Pricing Resources on Demand

Transcription

1 Pricing Resources on Demnd Costs Courcouetis, Sergios Soursos nd Richrd Weer Athens University of Economics nd Business Emil: courcou, University of Cmridge Emil: Astrct Trditionl contrcts for network nd computing resources re of sttic type where the customer is uying the right to use for given price fied mount of resources for long period of time. Typicl emples re the cse of contrcting ndwidth in ccess networks nd VPNs nd the cse of computing infrstructure tht customer leses (or uys) for fulfilling its IT needs. Current technology in ccess networks nd Grid computing llows suppliers to offer more fleile contrcts to their customers llowing them to choose dynmiclly the mount of resources they re llowed to use t given time. This fleiility my enefit the customers with ursty demnd since it llows them to otin resources only when they need them nd py only when they use them. We define contrcts where time is discrete nd customer is llowed to uy fied mount of resources hed of time for price, the sttic prt of the contrct, nd complement this t ech new time period y purchsing n etr mount t price, the dynmic prt of the contrct. We investigte the properties of such contrcts nd compre them with contrcts of purely sttic or dynmic type. Our results suggest tht in generl suppliers nd customers re oth etter off when using such mied contrcts, nd tht purely dynmic contrcts my not lwys e preferle compred to purely sttic ones. We lso show tht under price competition of suppliers using sttic contrcts ginst suppliers using dynmic contrcts, t the equilirium oth suppliers my secure some profit y segmenting the mrket. I. INTRODUCTION Trditionl ndwidth contrcts hve een for fied ndwidth pipes nd re long term, typiclly yer. We re thinking of network supplier tht leses high cpcity rodnd lines to other lrge compnies, such s nks or Internet service suppliers (ISPs). A nk might lese 55Mps line etween two loctions for one yer. An AOL suscrier might sign yer s contrct for the service known s AOL Brodnd Gold (Mps). Typiclly, such Internet ccess or VPN services do not offer ny fleiility to the customer to lter dynmiclly the size of the pipe. Similrly, when cquiring computing resources such servers nd PCs, compny s IT deprtment usully mkes yerly lesing contrcts for fied mount of computing resources, or uys these resources which is the equivlent of fied size contrct for n even longer period of time. The prolem customers fce in mking such contrcts is tht their demnd for resources is not constnt ut ursty nd in mny cses unpredictle. They hence fce the This work is prt of the project Economic nd Architecturl Aspects for Services of Differentited Qulity in high-speed Brodnd nd Wireless Access Networks, co-funded y the Europen Socil Fund & Ntionl Resources - EPEAEK II - IRAKLITOS Fellowships for reserch of Athens University of Economics nd Business. This work is lso supported y the IST project GridEcon (IST ) funded y the EU. risk of cquiring for long time period resources tht my not e effectively used, or cquiring fewer resources tht will fil to meet high short term demnd nd pek lod requirements. Emerging technology in ccess networks nd Grid computing llows suppliers to offer more fleile contrcts to their customers providing them the ility to choose dynmiclly the mount of resources they re llowed to use t given time. For instnce, DSLForum s BrodndSuite pltform ] nd IPsphere s frmework ] propose solutions tht will offer the ility to lter the size of the ndwidth pipe with which customer is provided on time scle tht is much finer thn yerly. They lso provide ccounting nd illing for such dynmic services. Grid computing offers direct nlogy to ndwidth provisioning. Current Grid utility computing rchitectures such s SUN Grid Compute Utility 3] llow customers to uy computing cycles on demnd while lso using their eisting computing fcilities. This fleiility should enefit the customers with ursty demnd since it llows them to otin resources only when they need them nd py only when they use them. If priced correctly, these services my increse the revenue of the provider y otining shre of the dded vlue to his customers. In this pper, we investigte the properties of new type of contrct, the mied contrct, which is composed y comining sttic nd dynmic contrcts. Under mied contrct customer is llowed to uy fied mount of resources hed of time for price, the sttic prt of the contrct, nd complement this t ech new time period y purchsing n etr mount t price, the dynmic prt of the contrct. Dynmic contrcts or the dynmic prt of mied contrct should not e confused with dynmic pricing pproches which im to control network trffic y mens of prices. One such pproch is pek-lod pricing 4], 5] where prices cn djust to the fluctutions of demnd nd my reflect the investment mde in order to serve the high-demnd periods. Due to the fct tht mied contrcts hve two prts, they my e considered s two-prt triffs. In the literture, two-prt triffs re used y the providers to otin greter portion -if not ll- of customer s utility. The sttic prt of such triff is lump sum tht the user must py in order to hve the right to use the resources 6] or wy for the provider to cover his fied costs. In some cses, this lump sum my give the right to use smll mount of resources. But in our pricing scheme, no lump sum nd no fied mount of resources is introduced. Our two-prt contrct ims to mke more fleile the customer in the wy he epresses his ndwidth needs. Throughout the pper we consider tht oth prices ( nd ) re fied, i.e. they

2 re not ffected y the demnd. On the other hnd, these prices cn e considered s usge-sed prices since customers re chrged sed on the volume of ndwidth they consume. An interesting question we nlyze in this pper concerns the position of the suppliers of dynmic services. Will they e lwys etter off y providing such services or they my cnnilize their lucrtive sttic contrcts? Will ndwidth consumption increse or decrese? Wht will e the result of price competition etween dynmic nd sttic service suppliers? Will one supplier get the whole mrket shre y displcing the other? We show tht mny interesting fcts my occur. For instnce, in monopolist sitution, customer my not lwys e etter off when the supplier offers purely dynmic contrcts. Also our models suggest tht resource consumption my drop when dynmic contrcts re used (which justifies the fers of the network opertors for llowing dynmic contrcts). Moreover, we show tht offering mied contrcts (the mied suppliers) is lwys eneficil to oth the customers nd the resource suppliers. Also, under price competition, the mrket will e segmented nd oth the dynmic nd sttic suppliers my mke profit t the equilirium. A simple emple Let us suppose, for emple, tht t the strt of yer customer cn sign contrct for fied line t cost of $ per Mps. Additionlly, he my purchse t the strt of ech week, dditionl cpcity for cost of $ per Mps per yer. Assuming <, there is the incentive to ook hed. This is esier for customers whose requirements re more predictle. Suppose tht the supplier hs no cpcity constrint. Consider now three customers nd their ndwidth requirements. For simplicity we ssume tht these requirements re inelstic. Customer needs Mps pipe throughout the yer. The most he would e prepred to py for this is $. Customer needs Mps pipe on hlf of the weeks of the yer nd Mps pipe on the other hlf of the weeks of the yer. These weeks re rndomly distriuted in the yer, ut re predictle one week hed. The most he would e prepred to py for this is $9. Customer 3 needs Mps pipe on 9/ths of the weeks of the yer nd 3 Mps pipe on the other /ths of the weeks of the yer, which re gin rndomly distriuted in the yer nd predictle one week hed. The most he would e prepred to py for this is $ 36. Customers nd 3 will only purchse contrct if it cn stisfy their requirements on ll weeks of the yer. The prolem for the supplier is to choose nd so s to solve the prolem 3 mimize y i, i= suject to y = if, otherwise, y = min, + / if min, + / 9, otherwise, nd y 3 = min3, +/ if min3, +/ 36, otherwise. If we llow only the yer-long contrcts, (so tht effectively = ), then y tking =95the revenue is mimized to = $85. Only the first two customers purchse contrcts. If we hve wnted to ensure tht ll three customers uy contrcts, we would hve needed to set = 35/3 =45 nd this would hve produced revenue of only $ 7. Now suppose we llow ookings to e mde just one week hed. It will e optiml to tke = nd = 8. All customers will mke purchses nd the revenue will e = $46. Note tht we re requiring customer to py $ more thn efore. If we wished to void this we might tke =95nd = 9. Now we cn e certin tht customers nd will purchse the sme s efore, nd the revenue will e = $48. The nice thing out this lst solution is tht we need not know efore introducing the new triff tht the third customer even eists. If he does, then we increse revenue y 46.7%. But if he does not eist then nothing hs een lost. If we re lucky, there will e even further customers, dditionl to customer 3, who will strt uying contrcts. This emple suggests tht provider my increse its profit y using mied contrct nd tht customers my otin t lest the sme net enefit they otined in the cse of using optiml sttic contrcts. But wht if the provider uses purely dynmic contrcts? In this cse the optiml revenue is $37, for =, which is lrger thn the optiml sttic revenue of $85. But this is not lwys the cse! If we hve different percentge of the sme customers, the optiml sttic revenue my e higher. For instnce, if no customer 3 is present, then the mimum revenue of sttic, dynmic nd mied contrcts re $85 ( = 95), $5 ( = ) nd $9 ( =, = 8) respectively. Hence dynmic contrcts my no longer e optiml for the provider. This pper is orgnized s follows. In Section II, we formulte the generl optimiztion prolem tht customer fces when provided with sttic, mied or dynmic contrct. Section III defines the utility function tht will e used throughout the rest of the pper nd in Section IV, we use this utility function to model the customer s net enefit mimiztion prolem. In Section V, we model the provider s revenue mimiztion prolem nd we show how the prices of the vrious contrcts re formed. We lso consider vrious cses of customer distriutions. Section VI models price competition gme etween providers of different type of contrcts, where we give n insight of how prices will djust. One cn prove tht, under certin conditions, t the equilirium the mrket will e segmented etween the providers. Section VII concludes our work nd provides some points for future work. II. THE OPTIMIZATION PROBLEM Let us strt from the position tht contrcts generte chrges over fied long time periods consisting of n shorter periods (slots). Prices re ssumed to e fied nd known to customers. They uy long-term contrcts where they secure certin fied mount of resources for the n slots, nd my comine these with short-term contrcts for cquiring dditionl resources in ech slot. For simplicity ssume tht the long time period is yer nd slots correspond to weeks, i.e., n = 5. There re N customers who re prepred to uy long-term contrcts given tht we re chrging sttic price corresponding to $ per

3 3 Mps per week (the ctul price is n for the long-term period ut we like to epress it per slot). Customer i uys ndwidth of i Mps in this long-term sttic contrct. The revenue is presently r(,= ) =n N i= i. Suppose now tht we introduce the possiility of uying further units of ndwidth, one-week-hed, for $ per Mps. Suppose tht for the coming week, customer i hs utility for ndwidth tht is prmeterized y θ, which he cn predict one week hed. If he lredy owns yer-long contrct for ndwidth, nd he dditionlly uys ndwidth y for the coming week, then his utility is u i (θ, + y). This customer will choose to uy yer-long contrct for i where this comes from his finding his mimum net enefit s ] n i (, ) = m ne mu i (θ, i + y i ) y i n i, i y i () where the epecttion is tken over θ nd n =5. Oserve first tht the solution of the optimiztion prolem is independent on n when oth prices re epressed on per slot sis. The verge revenue he genertes is r i (, ) = n i + nȳ i, where ȳ i = E rg m yi u i (θ, i + y i ) y i ] is the verge ndwidth he will uy in week in the dynmic ndwidth mrket, nd i is the mimizer in (). The totl revenue my e denoted s r(, ) = i r i(, ). We seek to mimize r(, ). The prolem fced y the supplier offering the triff (, ) ecomes more comple if we consider the possiility tht the customer my seek nother supplier offering the triff (, ) if his net enefit using this supplier is higher. For instnce, suppose the second supplier offers purely sttic contrct (, ). This suggests for mrket consisting of single customer the optimiztion prolem mimize r(, ) s.t. n(, ) n(, ). (), Intuition suggests tht t optimlity, nd >. We lso oserve tht for customers with constnt ndwidth requirements introducing the possiility of uying ndwidth on demnd does not increse the revenue of the supplier. But in the cse of customers with fluctuting requirements, () should led to n increse of revenue. We cn derive now n importnt property of sttic versus purely dynmic contrcts. Under the sme price =, dynmic contrcts re more eneficil for the customer. This is esy to see from () since ] m ne u i (θ, ) ne ] m u i(θ, y) y. y A simpler version of the prolem is one in which customers hve fluctuting ut predictle ndwidth requirements. This corresponds to their knowing the reliztion of the vlues of θ for the different weeks in dvnce. In this cse the customer solves n i (, ) = m i n j= ] mu i (θ j, i +y j y j i ) yj i n i, i (3) U Fig.. The utility function nd resulting demnd function when k =. where y j i is the ndwidth to e ought from the dynmic mrket during the jth period. Now the revenue generted is r i (, ) =n i + n j yj i. This model llows us to formulte some more interesting prolems. Oserve tht the dynmic demnd y j i of customer fluctutes over time nd its sttistics re ffected y the choice of,. If there is finite cpcity in the system tht must e procured t the eginning of the yer, then we must lso mke sure tht the totl demnd from customers rrely eceed the cpcity. This suggests tht we ssocite n effective ndwidth 7] with ech customer i cpturing the vriility of y j i. Lets denote this y e i(, ). Then relted optimiztion prolem is mimize, r(, ) s.t. e i (, ) C. (4) This cptures the fct tht incresing nd decresing my result in higher revenues ut it cretes more fluctuting demnd which my e hrder to provide. III. A MODEL FOR CONSUMER UTILITY We suppose tht t the time tht customer purchses sttic ndwidth he does not yet know his utility function for ndwidth. He knows only tht is will e of the form u k (), where k is prmeter, presently unknown, ut distriuted priori s rndom vrile with known distriution function F (k). In prticulr, we will illustrte ides with k u k () =, k, (5) k, k. Assuming this utility function, if the user fces sttic price of nd knows k then he will choose to mimize u k (). This gives the demnd function k () = mk, (See Fig. ). IV. THE CONSUMER S PROBLEM: MAXIMIZING NET BENEFIT Consider single consumer nd the purchses of ndwidth tht he will mke when fced with sttic, dynmic, or mied supplier, nd where the sttic nd dynmic ndwidth prices re nd respectively. If fcing sttic, dynmic or mied supplier, the customer optimizes over the quntity of ndwidth he uys nd otins net enefits (verge, per slot) i

4 4 of n S () = m E u k () ], ] n D () =E m u k(), E n M (, ) = m m u k( + y) y y ], where the epected vlues re tken over k. To illustrte the sort of things tht cn occur, we consider emples in which the utility function is of the form (5) nd k hs some specil distriutions. One of these is where k is ritrrily distriuted on, ] (see Section V-A). Let us consider for the moment only the cses of sttic nd dynmic sellers (not mied). We then hve, n S () = m u k () dk, n D () = m uk () dk. Another possile distriution for k is when k = k or k = k with proilities p nd p respectively (see Section V- B). We suppose k <k.wenowhve n S () = m ( p)u k ()+pu k (). For fied, this is conve function of p. For dynmic contrcts the user s net enefit is uk () ] + p m uk () ]. n D () =( p) m For fied, this is liner function of p. Thus, there will e some rnge p p,p ], ] for which the dynmic contrct will e preferred. To see this, note tht if = then the user will prefer to uy dynmiclly, ecept t p =nd p =, where the user is indifferent. As increses, n D () decreses t every vlue of p nd remins liner function of p. Thus s functions of p the liner function n D () crosses the conve function n S () t most twice. V. THE MONOPOLIST SUPPLIER S PROBLEM A. Identicl customers with ritrrily distriuted k Suppose there is single monopolist supplier who cn supply ndwidth t cost c nd who is ttempting to sell to popultion of identicl customers. Let us sk whether he cn mke more profit s sttic, dynmic or mied supplier. To discover the nswer, we must perform somewht complicted clcultion. We ssume tht the supplier knows F, the distriution of k nd cn determine typicl customer s demnd function, sy S () in the sttic cse. He then chooses the price to mimize the profit ( c) S (). The results of Tle I re otined numericlly under the ssumption tht k is uniformly distriuted on, ]. We suppose c =so tht the supplier s revenue nd profit re the sme thing. We show wht hppens for the three types of supplier in the columns leled (S), (M) nd (D). The figures for revenue nd men ndwidths ought nd sold re per customer. ] TABLE I (S) THE OPTIMAL PURELY STATIC CONTRACT HAS =/9AND THE OPTIMAL REVENUE IS /7. THE USER BUYS =/3.THE USER S NET BENEFIT IS 7/6. THE MEAN BANDWIDTH CONSUMED IS /3. (M) THE OPTIMAL MIXED CONTRACT HAS =6/5 AND =/5AND OPTIMAL REVENUE /5. THE USER BUYS =/5STATIC AND HIS NET BENEFIT IS /5. THE MEAN BANDWIDTH CONSUMED IS 7/5. (D) THE OPTIMAL PURELY DYNAMIC CONTRACT HAS =/3AND OPTIMAL REVENUE /7. THE USER S NETBENEFITIS4/8. THE MEAN BANDWIDTH CONSUMED IS /9. (S) (M) (D) seller revenue optiml. optiml.3333 ndwidth ought in sttic men ndwidth sold user net enefit These numericl results suggest some conjectures. If true in generl, they support the notion tht there is dvntge in offering mied contrcts. A. The revenue chieved y the seller is strictly greter in (M) thn in (S) or (D). B. The men ndwidth sold decreses from (S) to (M) to (D). C. The user s verge net enefit increses from (S) to (M) to (D). We lso notice the interesting fct tht the revenue chieved y the seller is the sme in (S) nd (D). This turns out to e true in generl, ssuming the utility is of form (5). We stte this s follows. Proposition : Suppose the utility function is of the form in (5). Then seller of sttic contrcts who mimizes his revenue y choice of otins the sme revenue s seller of dynmic contrcts who mimizes his revenue y choice of. Moreover, t these optimums the ndwidth tht is sold y the seller of sttic contrcts is equl to the optimum dynmic contrct price price. Proof: The theorem holds whtever the distriution of k, ut let us suppose for simplicity tht k is ritrrily distriuted over, ] with density function f(k). If the seller chooses price for sttic contrcts, the uyer will choose to mimize u k (k)f(k) dk + This is mimized where + u k ()f(k) dk. (k )f(k) dk =. Thus the seller mimizes his revenue y mimizing over (k )f(k) dk. (6)

5 5 Now if the price of dynmic contrcts is, then the mount of ndwidth purchsed is (k ) +. Thus the seller seeks to mimize over (k )f(k) dk. (7) The theorem follows y compring (6) nd (7). Note lso tht the optiml is where d (k )f(k) dk = (k )f(k) dk =. d So tking = s the solution, (k )f(k) dk =( F ( )) <. Thus s = we hve tht the ndwidth sold under the dynmic contrct is less thn the ndwidth sold under the sttic contrct, s in Conjecture B. B. Identicl customers with two-point distriuted k Since we would like to consider wht hppens when customers re not identicl we wish to find some simple model in which their differences re cptured y single prmeter. So let us suppose tht k = k or k = k with proilities p nd p. Now different vlues of p cn distinguish customers from one nother. Initilly, let us fi p nd consider wht hppens when monopolist is selling to popultion of customers, ll of whom hve the sme p. ) Sttic contrcts: The customer s net enefit is p u k ()+p u k (). This is continuous function of, nd it hs continuous first derivtive. Tking the first derivtive, we see tht the price t which the user will purchse is p u k ()+p u k () =p (k ) + + p (k ) +. In fct, we cn write p (k ) + + p (k ) + = m, k, p (k ). Or we cn lso invert this to write the demnd function s () = m, k, k /p. Thus the seller s mimum revenue is m m, k, p (k ) = m, m ( k ),p m (k ) = m k 4, 4 p k Now 4 k 4 p k if nd only if p p where p = (k /k ), since p = p. This gives the following. Proposition : The optiml (, ) re either ( k, k) or ( p k, k ) s p is less or greter thn p. Note tht k k = p = p p nd the optimum is t = k for ll vlues of p. Conversely, p >p = k <k. ) Dynmic contrcts: If the seller sets price then the user will uy verge ndwidth of p (k ) + + p (k ) +. We know y Proposition tht the optiml vlue of is k or k.ifp p we hve = k. Ifp p we hve = k >k. It turns out tht in oth cses the verge ndwidth is the sme s in the cse of the sttic contrct. Hence, Conjecture B is not true in strict sense for the cse of the two-point distriuted k. But under ritrrily distriuted k, Conjecture B is true in strict sense (s shown in Section V- A). Furthermore, we cn compute the difference in the user s net enefit under optiml purely dynmic nd purely sttic contrcts s n D n S = p p (k k ), p p p k, p >p This gives Proposition 3: If p >p the user strictly prefers to fce monopolist seller of sttic contrcts rther thn monopolist supplier of dynmic contrcts. If p <p his preference is reversed. Thus, we see tht wheres the optiml purely sttic nd purely dynmic contrcts produce the sme revenue for the seller, the type of contrct tht otins the greter net enefit for the user depends on whether p is less or greter thn p. Notice tht there is discontinuity in the user s net enefit t p = p. Conjecture C is not true in generl. It is only true if p p. Remrk: The ove proposition cn e generlized, in the sense tht for every price of the sttic provider, dynmic provider cn pulish price tht offers him t lest the sme revenue, while the customer cquires higher net enefit, if p p. The result is reversed in the cse of p >p.infig. we show how the user s net enefit cn vry with the seller s revenue s the seller vries his sttic price, or dynmic price. The sellers hve no cost. Notice tht the mimum revenue tht cn e otined y either method is the sme. 3) Mied contrcts: Consider the optiml mied strtegy, optimized over prices. Suppose the net enefit t the optimum is p u k ( + y )+p u k ( + y ) p y p y. Note first tht we cnnot hve y >, since s >the net enefit to the user could e improved y tking + y, y nd y y y. Therefore we my consider the prolem when the net enefit is of the form p u k ()+p u k ( + y ) p y. Now we must hve p <, otherwise the user does est to uy only sttic. Now lso note tht we cnnot hve >k.for if this were so, the user could mke the chnge ɛ> k nd y y + ɛ, nd his net enefit would increse y ( p )ɛ >. Thus the optimum must occur where k. Note lso, tht the optimum must occur where + y k.

6 6 NB REV () p =. NB () p =.5 REV Fig.. The user s net enefit (NB) ginst the seller s revenue (REV). The sttic seller is shown in lue nd the dynmic seller in violet. In this emple, k =, k =3.Ifp <p =.5, the seller of dynmic (violet) cn, t ny revenue vlue, offer the user greter net enefit thn cn the seller of sttic (lue). The reverse is true if p >p. For given, such tht <k, the user optimlly tkes y =(k ), nd so his net enefit is then TABLE II SUMMARY OF RESULTS p u k ()+p u k (k ) p (k ). The derivtive of this with respect to is + p + p (k ). p p r S = r D <r M n S <n M <n D m S = m M = m D p >p r S = r D <r M n D <n S <n M m S = m D >m M Since + p <, thisisnegtivet = k, nd so the optimum is t some <k, where = k +( )/p. Thus the seller s revenue cn e found y sustituting this vlue of into his revenue of + p (k ) nd optimizing over nd. This gives = k nd = k. Note tht these stisfy >>p, s required. The optiml purchses re = k nd y = (k k ). Thus the men mount tht is purchsed is + p y = k. Interestingly, we hve the sme sttic price, nd sme verge ndwidth purchsed, s in the sttic contrcts cse, provided we re in the regime where p p. As commented previously, specil cse is k k, which implies p =. The user s net enefit cn e clculted to e p u k (k /) + p u k (k /) (/4) ( p k + p k ) = 8( p k + p k ) Under the sttic contrct it ws 8 ( p k + p k ) = 8 k, which is less. The seller s revenue hs incresed from k 4 to 4 (p k + p k). It is interesting to note tht oth the seller nd user increse their profit nd net enefit, respectively, y the sme fctor of p k + p k (p k + p k ). The gretest possile vlue for this rtio in the region k k is.5. Remrk: Note tht the seller otins the sme revenue s he could otin if he knew k nd used personlized pricing, i.e., chrging price ρ = k if k = k, nd ρ = k if k = k. Let us summrize results. In Tle II, we write the seller s revenue s r S,r M,r D, the user net enefits s n S,n M,n D nd the men ndwidths sold s m S,m M,m D. These results suggest tht mied contrcts re lwys preferle to sttic contrcts, in the sense tht r M >r S, n M >n S nd m M m S. VI. PRICE COMPETITION AMONGST SUPPLIERS We now turn to investigte wht hppens when suppliers compete with one nother on price. Who wins? Is it the supplier of sttic, dynmic or mied contrcts? Is there n equilirium in which suppliers of different types cn oth mke positive profit? Let us suppose tht for given customer k =or k = k, with proilities p nd p. We suppose p, ] is distriuted cross the popultion of users with density function of f(p). As previously, the sttic provider sells t price, utthe contrct for purchse must e mde efore the customer knows whether his k is equl to or. The dynmic provider sells t price, nd with the fleiility tht the customer need not mke the purchse until he knows whether his k is equl to or. If uying from the sttic provider the customer chooses to mimize pu(,),

7 () () (c) Fig. 3. The rection curves () ginst (in lue), nd ginst () (in violet), for different distriutions of p. nd so optimlly uys m /p,. If uying from the dynmic provider the customer chooses to mimize pu(,) p, nd optimlly uys m,. Thus customer strictly prefers to uy from sttic rther thn dynmic if nd only if <p. Suppose the two providers hve unit cost of c nd c, respectively. Then the profits otined y the two providers re ( c f A (, ) = ) ( /p)f(p) dp, / <, ( c f B (, ) = ) / p(k )f(p) dp, < ( c ) p(k )f(p) dp, These re the pyoffs in Bertrnd gme of price competition. From these we cn compute the rection curves. () = rg m f A(, ), () = rg mf B (, ). To give some numericl emples, suppose c =. nd c =.. Fig. 3() shows the rection curves when p is uniformly distriuted, i.e., f(p) =. Fig. 3() shows these curves when f(p) =6p( p). Fig. 3(c) shows these curves when the distriution is more concentrted round p =/, with f(p) = 63p 4 ( p) 4. The point of intersection in ech grph is Nsh equilirium. In the equilirium point of Fig. 3(), =.5, =.3333 nd the respective revenues re.3 nd.77. As lredy mentioned in Section IV, t the equilirium point the mrket will e segmented etween the two providers. The customers tht reside in the edges of the p s distriution will prefer the sttic provider nd the rest customers will prefer the dynmic provider. VII. CONCLUSIONS &FUTURE WORK We hve presented model for nlyzing the different types of contrct in mrket of resources tht cn e provisioned on demnd. Our im ws to show tht, when using the pproprite pricing scheme, the provider s gol of revenue mimiztion is not opposed to the consumer s gol for net enefit mimiztion. We hve cptured the fluctuting demnd of the customers with single prmeter (p), leding to formultion of high nd low demnd periods. Bsed on this model, we hve studied the chrcteristics of such contrcts, under specific distriutions of k. We hve seen how the provider s revenue, the customer s net enefit nd the men ndwidth sold vry from one type of contrct to nother nd we hve provided the conditions under with mied contrct is etter from the rest for oth the provider nd the consumer. We hve lso proved tht the optiml revenue of sttic provider is the sme with the revenue chieved y dynmic provider, under ny distriution of k. Finlly, we hve provided some numericl results for price competition gme etween sttic nd dynmic provider tht shows tht Nsh equilirium point eists nd tht the mrket t this point is segmented, hence oth providers mke profits. Mny issues re open for further reserch. An interesting point is to see wht segment of the mrket is otined y mied provider when prticipting in price competition gme with sttic or dynmic provider. Furthermore, prt from the emple with the two-point distriution of k, results from more generic distriutions will e studied. The cse of non-identicl customers needs lso to e studied. Etensions with effective ndwidth will lso e considered. An importnt question tht my rise is how the shpe of the utility function ffects ll the forementioned results. Finlly, n open issue is how such model cn e etended for Grid resources, since the definition of the sic resource in Grids nd its chrcteristics re not yet well-defined. REFERENCES ] DSL Forum s BrodndSuite: Unleshing the Power of Brodnd, Technicl Pper, DSLForum.org. ] Creting Commercilly Sustinle Frmework for IP Services Relizing Net Genertion Revenues, White pper, IPSphere Forum, My 6. 3] GRID COMPUTING on SUN: Deploying Sclle nd Efficient HPC Grids in the Modern Commercil or Technicl Enterprise, White Pper, SUN Microsystems Inc., Mrch 6. 4] B.M. Mitchell nd I. Vogelsng, Telecommunictions Pricing Theory nd Prctice, Cmridge University Press, 99. 5] T. Bergstrom nd J.K. McKie-Mson, Some Simple Anlytics of Pel- Lod Pricing, The RAND Journl of Economics, vol. (), pp. 4-49, 99. 6] W.Y. Oi, A Disneylnd Dilemm: Two-prt Triffs for Mickey Mouse Monopoly, Qurterly Journl of Economics, vol. 85, pp , Fe. 97 7] F.P. Kelly, Notes on effective ndwidth, In: F. Kelly, S. Zchry nd I. Ziedins (Eds.), Stochstic Networks: Theory nd Applictions in Telecommuniction Networks, Volume 4 of Royl Sttisticl Society Lecture Notes Series, Oford, pp. 4-68, Oford University Press, 996.