Analyzing Mark et-basedResourceAllocation Strategiesfor theComputational Grid

�RichWolski

�JamesS.Plank

�JohnBrevik � ToddBryan

��Departmentof ComputerScience

Universityof Tennessee� MathematicsandComputerScienceDepartmentCollegeof theHoly Cross

Abstract

In this paper, we investigateG-commerce —computationaleconomiesfor controlling resourceallocationin ComputationalGrid settings.Wede-fine hypotheticalresource consumers (represent-ing users and Grid-aware applications)and re-source producers (representingresource ownerswho“sell” their resourcesto theGrid). We thenmeasure theefficiencyof resourceallocation un-der two differentmarket conditions:commoditiesmarkets and auctions. We compare both mar-ket strategies in termsof price stability, marketequilibrium, consumerefficiency, and producerefficiency. Our resultsindicatethat commoditiesmarketsare a betterchoice for controlling Gridresourcesthan previouslydefinedauctionstrate-gies.

1 Intr oduction

With theproliferationof theInternetcomesthepossibilityof aggregatingvastcollectionsof com-�

This work wassupportedin part by NSF grantsEIA-9975020,EIA-9975015,andACI-9876895.

putersinto large-scalecomputationalplatforms.A new computingparadigmknown asthe Com-putationalGrid [17, 3] articulatesa vision of dis-tributedcomputingin which applications“plug”into a “power grid” of computationalresourceswhen they execute, dynamically drawing whatthey needfrom the global supply. While a greatdealof researchconcerningthe softwaremecha-nismsthat will be necessaryto bring Computa-tional Grids to fruition is underway [3, 16, 20, 8,4, 24, 21, 1, 34], little work hasfocusedon theresourcecontrol policies that are likely to suc-ceed. In particular, almostall Grid resourceal-locationandschedulingresearchespousesoneoftwo paradigms:centralizedomnipotentresourcecontrol [18, 20, 28, 29] or localizedapplicationcontrol [9, 4, 2, 19]. The first is certainlynot ascalablesolutionandthe secondcanleadto un-stableresourceassignmentsas“Grid-aware” ap-plicationsadaptto competefor resources.

In this paper, we investigateG-commerce —theproblemof dynamicresourceallocationontheGrid in termsof computationalmarketeconomiesin whichapplicationsmustbuy theresourcestheyusefrom resourcesuppliersusinganagreed-uponcurrency. Framingthe resourceallocationprob-

lem in economicterms is attractive for severalreasons.First, resourceusageis not free. WhileburgeoningGrid systemsarewilling to make re-sourcesreadily available to early developersasa way of cultivating a usercommunity, resourcecosteventuallymustbe consideredif the Grid isto becomepervasive. Second,the dynamicsofGrid performanceresponseare, as of yet, diffi-cult to model. Application schedulerscanmakeresourceacquisitiondecisionsat machinespeedsin responseto theperceivedeffectsof contention.As resourceload fluctuates,applicationscanad-just their resourceusage,formingafeedbackcon-trol loop with a potentially non-linearresponse.By formulating Grid resourceusagein marketterms,we areableto draw upona large body ofanalytical researchfrom the field of economicsand apply it to the understandingof emergentGrid behavior. Last, if resourceownersareto beconvincedto federatetheir resourcesto theGrid,they mustbeableto accountfor therelativecostsandbenefitsof doingso.Any market formulationcarrieswith it aninherentnotionof relativeworthwhich canbeusedto quantify thecost-to-benefitratio for bothGrid usersandstake-holders.

While thereareanumberof differentplausibleG-commercemarket formulationsfor the Grid,we focus on two broad categories: commodi-ties markets andauctions. The overall goal ofthe ComputationalGrid is to allow applicationsto treat computational,network, andstoragere-sourcesas individual and interchangeablecom-modities, and not specific machines,networks,anddisk or tapesystems.ModelingtheGrid asacommoditiesmarket is thusa naturalchoice.Onthe otherhand,auctionsrequirelittle in the wayof global price information,andthey areeasytoimplementin a distributedsetting.Both typesofeconomieshavebeenstudiedasstrategiesfor dis-tributedresourcebrokering[11, 35, 25,6, 7, 10].Our goal is to enhanceour deeperunderstandingof how theseeconomieswill fareasresourcebro-keringmechanismsfor ComputationalGrids.

To investigateComputationalGrid settingsand

G-commerceresourceallocation strategies, weevaluatecommoditiesmarketsandauctionswithrespectto four criteria:

1. Grid-widepricestability

2. Market equilibrium

3. Applicationefficiency

4. Resourceefficiency

Pricestability is critical to ensureschedulingsta-bility. If the price fluctuateswildly, applicationandresourceschedulersthat basetheir decisionsonthestateof theeconomywill follow suit, lead-ing to poor performance,and thereforeineffec-tivenessof theGrid asacomputationalinfrastruc-ture. Equilibrium measuresthe degreeto whichpricesare fair. If the overall market cannotbebroughtinto equilibrium, the relative expenseorworthof aparticulartransactioncannotbetrusted,andagainthe Grid is not doing its job. Applica-tion efficiency measureshow effective the Gridis as a computationalplatform. Resourceeffi-ciency measureshow well the Grid managesitsresources.Poor applicationand/or resourceef-ficiency will meanthat the Grid is not succeed-ing asa computationalinfrastructure. Thus,weusethesefour criteria to evaluatehow well eachG-commerceeconomyworksasthebasisfor re-sourceallocationin ComputationalGrids.

Theremainderof thispaperis organizedasfol-lows. In thenext section,we discussthespecificmarket formulationswe usein this study. Sec-tion 3 describesthe simulationmethodologyweuseandthe resultswe obtainfor differenthypo-thetical market parameterizations.In Section4weconcludeandpoint to futurework.

2 G-commerce— Mark et Economiesfor the Grid

In formulating a computationaleconomyfortheGrid, we make two assumptions.#1: Therel-

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ativeworthof a resourceis determinedbyits sup-ply andthedemandfor it. Thisassumptionis im-portantbecauseit rulesout pricing schemesthatarebasedonarbitrarilydecidedpriorities.For ex-ample, it is not possiblein an economyfor anorganizationto simply declarewhat the price ofits resourcesare and then decreethat its userspay that price even if cheaper, betteralternativesare available. While thereare several plausiblescenariosin which suchDraconianpolicies areappropriate(e.g. usersare fundedto usea spe-cific machineaspart of their individual researchprojects),from theperspectiveof theGrid, there-sourceallocationproblemundertheseconditionshasbeensolved.

Further, weassumethatsupplyanddemandarefunctionsof price,andthat true relative worth isrepresentedattheprice-pointwheresupplyequalsdemand– that is, at market equilibrium. Con-versely, at a non-equilibriumprice-point(wheresupplydoesnotequaldemand),priceeitherover-statesor understatesrelativeworth.

#2: Resource decisionsbasedon self-interestare inescapablein anyfederatedresourcesystem.If we are to simulatea computationaleconomy,we must ultimately hypothesizesupply and de-mandfunctionsfor our simulatedproducersandconsumersrespectively. Individual supply anddemandfunctionsaredifficult to measureat best,particularly since thereare no existing Compu-tational Grid economieswhich we can observe.Our admittedly less-satisfactory approachis todefinesupplyanddemandfunctionsthatrepresenteachsimulatedproducerand consumer’s “self-interest.” An individual consumerbuys only ifthe purchaseis a “good deal” for that consumer.Analogously, producerssellonly whenasaleis intheirbestinterest.

In thenext section,we detail thespecificfunc-tions we investigate,but generallyour approachrelieson thesetwo assumptions.

2.1 Producersand Consumers

To comparethe efficacy of commoditiesmar-kets and auctions as Grid resourceallocationschemes,we define a set of simulated Gridproducersand consumersrepresentingresourceprovidersandapplicationsrespectively. We thenusethe samesetof producersandconsumerstocomparecommodity and auction-basedmarketsettings.

We simulatetwo differentkinds of producersin this study: producersof CPUs and produc-ers of disk storage. That is, from the perspec-tive of a resourcemarket, there are two kindsof resourceswithin our simulatedGrids: CPUsand disks. While the resultsshould generalizeto includea varietyothercommodities,networkspresenta specialproblem. Our consumermodelis that an application may requesta specifiedamountof CPU anddisk (the units of which wediscussbelow) andthattheserequestsmaybeser-viced by any provider regardlessof location ornetwork connectivity. Sincenetwork links can-not becombinedwith otherresourcesarbitrarily,they cannotbemodeledasseparatecommodities.We believe that network costcanbe representedin termsof “shipping” costsin morecomplicatedmarkets, but for the purposesof this study, weconsidernetwork connectivity to beuniform.

2.1.1 CPU Producer Model

In this study, a CPU representsa computationalenginewith a fixeddedicatedspeed.A CPUpro-duceragreesto sell to the Grid somenumberoffixed “shares”of the CPU it controls. The real-world scenariofor this modelis for CPUownersto agreeto hostafixednumberof processesfromtheGrid in exchangefor Grid currency. Eachpro-cessgetsa fixed, pre-determinedfraction of thededicatedCPU speed,but the owner determineshow many fractionsor “slots” heor sheis willingto sell. For example,in our study, the fraction is10%soeachCPUproduceragreesto sell a fixed

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number(lessthan 10) of 10%-sizedslots to theGrid. Whena job occupiesa CPU, it is guaran-teedto get 10% of the availablecycles for eachslot it consumes.EachCPU,however, differs inthetotal numberof slotsit is willing to sell.

To determinesupply at a given price-point,eachCPUcalculates����� � ������� � ������������������ !��"�#$��%&" (1)

where '�(�) (�*,+-( is the total amountof Grid cur-rency (hereafterreferredto as $G which is pro-nounced“Grid bucks”), */.�0 is an incrementingclock, and 1�23.5461 is the total numberof processslots the CPU owner is willing to support. The7 (589* :;'=<?>�( valueis theaverage$Gpertimeunitper slot the CPU hasmadefrom selling to theGrid. In our study, CPUproducerswill only sellif the current price of a CPU slot exceedsthe7 (589* :;'=<?>�( value,andwhenthey sell, they sellall unoccupiedslots.Thatis, theCPUwill sellallof its availableslotswhenit will turn aprofit (perslot)with respectto theaverageprofit over time.

2.1.2 Disk Producer Model

The modelwe usefor a disk produceris similarto that for the CPU producer, except that diskssell somenumberof fixed-sized“files” that ap-plicationsmayusefor storage.The 7 (589* :�'<@>�(calculationfor diskfiles is����� � ������� � �5���������������� !���A���B���A�3%DC (2)

where>�8�:�8E>�<�4GF is thetotalnumberof filesadiskproduceris willing to sell to theGrid. If thecur-rentpricefor afile is greaterthanthe 7 (589* :�'<@>�( ,adiskproducerwill sell all of its availablefiles.

Note that the resolutionof CPU slotsandfilesizesis variable. It is possibleto make a CPUslot equivalent to the durationof a single clockcycle, anda disk file be a singlebyte. Sinceourmarketstransactbusinessat thecommoditylevel,however, wehypothesizethatany realimplemen-tationfor theGrid will needto work with larger-

scaleaggregationsof resourcesfor reasonsof effi-ciency. For thesimulationsdescribedin Section3we choosevaluesfor theseaggregationsthat webelieve reflect a market formulation that is cur-rently implementable.

2.1.3 Consumersand Jobs

Consumersexpresstheir needsto the market inthe form of jobs. Eachjob specifiesboth a sizeand an occupancy durationfor eachresourcetobe consumed.Eachconsumeralsosportsa bud-getof $G that it canuseto pay for the resourcesneededby its jobs.Consumersaregivenaninitialbudgetanda periodicallowance,but they arenotallowedto hold$Goverfrom oneperioduntil thenext. Thismethodof budgetrefreshis inspiredbytheallocationpoliciescurrentlyin useat theNSFPartnershipsfor AdvancedComputationalInfras-tructure(PACIs). At thesecenters,allocationsareperishable.

Whenaconsumerwishesto purchaseresourcesfor a job, it declaresthe size of the requestforeachcommodity, but not theduration.Our modelis that job durationsarerelatively long, andthatproducersallow consumersoccupancy withoutknowing for how long theoccupancy will last.Atthe time a produceragreesto sell to a consumer,a price is fixed that will be charged to the con-sumerfor eachsimulatedtime unit until the jobcompletes.

For example,considera consumerwishing tobuy aCPUslot for 100minutesandadiskfile for300minutesto serviceaparticularjob. If thecon-sumerwishesto buy eachfor aparticularprice,itdeclaresto the market a demandof 1 CPU slotand1 disk slot, but doesnot reveal the 100 and300 minutedurations.A CPU producerwishingto sell at the CPU price agreesto acceptthe jobuntil thejob completes(asdoesthediskproducerfor thediskjob). Oncethesalesaretransacted,theconsumer’sbudgetis decrementedby theagreed-uponpriceeverysimulatedminute,andeachpro-ducer’s revenueaccountis incrementedby the

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sameamount.If thejob completes,theCPUpro-ducerwill haveaccrued100timestheCPUprice,thediskproducerwill haveaccrued300timesthedisk price, and the consumer’s budgetwill havebeendecrementedby the sum of 100 times theCPUpriceand300timesthediskprice.

In definingthis methodof conductingresourcetransactions,wemakeseveralassumptions.First,we assumethat in anactualGrid settingresourceproducersor supplierswill commitsomefractionof their resourcesto theGrid, andthat fraction isslowly changing. Oncecommitted,the fraction“belongs” to the Grid so producersarenot con-cernedwith occupancy. This assumptioncorre-spondsto thebehavior of somebatchsystemsinwhich,oncea job is allowedto occupy its proces-sors,it is allowed to run eitheruntil completion,or until its user’sallocationis exhausted.Produc-ersareconcerned,in our models,with profit andthey only sell if it is profitableon theaverage.Byincludingtime in thesupplyfunctions,producersconsiderpastoccupancy (in termsof profit) whendecidingto sell. We arealsoassumingthat nei-ther consumersnor producersaremaliciousandthat both honor their commitments.In practice,this requirementassuredlywill bedifficult to en-force.However, if consumersandproducersmustagreeto usesecureauthenticationmethodsandsystem-provided libraries to gain accessto Gridresources,thenit will bepossible.

2.1.4 ConsumerDemand

Theconsumerdemandfunction is morecomplexthan the CPU and disk supply functions. Con-sumersmustpurchaseenoughCPU anddisk re-sourcesfor eachjob they wish to run. If they can-not satisfythe requestfor only onetype, they donot expressdemandfor theother. Thatis, thede-mandfunctionsfor CPU and disks are stronglycorrelated(althoughthesupplyfunctionsarenot).This relationshipbetweensupply and demandfunctions constitutesthe most difficult of mar-ket conditions. Most theoreticalmarket systems

makeweaker assumptionsaboutthedifferenceincorrelation.By addressingthemoredifficult case,webelieveourwork morecloselyresembleswhatcanberealizedin practice.

To determinetheir demandat a given price,eachconsumerfirst calculatestheaveragerateatwhich it would have spent$G for the jobs it hasrunsofar if it hadbeenchargedthecurrentprice.It thencomputeshow many $G it canspendpersimulatedtime unit until thenext budgetrefresh.Thatis, it computes�=��H �5�%&� � IKJ %?��%@�=# L����M J�N � ������� J ���� (3)�O������PA#Q� �5�%&� � �5���R�����3;H PS��T5H���%U ����V�����"�WYXZ��� ![ (4)

where 4S.54S8E2 0\.�'�] J is the total amount ofwork performed so far using commodity < ,:�'<@>�( J is the current price for commodity < ,'=( 7 89<?*,<D*_^ `�+_ab^B(�4 is theamountleft to spendbe-fore thebudgetrefresh,'=(�c_'=(�1�d is thebudgetre-fresh time, and */.�0 is the currenttime. When>�8�:�8E`�23( '�8b4S( is greaterthanor equalto 89)E^ '=8b4S( ,aconsumerwill expressdemand.

Unlike our supplyfunctions,theconsumerde-mandfunction doesnot considerpastprice per-formancedirectlywhendeterminingdemand.In-stead,consumersusing this function act oppor-tunisticallybasedon themoney they have left tospendandwhenthey will receivemore.They usepastbehavior only asan indicationof how muchwork they expectto introduceandbuy whentheybelievethey canafford to sustainthis rate.

Consumers,in our simulations,generateworkasafunctionof time. Wearbitrarilyfix somesim-ulatedperiodto bea “simulatedday.” At thebe-ginningof eachday, every consumergeneratesarandomnumberof jobs. By doing so, we hopeto model the diurnal userbehavior that is typi-cal in large-scalecomputationalsettings. In ad-dition, eachconsumercangeneratea singlenewjob every time stepwith a pre-determinedproba-bility. Consumersmaintaina queueof jobswait-ing for servicebeforethey areacceptedby pro-ducers.Whencalculatingdemand,they compute

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89)9^ '�8b4S( and >�85:�8 `�23( '�8b4S( anddemandasmanyjobsfrom this queueasthey canafford.

To summarize,for our G-commercesimula-tions:e All entitiesexceptthe market-maker act in-

dividually in their respectiveself-interests.e Producersconsiderlong-termprofit andpastperformancewhendecidingto sell.e Consumersaregivenperiodicbudgetreplen-ishmentsandspendopportunistically.e Consumersintroducework loadsin bulk atthebeginningof eachsimulatedday, andran-domly throughouttheday.

We believe that this combinationof characteris-ticscapturesareasonablesetof producerandcon-sumertraitsin realGrid settings.

2.2 CommoditiesMark ets

In areal-world commoditiesmarket,commodi-ties areexchangedin a centrallocation. Impor-tant featuresof a commoditiesmarket are thatthegoodsof thesametypebroughtto market bythevarioussuppliersareregardedasinterchange-able, market price is publicly agreedupon foreachcommodityregardedasawhole,andall buy-ers and sellersdecidewhether(and how much)to buy or sell at this price. Contrastthis type ofcommercewith onebaseduponauctions,whereineachbuyerandselleractsindependentlyandcon-tractsto buy or sell at a price agreedupon pri-vately.

Since the goal of a computationalGrid is toprovideuserswith resourceswithoutregardto theparticularsupplier, it seemsverynaturalto modela Grid economyusingcommoditiesmarkets. Todoso,werequireapricingmethodologythatpro-ducesa systemof priceadjustmentswhich bringabout market equilibrium (i.e. equalizessupplyanddemand).

2.2.1 Pricing in Commodities Mark ets: Re-sultsof EconomicResearch

Ourmodelis anexampleof anexchangeeconomy,namelya systeminvolving agents(producersandconsumers),andseveralcommodities.Eachagentis assumedto controlasufficiently smallsegmentof themarket. In otherwords,the individual be-havior of any oneagentwill not affect thesystemasa wholeappreciably. In particular, priceswillbe regardedasbeyond the control of the agents.Givenasystemof prices,then,eachagentdecidesupon a courseof action, which may consistofthesaleof somecommoditiesandthepurchaseofotherswith theproceeds.Thuswe definesupplyanddemandfunctionsfor eachcommodity, whicharefunctionsof theaggregatebehavior of all theagents.Thesearedeterminedby thesetof marketpricesfor thevariouscommodities.

Naturally, we usethe languageof vectorsforprice,supply, anddemand;eachof thesewill bean * -vector, where * is thenumberof commodi-ties,of non-negative real numbers.Observe thatgivena commoditybundle, that is an *gfh) (5>�4S.�'of quantitiesikjml/n�o�pqprproAl�s of the commodities,and a price vector t the value of the bundle isequalto tvuAi . For givenpricevector t , definetheexcessdemandwxjyw-z�t|{ to be the differenceofthedemandandsupplyvectorsfor thispricelevel.Equilibrium for theeconomyis establishedwhensupplyis equalto demand;in otherwords,apricevector t is anequilibriumpricewhen w-z�t|{}j�~ .It shouldbenotedthat,for ourpurposes,currencywill be regardedasanothercommodity. Thusaproducerof a non-currency commodity(CPU ordisk for thepurposesof thispaper)will simplyberegardedas a “consumer”of currency; presum-ably, the currency will be usedin someway forthebenefitof theproducer.

In generalequilibrium theory, thereare threehypothesesmadeon thefunctionz: homogeneity,continuity, andadherenceto Walras’Law. Homo-geneitymeansthatonly theratiosbetweenpricesareimportantto how commoditiesareexchanged.

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That is, w-z?��t�{�j�w-z�t|{ for any positive number� . This relationshipis naturally true, sincecur-rency is regardedasa commodity. Continuity isthe propertythat excessdemandis a continuousfunction of the prices, which cannothold liter-ally in oursituation,dueto theindivisibility of thecommodities.However, we assumethatthenum-ber of agentsis large enoughthat all functionsmaybeapproximatedby continuousfunctionsofcontinuousvariables.Finally, Walras’Law statesthat for any price, w-z�t|{�uBt�j�� . This assump-tion is justified as follows: When eachagentissupplyingthesametotal valueasthatagentis de-manding,thevalueof thetotal supplybundle � isequalto thatof thetotal demandbundle � . Thus,asobserved above, t�uB��j�t�uB� , andthereforet�u w�jyt�u-zD�xf��5{�jm~ . Walras’ Law will ap-ply aslongasdemandis locally non-satiated, thatis, givena level of consumption,thereis alwaysapreferencefor greaterconsumption(pricenot be-ing anobject).

When theseassumptionshave been met, anequilibrium price vectorhasbeenproven to ex-ist via topologicalmethods,namelytheBrouwerfixed-pointtheorem (see[13], Chapter5, for theresult in its original form, or a remarkablyclearexposition in [15], Chapter6). Thesemethodsarenon-constructive,sothattheproblemremainsto find a methodof price adjustmentthat bringsaboutequilibrium or at leastapproximatesequi-librium within reasonabletolerances.

A few words on this last point are in order.From a purely “engineering”standpoint,reach-ing preciseeconomicequilibrium is surely im-possible. Thus we must contentourselves withthe moremodestgoal of producinga price vec-tor for which the excessdemandsare all closeto � . Since the excessdemandfunctions canbe quite general,it is alwayspossiblethat thereexists a price vector which producesexcessde-mandswhichareall within aprescribedtoleranceof � andyet is not closeto anactualequilibriumpoint; further, thereis no “engineering”methodwhichwill distinguishthis from apointwhich re-

ally is very nearto an equilibrium price. EvenScarf’s algorithm,describedbelow, which haser-roneouslybeencalleda “constructive versionofthe Brouwer fixed-pointtheorem,” is only guar-anteedto producepointswhich areapproximateequilibria in the first sense. Thus we will usethephrase“approximateequilibrium” to referto apricewhichmakestheexcessdemandsall closeto� without judgingwhetherit livesneara genuineequilibrium point. In any event, the theoreticalexistenceof an equilibrium price guaranteestheexistenceof approximateequilibria. Moreover,approximateequilibria arevaluable: If the mar-ket is approximatelycleared,thentheeconomyisdoingagoodjob of distributinggoods.

Walras in [37] suggesteda processcalledtatonnement(“groping”) by which real-worldmarketscometo equilibrium. With tatonnement,eachindividualpriceis raisedor loweredaccord-ing to whetherthatcommodity’sexcessdemandispositive or negative. Then,new excessdemandsaremeasured,andthe processis iterated. Whileit wassuggestedonly asa “behavioral” explana-tion asto how real-world marketsreachequilib-rium, tatonnementformedthe basisfor early at-temptsto prove theexistenceof equilibrium. It isnow known that tatonnementdoesnot in generalleadto a convergentprocess;Scarf in [30] pro-duceda very simpleexamplefor which thereis auniqueequilibriumbut for which,from almostev-ery startingpoint, the tatonnementprocessoscil-latesfor all time. In fact, tatonnementdoesbringaboutconvergenceto anequilibriumpricevectorunderthevery stronghypothesisof grosssubsti-tutes, which statesthat increasingthe �b��� pricewhile holdingtheothersconstantwill bringaboutan increasein excessdemandin all commoditiesotherthanthe �9��� . Unfortunately, for typicalGridapplications,the hypothesisof grosssubstitutesdoesnot hold, becausedifferentcommoditiesareoften complementary. (For example,an applica-tion mayneedbothCPUanddisk in orderto ex-ecute.If theprice for CPUsis too high, thentheapplication’s demandfor diskswill be lower in-

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steadof higher.)Thereare several different approachesto the

problem of finding an algorithm for adjustingprices which will lead to equilibrium. Scarf’salgorithm (see[31]) works roughly as follows:Supposethat thereare *���� commodities,andnormalizethe pricesso that their sum is alwaysequal to � . The set of possibleprice vectorsthusformsan * -dimensionalsimplex in � s��_n (theprice simplex). Scarf then divides this simplexinto a large numberof subsimplicesand showsthatthereexistsasubsimplex any of whosepointsprovides an approximateequilibrium price. Healso provides an explicit formula for how fineto make the subdivision in order to produceanexcessdemandwithin a pre-specifiedtolerance.Merrill [23] gives an important improvementtoScarf’s algorithmwhich makesit far moreattrac-tive from a computationalstandpoint.A differentsort of refinementof this idea is to be found inEaves’ algorithmwith “continuousrefinementofgrid size” [14].

A second approach, advocated by Smalein [32], is morein thespirit of multivariablecal-culus and is more dynamic in the sensethat itaimsto producea trajectoryfor thepricesto fol-low. In Smale’s method,the pricesare normal-ized by fixing one of the commodities(the nu-meraire) to have price � ; in our case,this com-moditywill bethecurrency. Further, supposethatthereare * othercommodities,so that the setofpossiblepricesforms the positive orthantin � s .Formthe *���* matrix� � z�t|{¡j ¢L£ wE¤£ t�¥¦ pNow definetheglobal Newtonordinarydifferen-tial equation � � z�t|{ �§t�§¨ j©f}��w-z�t|{ (5)

where � is a constantwhich has sign equal tozGfª�5{ s timesthesignof thedeterminantof��� z3t«{ .

(For contrast,notethatthetatonnementprocessis

encapsulatedin thedifferentialequationa9tab4 j¬w .

Thus the global Newton may be regardedas amoresophisticatedversionof tatonnementwhichtakes into accountthe interdependenciesof theway demandsfor the variouscommoditiesinter-act with the variousprices.) Smaleprovesthat,underboundaryconditionswhich are justifiableon the basisof the desirabilityof the commodi-ties,almostevery maximalsolutionof theglobalNewton equationstartingsufficiently nearto theboundaryof thepositiveorthantof � s (or to ­ )will convergeto thesetof equilibriumprices.

Notethatexceptunderstronghypotheses,mostcommonlygrosssubstitutes,the theorydoesnotguaranteethatthereis a uniqueequilibriumpricevector. However, there is a useful result alongtheselines as follows: Definea regular equilib-riumto beonefor whichthematrix

� � z�t|{ definedabove is nonsingular. Then accordingto [22],Theorem5.4.2,a regularequilibrium price is lo-cally uniquein thesensethat it is theonly oneinsomeopensubsetof thespaceof pricevectors.

2.2.2 Price Adjustment Schemes

Herein we examine the resultsof using severalprice adjustmentschemesin simulatedcomputa-tional market economies.Smale’s methodis notpossibleto usedirectly for a numberof reasons.First, any actualeconomyis inherentlydiscrete,sothepartialderivativesin equation5 do not ex-ist, strictly speaking.Second,giventhebehaviorof theproducersandconsumersdescribedabove,therearethresholdpricesfor eachagentthatbringaboutsuddenradicalchangesin behavior, sothata reasonablemodelfor excessdemandfunctionswould involve sizeablejump discontinuities.Fi-nally, the assumptionsin Smale’s modelarethatsupply and demandare functions of price onlyandindependentof time,whereasin practicetherearea numberof waysfor supplyanddemandtochangeover time for agivenpricevector.

Observe that taking ��j � and applying theEuler discretizationat positive integer valuesof

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4 reducesthis processto the Newton-Raphsonmethodfor solving w-z�t|{�j ~ ; this observationexplainstheterm“global Newton.”

Implementing Smale’s method: As observedabove,obtainingthepartialderivativesnecessaryto carryoutSmale’sprocessin anactualeconomyis impossible;however, within the framework ofour simulatedeconomy, we areableto get goodapproximationsfor the partialsat a given pricevector by polling the producersand consumers.Startingwith a price vector, we find their pref-erencesat pricevectorsobtainedby fixing all butonepriceandvaryingtheremainingpriceslightly,thusachieving a “secant-line”approximationforeachcommodity separately;we then substitutetheseapproximationsfor thevaluesof thepartialderivatives in the matrix

� � z3t«{ , discretizewithrespectto time, solve Equation 5 for the incre-ment aEt to getour new price vector, anditerate.Wewill refer, convenientlybut somewhatinaccu-rately, to thispriceadjustmentschemeasSmale’smethod.

The First Bank of ® : The drawback to theabove schemeis that it relieson polling the en-tire market for aggregatesupplyanddemandre-peatedlyto obtain the partial derivatives of theexcessdemandfunctions. If we wereto try andimplementSmale’smethoddirectly, eachindivid-ualproducerandconsumerwouldhave to beableto respondto the question“how much of com-modity l wouldyoubuy (sell) atpricevectorp?”In practice,producersandconsumersmaynot beableto make sucha determinationaccuratelyforall possiblevaluesof p. Furthermore,even ifexplicit supply and demandfunctionsare madeinto anobligationthatall agentsmustmeetin or-der to participatein anactualGrid economy, themethodologyclearlywill notscale.For theserea-sons,in practice,we do not wish to assumethatsuchpolling informationwill beavailable.

A theoretically attractive way to circumventthis difficulty is to approximateeachexcessde-

mandfunction ¯ J by a polynomialin :,n�o?:�°�o�pqprpqoD:;swhichfits recentpriceandexcessdemandvectorsandto usethepartialderivativesof thesepolyno-mials in Equation5. In simulations,this methoddoesnot, in general,produceprices which ap-proachequilibrium. The First Bank of G is aprice adjustmentschemewhich both is practica-ble andgivesgoodresults;this schemeinvolvesusing tatonnement(see above) until prices get“close” to equilibrium, in the sensethat excessdemandshave sufficiently small absolutevalue,and then using the polynomialmethodfor “finetuning.” Thus, the First Bank of G approxi-matesSmale’s methodbut is implementableinreal-world Grid settingssinceit hypothesizesex-cessdemandfunctionsandneednotpoll themar-ket for them. Our experienceis that fairly high-degreepolynomialsarerequiredto captureexcessdemandbehavior with the sharpdiscontinuitiesdescribedabove. For all simulationsdescribedinSection3, weuseadegree17 polynomial.

2.3 Auctions

Auctionshave beenextensively studiedasre-sourceallocationstrategies for distributed com-puting systems. In a typical auction system(e.g. [11, 35, 25, 6]), resourceproducers(typi-cally CPU producers)auction themselves usinga centralizedauctioneerandsealed-bid,second-priceauctions.That is, consumersplaceonebidwith theauctioneer, andin eachauction,thecon-sumerwith the highestbid receivesthe resourceat theprice of thesecond-highestbidder. This isequivalentto “just” outbiddingthesecond-highestbidder in an open,multi-roundauction,anden-couragesconsumersto bid what the resourceisworth to them(see[6] for further descriptionofauctionvariants).

Whenconsumerssimply desireonecommod-ity, for exampleCPUsin Popcorn[25], auctionsprovideaconvenient,straightforwardmechanismfor clearing the marketplace. However, the as-sumptionsof a Grid Computing infrastructure

9

posea few difficulties to this model. First, whenanapplication(theconsumerin aGridComputingscenario)desiresmultiple commodities,it mustplacesimultaneousbidsin multiple auctions,andmay only be successfulin a few of these.To doso,it mustexpendcurrency on theresourcesthatit hasobtainedwhile it waits to obtain the oth-ers. This expenditureis wasteful,andthe uncer-tainnatureof auctionsmayleadto inefficiency forbothproducersandconsumers.

Second,while a commoditiesmarket presentsanapplicationwith a resource’s worth in termsofits price, thus allowing the applicationto makemeaningfulschedulingdecisions,an auction ismoreunreliablein termsof both pricing andtheability to obtaina resource,andmaythereforere-sult in poorschedulingdecisionsandmoreineffi-ciency for consumers.

To gain a better understandingof how auc-tions fare in comparisonto commoditiesmar-kets,weimplementthefollowingsimulationof anauction-basedresourceallocationmechanismforcomputationalgrids. At eachtime step,CPUanddisk producerssubmittheir unusedCPUandfileslots to a CPU anda disk auctioneer. Theseareaccompaniedby a minimumsellingprice,whichis the averageprofit per slot, asdetailedin Sec-tion 2.1.1 above. Consumersuse the demandfunction as describedin Section2.1.3 to definetheir bid prices,andaslong asthey have moneyto bid on a job, anda job for which to bid, theybid oneachcommodityneededby theiroldestun-commencedjob.

Oncetheauctioneershave receivedall bidsfora time step,they cycle throughall thecommodi-tiesin arandomorder, performingoneauctionpercommodity. In eachauction,thehighest-biddingconsumergets the commodity if the bid priceis greaterthanthe commodity’s minimum price.If there is a second-highestbidder whosepriceis greaterthanthe commodity’s minimum price,then the price for the transactionis the second-highestbidder’s price. If thereis no suchsecond-highestbidder, then the price of the commodity

is theaverageof thecommodity’sminimumsell-ing price andthe consumer’s bid price. Whenaconsumerandcommodityhavebeenmatched,thecommodityis removedfrom the auctioneer’s listof commodities,asis theconsumer’s bid. At thatpoint,theconsumercansubmitanotherbid to thator any otherauction,if desired.Thissituationoc-curswhenaconsumerhasobtainedall commodi-ties for its oldestuncommencedjob, andhasan-other job to run. Auctionsare transactedin thismannerfor every commodity, andtheentireauc-tion processis repeatedat every timestep.

Notethatthisstructuringof theauctionsmeansthateachconsumermayhave at mostonejob forwhich it is currentlybidding. Whenit obtainsallthe resourcesfor that job, it immediatelystartsbiddingon its next job. Whena time stepexpiresandall auctionsfor thattimestephavebeencom-pleted, there may be several consumerswhosejobshave someresourcesallocatedandsomeun-allocated,asaresultof failedbidding.Thesecon-sumershave to pay for their allocatedresourceswhile they wait to startbidding in the next timestep.

While theauctionsdeterminetransactionpricesbasedon individual bids, thesupplyanddemandfunctionsusedby the producersand consumersto set ask and bid price are the samefunctionswe usein the commoditiesmarket formulations.Thus, we can comparethe market behavior andindividual producerand consumerbehavior inbothauctionandcommoditymarketsettings.

3 Simulations and Results

We comparecommoditiesmarkets and auc-tions using the producersand consumersde-scribed in Section 2.1 using two overall mar-ket settings. In the first, which we term under-demand, producers are capable of supplyingenoughresourceto serviceall of the jobs con-sumerscan afford. Recall that our markets donot includeresalecomponents.Consumersdonotmake money. Instead,$G aregiven to thempe-

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CPUs 100disks 100CPUslotsperCPU [2 .. 10]diskfilesperdisk [1 .. 15]CPUjob length [1 .. 60] timeunitsdisk job length [1 .. 60] timeunitssimulatedday 1440timeunitsallowanceperiod [1 .. 10] daysjobssubmittedat day-break [1 .. 100]new job probability 10%allowance �5��± $GBankof G PolynomialDegree 17� factor .01

Table 1. Invariant sim ulation parameter s forthis stud y

riodically muchthe in the sameway that PACIsdole out machine-timeallocations. Similarly,producersdo not spendmoney. Oncegathered,it is hoardedor, for the purposesof the econ-omy, “consumed.” The under-demandcasecor-respondsto a Grid economyin which thealloca-tions exceedwhat is necessary(in termsof userdemand)to allocateall availableresources.Sucha situationoccurswhen the rate that $G are al-locatedto consumersis greaterthan the rate atwhich they introducework to the Grid. In theover-demandcase,consumerswish to buy moreresourcethanis available.New jobsaregeneratedfast enoughto keep all producersalmost com-pletelybusy, therebycreatingawork back-log.

Table1 completelydescribestheinvariantsim-ulation parameterswe use for both under- andover-demandcases. For all ranges(e.g. slotsper CPU), uniform pseudo-randomnumbersaredrawn from betweenthe given extrema. For theunder-demandsimulation, we define �5�b� con-sumersto usethe �5��� CPUsanddisks.Eachcon-sumersubmitsarandomnumberof jobs(between� and �5��� ) at every day-break,and has a 10%chanceof submittinga new job every time unit.

Theover-demandsimulationspecifies²��b� of thesameconsumers,with all other parametersheldconstant.

Usingoursimulatedmarkets,wewishto inves-tigatethreequestionswith respectto commoditiesmarketsandauctions.

1. Do the theoretical results from Smale’swork [33] apply to plausibleGrid simula-tions?

2. Can we approximateSmale’s methodwithonethat is practically implementable?

3. Are auctions or commodities marketsa better choice for Grid computationaleconomies?

Question(1) is importantbecauseif Smale’s re-sultsapply, they dictatethatanequilibriumprice-point mustexist (in a commoditymarket formu-lation), andthey provide a methodologyfor find-ing those prices that make up the price-point.Assumingthe answerto question(1) is affirma-tive, we alsowish to explore methodologiesthatachieveorapproximateSmale’sresults,but whichare implementablein real Grid settings. Lastly,recentwork in Grid economies[1, 18, 28] andmuchprevious work in computationaleconomicsettings[12, 26, 5, 36] hascenteredon auctionsasthe appropriatemarket formulation. We wishto investigatequestion(3) to determinewhethercommoditiesmarketsarea viablealternative andhow they compareto auctionsasamarket-makingstrategy.

3.1 Mark et Conditions, under-demandcase

Figure 1 shows the CPU and disk prices forSmale’s methodin our simulatedGrid economyover �5��o��b��� timeunits.Thediurnalnatureof con-sumerjob submissionis evident from the pricefluctuations. Every 1440 “minutes” eachcon-sumer generatesbetween1 and 100 new jobscausingdemandand prices to spike. However,Smale’s method is able to find an equilibrium

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Figure 1. Smale’s prices for the under -demand case . Solid line is CPU price , anddotted line is disk price in $G

price for both commoditiesquickly, as is evi-dencedin Figure 2. Notice that the excessde-mandspikesin conjunctionwith thediurnalload,but is quickly broughtnearzero by the pricingshown in Figure1 whereit hoversuntil the nextcycle. Figure 3 shows excessdemandfor diskduringthesimulationperiod.Again,approximatemarket equilibrium is quickly achieved despitethe cyclic andnon-smoothaggregatesupplyanddemandfunctionsimplementedby theproducersandconsumers.

In Figure 4 we show the pricing determinedby our engineeringapproximation to Smale’smethod— theFirstBankof G. TheFirst BankofG pricing closelyapproximatesthe theoreticallyachievable resultsgeneratedby Smale’s methodin oursimulatedenvironment.TheBank,though,doesnot requirepolling to determinethe partialderivativesfor the aggregatesupplyanddemandfunctions.Instead,it usesaniterativepolynomialapproximationthat it derivesfrom simpleobser-vationsof purchasingandconsumption.Thus itis possibleto implementthe First Bank of G forusein arealGrid settingwithoutpolling Gridpro-

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Figure 2. Smale’s CPU excess demand for theunder -demand case . The units are CPU slots.

ducersor consumersfor their supplyanddemandfunctionsexplicitly. Figures5 and 6 show ex-cessdemandmeasuresgeneratedby First Bankof G pricing over the simulatedperiod. Whiletheexcessdemandsfor bothcommoditiesarenotastightly controlledaswith Smale’s method,theFirst Bank of G keepspricesvery nearequilib-rium.

Thepricingdeterminedby auctionsis quitedif-ferent, however, as depictedin Figures7 and 8(we show CPU anddisk price separatelyastheyarealmostidenticalandobscurethe graphwhenoverlayed). In the figure, we show the averagepricepaidby all consumersfor CPUduringeachauctionround. We usethe averageprice for allauctionsas being representative of the “global”market price. Eventhoughthis priceis smoothedas an average(someconsumerspay more andsomepay lessduring eachtime step), it showsconsiderablymorevariancethanpricessetby thecommoditiesmarket. Thespikesin workloadarenot reflectedin theprice,andthevarianceseemsto increase(i.e. the price becomeslessstable)over time.

Excessdemandfor anauctionis moredifficult

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Figure 3. Smale’s disk excess demand for theunder -demand case . The units are sim ulatedfile units.

to measuresincepricesarenegotiatedbetweenin-dividualbuyersandsellers.As anapproximation,we considerthe sumof unsatisfiedbids and thenumberof auctionsthat did not make a saleasa measureof market disequilibrium. Under thisassumption,the market is in equilibrium whenall bidsaresatisfied(demandis satisfied)andallauctionedgoodsare sold (supply is exhausted).Any surplusgoodsor unsatisfiedbids are “ex-cess.” While is doesnot make senseto assignasignto thesesurpluses(surplussupply, for exam-ple, may not be undemandedsupply)in the waythatwecanwith aggregatesupplyanddemandina commoditymarket, in absolutevaluethis mea-surecapturesdistancefrom equilibrium. Hencewe termit absoluteexcessdemand.

In Figure9 weshow thismeasureof excessde-mandfor CPUsin the under-demandedauction.Figure 10 shows the samedata as in Figure 5from the First Bank of G, but in absolutevalue.While theFirst Bankof G shows morevariance

in absoluteexcessdemand,it achieves approxi-mateequilibrium and sustainsit over relativelylong periods.By contrast,theauctionsetsprices

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Figure 4. First Bank of G prices for the under -demand case . Solid line is CPU price , anddotted line is disk price in $G

thatnever satisfythemarket. Strangely, theauc-tion comesclosestto equilibrium whendemandspikesat eachday-break.We areworking to un-derstandthisbehavior andwill reporton it aspartof our futurework.

From thesesimulationdatawe concludethatSmale’smethodis appropriatefor modelingahy-potheticalGrid market andthat theFirst BankofG is a reasonable(and implementable)approxi-mationof this method. Theseresultsaresome-what surprising given the discreteand sharplychangingsupply and demandfunctionsusedbyour producersand consumers. Smale’s proofsassumecontinuousfunctions and readily avail-able partial derivatives. We also note that auc-tioneering,while attractive from an implementa-tion standpoint,doesnot producestablepricingor marketequilibrium.If Grid resourceallocationdecisionsare basedon auctions,they will sharethis instability andlack of fairness.A commodi-ties market formulation, at least in simulation,performsbetterfromthestandpointof theGrid asa whole. Theseresultsagreewith thosereportedin [36] which indicatethat auctionsare locally

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Figure 5. First Bank of G CPU excess demandfor the under -demand case . The units areCPU slots.

advantageous,but may exhibit volatile emergentbehavior systemwide.

3.2 Mark et Conditions, over-demandcase

For theover-demandmarket case,we increasethenumberof consumersto 500leaving all otherparametersfixed. As in the under-demandcase,Smale’s method producesa stable price serieswhich the Bank of G is able to approximatebutwhich auctionsareunableto match.We omit thebulk of the resultsin favor of examiningthe be-havior of both Smale’s methodand the Bank ofG asthey converge to an approximateeconomicequilibrium.

Figure 11 shows the pricing information us-ing Smale’s methodfor theover-demandmarket,andFigure12showsthepricesdeterminedby theFirstBankof G. NotethatSmale’smethoddeter-minesa higherprice for disk thanCPU andthattheFirstBankof G choosesasignificantlyhigherprice for CPU, but a lower price for disk. Intu-itively one expectsa higher price for CPU thandisk sinceCPU is the “rarer” commodityin our

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Figure 6. First Bank of G disk excess demandfor the under -demand case . The units aresim ulated file units.

simulation. The Bank of G would seemto cor-rectly identify CPUasthescarcercommoditybysettinga higherprice for it. Nonetheless,excessdemandgraphs(Figures13 and 14) for CPUin-dicatethatbothsolutionmethodsarecenteredonmarket equilibrium. While it is difficult to readfrom the graphs(we usea uniform scaleso thatall graphsof a certaintype in this studymay becompared),themeanexcessdemandfor thedatashown in Figure13is ²b· p¹¸ , andthetheFirstBankof G datain Figure14, themeanexcessdemandis ·b²Bp¹º . Both of thesevaluesarenearenoughtozeroto constituteapproximateequilibria for ourpurposes.

3.3 Multiple Equilibria

We wish to examine more closely the phe-nomenonof apparentmultiple economicequilib-ria within our simulatedmarket. In particular, weclaimthatboththesolutionsarrivedatby Smale’smethodandby the Bank of G arevalid approxi-mationsof economicequilibria and may in factbeapproximationsof actualequilibria. To facili-

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Figure 7. Auction prices for the under -demand case, average CPU price onl y, in $G

tateour examination,we will examinetheaggre-gatesupply and demandfunctionsover all pro-ducersandconsumersat particularpoints in thesimulation. To do so, we freezethe simulationafter it hasreachedapproximateequilibriumandthenquerytheproducersandconsumersfor sup-ply and demandvaluesover a rangeof prices.This techniqueproducesa profile of themacroe-conomicsupplyanddemandcurveswhichshouldrevealequilibriaat their intersectionpoints.

Recall that, in our simulatedeconomy, CPUand disk are highly complementary. Since de-mandfor one commodity is not independentofdemandfor the other, we mustgeneratefamiliesof aggregatedemandcurves, in which the priceof onecommodityis heldconstantwhile thepriceof the othercommodityis variedover the spec-ified range. Eachgenerateddemandcurve in afamily is associatedwith a singlefixed price fortheothercommodity. Then,thefixedprice is in-crementedandanotheraggregatesupplycurve isgenerated.This processcontinuesuntil thefixedprice also reachesthe upper limit of the speci-fied pricerange.If generatingaggregatedemandcurvesfor theCPUcommodity, for example,the

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Figure 8. Auction prices for the under -demand case, average disk price onl y, in $G

simulatorproducesonecurveperpriceof thediskcommodity.

Note that, together, thesefamilies of curvesform a three-dimensionalsurfacefor eachcom-modity in which the axes are CPU price, diskprice,anddemand.That is, for eachorderedpairof CPU anddisk pricesthereis a correspondingCPUdemandvalue.Similarly, asecondsurfaceisformedfrom the CPUprice,disk price,anddiskdemandcoordinates.

In contrast,the supplyof a commodityin oureconomyis never correlatedwith the supply ofanothercommodityandvariesonly with price,soit is not necessaryto producefamiliesof aggre-gatesupply curves. Instead,we producea sin-gle supplycurve by freezingthe simulationandvaryingthepriceof acommodityoversomerangewhile queryingfor aggregatesupplyat eachnewpricevalue.

Figures 15, 16, 17 and 18 show aggre-gatesupplyanddemandcurvesfor CPUanddiskin the over-demandcase. Both Smale’s methodand the Bank of G are shown. The simulationfreezesat time slice 2000 and producesaggre-gatecurves. Ratherthanrepresentingthe three-

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Figure 9. Auction absolute excess demand forCPU in the under -demand case . The units areCPU slots.

dimensionalsurfaceof pricesanddemand(whichis difficult to representwithout theuseof color),we depict the relationshipsin termsof a labeledtwo-dimensionalprojection.

In Figure15, the l axis representsCPU priceand the F axis correspondsto CPU units (eitherof supplyor demand).Eachnearlyverticalcurveis a CPUdemandfunction relatingCPUprice toCPU demandfor a given disk price (shown asalabel on eachcurve at the top of the graph). Weonly show CPU demandcurvesat 10 $G incre-ments,althoughoneexistsfor eachpossibleprice.As a thick gray line, we show the CPU demandcurvethatcorrespondsto thediskprice($G211.4in the figure) that Smale’s methoddeterminedatthe time we froze thesimulation. The thick dot-ted line nearthe bottomof the graphshows theCPU supplycurve asa function of price. The lcoordinateof the price point wherethe CPU de-mandcurve (shown in thick gray) intersectstheCPU supplycurve (dottedblack) correspondstotheapproximateequilibriumpricefor CPUwithinsimulatedeconomyat the given time step. Thesolid circle on the graph shows the price-point

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Figure 10. First Bank of G absolute excessdemand for CPU in the under -demand case .The units are CPU slots.

thatSmale’smethoddeterminedfor thesametimestep. If the circle covers the intersection(as itdoesin Figure15) the price adjustmentstrategyhascorrectlydeterminedanapproximateequilib-rium pricefor theeconomy.

Similarly, in Figures16, 17, and 18 the de-mandcurves are labeledwith the fixed price ofthe othercommodityusedto producethe curve:for example,oneCPUdemandcurve shown cor-respondsto holding the price of disk to $G 200while varying the price of CPU. Sincedemandfor onetypeof commodityis tied to demandfortheother, thedemandcurvefamiliesfor bothdiskandCPUtendto besimilar. Only a few demandcurves in the family are shown, but it is impor-tant to notethat an infinity of suchcurvesexist,forming a demandcurve surface. Also shownin Figures 16, 17 and 18 aretheaggregatesup-ply curvesfor eachcommodity, shown in a thickdottedline. Supplyof bothcommoditiesremainsconstantacrossthepricerangeshown,becauseallsimulatedsuppliersare“producing” at maximumcapacity. No matterhow high the price may beset,no moreCPU or disk is availablewithin the

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Figure 11. Smale’s CPU and disk prices for theover-demand case . Solid line is CPU price ,dotted line is disk price , and the units are $G.

economy.Figures 15and 16 havebeenobtainedby run-

ning Smale’s methoduntil it reachesan approx-imate equilibrium at a CPU price of about $G161.8anda disk priceof about$G 211.4,whicharemarkedasheavy dotsontherespectivegraphs.For Figure 15, the disk priceswere then artifi-cially fixedatvariousvaluesandtheCPUdemandcurves,labelledby diskpriceacrossthetopof thegraph,weregeneratedby polling theconsumers.Again, in principlethereexist demandcurvesforall possiblediskprices;wehaveshown only mul-tiples of $G 10. For Figure 16, the rolesof thecommoditiesare reversed. Note that supply ofeachcommodityis afunctionof thatcommodity’sprice alone,so that only onesupplycurve existsoneachof thegraphs.

Figure 15 shows that the CPU market isclearedfor a CPU price of about$G 161 (readfrom thehorizontalaxis)andadiskpriceof about$G 211 (readfrom the family of curves). Sim-ilarly, one finds from the heavy dot in Figure16 that the disk market is clearedfor about thesamerespective pricesfor disk and CPU. How-

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Figure 12. First Bank of G CPU and disk pricesfor the over-demand case . Solid line is CPUprice , dotted line is disk price , and the unitsare $G.

ever, from the graphsit is possibleto find otherpricecombinationswhich cleareachmarket sep-arately. For example, it is evident from Figure15 that a CPU price of about$G 175 anda diskprice of $G 200 will alsoclearthe market, sincethe CPU demandcurve correspondingto a diskprice of $G 200 intersectsthe supplycurve at apoint wheretheCPUpriceis about$G 175.Nowlook at Figure 16. It seemsthat a disk price ofabout$G 200 and a CPU price of $G 175 willclear the disk market aswell! Moreover, withinthe rangeof pricesshown on the two graphs,itlooksasthoughanypricevectorwhichclearsonemarket alsoclearstheothermarket aswell, or atleastvery nearly so. Thus it would appearthatthereis a wholeconnectedcurve of market equi-libria for our economy.

Froma“behavioral” standpoint,thissetof rela-tionshipsbetweensupply, demand,andpricemaybe explainedas follows: The two commoditiesareextremelycomplementary, meaningthat theyareusedtogetherratherthanin competitionwithoneanother. As longastheconsumershavesome

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Figure 13. Smale’s CPU excess demand forthe over-demand case . The units are CPUslots.

choiceas to which jobs to perform (as they doin the overdemandcase,sincejob queuesneverclear),andaslong asthepriceof onecommodityis loweredin conjunctionwith arisein thepriceoftheother, it is alwayspossiblefor theconsumersto make purchasingdecisionswhich allow themto spendtheir allotment,choosing,if the pricesaredifferent,to completejobswhicharemorein-tensivein thecommoditywhich is lessexpensive.

It is interestingto notethatin thiscaseonecanfind the point in the theory wherethe hypothe-seswhich rule out non-locally-uniqueequilibriabreak down. It is apparentthat in our experi-mentsthe two commoditiesare so complemen-tary that the demandfunctionsshift in the sameway in responseto increasesin eitherprice.Thusthecolumnsof theJacobianmatrix

��� z3t«{ of par-tial derivativesof theexcessdemandwith respectto priceare(approximately)linearlydependentatequilibrium. By definition, then,theequilibriumis not regular, and thereforeit neednot be lo-cally uniqueaccordingto thetheory(Cf. Section2.2.1).

In any event, it would seemthat theseappar-

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Figure 14. First Bank of G CPU excess de-mand for the over-demand case . The unitsare CPU slots.

ent multiple equilibria arisenot becauseof anyanomaliesin our methodper se, but ratherbe-causeour experimentaleconomyis so very sim-ple as to consistof only two commodities(pluscurrency) whichareessentiallyin perfectcomple-mentarity. Onewould expect that, as the modelbecomesmorecomplex, thisparticularsortof dif-ficulty will vanish. Further, even in thepresenceof multiple equilibria, eachof our price adjust-mentschemescontinuedto behave in sucha wayasto producelong-termstabilityandapproximatemarket-clearing. This is all that onecan practi-cally hopefor, sinceevenin well-behaved(“regu-lar”) economies,theremaybemultiple (isolated)equilibriawith norationalbasisfor choiceamongthem.

Our implementation of Smale’s technique,then,findsa valid equilibriumprice from amonga spaceof possibleequilibria. The Bank of Galsofindsa valid pricesolution,albeita differentonefrom Smale’s technique.In Figures 17 and18, we show the supplyanddemandcurve fam-ilies aswell astheir price solutionsfor theBankof G. Note againthat the pricescorrespondto a

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Figure 15. CPU aggregate suppl y and demandcur ves for Smale’s method, over-demandcase, iteration 2000.

global equilibrium; the CPU price point lies atthe intersectionof theCPUsupplycurve andtheCPU demandcurve correspondingto disk priceof $G166.Sincethemarket is in anover-demandsituation,resourceconsumershave no choiceinthe mix of jobs they run. Rather, they can runonly jobs for which some supply is available.Consumers’jobs queuewaiting to be serviced,and this queuecontainsa mixture of CPU- anddisk-intensive jobs. Thus,from thestandpointofglobalequilibrium,additionaldisksupplyandad-ditional CPU supply are interchangeable;thereis ample demandto utilize either. The marketis free to chooseany balancebetweenCPU anddisk price so long asthe aggregatesupplyof ei-thercommodityremainsfully utilized.

Fromthis basisthepriceinversionof CPUanddisk betweenthe Smale and Bank of G over-demandsimulationsis easyto understand.Bothmethodsclearthe market andcontrol excessde-mand. Valid price solutionsarenecessaryto ac-complishsuchcontrol, andboth techniquesfind

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Figure 16. Disk aggregate suppl y and demandcur ves for Smale’s method, over-demandcase, iteration 2000.

suchsolutions.It is intuitively uncomfortableforSmale’s techniqueto arrive at higher pricesformoreplentiful commodities,but suchbehavior issoundfrom aneconomicstandpoint.

Note that in every case(Figures15, 16, 17,and 18) the respective method(eitherSmaleorBank of G) determinesa price that is at or veryclose an approximateequilibrium price for theeconomy.

As notedabove,thepricevectorsolutionspacefor two commoditiescaneffectively beviewedasa 3 dimensionalplot of total absoluteexcessde-mandversusthepriceof bothcommodities.Totalabsoluteexcessdemandis in this casedefinedasthe sum of the absolutevalue of the excessde-mandfor bothcommodities,andcanbeusedasameasureof closenessto economicequilibrium.InFigures19and 20weshow thisspaceof priceso-lutionsfor theover-demandcase.For clarity, onlythepointof minimumexcessdemandfor eachde-mandcurve is shown. Thesepointsform a line inprice/excessdemandspacealongwhich approxi-

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Figure 17. CPU aggregate suppl y and demandcur ves for the Bank of G, over-demand case,iteration 2000.

matemarket-clearingsolutionsmayfall. We alsoshow theprojectionof this line of equilibriaontothe price plane,andnotethat the price solutionsindeedfall very near or upon this line of min-ima. Also importantto noteis thattheprojectionis nearlinear with slope j fÙ� . This serves asfurtherconfirmationthatthetwo commoditiesarealmostperfectly complementary. We conclude,basedon this further evidence,that both our im-plementationof Smale’s method and the FirstBank of G arefunctioningcorrectlyandachiev-ing theresultsexpectedby thegeneraltheoreticalformulationadvancedby Smaleasappliedto oursimpleGrid economy. Theresultsareparticularlyencouragingsincethey do notdependupongrosssubstitutabilityrestrictionsandbecausethey canbe achieved via an implementablesystemwhichdoesnot requiremarket-widepolling.

3.4 Revisiting under-demand

Having seenthatour simulatedeconomycon-vergesto real equilibria in the overdemandcase,

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Figure 18. Disk aggregate suppl y and demandcur ves for the Bank of G, over-demand case,iteration 2000.

we canre-examinetheunder-demandcaseagainusingour characterizationsof its macroeconomicbehavior. Figures21and22show theeconomicstateof the simulationusingSmale’s method,it-eration3119. This timesliceoccursjust after thebeginningof asimulated“day”, whenjobsarein-jectedinto the system. The stateof the systemat this point is similar to the over-demandcase,andthis is reflectedby thesimilarity of Figures21and22 to Figures15 and16.

However, oncetheconsumers’jobsfor thedaybecomeserviced, the systementersan under-demandedstate. Consumersget new jobs at anaveragerateof oneevery tentime steps,andtheytypically haveplentyof $G with which to servicejobs.Producersontheotherhand,aremostlyidle.However, sincethey basetheir supply functionson averageprofit, they still refuseto sell until acertainthresholdpriceis met.Thestateof thesys-temduringiteration4000is plottedin Figures23and24,usingthesamelinearscalefor they-axesasin theothergraphs,andin Figures25 and26,

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Figure 19. Total absolute excess demand min-ima, Smale’s Method, overdemand case . Theprojection upon the price plane is also sho wn.Filed cir cles represent equilibrium price solu-tions at this iteration.

usinga morereadablelog scale.

Although it is difficult to discernfrom thefig-ures,thereis no equilibriumpoint for both com-moditiesin thisgraph.This is becausethesystemat thispoint is notawell-behavedeconomy, sincethe loweringof pricesdoesnot necessarilybringaboutanincreasein demand.Putanotherway, thedemandis solow thattheassumptionthatindivid-ual agentsdo not make a significantdifferenceisviolated. Regardless,both Smale’s methodandthe Bank of G default to a “normal” price. Themarket is not cleared– there is a supply glut –but pricesdo not becomeabnormallydepressed.Theseresultsindicatethat both Smale’s methodand the First Bank of G will be reasonablyro-bust with respectto degenerationin the underly-ing economicbehavior of the systemsto whichthey areapplied.

Probingfurther, the behavior of the banksinthis casecanbe accountedfor by looking at thesupplyanddemandcurves;notethatthepricethateachbankfinds is onewherethesupplycurve isalmostverticalandthedemandcurve horizontal,indicatinga largejump in producerbehavior ator

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Figure 20. Total absolute excess demand min-ima, Bank of G, overdemand case .

nearthis price. This meansthat the excessde-mandfunction for eachcommoditywill locallydependonly on that commodity’s price andwillbe extremelysensitive to small changesin price.Thus the Jacobianmatrix

� � z�t|{ will have theformÚÛÛÛÜ very large �

negativenumbervery large� negativenumber

ÝSÞÞÞßThelargediagonalentrieswill produceextremelysmall valuesof à�t for either price-adjustmentscheme. Note in this casethat Smale’s methodreducesto tatonnement(Cf. Section2.2.1)duetotheoff-diagonalzeros.

It is reasonableto expect that in more realis-tic simulationswheretruemarketbehavior holds,and in any meaningfulimplementationof eitherof theseprice adjustmentschemes,the behaviorof theagentswill besufficiently heterogeneousasto precludethe existenceof suchlarge jumps incumulativesupply.

3.5 Efficiency

While commodities markets using Smale’smethodof pricedeterminationappearto offer bet-

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Figure 21. CPU aggregate suppl y and demandcur ves for Smale’s method, under -demandcase, iteration 3119.

ter theoreticalandsimulatedeconomicproperties(equilibriumandpricestability) thanauctionsdo,we also wish to considerthe effect of the twopricing schemeson producerandconsumereffi-ciency. To do so,we reportthe averagepercent-ageof timeeachresourceis occupiedasautiliza-tion metricfor suppliers,andtheaveragenumberof jobs/minuteeachconsumerwasable to com-pleteasa consumermetric. Table2 summarizesthesevaluesfor boththeover- andunder-demandcases.

In termsof efficiency, Smale’s methodis bestandtheFirst Bankof G achievesalmostthesameresults.Both aresignificantlybetterthantheauc-tion in all metricsexcept disk utilization in theover-demandedcase.SinceCPUsarethe scarceresource,diskpricemayfluctuatethroughasmallrangewithout consequencewhen lack of CPUsupplythrottlesthesystem.Theauctionseemstoachieveslightly betterdiskutilizationundertheseconditions.In general,however, Smale’s methodandtheFirst Bankof G approximationbothout-

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Figure 22. Disk aggregate suppl y and demandcur ves for Smale’s method, under -demandcase, iteration 3119.

performtheauctionin thesimulatedGrid setting.

4 Conclusionsand Futur eWork

In this paper, we investigateG-commerce—computationaleconomiesfor controllingresourceallocationComputationalGrid settings. We de-fine hypotheticalresourceconsumers(represent-ing usersand Grid-aware applications)and re-sourceproducers(representingresourceownerswho “sell” their resourcesto the Grid). Whiletherearean infinite numberof waysto representindividual resourcesupplyanddemandin simu-lated setting,and noneare completelyaccurate,we have identifieda setof traits that we believearerealistic.e All entitiesexceptthe market-maker act in-

dividually in their respectiveself-interests.e Producersconsiderlong-termprofit andpastperformancewhendecidingto sell.

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efficiency metric under-demand over-demand

Smaleconsumerjobs/min 0.14j/m 0.05j/mB of G consumerjobs/min 0.13j/m 0.04j/mauctionconsumerjobs/min 0.07j/m 0.03j/m

SmaleCPU utilization% 60.7% 98.2%B of G CPU utilization% 60.4% 93.9%auctionCPU utilization% 35.2% 85.5%

Smaledisk utilization% 54.7% 88.3%B of G disk utilization% 54.3% 84.6%auctiondisk utilization% 37.6% 85.1%

Table 2. Consumer and Producer efficiencies

0å

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Figure 23. CPU aggregate suppl y and demandcur ves for Smale’s method, under -demandcase, iteration 4000.ç Consumersaregivenperiodicbudgetreplen-

ishmentsandspendopportunistically.ç Consumersintroducework loadsin bulk atthebeginningof eachsimulatedday, andran-domly throughouttheday.

Using simulated consumersand producersobeying these constraints,we investigate twomarket strategies for setting prices: commodi-ties markets and auctions. Commoditiesmar-

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Figure 24. Disk aggregate suppl y and demandcur ves for Smale’s method, under -demandcase, iteration 4000.

kets are a natural choice given the fundamen-tal tenetsof the Grid [17]. Auctions, however,aresimpleto implementandwidely studied.Weareinterestedin which methodologyis mostap-propriatefor Grid settings. To investigatethisquestion,we examinethe overall price stability,marketequilibrium,producerefficiency, andcon-sumerefficiency achieved by three methodsinsimulation. The first implementsthe theoreticalwork of Smale[33] which describeshow to ad-

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200é 190ê 180ë 170ìFigure 25. CPU aggregate suppl y and demandcur ves for Smale’s method, under -demandcase, iteration 4000, log y axis scale .

just prices in a commoditiesmarket to achieveequilibrium. It is viable in simulation,but im-practicalin the “real-world” asit relieson beingableto poll reliably producersandconsumersforsupply and demandinformation. Often they donot know, or will not saywhattheir responseto agivenpricewill be.Thesecondmethod(TheFirstBank of G) is an implementableapproximationto Smale’s method. It usesa large-degreepoly-nomial to approximateexcessdemandfunctionsinsteadof polling making it parameterizablebyobservedmarket behavior only. Lastly, we simu-lateauctionsin thestylethathasbeeninvestigatedpreviously.

Our resultsshow that Smale’s resultshold forour simulatedGrid environment, despitebadlybehaved excessdemandfunctions, and that theFirstBankof G achievesresultsonly slightly lessdesirable. In all cases,auctionsare an inferiorchoice.

As part of our future work, we plan two par-allel thrusts.First, we areexploring thespaceofplausibleG-commerceformulations.Our goal isto identify andtest,in simulation,differentpossi-

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Figure 26. Disk aggregate suppl y and demandcur ves for Smale’s method, under -demandcase, iteration 4000, log y axis scale .

ble economiesfor theGrid. Secondly, we plantoconstructa working versionof the First Bank ofG. Our previouswork with theNetwork WeatherService[38, 39] andIBP [27] leavesuswith theinfrastructurenecessaryto build alargescalesup-ply anddemandinformationrepository. UsingtheFirst Bankof G, we cangeneratepricesbasedon“li ve” supplyanddemandinformation.

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