Pricing under Constraints in Access Networks: Revenue Maximization and Congestion Management

Transcription

1 Pricing under Consrains in Access Neworks: Revenue Maximizaion and Congesion Managemen Prashanh Hande 1,2, Mung Chiang 1, Rober Calderbank 1, Junshan Zhang 3 1 Deparmen o Elecrical Engineering, Princeon Universiy, NJ 8544, USA 2 Qualcomm Inc, NJ 887, USA 3 Arizona Sae Universiy, AZ 85287, USA Absrac This paper invesigaes pricing o Inerne conneciviy services in he conex o a monopoly ISP selling broadband access o consumers. We irs sudy he opimal combinaion o la-rae and usage-based access price componens or maximizaion o ISP revenue, subjec o a capaciy consrain on he daarae demand. Nex, we consider ime-varying consumer uiliies or broadband daa raes ha can resul in uneven demand or daa-rae over ime. Pracical consideraions limi he viabiliy o alering prices over ime o smoohen ou he demanded daarae. Despie such consrains on pricing, our analysis reveals ha he ISP can reain he revenue by seing a low usage ee and dropping packes o consumer demanded daa ha exceed capaciy. Regulaory aenion on ISP congesion managemen discourages such echnical pracices and promoes economics based approaches. We characerize he loss in ISP revenue rom an economics based approach. Regulaory requiremens urher impose limiaions on price discriminaion across consumers, and we derive he revenue loss o he ISP rom such resricions. We hen develop parial recovery o revenue loss hrough non-linear pricing ha does no explicily discriminae across consumers. While deerminaion o he access price is ulimaely based on addiional consideraions beyond he scope o his paper, he analysis here can serve as a benchmark o srucure access price in broadband access neworks. I. OVERVIEW This paper sudies he impac o access prices on he congesion managemen pracices and revenue o a monopoly ISP, operaing a single boleneck link wih ixed capaciy. Alhough Inerne daa lows along muliple links on a roue beween source and desinaion, he end-user access link is ypically he mos consrained or capaciy, and he major conribuor o he conneciviy price. Consumer daa rae allocaion can be deermined by socially opimal prices in a compeiive marke on he one hand, or he revenue maximizing prices in a monopoly ISP marke on he oher hand. The analysis in his paper explores he laer and represens a benchmark: he mos avorable oucome o he ISP and possibly he leas avorable oucome o he consumers. The analysis will serve as a basis o undersand he pricing o shared access links in wireline and wireless broadband. The analysis has paricular signiicance o pricing wireless broadband as consumers demand high-rae wireline-like applicaions and conen over relaively lower-rae wireless broadband. Access pricing is ypically in he orm o a la rae ha is independen o usage, or a usage based price, or some combinaion o he wo pricing schemes [1], [2]. We quaniy ha a signiican componen o he monopoly ISP revenue is rom la price i consumer price sensiiviy is low and hrough usage price i consumer price sensiiviy is high. Fla pricing is generally considered as he preerred choice o consumers [3], bu our analysis indicaes ha la pricing can lead o a signiican loss o consumer ne-uiliy, paricularly when he consumers have low price sensiiviy. Consumer demand or daa changes over hours o he day and days o he week, resuling in peak usage o neworks ha can be signiicanly high compared o average usage. Access ISPs ace a mismach beween heir revenue rom average usage and cos incurred rom peak usage o neworks. Consideraions on billing managemen and price simpliciy discourage requen changes in prices over ime [3]. This limiaion on ISP s abiliy o manage peak aggregae demand hrough price variaions can resul in poenial loss o revenue. Our analysis reveals ha, despie he lack o lexibiliy o aler he ime-dependen consumpion o consumers hrough price variaions, he ISP can reain he revenue hrough congesion managemen by dropping packes o consumer demanded daa ha exceed available capaciy. We quaniy he inuiion ha he revenue reenion can be achieved hrough a combinaion o low usage ee ha ensures suicien consumer demand a all imes, and ahigh la ee ha capures he remaining consumer ne uiliy rom he served daa raes. However, ISPs ace regulaory hurdles [4], [5], including nework neuraliy concerns [6], ha do no encourage congesion managemen by selecively dropping packes o consumer demanded daa. The recen FCC noice [4] proposes dra language o codiy a principle ha would require a broadband ISP o rea lawul conen, applicaions, and services in a nondiscriminaory manner. In a relaed decision, he CRTC recognizes [5] ha economic pracices are he mos ransparen Inerne raic managemen pracices. I urher noes ha such economics based congesion managemen pracices mach consumer usage wih willingness o pay, hus puing users in conrol and allowing marke orces o work. This suggess an alernaive approach or he ISP: se suicienly high access prices ha ensure ha he peak aggregae demand does no exceed he available capaciy. In his case, he lack o lexibiliy o aler consumer daa rae consumpion hrough price variaions resuls in a revenue loss. We show ha he revenue loss is due o he inabiliy o he ISP o adap he usage price o capaciy usage and quaniy he dependency o he revenue loss on consumer price sensiiviy by showing ha he loss is higher i he consumers are highly price sensiive. We sudy he pricing o a less han bes-eor

2 2 conneciviy service ha he ISP can oer over he unused capaciy o miigae revenue loss. Regulaory aenion urher discourages ISPs rom pracicing price discriminaion across consumers. We show ha he revenue loss rom imposing uniorm pricing across lows is due o he inabiliy o he ISP o charge he consumerdependen la price o capure he consumer ne-uiliy. Furher, we quaniy he dependency o he revenue loss on consumer price sensiiviy and show ha he loss is higher i consumers are less price sensiive. The revenue loss can be miigaed i he ISP can oer prices non-linear in daa-volume or daarae consumpion, wih discoun on higher daa-rae demand. The choice o he daa consumpion is le o he consumer, hus prevening explici price discriminaion among consumers. This problem is sudied as second-degree price discriminaion in economics. We use relaed mehods o quaniy he loss recovery hrough non-linear pricing and show ha he recovery is less when consumer price sensiiviy is low. The pricing analyses in he paper are applicable o real scenarios in broadband pricing, and we lis below wo examples: Example 1: An ISP currenly oers ixed monhly prices or broadband. Wih increasing broadband demand, he ISP noices raic peaks during evenings in residenial areas, and during mid-day in commercial locaions. One opion or he ISP o address he likely deerioraion in cusomer experience is o spread he demand more evenly by charging less during nonpeak imes. The ISP can work ou he addiional cos o billing and logisics bu is unsure how o price he non-peak raic and i he revenue gain can ose he added billing coss. A second opion is or he ISP o suppor, during non-peak hours, bulk raic ha can olerae large delays. The revenue gain rom his bulk-raic may be used or capaciy expansion o alleviae he cusomer experience problems during peak imes. Example 2: An ISP oers broadband services a a uniorm price o all cusomers bu noices signiican variaions in cusomer valuaion o heir services. One cusomer, a smallbusiness owner, relies on her broadband o conduc business, and is willing o pay more o reain he ISP services. A dieren cusomer is an occasional user o broadband, and does no consider he service worh he curren price. One opion or he ISP is o explicily price he wo cusomers dierenly, bu realizes ha such discriminaory pricing could draw regulaory aenion. Anoher opion or he ISP is o implicily price he wo cusomers dierenly by oering non-linear pricing, wih he cusomers choosing he consumed quaniy. A number o access nework pricing models have been proposed, and an overview o he various models can be ound in [7]. In economics, he wo-par pricing model in [8] and he lieraure on club goods [9] have sudied la ee membership ee and usage based ee exensively. The work in [1] draws upon his lieraure and sudies congesion pricing in neworks. I is shown ha an ISP wih marke power e.g.monopoly ISP can exrac all o he ne-uiliy rom he consumers hrough a combinaion o la and usage based pricing, resuling in daa-rae allocaion and overall welare similar o he case o a compeiive ISP marke he social opimum allocaion. We derive an equivalen resul or our model ha serves as he basis or urher analysis o resricions on he access prices. Revenue maximizaion is considered under a combinaion o access price and congesion price wih ixed uiliy uncions in [1] and he ISP incenive o increase capaciy is sudied. We analyze congesion managemen hrough access price under general uiliy uncions or ixed capaciy. Time-dependen pricing is considered in [11], where he auhors propose a model wih ime-dependen consumer uiliies and analyze he revenue loss o he monopoly ISP due o insuicien inormaion abou he consumer uiliies. We assume complee inormaion abou consumer uiliies and analyze he revenue loss o he ISP due o consrains on ime variaions o prices. The res o he paper is organized as ollows. Secion II presens he seup or he revenue maximizaion problems, based on which we sudy in Secion III he opimal choice o la and usage based componens in access pricing. Secion IV imposes consrains on price variaions in ime and quaniies he resuling revenue loss. Secion V derives he revenue loss rom imposiion o he requiremen o uniorm pricing across lows and develops recovery o loss hrough non-linear pricing. We conclude wih secion VI. II. SYSTEM SETUP AND BASIC NOTATION The congesion consrain aced by an ISP is on he peak aggregae consumer daa-rae. In conras, he reail access price is based on he volume o daa measured in megabyes or MB, consumed over a speciied ime period T ypically a monh. The pricing on daa volume is equivalen o pricing on average daa rae over ime T. Thereore, ISPs ace a mismach, where revenue is accrued on average daa rae bu congesion cos is incurred on he peak daa rae. To address his dispariy, we noe ha he dierence beween peak daa rae and he average daa rae is reduced when measured over smaller ime periods. Consider a uni ime inerval ha is suicienly small so ha he peak daa rae demand o a consumer in ha ime inerval is a close approximaion o he average daa rae or ha consumer in ha inerval 1. Le F be a consumer daa low wih F denoing he se o all lows. Le daa rae or low in he inerval [ 1,] be given by x. The daa volume consumpion over ime T is hen given by x = T =1 x, and he capaciy consrain applies a every ime insan : x C, A qualiaive observaion is ha he shape o he uiliy uncion depends on he response o he conen or applicaion o varying daa-raes, and he uiliy level represens he consumer s need or he applicaion or conen. This moivaes us o assume ha he consumer s uiliy level varies in ime, bu he shape o he uiliy uncion does no. The observaion will have o be veriied hrough analysis o real daa. Le σ u x be he uiliy o a consumer associaed wih low a ime insan 1 In pracice, measuring daa usage in very shor ime inervals migh no be amenable due o high implemenaion coss. I suices o use an inerval over which he peak usage can be esimaed as a linearly scaled value o average usage wih high conidence.

3 3, wih acor σ denoing he ime dependency o consumer s uiliy level. We assume ha he revenue maximizing ISP, hrough nework measuremens, has complee knowledge o he eiher he consumer uiliy uncion parameers or he probabilisic disribuion o he parameers. A ypical ISP pracice is o collec ime snap-shos o aggregae daa-rae demand, which moivaes us o assume a deerminisic model o ime variaions in he uiliy level in secion IV. Measuring low-dependen parameers is more diicul, and we assume a probabilisic model o he uiliy level variaion across consumers in secion V. We noe ha he analysis in secion IV can be easily exended o a probabilisic model o he ime variaions o uiliy levels. Faced wih ime-varying consumer uiliies, he ISP can charge a ime-dependen price or conneciviy r x as a uncion o he allocaed daa rae x. Consumers he ne uiliy or each low : σ u x r x variable x 1 The consumer demand uncion is he daa rae ha s he ne-uiliy in 1. One orm o he price wih linear increase in daa-rae is given by: r x =g + h x. The la price g is ixed or he duraion o he ime inerval, irrespecive o he allocaed daa rae. The usage based componen involves a price h per uni daa consumpion. The demand uncion or his orm o he price can be shown o be given by: { y g,h u 1 = h /σ i g σ u y h y, oherwise 2 The condiion σ u x g h x ensures ha consumers have non-negaive uiliy. To simpliy noaion, we oen use y h =u 1 h /σ wih he implici assumpion ha he la price is low enough o ensure non-negaive consumer ne-uiliy. We do no consider an explici penaly o consumer uiliy, as considered in [1], [11], rom congesion due o aggregae daa-rae demand. However, we noe ha he such a penaly can be easily incorporaed ino he consumer uiliy uncion, in which case y g,h is he bes-response updae a a Nash-equilibrium o consumer daa-rae demands. The elasiciy o demand η is a sandard measure o he sensiiviy o he consumer demand o price lucuaions [2], and is deined as η = y h h y 3 Oen, we specialize he uiliy uncion o he sandard alphaair uiliy orm: { logx i α =1, u x = 1 α 1 x 1 α 4 i α < 1 or which we have η =1/α : he price sensiiviy is inversely proporional o he parameer α and independen o uiliy level σ. III. BASIC PRICE STRUCTURES We irs analyze he srucure o he price r x =g + h x ha allows he ISP o manage congesion while maximizing revenue. A. Price Srucure The revenue maximizaion problem or he monopoly ISP can be deined by he ollowing: g + h x subjec o x C, x u 1 h /σ 5 σ u x g h x, variables {g,h,x } Obviously, he revenue increases wih higher la ee componen g, which can be se so ha he consumer ne uiliy is zero. The revenue rom each low is hen g + h x = σ u x, which can be realized by any combinaion o la and usage ee ha can suppor a daa-rae o x. I he usage ee h is such ha he consumer demand y h is sricly greaer han he ISP provisioned daa rae x, hen low daa packes have o be dropped. However, he ISP can avoid packe drops by seing a suicienly high usage price o reduce he consumer demand so ha he aggregae demand is wihin he available capaciy. I ollows ha x = y h and he ISP revenue maximizaion problem 5 can be re-wrien as: σ u y h subjec o y h C, 6 variables {h } Le he d revenue in problems 5 and 6 rom unresriced pricing be Ru, which will be conrased wih d revenue under resricions on pricing in laer secions. Problem 6 can be easily decomposed ino sub-problems a each ime insan, and has a soluion given by he ollowing Theorem 1: An opimal pricing scheme ha achieves he maximum in 5 is given by seing or each : h = μ x x g = u 1 μ /σ = C = σ u x μ x The proo ollows direcly rom he observaion ha he opimal price srucure in 7 represens he KKT condiions [21] or he decomposed sub-problems o he opimizaion problem in 6. The opimal usage ee h is he ime dependen congesion price μ, which is he same across all lows, and he opimal la ee g = σ u x μx is low dependen, allowing he ISP o ully exrac he consumer ne-uiliy. Le R be he revenue rom la componen o he price and Rs he revenue rom usage componen so ha Ru = R + R s. I can be shown ha R, Rs = σ u x, σ u x x 1 8 In an exemplary special case, we have he ollowing resul, whose proo is sraighorward and is omied. Theorem 2: I uiliy uncions are alpha-air 4 wih α = α or all, he raio o la revenue o usage dependen revenue is given by R = α 9 1 α R s 7

4 4 Remarks: Theorem 2 reveals ha usage dependen revenue dominaes wih linear uiliies α, while revenue rom la rae componen dominaes wih log uiliies α 1. I is apparen ha he la price is a signiican componen in he exracion o consumer ne-uiliy, i he consumer price sensiiviy is low. R, can be wrien as: g + h x subjec o x C, x u 1 h /σ σ u x g h x variables {g,h,x } 1 B. An Illusraive Example Consider a monopolis ISP providing conneciviy service o 1 lows over an access link o capaciy C = 1Mbps. Consumers associaed wih each low have uiliies o he orm given in 4 wih α = α, and η =1/α he common elasiciy o demand across all lows. We generae he uiliy levels {σ } as random variables uniormly disribued in [,σ 1 ]. Revenue $ R * /Rs * Toal Fla Usage Inverse elasiciy: α Inverse elasiciy: α Fig. 1. Comparison o revenue rom la pricing o usage based pricing a dieren consumer demand elasiciy. Figure 1 illusraes he average revenue per uni ime per uni low generaed by he monopolis ISP wih no consrains on he prices. The la componen o he revenue, which enables he monopolis o compleely exrac he consumer ne-uiliy, increases wih decreasing elasiciy o demand. The usage componen o he revenue decreases wih decreasing elasiciy o demand. The lower par o Figure 1 plos he raio o he la componen o he usage componen o he revenue as given by 9, demonsraing he increased reliance on revenue rom la price a low consumer demand elasiciy. The ISP pricing lexibiliy, in pracice, is resriced along ime and across lows. In he ollowing secions, we exend our model o quaniy he revenue loss o he ISP and develop pricing schemes o miigae he loss. IV. TIME CONSTRAINED PRICING As discussed in Secion I, he common pracice oday is or he ISP o price he oal daa volume, or equivalenly, he average daa rae over he longer ime horizon T. We model his by resricing ime variaions in r x =g + h x and allow a single price r = g + h x per low across all imes [1,,T], wih x = x. The ime-consrained ISP revenue maximizaion problem, whose opimum we denoe by The lack o lexibiliy in varying he price over ime can poenially reduce he ISP revenue. One approach or he ISP is o se he ime-independen usage ee h low enough o ensure ha he aggregae consumer demand y h is higher han he available capaciy C. The ISP can hen choose o serve daa raes x y h ha s he revenue. Consider he opimal usage prices in 7 or each ime insan ha was obained as a soluion o he revenue maximizaion problem in 5 wihou ime consrains. The soluion o he ime-consrained pricing problem 1 can be obained by irs seing he usage price o he minimum across ime o he opimal usage prices as given in 7: h = min{h }.Lehe daa raes served be given by x = x, he opimal daa raes served in 7 so ha he capaciy consrain is saisied. I he ime-independen la ee g can be se such ha g = σ u x min{h } x, he ISP can generae a revenue o σ u x, which is he same as in he case wihou ime consrains. This implies ha R = Ru, wih he ime-consrain on pricing no impacing he ISP s revenue. Noe ha he ISP could reain he revenue even wih imeconsrain on pricing by ensuring ha consumer demands exceed capaciy a all imes. The downside is ha ISP has o drop consumer daa packes o limi he daa raes o wihin he capaciy. The conrol exered by he ISP in packe-drop decisions wih his approach has araced regulaory aenion [4] and is increasingly ineasible. A. Time Consrained Pricing wihou Packe Dropping Alernaively, he ISP can adop economics based congesion managemen pracices ha are avored by some regulaors [5]. One such approach is o se he usage price o h = max{h }, so ha he wors case demand is wihin he capaciy consrain. This ensures ha he consumer aggregae demand is always wihin he capaciy limi, hus avoiding he need or an explici packe-drop decision by he ISP. However, capaciy is no ully uilized a non-peak imes, incurring a revenue loss. To characerize he revenue loss, consider he revenue maximizaion problem under his approach, wih he corresponding opimal revenue denoed by Rp: g + h y h subjec o y h C, σ u x g h x 11 variables {g,h } Inuiively, i uiliy levels {σ } are well spread, hen he revenue generaed by he ISP will be close o ha o an ISP wih no resricion as in 5. A more likely paern o daa

5 5 usage is one where muliple lows overlap in ime. The pricing hen will be such ha he daa usage is wihin capaciy limis during hose imes, and he capaciy is wased during oher imes, resuling in a loss o revenue compared o he case wih no resricion. To evaluae he eec o he spread in uiliy levels {σ }, consider low-independen {σ }.Le = max{σ } be he maximum uiliy level or any low over ime. In an imporan special case, he raio o he opimal revenue Rp o he revenue Ru wihou ime-consrained pricing is given by he ollowing heorem proved in Appendix A. Theorem 3: Assume uiliy uncions are alpha-air 4 wih α = α or all and low-independen uiliy levels {σ }.The raio o opimal revenue Rp rom problem 11 o he revenue Ru rom problem 5 is given by R p R u = σ n 1/α T σm 1/α σ σ 12 Remarks: I is easy o see ha Rp/R u 1. The revenue ineiciency sems rom he inabiliy o he ISP o charge he ime-dependen usage ee o ensure ull uilizaion o he capaciy. The ineiciency is higher wih lower values o α, which is equivalen o high consumer price sensiiviy. This is consisen wih he observaion ha a larger racion o he revenue is dependen on he usage ee wih lower α, asshown in Theorem 2. We can develop a lower bound in erms o he minimum uiliy level σ n = min{σ } by noing ha R σ 1/α p R = σm σn σm 1/α u σ T σ 13 n σ 1/α I σ can be modeled as realizaions o a random variable, hen we can invoke Jensen s inequaliy o speciy a igher bound. R p R u = 1 1/α Eσ1/α Eσ 1 1/α Eσ1/α Eσ 1/α 1 Eσ 14 The bounds indicae ha he revenue ineiciency is limied i he spread o uiliy levels is small. However, revenue can be highly ineicien i uiliy levels have a large spread, which is equivalen o consumers having similar and srong ime preerence or daa consumpion. B. Pricing Scavenger Class or Revenue Loss Miigaion When pricing is ime-consrained and he packe-dropping approach is ineasible, i was demonsraed in Secion IV-A ha he capaciy is underuilized during uncongesed ime. The underuilized capaciy can be used o serve a class o raic called scavenger class [12], [13], which is under-prioriized compared o even he bes-eor raic. The scavenger raic ges lile delay assurance, and is served using he spare capaciy. We analyze he pricing o such raic as a way o miigae he impac o ime-consrain resricion on access price. Le S be a se o scavenger class lows served in addiion o he bes-eor lows F by he ISP. Le he scavenger lows have a uiliy σ S 1 β 1 z 1 β as a uncion o he daa-volume z allocaed o each low in he class. A naural comparison o revenue rom scavenger class R S is wih he addiional revenue Ru Rp ha he ISP would have earned rom bes-eor lows Fwih ime-lexible pricing, allowing ull uilizaion o capaciy a all imes. A suicienly high scavenger class low uiliy level σ S allows he ISP o regain he revenue loss due o ime-consrained pricing as shown by he ollowing heorem, proved in Appendix A. Theorem 4: Assume uiliy uncions associaed wih beseor lows are alpha-air 4 wih α = α or all F and low-independen uiliy levels {σ }. Assume uiliy uncions associaed wih scavenger lows are alpha-air 4 wih α = β or all S and low-independen uiliy level σ S. Then, revenue rom scavenger class R S is a leas as much as he revenue gain rom ime-lexible pricing Ru Rp i 1/α σ σ σ m σ S F α S β 1 β 1 α Cβ α C. An Illusraive Example Revenue Raio Revenue Raio σ 1/α 1 β 15 α=.25 α= Spread α=.5 α=.9 α= Inverse elasiciy: α Theoreical Simulaed Fig. 2. Comparison o revenue loss rom ime consrained pricing a dieren spread in uiliy levels and dieren consumer demand elasiciy. Consider he example in Secion III-B. The upper par o Figure 2 demonsraes he revenue loss rom ime-consrained pricing by ploing he revenue raio R p/r u agains he spread in uiliy levels, represened by σ 1. Uiliy levels were realized hrough a uniorm disribuion or 1 lows. Higher spread resuls in higher reducion in revenue, wih he loss sauraing a higher values o he spread. For σ wih a uniorm disribuion in [,σ 1 ],wehave Eσ 1/α = σ1+1/α 1 σ 1+1/α, Eσ = σ2 1 σ 2 1+1/α 2 A high values o σ 1, he revenue raio can be shown o saurae o R p R u = σ 1Eσ 1/α σ 1/α 1 Eσ 2 1+1/α

6 6 The revenue loss is ploed agains α in he lower par o Figure 2. Low values o α, indicaing high price sensiiviy, resuls in high losses. V. NON-DISCRIMINATORY PRICING ACROSS FLOWS In Secion IV, we considered resricion over ime on he prices r charged by he monopolis ISP. However, he ISP could discriminae across lows, pracicing irs-order price discriminaion [2] o exrac he maximum revenue. In pracice, he abiliy o a monopoly ISP o pracice such complee price discriminaion across lows is resriced. The possibiliy o regulaory scruiny and demand or nework neuraliy [6] prevens monopoly ISPs rom pracicing exensive price discriminaion. We invesigae he revenue loss rom such price resricion across lows and miigaion o he loss hrough nonlinear pricing. A. Revenue Loss rom Uniorm Pricing across Flows The revenue loss rom price resricion across lows is incurred due o variaions in he consumer uiliy uncions. We simpliy he model by removing he ime-dependency o he uiliy levels. As explained in secion II, we consider a probabilisic model o uiliy levels across consumers, governed by he densiy disribuion uncion σ deined over a range [,σ 1 ] wih he corresponding cumulaive disribuion uncion F σ = σ σ. Wih no price resricion, le he price rσ =gσ +hσxσ be he σ dependen price charged o a low wih uiliy level σ. The demand uncion o he consumer acing his price is given by yσ =u 1 hσ/σ. The revenue maximizaion problem or he ISP in his case is similar o 6, wih he ISP capuring he enire consumer neuiliy by seing he la ee gσ =σuyσ hσyσ. The revenue rom unresriced pricing, which we denoe by Π u,is hen deermined by he appropriae usage ee, and is given by σuyσσ dσ subjec o σ 1 yσσ dσ C variables {hσ} 16 For alpha-air uiliy uncions 4, he demand uncion is given by yσ =σ/hσ 1/α 17 The ollowing heorem saes he soluion o problem 16 in his case. Theorem 5: Assuming uiliy uncions are alpha-air 4 wih α = α or he coninuum o lows, he soluion o problem 16 is given by h 1/α = Bσ,σ1 C B,σ 1 = σ 1 σ 1/α σ dσ Π u = 1 α 1 h 1 1/α B,σ 1 =1 α 1 hc 18 The above resul can be readily shown: The opimal price srucure in 18 represens he KKT condiions [21] or he opimizaion problem in 16. Wih he ISP resriced o charge a uniorm price wih linear increase in usage, he la componen g and he usage rae h do no change wih he consumer uiliy level σ. Theσindependen usage and la rae pricing does no allow he ISP o capure he consumer ne-uiliy a every uiliy level, resuling in a revenue loss. The ne-uiliy o a consumer wih uiliy level σ given by σuyσ hyσ can be shown o be increasing in σ. Given he resricion ha he user ne-uiliy canno be negaive, he ISP has o se he σ-independen larae componen g o he ne-uiliy o he consumer wih he minimum uiliy level [,σ 1 ] ha he ISP is willing o serve. Consumers wih uiliy level σ< are no served by he ISP. The hreshold uiliy level is hen par o he revenue opimizaion problem, wih higher providing a higher laee componen rom all consumers wih uiliy levels higher han, bu also resuling in zero revenue rom all consumers wih uiliy level less han. The revenue maximizaion problem wih his price resricion, whose opimum we denoe by, is given by T g + hyσσ dσ subjec o σ 1 yσσ dσ C, 19 g = uy hy variables {,h} For alpha-air uiliy uncions o he orm in 4, he marginal uiliy a zero daa allocaion is ininiy, resuling in ininie penaly or no serving a low o non-zero uiliy level. Indeed, we show in Appendix B ha =, so ha he ISP is beer o serving all lows, and urher prove ha he soluion o 19 is given by he ollowing heorem. Theorem 6: Assuming alpha-air uiliy uncions o he orm in 4 wih α = α or he coninuum o lows, he revenue, rom uniorm price across lows wih linear increase in usage, is given by = hc 1+ 1/α α 1 αb,σ 1 h 1/α = Bσ,σ1 C B,σ 1 = σ 1 σ 1/α σ dσ 2 and he raio o revenue rom low-resriced pricing o revenue rom unresriced pricing Π u is given by =1 α 1 1/α 21 Π u B,σ 1 Remarks: Noice ha σ 1/α 1 B,σ 1 σ 1/α. Thereore, Π u wih equaliy when all consumers have he same uiliy level σ 1 =. The revenue ineiciency sems rom he inabiliy o he ISP o charge he low-dependen la ee o capure he ne-uiliy o each low. The ineiciency is higher wih higher values o α since a larger racion o he revenue is dependen on he la ee wih higher α, as shown in Theorem 2. The impac o he spread in uiliy levels on he revenue can be esimaed by noing he ollowing lower bound on he revenue ineiciency. 1 α 1 1/α 22 Π u σ 1/α 1 The lower bound can be inerpreed as he inverse compeiive raio o he resriced pricing scheme. The revenue ineiciency is low i he spread in uiliy levels is low. A large spread in

7 7 uiliy levels among lows can, however, resul in high revenue ineiciencies. B. Non-linear Pricing or Revenue Loss Miigaion The price resricion across lows has hus ar been modeled as a combinaion o la-ee and usage ee, resuling in a price linear in usage rσ =g + hyσ. The ISP can incur a higher revenue by oering a price ha is non-linear in usage wihou discriminaing on a per low basis. The nonlinear price ypically akes he orm o quaniy discouns where he price per uni daa rae is discouned or higher daa rae purchases. We sudy he revenue oucome o non-linear pricing in his secion by invoking resuls rom second-order price discriminaion in economics [2]. The non-linear price oering is in he orm o a package consising o he daa rae yσ and he price rσ oered o he consumers. The package has o be srucured in such a way ha he package {yσ,rσ} or a given σ, inended or a consumer wih uiliy level σ, should indeed be he mos desirable choice or he consumer. This requiremen is reerred o as he incenive compaibiliy and he requiremen can be expressed in erms o he ne-uiliy o consumers as σuyσ rσ σuy σ r σ, σ [,σ 1 ] 23 In addiion, he package has o be srucured so ha he consumers derive non-negaive ne-uiliy rom he package, a requiremen ha is reerred o as he individual raionaliy, expressed in erms o ne-uiliy o consumers as σuyσ rσ, σ [,σ 1 ] 24 The revenue maximizaion problem or he ISP, whose maximum is denoed by Π n, can hen be wrien as rσσ dσ subjec o σ 1 yσσ dσ C σuyσ rσ σuy σ r σ, σ [,σ 1 ] σuyσ rσ, variables {yσ,rσ} σ [,σ 1 ] 25 In general, we expec Π u Π n. Consider a special case when consumer uiliy uncion is uniormly disribued in he range [, 1] so ha σ =1, Fσ =1 σ, =, σ 1 =1 26 For his special case, we can sae he ollowing resul, proved in Appendix C, on he revenue raios. Theorem 7: Assuming uiliy uncions are alpha-air 4 wih α = α or he coninuum o lows, and he uiliy levels disribued uniormly as in 26, we have =2 α 1 α, =1 α, Π n Π u C. An Illusraive Example Π n Π u =2 α 27 Consider he example in Secion III-B. The upper par o Figure 3 plos he revenue rom low-resriced pricing, unresriced pricing, and non-linear pricing. The lower par plos he revenue raio beween resriced and non-linear pricing, resriced and unresriced pricing, and non-linear and unresriced Revenue Revenue Raio Resriced Unresriced Non linear /Π n /Π u Π n /Π u Inverse elasiciy: α Inverse elasiciy: α Fig. 3. Comparison o he ISP revenue rom hree cases: unresriced, resriced and non-linear pricing. pricing. Non-linear pricing allows he ISP o recover losses rom resricions on price diereniaion beween lows, wih he loss no worse han 5%. VI. CONCLUSION The quesions o whom o price? and how o price? are wo relaed aspecs o conneciviy economics, ha has bearing on he expansion o broadband access or ubiquious availabiliy o Inerne conneciviy and preservaion o open Inerne. The irs quesion is addressed in previous works, including our recen one [19], in he conex o wo-sided pricing by invesigaing he appropriae price spli beween end-users and conen or applicaion providers. This paper addressed he second quesion in he conex o a monopoly ISP wih complee pricing power, charging one-sided access price o consumers o conneciviy ha can be eiher he endusers or he conen providers. The work can be exended or urher analysis o beneis disribuion under marke scenarios where ISPs have less pricing power, consumers are price anicipaing [15], or ISPs have insuicien inormaion on consumer uiliies [11]. Furher analysis is required o incorporae iner-isp ineracions [16] when consumer daa packes are roued hrough muliple boleneck links operaed by dieren ISPs. A dieren line o work is he analysis o access price when he ISP oers iered conneciviy services wih qualiy o service diereniaion across iers. Relaed works have analyzed he need or service iering [2] or analyze he revenue gain rom a speciic orm o service iering [17], [18]. A opic o uure ineres is he opimizaion o key parameers o service diereniaion joinly, wih pricing. Alhough he analysis in his paper is applicable o wireless neworks wih he ixed capaciy represening he average shared daa-rae, urher reinemen is required o analyze he eec o inererence and wireless channel variaions across consumers and over ime. A relaed line o work is he analysis o pricing schemes or secondary specrum access hrough cogniive radio. Beyond revenue maximizaion, access prices are ulimaely deermined by addiional acors, including he need o recover

8 8 he subsanial capial cos o provisioning he access link. A clearer picure on access pricing will emerge wih urher analysis ha akes he ISP cos model ino accoun. APPENDIX A: PROOFS OF THEOREMS 3 AND 4 The irs-order KKT condiions or problem 11 resul in h = μ x h = x h μ η x η x 28 where μ is he dual congesion price associaed wih he capaciy consrain a each insan and η is he elasiciy o demand 3. Wih uiliy uncions specialized o he orm in 4, we have h = μ σ 1/α σ 1/α 29 The complemenary slackness condiion implies ha μ is such ha μ = i σ /h 1/α <C μ > i σ /h 1/α 3 = C Le T be he subse o imes wih peak aggregae usage, so ha we have σ /h 1/α = C, T.Iollowsha h = T μ σ 1/α σ 1/α To evaluae he eec o he ime acors σ, consider lowindependen ime acors σ, and low independen uiliy uncion o he orm in 4 parameerized by α. Flows are idenical so ha he capaciy share is idenical. Wih = max{σ } he maximum uiliy level or any low over ime, we have σ =, T. The ime-consrained price h will have o be se so as o limi he daa raes a hese imes o be wihin capaciy, resuling in h = C/ F α 31 where F is he number o lows sharing he capaciy. The daa rae or each low a any ime is hen given by σ x 1/α = C/ F 32 Wih no ime-consrain on pricing, he usage price a each ime insan h can be se so ha he daa rae or each low is x =C/ F 33 The raio o revenue Rp o he revenue Ru wihou imeconsrained pricing is hen given by = σ ux σ x 1 α R p R u = = yielding he resul in 12. σ ux = σ x 1 α σ σ σm 1/α1 αc/ F 1 α σ C/ F 1 α σ σ 1/α1 α σ 34 For proo o Theorem 4, noice ha daa rae allocaion or bes-eor raic lows is given by 32. Wih he spare capaciy rom ime-consrained price equally allocaed across scavenger class raic, he capaciy share o each raic low is given by z = C x S = C S σ 1/α 1 35 A combinaion o la and usage ee allows he monopoly ISP o exrac all he ne-uiliy rom he scavenger class users, generaing a revenue R S o 1 β S σ S 1 β 1 C σ 1/α 1 β 1 S 36 The revenue Ru Rp is given by 1 α F 1 α 1 C F rom which he resul in 15 ollows. σ APPENDIX B: PROOF OF THEOREM 6 Consider he Lagrangian or he problem 19 Lh,,μ= σ 1 +μc σ 1 yσσ dσ σ 1/α 37 uy hy +hyσ σ dσ = σ 1 1 α 1 h 1/α 1 h σth 1/α + h σ h +μ C σ 1 σ 1/ασ h dσ 1/α σ dσ The usage price h should be such ha L h =which ranslaes o μh 1 1+α σ 1/α σ dσ = ασ 1/α T 1 F 38 Since σ 1,wehaveμh 1 1+α α or h μ. L The hreshold uiliy level should be such ha. We have L = 1 α 1 h 1/α 1 + μ σth 1/α 39 which resuls in he requiremen ha h μ1 α 4 For < α < 1, his conradics he requiremen ha he opimum usage ee be no less han he congesion price: h μ. Tha is, here is no benei o seing he hreshold uiliy level o any value higher han, since he loss o revenue rom no serving a consumer o uiliy level is more han he gain in revenue rom disribuing he resuling spare capaciy o higher uiliy level consumers. Seing =, he revenue rom price resricion across lows,, is given by 1 α 1 h 1/α 1 σ 1/α σ 1/α h + h σ dσ h h 41 wih he usage price h se o saisy he capaciy consrain. The resul in Theorem 6 ollows.

9 9 APPENDIX C: PROOF OF THEOREM 7 We irs invoke a resul on second-degree price discriminaion [14] rom economics. Lemma 1: Under cerain echnical condiions on consumer uiliy uncion vx, σ, he daa rae allocaion yσ under second-degree price discriminaion saisies vyσ,σ μ + 1 F σ 2 vyσ,σ = 42 x σ x σ where μ is he capaciy price. Furher, he orm o he price srucure rσ is given by σ vy, rσ =vyσ,σ d 43 σ For vx, σ = σux where ux is he alpha-air uiliy uncion as deined in 4 wih α = α, wehave vx,σ x = σx α 2 vyσ,σ 44 x σ = x α Plugging in 42, we ge yσ = μ 1 σ 1 F σ 1/α 45 σ where μ is he congesion price such ha yσ dσ = C 46 We see ha in comparison o demand in he linear pricing case 17, he demand in he non-linear pricing case is disored downward or every uiliy level σ excep he highes σ = σ 1, a which poin we have 1 F σ 1 =. The relaive share o capaciy has been adjused in avor o consumers wih higher uiliy levels. Consider he case when consumer uiliy uncion is uniormly disribued in he range [, 1] wih resuling disribuion uncions give by 26. In his case, we have { 1/α 2σ 1 yσ = μ i σ [.5, 1] 47 oherwise where μ =2C 1 + 1/α α 48 We see ha consumers wih uiliy level σ [,.5] are no being served by he ISP charging a non-linear access price. Plugging 47 ino 43, he price srucure rσ or σ [.5, 1] can be easily shown o be given by 2C 1 + 1/α1 α rσ = 2σ 1 1/α σ 1 α 2σ 1 α/2 49 The resuling revenue rom non-linear pricing is given by α α C Π n = 5 2C1 + α 1 α Plugging in parameers rom 26 o he assumed uniorm disribuion ino 2, we obain he revenue rom resriced pricing as α α = C 51 C1 + α Similarly, plugging parameers rom 26 o he assumed uniorm disribuion ino 18, we obain he revenue rom unresriced pricing as Π u = hereby concluding he proo. α C1 + α α C 1 α ACKNOWLEDGMENT 52 The auhors would like o acknowledge helpul discussions wih Sundeep Rangan a Qualcomm Inc. This work has been in par suppored by NSF CNS-9586 and CNS REFERENCES [1] J. K. Mackie-Mason and H. Varian, Pricing Congesible Nework Resources, IEEE Journal on Seleced Areas in Communicaion, vol. 13, no. 7, pp , [2] G. Kesidis, A. Das and G. de Veciana, On Fla-rae and Usage-based Pricing or Tiered Commodiy Inerne Services, Proc. CISS, March 28. [3] A. M. Odlyzko, Inerne pricing and he hisory o communicaions, Compuer Neworks vol. 36, pp , 21. [4] FCC, Open Inerne Noice o Proposed Rulemaking, hp://www. openinerne.gov/abou-he-nprm.hml, Ocober 29. [5] CRTC, Review o he Inerne raic managemen pracices o Inerne service providers, hp:// Ocober 29. [6] C. Yoo, Nework Neuraliy and he Economics o Congesion, Georgeown Law Journal, Vol. 94, June 26. [7] M. Falkner, M. Devesikiois, and I. Lambadaris, An overview o pricing conceps or broadband IP neworks, IEEE Communicaions Surveys, vol. 3, no. 2, 2. [8] W. Oi. A Disneyland dilemma: wo-par aris or a Mickey Mouse monopoly, Quarerly Journal o Economics, [9] S. Scochmer, Proi-maximizing clubs, Journal o Public Economics, 27, 2545, [1] T. Basar, R. Srikan, Revenue-Maximizing Pricing and Capaciy Expansion in a Many-Users Regime, Proc. o IEEE Inocom, 22. [11] L. Jiang, S. Parekh, J. Walrand, Time-dependen Nework Pricing and Bandwidh Trading, IEEE Inernaional Workshop on Bandwidh on Demand, 28. [12] Inerne 2, QBone Scavenger Service QBSS, hp://qbone.inerne2. edu/qbss/ [13] R. Bless, K. Wehrle, A Lower Than Bes-Eor Per-Hop Behavior, Inerne dra a hp://ools.ie.org/hml/dra-bless-diserv-lbe-phb-, Sepember [14] E. Maskin, J. Riley, Monopoly wih Incomplee Inormaion, RAND Journal o Economics, Vol , pp [15] R. Johari, J. N. Tsisiklis, Eiciency o Scalar-Parameerized Mechanisms, Operaions Research, Vol. 57, No. 4, pp , July-Augus 29. [16] R. Ma, D. M. Chiu, J. C. S. Lui, V. Misra, D. Rubensein, Inerconnecing Eyeballs o Conen: A Shapley Value Perspecive on ISP Peering and Selemen, Proc. ACM NeEcon, 28. [17] D. Acemoglu, A. Ozdaglar, and R. Srikan, The marginal user principle or resource allocaion in wireless neworks, Proc. 43rd IEEE Conerence on Decision and Conrol December 24. [18] S. Shakkoai, R. Srikan, A. Ozdaglar and D. Acemoglu, The Price o Simpliciy, IEEE Journal on Seleced Areas in Communicaion, Vol. 26, No. 7, Sepember 28. [19] P. Hande, M. Chiang, R. Calderbank, S. Rangan, Nework Pricing and Rae Allocaion wih Conen Provider Paricipaion, Proc. o IEEE Inocom, April 29. [2] A. Mas-Colell, M. D. Whinson, and J. R. Green, Microeconomic Theory, Oxord Universiy Press, Oxord, Unied Kingdom, [21] S. Boyd and L. Vandenberghe, Convex Opimizaion, Cambridge Universiy Press, 24.