectionfirstprovestheobjectivefunction’sconvexity
eachnodecaredescribedbyaPoissonprocess.Therequests andshowsthatitisnotsimpler,e.g.,quadratic.Afterwards,
(cid:80)
canbeassignedtomultiplefacility( fxcf)and,there,the itdescribeshowweusedtheconvexsolver.
individualassignmentfromdifferentnodesareaggregated
(cid:80) Definition1. Afunctiong isconvexifitsdomaindom(g)is
( cxcf). The resulting process is also a Poisson process, aconvexsetandifg(cid:48)(cid:48)(x)≥0holds∀x∈dom(g)[12].
becausesplittingandjoiningdoesnotchangetheunderlying
randomdistribution.Asaresult,wehaveaM/M/1-queuing Lemma 1. Function g=(cid:80)iwigi, g:Rn→R, is convex, iff
model[11].Thefunctionforthetimeinqueuingsystem(TiS) ∀i:gi:Rn→Randwi∈R>0 isconvex[12].
computes the processing time plus the queuing delay (??),
Tfoµr(mλu)=laµti−1oλn.oPfuttthinisgqeuveeruyitnhgi-negxtteongdetehderp,-tmheecdoirarnespproonbdlienmg wThitehofruemncti1o.nTThµecoombjepcutitvinegftuhnecstoiojonur(8n)tiomfeQinPainsMco/nMve/x1
queuingsystem.
QP(G, λ, µ, p)is:
mx,iyn (cid:80)c(cid:80)fxccλfclcf +(cid:80)f((cid:80)cx(cid:80)cfc)λµcf−(cid:80)1cxcf (8) 0co≤nλv<exµPsreeonto.ffoT:rhTceehdseebcdoyonmcdoandisnetrrioavifantTtiv(µe1(0λT));(cid:48)µ(cid:48)=a(λn1/)iµ=n−t2λe/(rµvis−aλlt)hi3seiasilnwatlaweyravsyaasl
(cid:124)avera(cid:123)g(cid:122)eRTT(cid:125) (cid:124) aver(cid:123)ag(cid:122)eTiS (cid:125) larger0withinitsdomain.By??,Tµ(λ)isaconvexfunction.
(cid:88) Forafixedf intheobjectivefunction,0<Λf=λ<µand
s.t. xcf =λc, ∀c (demand) (9) Tµ(Λf)isconvex.Then,thenon-negativeweightedsumof
(cid:88)f convexfunctions(cid:80)fΛfTµ(Λf), Λf=(cid:80)cxcf isalsoconvex
xcf <yfµf, ∀f (capacity) (10) (??).Thetermremainsconvexafter1/Λ>0ismultiplied.The
c lefttermoftheobjectivefunctionislinearandalsoconvex.
(cid:88)
yf =p (limit) (11) Sincethesumoftwoconvexfunctionsisconvex,theobjective
f function(8)isconvex.
Withtheknowledgeofaconvexobjectivefunction,we
Thenewobjective(8)istominimizetheaverageresponse
canignorelessefficientsolversformoregeneral,non-convex
time,whichisthesumoftheaverageroundtriptimeand
problems.Thenextmoreefficientsolverclassisquadratic,
theaveragetimeinsystem(??).Constraint(9)isthesameas which need objective functions of the form xTMx with
Constraint (5); all demand must be served. Constraint (10) symmetric matrix M∈Rn. But our objective function is
assures the steady state (λ<µ, c.f. [11]) for each queuing
not of this form, making quadratic solvers inapplicable.
system. Finally, Constraint (11) mandates to use exactly p
Consequently,wehavetouseaconvexsolver.
locations,justlikeConstraint(7).
Implementation:Wechoosetheoptimizationframework
CVXOPT[5]fromtheauthorsof[12].QPisamixedinteger
3.2 Systemdesign problem, which is not directly supported by CVXOPT.
Continuouslyrelaxingtheproblemisnotpossible(??).We
The presented optimisation problem QP is part of a large
system which dispatches requests of a certain origin to 5.Theassignmentxcf istherequestratedispatchedfromctof,in
sites as decided. Examples of such a system ranges from shortrequestassignment.
7
decomposedQPintosolvingmultiplesubsetsF(cid:48) ofF with afterwardssummingalltermsup.Twotypesoftermsexist
|F(cid:48)|=p: (18)withdifferentderivatives(19,20).InJacobimatrix(17),
QP(G=(C∪F,E), λ, µ, p, τ)= eachpartialderivativeacf isg1(cid:48),cf(x)+g2(cid:48),cf(x).
F(cid:48)∈Fm,|iFn(cid:48)|=p{PQP((C∪F(cid:48),E), λ, µ, τ)} (12) g1(x)= lcfΛxcf g2(x)=Λ(ΛΛ−f µ ) (18)
f f
withthepurelyconvexsub-problemPQP(G, λc, µf, τ):
min (cid:80)c(cid:80)fxcflcf + (cid:80)f (cid:80)µf(cid:80)−c(cid:80)xccfxcf (13) g1(cid:48),cf(x)= dxdcf lijΛxij =(cid:40)0lΛij ieflscef=ij (19)
x cλc cλc d Λ
(cid:88) g(cid:48) (x)= j
s.t. xcf =λc, ∀c (demand) (14) 2,cf dxcf Λ(Λj−µj)
(cid:88)f xcf ≤µf −τ, ∀f (capacity) (15) =(cid:40)0Λ(µjΛ−jΛj)2 + Λj(µj1−Λj) ieflsfe=j (20)
c
ThedecompositionoptimallysolvesQPbysolvinganon- Similarly, the Hessian matrix (21) contains second-order,
integerconvexsubproblemPQPseveraltimesfordifferent partial derivatives which are first derived in xcf direction
configurations of the binary variables yq; these variables (rows)andtheninxde direction(columns).
indicatewhichfacilityisused.First,problemPQPissolved
H (x)=(a ) ,
withallfacilitysubsetsQ(cid:48)⊆Qwith|Q(cid:48)|=t.Then,oneofall f cfde cf∈C×F,de∈C×F

pvreeqersoucptbaoollrsenmtsyheeiPstQiQmrPePep’’srs(ed1ss3eeo)cnl.iustIietnoidontnbhvyieisscstsuosoerbllusxeetciattoenQndd,(cid:48),ptth∀hroaqebt∈dlQheemac(cid:48):sisQyitoqhP=ne’s1vmd.eeIicnnctiiosmtrihoaixnsl acfde =0Λ(Λ2fΛ−Λf(µΛff)3−2+µf)2 ieflsfe=e (21)
wayC,VthXeOoPpTtimsaollvseosluPtioQnPfobryprcohbelcekminQgPthisefoduonmda.in (con- Eachcellacfde (21)isg1(cid:48)(cid:48),cfde(x)+g2(cid:48)(cid:48),cfde(x)from(22,23).
straints) and iterating towards the optimum by using the g(cid:48)(cid:48) (x)= d lijxij =0 (22)
JacobiandHessianmatrix(firstandsecondorderderivatives) 1,cfde dx dx Λ
cf de
oftheobjectivefunction(13).Hardcodingsuchmatricesis d Λ
g(cid:48)(cid:48) (x)= j
notfeasibleforalargenumberofparameterconfigurations. 2,cfde dx dx Λ(Λ −µ )
cf de j j
Wmeatrwicaenstattorhuanvteimaen. aAulgtoembraatesdysstoelmutsiolnikeobMtaainxiinmga6thceasne =(cid:40)Λ(Λ2jΛ−jµj)3 + Λ(Λj−2µj)2 iff=j∧e=j (23)
be used, but need a detour through another system and 0 else
computingderivativesofmulti-dimensionalfunctionstakes
time;forsmallinput,moretimethansolvingtheproblem.To
3.4 LinearApproximation
obtainthesematricesfaster,wefound,nottoosurprisingly,
WhileCVXOPTsolvestheproblemoptimally,ithastotestall
thatthestructureof(13)anditsderivativesarethesamefor
different|C|, |F|. subsetsF(cid:48),whichtakestime.Asanalternative,theconvex
objective function is linearised. This way, well researched
Exploiting this property, we were able to deduce a
linearsolverscanbeusedtoobtainsolutionsfaster.
construction rule for both matrices. Using this rule, we
construct our Jacobi and Hessian matrices at runtime for
3.4.1 Piece-wiselinear
differentinputswithoutnotableoverhead.
Indetail,weconstructedtheJacobiandHessianmatrices Any non-linear function g(x):R→R over a finite interval
from the objective function (13); here, reintroduced as a [α0, αm−1]⊂Rcanbeapproximatedbyapiece-wiselinear
convenientcopyf (16),f(x11,...,xcf). (PWL)functiong˜[24].Thisfunctionconsistsofmbasepoints
f(x)=Λ1 (cid:88)lcfxcf + Λ1 (cid:88)ΛΛ−fµ , αan0d,..i,sαdse,fi.n.,eαdmin−1(2,4co)rfroerspαos≤ndxi≤ngαfsu+n1c.tionvaluesβs=g(αs),
f f
cf f (β −β )
(cid:88) (cid:88) g˜(x):=(x−α ) s+1 s +β ,
Λ= λc, ∀f:Λf = xcf (16) s (αs+1−αs) s
c c αs ≤x≤αs+1 ∀s∈[0,m−2] (24)
TheJacobimatrix(17)foronefunctionisavectorofpartial
derivatives(a11,...,acf)foreachvariablexcf. As an example, let us consider the part λTµ(λ) of (8)
for µ=1.0. Then g(ρ):=λT1(λ)=λ/1−λ is our example func-
Jf(x)=(acf)cf∈C×F , tiontolinearise.??ashowsg andtwodifferentlinearisations
acf = Λ(µfΛ−fΛf)2+Λ(µf1−Λf) +(cid:88)lΛcf (17) gt˜h1eavnedrtig˜c2a.lTahxieshshooriwzosnthtaelcaoxrirsessphoonwdsintgheTiaSr.r?i?vbalshraotwesatnhde
c
Structurally,f(x)isasumofterms,anddifferentiatingf(x) absolute differences between g and either linearisation g˜1
or g˜2. These differences denote the linearisation accuracy:
can be done by differentiating the terms individually and
Thesmallerthedifferencesare,thetighterthePWLfunction
resembles the original function. We use the maximum of
6.Maximamanuel:http://maxima.sourceforge.net/docs/manual/
maxima.pdf all absolute differences (cid:15)g˜, defined in (25), to measure the
8
3.4.2 Linearisationalgorithm
50 g(¸)= ¸ =Tw (¸) ¯
1 ¸ 1:0 4 Our algorithm obtains basepoints for convex functions
g~ (¸), i¡mamoto
40 1 ¯ with low error. It is an extended version of Imamoto’s al-
g~ (¸), uniform 3
2 gorithm [29]. Imamoto’s algorithm iteratively refines m
S 30 basepointsbymovingthemindividuallyalongtheabscissa
Ti 20 ¯ to reduce the error (cid:15)g˜. Each basepoint’s adjustment ∆s
2 along the abscissa, αsneu=αsalt+∆s, is computed from the
10 ¯1 bbaasseeppooiinntt’dsifistrasnt-coerddse=rαdse−rivαast+iv1.e dαdasltg(αsalt) and the inter-
0 ¯0 Thepaper’s[29]statementisthatthealgorithmcomputes
®
®0 ®1 ®4 basepointswhichhavethemaximallinearisationaccuracy
® 3
2 for the given number of used basepoints. However, the
30
algorithmrunsinnumericalissuesrenderingthealgorithm
g~ (¸) g(¸), imamoto
S 20 1 ¡ useless for some convex functions. When fixing7 them, it
¢ Ti 10 g~2(¸)¡g(¸), uniform cannotbeguaranteedanymorethattheresultingbasepoints
0 formanlinearisationwithmaximalaccuracy(minimalerror).
But it is still very small – still a good and fast option to
0.0 0.2 0.4 0.6 0.8 1.0
lineariseconvexfunctions.
arrival rate ¸
Moreindetail,weextendImamoto’salgorithm[29]and
fixedthefollowingtwocases:First,thealgorithmiteratively
Figure6:Thetopplotshowsanexamplefunctiongwithtwo adjusts the current set of basepoints so that the error is
successively reduced. These adjustments are weighted in
possiblelinearizationswiththesamenumberofbasepoints.
ordertoallowgraduallyfinerchangessothattheerrorafter
The bottom plot shows absolute differences between the
linearizationsandg.Imamoto’slinearisationhasasmaller eachiterationconvergestotheminimumerrorintheory.In
practice, floating-point accuracy is limited and sometimes
maximumdifference.
valuesaretoosmall,changesnotapplied,andthealgorithm
iteratesinfinitely.Wefixedthatbyadditionallyabortingif
nofurtherbasepointchangesareobserved.
linearisationaccuracy.Weseekbasepointsαi thatminimize Second, for special functions the algorithm terminates
thiserror.
with a division by zero. The cause is computing a base-
(cid:15)g˜:= max |g˜(x)−g(x)| (25) pointαs’sadjustment∆s dependingonoriginalfunction’s
Given a set of bxa∈s[eαp0,oαimnt−s1]resulting in a certain error, dtheeridviaftfievreengc(cid:48)e(αosf)t=wdoαdvsgal(uαess).gT(cid:48)(hαei)dnivuimsioenricbayllzyeerqouoaclcsuzresroif,
this error is reduced by placing an additional basepoint ∃i(cid:54)=s : g(cid:48)(αs)−g(cid:48)(αi) = 0. That is if g resembles a linear
functionoversomeinterval.Wefixedthatbyremovingall
at a point where the absolute difference equals the error.
However, more basepoints also increase the number of basepoints αs with g(cid:48)(αs)=g(cid:48)(αi), i<s and inserting these
basepoints between basepoints whose g(cid:48)(α) values differ
necessary variables for the optimization, which increases
fromeachother.Thisassuresthattheerrorneverincreasesor
searchspaceandsolvingruntime.
canbereduced:Forthoseintervalsofthefunctionwhichare
Some functions are hard to approximate with linear
nearlylinear,alinearisationoverthewholeintervalyieldsa
segments,e.g.,functionswithlargesecond-orderderivative
lowerror;thus,removingbasepointswithinthisintervalhas
values. If their values are large within the linearisation
littleimpact.Insertingthesebasepointsatanothernon-linear
interval[α0, αm−1],theerrorwillbelarge.TheTiSfunction’s
partofthefunctionimprovesthelinearisationaccuracyas
asymptote limλ→µTµ(λ)=∞ approximated by linear seg-
thePWLfunctionbecomestighter.
mentsresultsinsuchalargererror.Onepossiblecontrolknob
is to adjust the interval αm−1<µ. But this also introduces
3.4.3 Formulationoflinearisedproblem
an artificial capacity limit: Small values (e.g. αm−1=0.8µ)
resultinfewerrequestsservedthanpossible(c.f.??b’sS2). This section describes the problem reformulation using a
Consequently,thetotalarrivalrateforwhichsolutionsare PWL function. From existing alternatives [45], we used a
feasibletoobtainissmaller,(cid:80)cλc/p≤αm−1<µwithpused Special Ordered Set (SOSk) of type k = 2 (SOS2) [9]: In a
resource. set of continuous variables, at most k of them, adjacent to
eachother,maytakenon-zerovalues.Currentlinearsolvers
BothPWLfunctionsin??b,uniformandimamoto,have
directlysupportSOS2.
thesamenumberofbasepointsbutatdifferentpositions.As
A PWL function y˜=g˜(x) is represented by a set
shown,uniformlydistributingthebasepointscandramati-
callyincreasetheerror(g˜1).Incontrast,thegreybasepoints of m continuous decision variables 0≤zs≤1 with a
havesmallerrors(g˜2).Thosebasepointswerecomputedby SOS2(z0(cid:80),..zs,..zm−1) constraint and a convex combina-
ouralgorithmsdetailedin??. tion 1= szs. This way, two adjacent values sum up to
1 = zs+zs+1. These values are then used as weights
Weevaluatethefirsttwocontrolknobs,thenumberof
for the basepoints (αs, βs) obtained previously by the
basepointsandthelinearisationinterval’supperbound,in
??.Forthethirdcontrolknob,thebasepointpositions,our
7.This paper’s extended version details our improvements of
algorithmdeterminesbasepointswithlowerror. Imamoto’salgorithm.
9
linearisation process (??). This way, the weighted sum of areusuallysolvedfasterthannon-linearproblems.Which
all basepoints results in the piece-wise linear problem, problem is solved faster? The answers is not obvious, e.g.,
(cid:80) (cid:80)
x= szsαs,y= szsβs. Q(cid:102)Pislinearbuthasalargersearchspace.Theirruntimesare
Using this representation, we linearise the convex part experimentallyevaluatedin??.
(cid:80)
of the objective function (8) ΛfTµ(Λf), Λf= cxcf, and The maximal linearisation error (35) of the objective
substituteitbycorrespondingweightedbasepointsums,the function(29)dependsontheerrorsofthelinearisedparts,
SOS2constraint,andaconvexcombination.First,wefocus whichisthesumofusedresources,yf=1,andtheirmaximal
on one facility location and then add indexes to model all linearisationerrors(cid:15)T˜w.
locations.Forlocationf,functionTµ computestheTiS(26). µ
Wobijtehctiitvselifnuenacrtiisoend,ΛvefrTsµio(nΛfT˜)µ,b(e2c7o)mtheescΛofn(cid:80)vesxβpsazsrt.AofstΛhfe Λ1 f,(cid:88)yf=1(cid:15)T˜µwf ≤ Λp mT˜awx{(cid:15)T˜w} (35)
dependsondecisionvariablexcf,multiplyingxcf withzs
turnsthereplacementtermtobequadratic;onlyfunctionTµ The linearisation accuracy drops if more resources are
allowed to open (p). To maintain the same linearisation
waslinearised,notthewholeobjectivefunction.However,
havingalinearandnotquadraticobjectivefunctionwould accuracywhiledoublingpthelinearisationerror(cid:15)T˜w hasto
behalved.Thiscanbeachievedbyusingmorebasepointsfor
reduce problem complexity and speeds up solving. The
quadratic term Λf(cid:80)sβszs needs to be replaced with an thelinearisation.Evenif??indicatesthatlessthantwicethe
basepointsarenecessary,increasingthenumberofbasepoints
equivalent linear term. This is achieved by “moving” the
weightΛf intofunctionTµ,whichbecomesTwµ (28).Using and,hence,increasingtheproblem’ssearchspaceincreases
T˜w’sbasepointswilltransformthequadraticintothelinear theruntime.
µ
(cid:80)
term sβszs; the ordinate basepoints are now already
weighted. Since the other parts of the objective function 4 EVALUATION
werelinear,thewholeobjectivefunctionisnowlinear. This section has four parts. First, it presents different TiS
1 function linearisations to find a balance of two conflicting
Tµ(x)= µ−x =y, withx=Λf (26) goals:Smallobjectivefunctionapproximationerrorandfew
T˜µ :x=(cid:88)zsαs, y˜=(cid:88)zsβs (27) bthaesecpoonivnetsx(amnd) floinrefaarstpcroobmlepmuta(QtioPnv.sS.eQc(cid:102)oPn)da,rseocluomtiopnasreodf
s s
x for different real networks. Third, the trade-off between
Twµ(x)=xTµ(x)= µ−x =y, withx=Λf (28) the number of used locations and resulting response time
is discussed. Finally, application and network properties
To model all locations, index f is added for each fa-
are presented for which considering the QD yields better
cility location forming the decision variables zfs and responsetimesthanignoringQD(Q(cid:102)Pvs.P).
basepoint variables αfs, βfs. The linearised version
Q(cid:102)P(G, λc, µf, p, αs, βs)ishence: 4.1 WeightedTiSLinearization
1 (cid:88) 1 (cid:88)
xm,yi,nz Λ xcflcf + Λ βfszfs (29) TpohiinstssefcotrioTnws(dλe)s(c2r8ib)einsthhoewevwaleuaotbiotani.nFothrethcios,nwcreetsehobwasea-
cf fs µ
(cid:88) simplificationwithonesetofbasepointsadaptedatruntime
s.t. xcf =λc, ∀c (demand) (30) fordifferentµvalues.Afterwards,wediscussthetrade-off
f
betweenafastsolvingtimeandlowapproximationerror.
(cid:88) (cid:88)
xcf = αfszfs, ∀f (capacity) (31) FunctionTwµ(λ)dependsonµandneedsindividuallin-
c s earizationsfordifferentµ;letαsµ, βsµ betheirbasepoints(36).
(cid:88)(cid:88)s zxfcfs =≤1y,f,SOS2(zf·,..),∀∀ff(force(pflwipl)) ((3323)) ActceoalnnrtreaebrssnepTarowteinw(vdλe/rilniµyt)gt,e=fbnuTansswµicemt(piλoiol)nian(r3Ttlsy8w):α(a∀ρsns,)dβ:(3tsαh7.sµe)F=icusoµnircαnrtedsiso,epnβposeTµnn=dwµdiβncesagn.ntbobafseeµrpeowwinirtithts-
c
(cid:88) λ (cid:88) (cid:88)
yf =p (limit) (34) Twµ(λ)= µ−λ : αsµzs=λ, βsµzs=Twµ(λ) (36)
f s s
ρ (cid:88) (cid:88)
The demand-weighted TiS is represented by term Tw(ρ)= 1−ρ : αszs=ρ, βszs=Tw(ρ) (37)
((cid:80)(cid:80)31ss).βαTffshszzisffssc=aap(cid:80)nadcfityΛthfceo=n(cid:80)csotrrcarxiencsftp;aoltnshodeiinmngepwlaicrcirtialvypaaalcsisrtuyatreecsoanthtsatrftatihniest Tw(µλ)= 1−λ/λµ/µ : (cid:88)ss αszs=µλ, (cid:88)ss βszs=Tw(µλ) (38)
queuingsystemisinasteadystatethroughtheupperbound As the ordinate basepoints βs remain unchanged, the
of the linearization interval α(m−1) < µ; τ from the old basepoints’ approximation error is also not affected. With
constraint(10)becomesobsolete. this handy transformation, we only need to precompute
ThesearchspaceofQPconsistsof|F|binaryand|F||C| basepoints of Tw instead of basepoint sets of Tw for each
µ
realvariables.InadditionQ(cid:102)Phasm|F|real,restrictedSOS differentµinthemodel,whichspeedsupthemodelsetup
variables. If both problems were linear we could guess process.
that solving the second problem Q(cid:102)P takes longer than QP In the remaining section, we investigate the trade-off
becausethesearchspaceislarger.However,linearproblems between a fast solving time and a low-error linearisation.
10
??comparesthesolutionquality(a)andsolvingtime(b).
7 ® =0:80 (¯ =4:00)
m 1 m 1 The horizontal axis lists the groups of different topologies
w~Trror of 465 ®®®®mmmm¡¡¡¡1111====0000::::99990689 ((((¯¯¯¯mmmm¡¡¡¡1111====9249:4990:::0000)000))) aarQenxPsidpsot(ihcnnreso(eaus)stsies)mdhaoennswudasmsQt(cid:102)bthhPeeer9oq(5lfui%nraeelsic)tooyufnoorfircfdeseesoanflucochertieoionanfctsehtrohvgbeartoals5uin0opef.drTtehbhaeyeliazvsvoaeeltrvritoaiicnngagesl
e ¡ ¡
x. foreachgroup.Similarly,theverticalaxisin(b)showsthe
o 3 95%confidenceintervalsofthesolvingtimeforeachgroup.
r
p
p 2 TheQ(cid:102)P’ssolutionshaveaverysimilarqualitytoQP’ssolutions.
a
bs. 1 obtaLionoinkgintgheatQ(cid:102)(bP)’,sosbotlauitniionngs.QHPo’swseovleurt,iothnossteoovkalluoensghearvtheatno
a
0 beinterpretedwithcare.OurimplementationofQPhasto
processallpossiblecombinations(F(cid:48)),whereasQ(cid:102)Pbenefits
5 10 15 20 25
fromtheMIPsolver’sbranch-and-cutalgorithmtoreducethe
number of basepoints (m)
searchspace.Tocomparethisstructurallydifferentproblems,
werestrictthenumberofcandidatefacilitiesto10andthe
Figure7:TheerroroflinearisingTw isshowndependingon number of possible combinations; the major cause of the
µ highersolvingtimeofQP.Nevertheless,theabsolutesolving
thenumberofbasepointsanddifferentlinearizationintervals
[0,..,αm−1]. timesofQ(cid:102)Pareveryshortforallgroups.
In conclusion, the linearised problem Q(cid:102)P is an adequate
substitutionforouroriginalproblemQP:fastandaccurate.
For the first, we need to minimize the number of decision
variableszfs or,equivalently,thenumberofbasepointsused 4.3 ResponseTimeReduction
for the linearisation. For the second, we investigate two ??alsoshowshowresponsetimeimproveswhenaddinga
control knobs (??): many basepoints or small interval end resource.Wecouldverifytwoeffectsdecreasingtheresponse
αm−1.??showstheerrorofT˜w dependingonthenumber times:First,usingmorelocationsallowsbetterloadbalanc-
ofbasepointsmfordifferentαm−1 values.Weneedasmall ing,whichreducesthequeuingdelays.Thesereductionsare
error (down the vertical axis) with small m (left on the largerforhighlyutilizedlocationsthanforlessutilizedones.
horizontalaxis)withlargeαm−1.Thelatteralsoartificially Second,morelocationsallownearerlocations,reducingthe
limits the resource capacity and renders solving an input roundtriptimes.Inconclusion,theaverageresponsetimeof
infeasiblethatcouldinfactbesolvedwithlargerαm−1. QP(..., p)decreasesmonotonicallyinnumberofresources.
For our evaluation, we set αm−1=0.96 and m=6 with Then,serviceprovidersearningmoremoneybyconnecting
error(cid:15)Tw=2.67asagoodcompromisebetweenthenumber userswithlowerresponsetimesfaceadiminishingreturn.
µ
of decision variables, approximation error, and artificial Atonebreakingpointp∗ thecostforaddingaresourcewill
capacitylimit. exceedtheadditionallyearnedmoney.Thispointdepends
onthetopology,servicetimes,andservicemonetization.By
4.2 Comparison:Convexvs.Linear using QP with different p values, the service provider can
determinep∗ inadvancetoavoidprofitloss.
We choose the following structurally different topologies
Foracloserlook,??showsnotonlytheaverageresponse
fromSndLib[44]:ta2,zib54withmanynodes(around50);
time but also the time in system and round trip times
yuan,bwinwithfewnodes(around10);atlanta,norway
groupedalongthehorizontalaxisthesamewaylikein??.
fordensenetworks(node:edgeratio1:2).Alltopologiesare
Wecantracethetwoeffectsofresponsetimereductions:First,
connected. We approximate the latency between nodes by
loadbalancingacrossmoreresourcesreducesthequeuing
theirgeographicaldistance[32].Weassumethatdatacentres
delay and with it the time in queuing system. Second, the
arebuiltatwellconnectednodes/routers,soweselectedthe
roundtriptimeisreducedasnearerresourcesareused.
10 nodes with the highest degree8 to be data centres. We
setarelativelylowservicerateofµ=100req./storeflectour
4.4 ConsideringQueueingDelays
computation-intensive example applications [7], [37], [54].
Thispaperpresentsarefinementoftheassignmentproblem
The service rates were the same for all data centres. User
requestsarriveatallnodesandthearrivalrateλc forusers (Pin(4))byadditionallyconsideringqueuingdelays.This
at site c is randomly generated: Each value is uniformly refinement increases problem’s complexity, accuracy, and
drawnfroman[0,1]intervaland,afterwards,allvaluesare solvingtime.Onlyasignificantlylowerresponsetimewould
normalized to (cid:80)cλc=470req./s. This value, together with make these drawbacks worthwhile; for instances, small
theservicerateµ=100req./s,ensuresfeasibilityfor5ormore queuing delays compared to large round trip times will
facilities.Werandomlygenerated50differentrealizationsfor rendertherefinementunnoticeable.Thissectioninvestigates
allarrivalrates.Foreachofthese50setsofarrivalrates,we multiple scenario factors influencing queuing delays and
consideredp∈[5, 10]facilities,resultingin300configurations judgestherefinementgainbycomparingsolutions’response
pertopology.EachconfigurationwassolvedusingeitherQP times obtained by either ignoring or considering queuing
orQ(cid:102)P. delays.
Forthis,weperformasecondexperimentwithaslightly
8.Forsamedegreesthenodeidisthetiebreaker. differentconfigurationasin??.Thefirstexperimentshowed