An Economic Analysis of Interconnection Arrangements between Internet Backbone Providers

Transcription

1 An Economc Analyss of Interconnecton Arrangements between Internet Backbone Provders Yong Tan Unversty of Washngton Busness School, Box 353, Seattle, Washngton I. Robert Chang Unversty of Connectcut, School of Busness, U-4IM, Storrs, Connectcut Vay S. Mookeree Unversty of Texas at Dallas, School of Management, JO4.64, Rchardson, Texas Transt and eerng arrangements among Internet Backbone Provders IBPs are essental for the global delvery of communcaton servces on the Internet. In addton, to suort delay senstve alcatons e.g., streamng and multmeda alcatons t s mortant for IBPs to mantan hgh servce ualty even f the network s congested. One romsng aroach s to establsh nterconnecton agreements among rovders to dynamcally trade network caacty. To make such nterconnectons ossble n a comettve settng, we roose a rcng scheme that consders factors such as network utlzaton, lnk caacty, and the cost structure of the nterconnectng artcants. Our analyses show that the common SKA Sender Kees All mode of settlement does not rovde adeuate ncentves for collaboraton; rather, the rovder that delvers the ackets should be sutably comensated at an eulbrum rce. Two rce eulbra are dentfed: the frst favors slower IBPs whereas the other s congeston-based and can be more benefcal for faster IBPs. When cost asymmetres exst, the lower cost IBP needs to offer a rce dscount to nduce artcaton. We show that a usage-based, utlzaton-adusted nterconnecton agreement could algn the costs and revenues of the rovders whle allowng them to meet more strngent os reurements. Key words: Internet backbone, network connectvty, ueueng, ualty of servce, game theory, eulbrum strateges

2 . Introducton The Internet s a global, comlex, yet loosely organzed collecton of data networks runnng on the TCP/IP rotocol famly. A communcaton sesson on the Internet could traverse through network segments owned by several cometng servce rovders. Thus, collaboraton among rovders s needed for the end-to-end delvery of network servces. A network oerator can be of two categores: Internet Servce Provder ISP or Internet Backbone Provder IBP. ISPs offer retal network access for ndvduals and nsttutons, whle IBPs rovde hgh-seed, long haul communcaton lnks for ISPs. These categores can overla: an ISP may sometmes oerate wde area lnks and an IBP may serve retal clents; however, the overall structure of the Internet s generally consdered to be herarchcal n nature. Of the three ossble modes of nterconnecton among rovders between ISPs, between an ISP and IBP, and between IBPs, the focus of ths aer s on nterconnecton arrangements between IBPs. Snce the commercalzaton of the Internet, several network access onts NAPs and Metrooltan Access Exchanges MAEs have been set u for IBPs to exchange or eer traffc wth one another. At ublc exchanges, current nterconnecton arrangements are relatvely smle: redomnantly the orgnator of the traffc ays no fee to the recent; thus the name: Sender Kees All SKA. Free swang of traffc worked well when traffc flows were lght and nearly symmetrcal Freden 998; the low network utlzaton allowed rovders to accommodate eered traffc wth lttle ncremental congeston. Furthermore, traffc symmetry mled near-zero net volume between rovders, makng sohstcated network montorng and accountng for bllng uroses unnecessary. Wth the emergence of delay senstve servces such as streamng meda, VoIP, and Vrtual Prvate Network rovson however, barter arrangements such as SKA have become nadeuate, exosng eerng artcants to the oortunstc behavor of ther artners. For examle, source routng Steenstru 995 and shortest ext routng Jew and Ncholls 999, Cuker 998, Freden 998 allow offloadng atterns of network traffc that lead to subotmal network utlzaton, data loss, and delay. Another serous conseuence s that there s lttle ncentve for nfrastructure

3 exanson a carrer wllng to exand caacty has no way of reventng further exlotaton by others. To allevate or even solve ths roblem, ad eerng seems to a natural choce. In an attemt to recou ther nvestments, many establshed backbone rovders have oted to remove ther resence at ublc NAPs and nstead negotate nterconnecton arrangements at rvate exchange onts for examle, see Lambert 997 for Srnt s and UUnet s decsons. Whle the exact terms have not been made ublc, rvate nterconnecton agreements are currently based on several techncal and oeratng crtera, such as: network caacty, geograhc coverage, number and dserson of nterconnecton onts, volume of traffc exchanged, traffc flow symmetry, and network management caablty for examle, see stated crtera by WorldCom and Level 3. Blateral relatonshs at rvate exchanges range from settlement-free eerng agreements to transt contracts that have usage-based and congestonadusted comonents. The ueston of how nterconnecton can be establshed n an economcally otmal manner has attracted consderable nterest from academa, ndustry, and government agences Govannett, Huston 998, Jew and Ncholls 999, ACCC. However, whle eerng and transt agreements are wdely used among network rovders, there have been very few studes Crèmer et al., Foros and Hansen that address the otmal desgn of nterconnecton contracts at the Internet backbone level usng network engneerng arameters. As ndcated earler, an obvous beneft of nterconnectng backbone networks s that seral nterconnecton enables the global delvery of communcaton servces even when no sngle network s omnresent. Another beneft the focus of the resent study s that arallel nterconnecton that allows rovders to guarantee better ualty of servce os through dynamc route selecton among several ossble routes. The ntuton here s that nterconnecton could hel reduce network delays or acket loss by dvertng traffc to alternatve network aths when the rmary one s under heavy load. However, to acheve caacty sharng n a comettve market reures carefully desgned rcng mechansms, as smle barter agreements do not rovde adeuate ncentves for such collaboraton. 3

4 The maor contrbutons of ths research are:. We evaluate the erformance of three oeratng modes among IBPs: ndeendent, centrally managed, and nterconnected. The analyss demonstrates the extent of mrovement n network erformance that can be acheved through nterconnecton. We also show that, wthout adeuate comensaton, cometng IBPs do not share caacty wth one another.. We next roose a usage-based nterconnecton arrangement and derve the nterconnecton rce that an IBP should offer ts eerng artner for acket delvery. We also analyze the crcumstances under whch an IBP should agree to delver an nterconnected acket. Usng ueung and game theoretc analyses, we fnd that an IBP should agree to serve a artner s acket only f the ayment t receves covers or exceeds the cost of losng ts own acket whle the other network s acket s beng delvered. Two rce eulbra are dentfed. In the frst eulbrum, the recevng IBP extracts the entre surlus whch s n drect contrast to SKA. Ths eulbrum awards the slower IBP a larger roorton of the surlus generated from nterconnecton. The second rce eulbrum secfes a congeston-adusted rce that awards a greater share of the surlus to the IBP wth hgher network caacty. 3. We also examne how the above rce eulbra shft under cost asymmetry. An IBP that charges a lower rce to ts customers e.g., by offerng a lower os guarantee also needs to offer ts servces to the other IBP at a rce dscount so as to nduce artcaton. We derve a threshold dscount rate below whch a low-cost IBP may be better off wthout nterconnecton. The rest of ths study s organzed as follows. In Secton we summarze related work on the techncal, economc, and oltcal asects of Internet nterconnecton. In Secton 3 we aly ueung theory to assess the network os under dfferent nterconnecton decsons. Secton 4 ncororates the results from Secton 3 and dentfes the otmal rcng scheme and the nterconnecton decson. In Secton 5 we consder the effect of cost asymmetres among IBPs. Secton 6 dscusses research mlcatons and ossble extensons whle Secton 7 concludes and summarzes the aer. 4

5 . Related Work In ths secton, we summarze the lterature on Internet nterconnecton. We frst cover techncal advancements on Internet routng effcency and then revew work on Internet eerng and transt agreements from economc and oltcal ersectves... Techncal Asects Imrovng routng effcency n large-scale communcaton networks n general Ball et al. 995, Steenstru 995 and n the Internet n artcular Hutema 995, Halb s a well-studed area. The research on network routng ncludes algorthmc route selecton Halb and methods to exchange routng nformaton n the network Chnoy 993. Savage et al. 999 have dentfed several lmtatons n Internet routng and roosed nformed routng mechansms. Transmsson delays can also be reduced by shortenng the dstance of transmsson by cachng oular content at roxy servers and surrogate servers stuated at key locatons n network Kwan et al. 995, Hatlestad. However, wth cachng, synchronzng dstrbuted cache contents becomes an ssue Jarke et al., Chaudhur and Dayal 997 and ths aroach s less alcable f the content beng demanded changes radly. In general, techncal solutons alone are not suffcent to otmze erformance. In the absence of arorate ncentves, rovders may not select the otmal route to delver ackets. Hence, n addton to techncal consderatons, economc ncentves are needed to mrove erformance... Economc and Regulatory Consderatons Snce the Internet heavly reles on nterconnecton for the delvery of global communcatons, the economcs of network nterconnecton have receved consderable attenton Ferguson and Huston 998, Huston 999, ACCC, Srnagesh 996, Baley 997. Deendng on the settlement.e., ayment structure, network nterconnecton s tradtonally descrbed ether by a eerng agreement between ISPs or between IBPs, or by a transt agreement n whch an ISP s regarded as a customer of an IBP. Our nterest s to analyze nterconnecton between IBPs 5

6 Crèmer et al., Wess and Shn 4, wth a secal focus on how such nterconnecton affects the overall os of the network. There have been several studes that focus on the nterconnecton among ISPs Baake and Wchmann 999, Jew and Ncholls 999. These roosals nclude models that consder the value of the subscrton base as well as the externalty generated from nterconnecton. For examle, Baake and Wchmann 999 assume that two ISPs could engage n Cournot cometton by choosng the caacty of the nterconnecton lnk so as to maxmze revenue. Govannett found that nterconnecton creates surlus n stuatons where there s suffcent dfferentaton e.g., wth resect to the content hosted between the markets served by the dfferent ISPs. Foros and Hansen use game theory to derve the caacty of a shared connectng lnk at a rvate exchange for ISPs. The nterconnecton at the Internet backbone, on the other hand, has drawn less attenton. A notable exceton s the study by Crèmer et al. that rovdes an excellent summary on the techncal and economc structure of the Internet backbone. It also analyzes the ossblty for a domnant IBP to ntentonally degrade lnk ualty when connectng wth a targeted grou of smaller rovders. An mortant ssue that has not been addressed n the exstng lterature s the rcng of backbone nterconnecton. Whle there have been numerous studes on congestonbased network rcng for examle, see Guta et al. 999, Johar and Tstskls 4, the focus has been manly on rcng ssues between rovders and subscrbers rather than among backbone eers. Thus the results of many network rcng studes do not drectly aly here. Whle SKA Freden 998 s smle to mlement and facltates unversal backbone access hgh cost rovders are mlctly subsdzed, t ermts free rdng and s consdered a subotmal nterconnecton arrangement at the backbone level Laffont et al. 3. The nadeuacy of exstng nterconnecton mechansms s further exemlfed by the dffcultes faced by a former network rovder Exodus Communcatons now art of Cable & Wreless to gan backbone access Cook 998, InternetNews. From an nternatonal ersectve, a varety of roosals have called for governmental 6

7 nterventon Cuker 998, Freden 998, Roehrch and Armstrong, Cave and Mason wth the goal of ensurng oen, far, and ubutous nterconnecton. Foregn rovders, because they erceve to bear a dsroortonate share of the nterconnecton cost, have voced the need for regulatory acton aganst ther U.S. counterarts Roehrch and Armstrong, ACCC. Currently, Internet servces reman exemt from the regulatons mosed on the utlty and telecommuncaton sectors Kende. To the best of our knowledge, the current study s among the frst of ts knd that ontly nvestgates how oeratonal and economc factors e.g., network lnk seed, network congeston and nterconnecton rce could lay an mortant role n the successful establshment of backbone level network nterconnecton. Our am s to suggest a market mechansm that retans the dynamc asects of the Internet and reles on techncal and economc means rather than on regulatory nterventon. We formalze our analyss next. 3. Interconnecton wthout Prcng In ths secton, we show how nterconnecton decsons could affect the ndvdual and overall network os. Here, we use arallel ueues, a well-known area n ueung theory, to model the nterconnecton between cometng backbone rovders servng the same geograhcal regon. Parallel ueues have been extensvely studed, datng back to the 95 s Haght 958. Koengsberg 966 studed ockeyng n arallel ueues; a stuaton smlar to the nterconnecton roblem studed here. Consder two IBP s IBP- and IBP- wth demand arrval rates of and and lnk transmsson seeds.e. servce rates of and. We consder arallel M/M// ueues where both the arrval and dearture rocesses are Posson, and ths ales at the level of network nodes where the routng decsons are made. If nterconnected, a acket arrvng at a busy IBP s node can be redrected to the other IBP. Note, however, that the second IBP may declne to serve a redrected acket. In ths aer, we assume bufferless ueues. Under ths assumton there s no need to analyze the system for ueung delays; rather, the key metrc s the rate of acket loss. The 7

8 assumton of bufferless ueues can be ustfed as follows. Frst, the nterconnecton studed here arses manly when there are tme constrants on acket delvery. Wthout tme constrants, ackets can n be ndefntely stored n a buffer, and rovded that the servce rate s greater than the arrval rate, the stored ackets wll eventually be delvered. However, wth tme constrants mosed on delvery, buffer szes need to be ket small and can be reasonably aroxmated by bufferless ueues. Bufferless networks are common n otcal networks that use hot-otato routng Duato 996. We next analyze three ossble modes of oeraton between two arallel networks: ndeendent, centrally managed, and fee-ayng nterconnecton. 3. Two Indeendent IBPs In steady state, the robablty that there are ackets n IBP-k s system, k, s gven by k k, k k ρ k k k, k k where the suerscrt k, are ndces for the IBPs. The subscrt dects the status of the network: for an dle network lnk and for a busy one. For ths artcular settng, k s both the loss robablty of ackets and the level of network utlzaton for IBP-k. To comare wth the next two modes of oeraton, the average loss rato s n LR ρ ρ, where s the total arrval rate. Ths s a weghted average based on the fact that arrvals to each IBP follow a Posson rocess and that a fracton k / of all arrvals comes from the IBP-k s subscrber. 3. Centralzed IBPs We now show how centralzed oeraton can reduce the loss rato for both rovders. The ntuton for ths les n the fact that when a acket arrves at a busy network, the other network, f dle, can assume the delvery. The centralzed case extracts the maxmum beneft ossble from 8

9 nterconnecton. Denote the state robabltes for centralzed oeraton as c, where,,. The subscrts reresent system states for IBP- and IBP-. A zero value for a subscrt ndcates that the corresondng IBP s dle. A central admnstrator observes an aggregated arrval rate,, and assgns each arrval to a secfc IBP. The loss rato assocated wth centralzed oeraton s descrbed n the followng rooston. PROPOSITION. The loss rato for centralzed oeraton s where mn mn,. c mn, mn The roof of ths rooston s rovded n the Aendx, where the roofs for all the roostons, lemmas, and corollares can be found. When both IBPs are dle, t can be shown that a new arrval should be routed to the network wth the hgher servce rate. COROLLARY. The loss rato for centralzed oeraton s strctly less than that for ndeendent oeraton, n c < LR. The above corollary asserts that centralzed oeratons wll always lead to better overall os, as measured by the loss rato. However, ths does not mly that both IBPs wll be better off. In fact, a large rovder wth low network utlzaton wll be worse off wll exerence a hgher loss rato f t s co-managed wth a slower but buser one. Therefore, whle such sharng of caacty s desrable, t may not be a natural outcome n a comettve market. to, When the IBPs have the same utlzaton ρ, the loss rato n Prooston can be reduced c where S s the rato of the lnk seeds, namely, ρ ρ ρ S ρ S ρ, 3 ρ ρ S ρ S 9

10 S /. The loss rato n s tycally smaller than that for the ndeendent case shown n excet for ρ or, or S ; when they are eual. It s worth notng that the benefts from centralzed oeraton,.e., the dfference between and, decreases f both IBPs become buser hgher ρ. We also note that c / S ρ S, mlyng that the beneft ncreases.e. c decreases wth S but starts to dmnsh once the dervatve becomes ostve,.e., when S / ρ. The ntuton s that the slower rovder s contrbuton s margnalzed once S ncreases beyond the above threshold. 3.3 Interconnected IBPs Now consder a stuaton when the two IBPs oerate searately, but when an IBP s busy t offers to ay the other IBP to delver ts ackets. The recevng IBP can, n turn, decde to accet or reect such reuests. An dle IBP would choose to reect a routed acket and forgo the ayment f t s not suffcently comensated for the rsk of losng ts own ackets. Assume that each IBP accets a comettor s acket wth robablty k, k,. The state robabltes,,,,, are solved and exlctly exressed n the roof of the followng rooston. The rooston consders a stuaton when the two IBPs always agree to delver each others ackets. PROPOSITION. The loss rato for nterconnected IBPs, when,,, s. Corollary below comares the three modes of oeraton. COROLLARY. The followng neualty holds, c <. n LR It s clear that nterconnecton mroves the erformance of both IBPs, from the case where the two rovders oerate ndeendently. However, as exected, nterconnected erformance s strctly domnated n most cases by centralzed oeraton. To comare the loss rato, we restrct ourselves to the case where both IBPs have the same utlzaton. The loss rato s,

11 ρ ρs ρ S ρ. 3 3 ρ ρ ρ ρ S ρ S The beneft of nterconnecton over the ndeendent case s the dfference between 3 and. Ths dfference dmnshes when IBP- domnates because / S S. Comared wth centralzed oeraton, the loss rato n 3 s hgher than n, excet for ρ or, or, S or S, when they are dentcal. In Table, we show the sgn of frst dervatve of wth resect to the routng decson varables and. The exlct exressons for these dervatves can be found n the roof of Corollary 3. TABLE. Frst dervatves of state robabltes wth resect to routng decson varables As a functon of the nterconnecton decsons,.e.,,, the loss rato can be exressed as, L,, for IBP-; and smlarly, L,, for IBP-. COROLLARY 3. The loss rato for IBP- satsfes the followng neualtes, L / > ; and, L / <, where,. Intutvely seakng, accetng ackets from a eerng artner wth a hgher robablty ncreases the rsk that one s own ackets get droed. Ths suggests a trade-off between the extra revenue from delverng routed ackets and mantanng servce ualty for one s own customers. We wll analyze ths tradeoff further n the next secton. It s mortant to note that the ont,, s not an eulbrum. However, f ths routng strategy were somehow mandated, both rovders wll be able to ncrease os. Although

12 such enforcement s dffcult to vsualze n a commercal settng, t may be consdered as a welfare olcy choce for ublc exchanges. In the next secton, we dscuss how artcaton n eerng arrangement could occur.e., s an eulbrum n the resence of a rcng scheme. Wthout rcng, the IBPs wll lkely ot out of eerng a subotmal outcome smlar to the one that occurs n a rsoner s dlemma game. Ths outcome.e., rsoner s dlemma occurs under the usual assumtons of self nterested artcants. If the artcants consder global as oosed to myoc mlcatons of ther actons, then other sueror outcomes may be realzed. 4. Routng and Prcng Strateges In ths secton, we ontly analyze the two elements of the nterconnecton decson: the routng robabltes.e., and and the nterconnecton rces denoted by and. The temoral structure for the two-stage game s as follows. Frst, both IBPs smultaneously decde on the nterconnecton fees; second, they smultaneously choose ther nterconnecton strateges. As a standard rocedure, we use backward nducton to dentfy the subgame erfect Nash eulbra. 4. Routng Eulbrum Wthout loss of generalty, we assume that each IBP charges ts customers a unt rce for each acket. Also, the other IBP s charged a transfer fee of by IBP- IBP- for each routed acket. It s exected that. More recsely, we defne a set for transfer rces, The roft functon s, for IBP-, and smlarly,, {, R R :, } Ω. π, 4 π, 5 for IBP-. Each roft exresson above contans three terms; the frst s the roft from servng an IBP s own customers, the second s the roft from servng ackets comng from the other rovder, and the thrd the roft that comes from one s own customers that were servced by the other IBP. The obectve s to maxmze π by choosng.

13 PROPOSITION 3. IBP- s best routng resonse functon, to IBP- s decson, s,, f f < θ ; > θ, 6 where, θ ; 7 and s an arbtrary value between and f θ. The routng resonse functons are lotted n Fgure. In ths fgure, IBP- s resonse s lotted as a dashed curve and IBP- s resonse as a sold curve. Tycally, the two curves cross at three onts {,, θ,θ,, }, suggestng three ossble eulbra. θ θ FIGURE. Routng resonse functons 4.. Interconnecton Strateges n Prce Doman Next we examne how rces would nfluence the routng decson. It s straghtforward to observe from 7 that the threshold θ decreases wth and becomes negatve f, > ρ. 8 Ths ndcates that IBP- wll always artcate n nterconnecton f t can charge a rce above ρ, euvalently,, regardless of IBP- s strategy.e., the value of. The threshold ρ s smly, the value of the loss rato n the stuaton where the two IBPs oerate ndeendently. 3

14 Intutvely, a rovder would choose to nterconnect f the roft from servcng ackets from the eerng artner more than comensates the loss from drong one s own ackets. From 7 t s also clear that the value of θ ncreases wth. When θ goes beyond, IBP- would ot not to nterconnect.e.,, agan regardless of IBP- s strategy. The corresondng when θ lnear relatonsh between and, can be found by settng the rght hand sde of 7 to, yeldng a α ϕ 9 where the sloe s α ; and the ntercet the value of when s It s clear from that ϕ > and from 9 that when ρ. ϕ. α ρ,, ϕ, 3, ϕ ρ FIGURE. Eulbrum routng robabltes at dfferent rces To grahcally llustrate the deendence of the routng robabltes on rces, we lot 8 wth an eualty sgn and 9 n Fgure. There are four lnes that dvde Ω nto 9 regons. We frst focus on the followng 3 regons, defned as, {, : ρ, ρ, α } ϕ Ω; 4

15 {, : ρ, α ϕ, < ρ, α } Ω{, : ρ, > α ϕ, α }. ϕ 3 ϕ Ω; These three regons are ndcated n Fgure. In these regons, the value of θ vares from to. We ntroduce two lemmas. LEMMA. The eulbrum routng robabltes are,,,,,, ;,. 3,, the value of θ changes from to. Snce > ρ, as dscussed PROOF. For earler, IBP- should always nterconnect,.e., mlyng that > θ. Conseuently, the best nterconnecton resonse as stated n Prooston 3 s. Smlarly, for, 3, IBP- does not artcate, and nether should IBP-. LEMMA. For,, there exst three eulbra: two ure states θ,,,, and a mxed state, θ,. The eulbrum,, s the domnant strategy. Next we summarze the results n the Prooston 4 and reresent the nformaton grahcally n Fgure. There are four dstnct regons n the rce doman wth corresondng nterconnecton eulbra. ΩPROPOSITION 4. In the rce doman,, and gven the domnant strategy n, the otmal eulbrum nterconnecton strategy s,,, Φ, ;,,, Φ, ;,,,, Φ, ;,,, Φ,, where the four regons are defned as, Φ Φ {, : α ϕ, α } Ω{, : ρ, > α }; Ω;, ϕ, ϕ 5

16 Φ Φ {, : ρ, > α }, ϕ,, :, Φ, Φ, Φ, Ω; { } Ω. PROOF. Prooston 4 follows Lemmas and drectly. 4. Prce Eulbrum ΩIn the revous secton, we have dentfed four regons n, and showed that each regon has an otmal eulbrum nterconnecton strategy. These four regons are lotted n Fgures and 3. Based on ths, we next dentfy ossble eulbrum rces.,, R ρ X η κ,, ρ FIGURE 3. Prce resonse functons As usual, we start wth the dervaton of the best rce resonse functons. Recall that the roft functons, gven by 4 and 5, deend lnearly on the rces; therefore, t s exected that the best resonses would le on some boundares of the above four regons. Ths s ndeed the case, as formally stated n Prooston 5, whch can be roved wth the hel of Lemmas 3 and 4. The results n these two lemmas hel to show that some arts of the boundares of Regon-, n Fgure 3 gve the best resonses. Wth a more detaled notaton for IBP- s roft functon, π, ;,, we have the followng lemmas. LEMMA 3. The followng neualtes hold, for IBP-; and, for IBP-. π, ;, ρ > π, ;,, ρ π,; ρ, > π,;,, ρ 6

17 If IBP- s rce s and IBP- charges at ts utlzaton level, IBP- wll always be better off have a hgher roft f IBP- accets routed ackets wth robablty. In Fgure 3, the frst neualty s shown to hold at Pont-X marked as a hollow trangle. LEMMA 4. The followng neualty holds for IBP-, where, dπ,;, /, d > R α ϕ, for ϕ ρ. Lemma 4 states that when both IBPs artcate n nterconnecton, IBP- s roft wll strctly ncrease wth IBP- s rce, f, n resonse, IBP- charges a rce accordng to. An examle for IBP- s shown n Fgure 3, where art of s drawn see Lne-R. Lemmas 3 and 4 together show that, for IBP- or smlarly IBP-, π, ; R, π, ;, for η ρ, where the eual sgn holds at η. Ths ndcates that a contnuous segment Lne-R, startng from Pont-X and endng at η, renders a better rce resonse, than settng, for IBP- f IBP- chooses ts rce n the range η ρ. PROPOSITION 5. IBP- s best rce resonse functon s, α ϕ, f < max η, κ or otherwse, ρ ; where η and η solve,, ; η, η π, ;,, ; η, η π, ;, π R ; π R. η η resectvely; and, κ and κ solve κ R and κ R smultaneously. κ κ The rce resonse functons are lotted n Fgure 3, where the thck sold dashed lne reresents the IBP- IBP- s best rce resonse to the rce offered by IBP- IBP-. If IBP- s rce s above a threshold value η, IBP- has to charge a lower rce than to nduce IBP- s artcaton n nterconnecton. The locatons where the two best rce resonse curves 7

18 cross yeld the eulbrum rces, as descrbed n Prooston 6. PROPOSITION 6. The rce ar,, s a Nash eulbrum. There exsts another κ eulbrum, κ,, where, under the condton that and that ρ ρ α κ ρ, 3 α α π,; κ, κ π, ;,, 4 κ π,; κ, κ π,;,. 5 κ The Nash eulbrum,, ndcates that each IBP wll treat a acket sent by the other IBP as ts own and charge a full rce. Our result suorts the off-net-cost rcng rncle Laffont et al. 3 under whch the usage charge for on-net ackets generated by an IBP s clent euals the off-net.e., routed from the other IBP rate. The κ-eulbrum at, κ, κ, s a congeston-based rce where the routng fees deend on the utlzaton of each network. It s worth notng that the κ-eulbrum s unstable Trole 988. Two IBPs can start from any ntal ar of rces, and seuentally adust ther resect rces accordng to the resonse functons dected n Fgure 3. Ths rocess of rce adustment wll converge to,,, the stable eulbrum ont. Of course, ths defnton of stablty s under the assumton that both IBPs are myoc wth resect to the tme horzon they use for roft maxmzaton. 4.3 Congeston Based κ-eulbrum We frst numercally nvestgate the regons where the congeston-based κ-eulbrum exsts. Fgure 4 lots the regons of exstence for dfferent settngs. In Fgure 4a, two IBPs have the same lnk seed S. The condton n 4 s true n the regon above the lower curve. Ths ndcates that when IBP- s more congested than IBP-, t should charge a lower rce κ < 8

19 to nduce IBP- to oen u ts network. Smlarly, below the hgher curve n Fgure 4a the condton n 5 s satsfed. The κ-eulbrum therefore exsts n the regon between these two curves. Snce the regon s a relatvely small, n Fgure 4a the dfference ρ ρ s lotted. As the caacty rato S ncreases, the regon of exstence becomes larger and gradually moves below ρ ρ, as shown n Fgure 4b. ρ ρ.. S ρ S S -. a. b ρ ρ FIGURE 4. Regon of exstence of κ-eulbrum We have also obtaned exlct exressons for two secal cases of κ-eulbrum: a domnant IBP Fgure 4b, and two dentcal IBPs Fgure 4a. COROLLARY 3. When IBP- domnates IBP- n lnk seed, namely, S, κ-eulbrum exsts n the regon where, The eulbrum rces are PROOF. Omtted. ρ. ρ κ ρ ; and, ρ ρ ρ κ ρ ρ ρ. Here IBP- the faster IBP charges at ts utlzaton level, ndeendent of IBP- s congeston level as IBP- becomes margnalzed. The slower IBP IBP- charges a rce lower than ts utlzaton as t benefts from the revenue by transmttng IBP- s ackets; ths rce ncreases when IBP- becomes more congested and more ackets are routed to IBP-. COROLLARY 4. For two dentcal networks S and ρ ρ ρ, κ-eulbrum exsts and, 9

20 ρ ρ κ κ ρ. ρ PROOF. Omtted. In ths case, two IBPs charge close to ther utlzaton level when ther traffc s ether very lght or very heavy ρ near or ; and when ρ.3, κ and κ devate most sgnfcantly from ρ. Ths non-lnear behavor reflects the fact that the mact of nterconnecton on both IBPs s mnmal when ther traffc s lght small robablty of routng ackets or heavy small robablty of accetng ackets. 4.4 Dstrbuton of Interconnecton Surlus In ths subsecton, we attemt to fnd out whch IBP benefts more from nterconnecton. We frst notce that, when both IBPs artcate n nterconnecton, or,,,, π π, that s, the total roft s ndeendent of the nterconnecton rces. ρ S S S S ρ S S S S. a b ρ ρ FIGURE 5. Regons where faster IBP has smaller a absolute or b ercentage ncrease n roft We examne the dstrbuton of surlus at the stable eulbrum ont,,. Defne the ncrease of roft over non-nterconnecton as, π π,;, π,;. k k k In Fgure 5a, the regon above each curve ndcates when IBP- has a smaller share of the overall surlus, or when π < π. For S, the boundary s gven by ρ ρ. The less busy network

21 also gets a larger share of the surlus. The regon where π < π exands as S becomes larger, an ndcaton that IBP- receves a dmnshng roorton of the overall surlus. As shown n Fgure 4b, IBP- does even worse f the ercentage ncrease, π / π,;, s concerned. On the other hand, IBP- can be better off at the κ-ont eulbrum. For examle, the mddle lne n Fgure 4a shows the boundary, above whch, π,; κ, κ,;,. π However, as S gets larger, ths lne aroaches the uer bound of the exstence regon for examle, n Fgure 4b. To conclude ths secton, t aears that an nterconnecton fee whether full rce or the amount rescrbed by the congeston based κ-eulbrum s necessary to nduce nterconnecton n order to acheve better os, and hence hgher rofts. Another ssue s the dstrbuton of surlus. Followng the full rce eulbrum,, the faster IBP tycally gets a smaller share of the surlus. However, n the congeston eulbrum κ, κ the faster IBP s at a relatve advantage. In the next secton, we consder two rovders that charge ther customers dfferent rces and see how the nterconnecton and rcng decsons are affected. 5. Interconnecton between Asymmetrc Provders So far, we have assumed that both IBP s charge ther customers the same rce unt normalzed to er acket. In realty, dfferent IBPs may have dfferent cost structures, os, and rcng schemes. We analyze the roertes of the eulbrum n the resence of such asymmetres among the IBPs. Let the rces charged by IBP- and IBP- be c, resectvely. Here we assume c ; the analyss for c > s smlar. The roft functon for IBP- changes to, π c c c,, ; whle the roft functon for IBP- remans the same as the one defned n Euaton 5. It s easy to observe that, π, ; c c π / c, / c;. k k

22 Due to scalng, most of the revous analyses hold wth the rescaled rces. Ths allows us to easly redraw the otmal nterconnecton strateges as shown n Fgure 6. We summarze the followng effects due to asymmetrc rces.. c. The rce that IBP- charges IBP- s bounded by c, the rce that IBP- charges ts customers;. Euaton 9 changes to cρ, dected by Lne-B n Fgure 6a. Euaton 8 becomes α c ; ths s Lne-R n Fgure 6a. ϕ 3. There are four eulbrum nterconnecton strateges: f c s above ρ Fgure 6a. Otherwse the strategy,, no longer exsts Fgure 6b. In addton, we have the followng lemma. LEMMA 5. η decreases lnearly wth c, where η >. η, 6 η ηc c ρ ϕ R,,, R, c ϕ c ρ B a η c,,, ϕ c ρ c ϕ b η FIGURE 6. Imact of cost asymmetry c on eulbrum strateges for a c > ρ and b c ρ In Fgure 6, the best rce resonse functons are lotted. For IBP-, the resonse functon remans the same. That s, for the case of c > ρ shown n Fgure 6a, the best resonse curve s descrbed by the lne c, excet for a segment aearng on Lne-R. If c ρ, Fgure 6b shows that the best rce resonse functon s c or R, but only for a set of that nduce nterconnecton strategy,,. Outsde ths regon, IBP- sets rces

23 arbtrarly as t no longer artcates n nterconnecton. Fgure 6 also draws the best rce resonse functon shown as a thck lne for IBP-. We are now n a oston to derve the rce eulbrum. Here, we focus only on the stable eulbrum. PROPOSITION 7. The rce eulbrum s and, α c ρ, f cl c ρ;, otherwse, c, arbtrary, f c c L ; otherwse, where, c η. L / η The corresondng eulbrum rces, and, are lotted n Fgures 7 aganst c. When c dros below ρ, IBP- starts to charge a rce lower than so as to nduce IBP- to artcate n nterconnecton. As c goes below a threshold c < c L, IBP- swtches back to the rce of as t s no longer benefcal for IBP- to route ackets when the acket value s very small. c L ρ c FIGURE 7. Effect of c on eulbrum rces c L c Fgure 8 lots the wdth of c where IBP- s rce s dscounted, aganst ts utlzaton level. The regon s relatvely small, but t ncreases wth the relatve network sze S. However, the magntude of rce dscount from a rce of s ute sgnfcant. More dscounts wll be needed f IBP- s faster. Also, IBP- gves a hgher dscount when ts network becomes buser. 3

24 Fnally, Fgure 9 shows the mact of IBP- s congeston level; when IBP- gets buser, the dscount rce that IBP- rovdes wll dro. ρ cl S S 5 S ρ α ρ cl S S 5 S ρ FIGURE 8. a Range of c for rce dscountng and b maxmum rce dscount aganst ρ ρ cl S S 5 S α ρ cl S S 5 S ρ ρ FIGURE 9. a Range of c for rce dscountng and b maxmum rce dscount aganst ρ 6. Dscusson We have shown how rcng contracts between nterconnected IBPs can be drawn based uon uantfable attrbutes such as network caacty and utlzaton. Currently, subscrbers that demand hgh ualty Internet connectons often resort to mult-homng Morrssey 3 or thrd arty routng. Alternatvely, backbone rovders could rovde more streamlned servce and better os usng the nterconnecton models studed here. Plannng for network caacty s dffcult: nsuffcent network caacty results n sotty coverage, low relablty Paxon 999, Chnoy 993, and long delays, whereas nvestng blndly on nfrastructure could mean huge losses Mehta. Even though we do not secfcally 4

25 address the roblem of caacty lannng, nterconnecton offers greater flexblty wth resect to caacty decsons, thereby reducng the cost of over and under caacty. Essentally, rovders can buy sot caacty when the demand surges, and sell extra caacty n exchange for a ayment durng erods of low demand a common busness ractce n the utlty ndustry. We antcate that once usage-based rcng schemes become revalent at the Internet backbone, more ntense collaboraton could take lace among regonal ISPs. If the ayment for backbone access s roortonal to the amount of data transorted, regonal ISPs have a hgh ncentve to byass the backbone for local traffc. Such cooeratve use of local routes could further mrove the overall network resource utlzaton. Ths vew concdes wth redctons that, as the ndustry matures small ISPs wll start to form allances Cuker 998 or consoldate Jew and Ncolls 999. Our analyss reures that traffc volume needs to be measured,.e., the number of ackets sent and receved needs be accounted for. Snce exstng communcaton networks such as the wreless 3G have already mlemented sohstcated network montorng and management features, acket level accountng should not hnder the alcablty of our model. Also, whle our model s ntended to rce a sot nterconnecton fee based on real-tme network condtons, t s ossble to derve a more stable rcng scheme based on smoothed hstorcal data. Provders from dfferent classes ossess dvergent vews on network traffc. Front-end rovders such as retal and web hostng ISPs favor hgh levels of traffc, for t ndcates ether the sze of ther customer base and/or the oularty of the content they host. Back-end rovders, however, are manly concerned the cost of carryng the traffc. Ths could exlan why IBPs are reluctant to eer wth lower-ter rovders, snce under SKA the cost of the data transorted by the backbone rovder cannot be recoued. Ths also hels exlan the trend that nterconnecton across dfferent classes of rovders have become fee-ayng transt agreements. The nteracton between value-centrc and cost-centrc rovders, however, reures further analyss and s only artally addressed by our model. 5

26 7. Concludng Remarks The enormous network externaltes generated by the Internet are, to a large extent, due to ts oen archtecture that allows ndvdual networks to easly nterconnect wth one another. The ast decade has wtnessed the transton of the Internet from a research exerment among a handful of unverstes to an ndsensable tool n areas such as scence, educaton, and commerce. An ntegrated aroach addressng the techncal, economc, and oltcal asects of network nterconnecton s therefore crtcal to ensure the contnued growth and utlty of the Internet. In ths aer, we nvestgate nterconnecton arrangements at the Internet backbone wth the am to dentfy otmal nterconnecton strateges whch could hel IBPs mrove routng decsons and servce ualty. Parallel M/M// ueues were used to model eered networks. The model catures the os mact of the nterconnecton strategy of the artners. Results from the ueung model are examned usng game theory to derve varous ncentve comatble eerng arrangements. We also extend our analyss for asymmetrc rovders,.e., the rovders that charge dfferent rces to ther customers. Here, the lower rced rovder offers a dscount to the hgher rced one so as to nduce nterconnecton. Several ossble extensons exst on the ueung front. There could be more than two arallel ueues; the os can be measured by not ust the loss rato but also by a comoste factor consstng of ueung delay, loss rato, and so forth Paxson 999, Bolot 993. These extensons, however, wll sgnfcantly comlcate the analyss, makng smulaton or other numercal methods necessary. Our assumton of arallel lnks s vald when dealng wth the nterconnecton of maor backbone networks wth extensve network nfrastructure. However, when dealng wth nterconnectvty among front-end rovders, arallel routes may not always exst. Another comlcaton arses from the fact that front-end rovders may value network ackets dfferently from backbone rovders, thus consderng only lnk and traffc characterstcs may not be suffcent to analyze nterconnecton n the ISP settng. Work s underway to study a more 6

27 comrehensve stuaton that consders both the value and cost of network traffc. References ACCC, Internet Interconnecton: Factors Affectng Commercal Arrangements between network Oerators n Australa, February. Baake, P., T. Wchmann, On the Economcs of Internet Peerng, Netnomcs,, 89-5, 999. Baley, J. P., The Economcs of Internet Interconnecton Agreements, In McKnght and Baley, eds., Internet Economcs, MIT Press, , 997. Ball, M.O., et al., eds., Network Routng, Elsever, New York, 995. Bolot, J.-C., End-to-End Packet Delay and Loss Behavor n the Internet, SIGCOMM, 89-98, 993. Caruso, D., Advanced Peerng A Better Alternatve to the Tradtonal Internet Peerng Model, Level 3 Communcatons, 3. Avalable at: htt:// Cave, M. and R. A. Mason, The Economcs and Regulaton of the Internet, Oxford Revew of Economc Polcy, 7, 88-,. Chen, C., The Last-mover Advantage, Fortune, July 9,, 984. Chaudhur, S., U. Dayal, An Overvew of Data Warehousng and OLAP technology, SIGMOD Record, 6, 65-74, 997. Chnoy, B., Dynamcs of Internet Routng Informaton, SIGCOMM 93, 89-98, 993. Cook Reort, The, A Content versus Carrer Peerng Battle n Hgh Stakes BBN vs. Exodus Dsute. Avalable at: htt:// Crèmer, J., Rey, P., Trole, J., Connectvty n the Commercal Internet, Journal of Industral Economcs, 48, ,. Cuker, K., Peerng and Fearng: ISP Interconnecton and Regulatory Issues, 998. Avalable at htt://ksgwww.harvard.edu//comol/paers/cuker.html. Dewan, R., M. Fremer, P. Gundeud, Evoluton of Internet Infrastructure n the st Century: The Role of Prvate Interconnecton Agreements, Proceedngs of ICIS, 999. Duato, J., A Necessary and Suffcent Condton for Deadlock-Free Routng n Cut-Through and Store-and-Forward Networks, IEEE Transactons on Parallel and Dstrbuted Systems, 7, 8, ,

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30 3 Aendx Proof of Prooston c c c c r r FIGURE A. State transton dagram for centralzed IBPs Here, we assume that when a acket arrves and both networks are dle, the central admnster wll assgn t to IBP- wth robablty r. We can reresent the transton dagram wth a set of balance euatons n the steady state. In the matrx form, c c c c r r. The above euatons can be readly solved to gve the loss rato for the centralzed IBPs, c, s, r r r r c. The senstvty of the loss rato wth resect to the routng decson r s / r c. Ths ndcates that r mnmzes c f >. The ackets thus should always be routed to the faster ath when both are dle. The mnmal loss rato can be acheved by settng r f >. Proof of Corollary Frst, gnorng the second term n the denomnator of c, ths gves, < c. Then we examne the dfference,

31 3 ρ ρ. Corollary follows. Proof of Prooston FIGURE A. State transton dagram for nterconnected IBPs The state transton dagram s dected n Fgure A. When IBP- s busy and IBP- dles,.e., n the state,,, IBP- wll become busy f ts own acket arrves wth robablty, or f t decdes to accet a acket routed from IBP- wth robablty. Ths leads to a state where both IBPs are busy, or,,. The rest of the transton arcs are self-exlanatory. We can agan wrte all the balance euatons n a matrx form,. State robabltes for the case of nterconnecton,,,, where can be obtaned by normalzaton. Exlctly,

32 3. Proof of Corollary The frst neualty follows snce mn. The second neualty can be shown followng the same ste n the roof of Corollary. Proof of Corollary 3 We start wth the frst dervatves of state robabltes, wth resect to, Γ ; Γ ; Γ ; Γ, where the non-negatve term Γ s gven by, Γ. Due to the symmetry, the frst dervatves wth resect to can be found by swang the ndces for arameters,,, Γ; and also. The sgns of these frst dervatves are summarzed n Table. It s obvous that the frst neualty holds as the frst dervatve of both terms s ostve. To rove the second neualty, we exlctly carry out the dfferentaton. Ths gves, L Γ,

33 33 where the ndces and take values of and, but. The above exresson s strctly negatve snce. Proof of Prooston 3 The frst dervatve of IBP- s roft functon, wth resect to ts nterconnecton decson, s, π Γ A where. Notce that the rght hand sde of A s ndeendent of, suggestng extreme-ont solutons that deend on the routng fees as well as the nterconnecton strategy of IBP-,. Prooston 3 follows an examnaton to the sgn of A θ s the value of when A becomes zero. Proof of Lemma To dentfy the domnant strategy n the Regon-, the rofts for a gven rce ar,, are,, π ; and,, θ θ π. By adotng the mxed state nterconnecton strategy, each IBP s roft decreases wth the transfer rce t charges. Nevertheless, n the Regon-,,, π θ θ π ; the eual sgn holds only when ρ or ρ for IBP-. We frst show that,,, θ θ π θ π. Ths s evdent as,

34 34,, > Γ θ θ θ π namely, π strctly ncreases wth along the lne θ. Next, we have,, > π, snce > θ. Therefore,, θ π π. Proof of Lemma 3 We exlctly calculate, ;,, > Γ ρ π. Therefore, when s changed from to, π ncreases. Proof of Lemma 4 From, we fnd that / α R, Along Lne-R, we have,,,; > α π R. The roof for IBP- s smlar. Proof of Prooston 5 We frst notce that, wthn each of the four regons defned n Prooston 4 and drawn n Fgure 3,.e.,, Φ,, Φ,, Φ, and, Φ resectvely, /, ;, π. A Here,, ;, π s the roft functon for IBP-, gven a rce ar, and the corresondng otmal nterconnecton strateges,. Ths neualty s vald from the

35 roft functons π whch ncrease lnearly wth, holdng other factors constant. Euaton A mles that the best resonse functon wll be located on the boundares. In the followng, we derve the best resonse functon for IBP-. For any gven, we ncrease from and try to fnd a value of where π s the maxmum. Suose that ρ, that s, we start n Φ,. can be ncreased, due to A, to reach the boundary wth Φ,. On that boundary, π s ndfferent to the value of. Ths s ndcated by Prooston 3, snce on the boundary between Φ, and Φ,, θ. We can contnue ncrease across the boundary, and eventually end u n. Smlarly, for low value of, the otmal resonse rce s. When κ ρ, wll be ncreased to enter Φ, and reach ts bottom border,.e., the Lne-R drawn n Fgure 3. can be further ncreased to cross Lne-R, enter Φ,, and reach. For ths to haen, the neualty π,; R, < π, ;, has to be vald. However, accordng to Lemmas 3 and 4, for close to ρ, the above neualty does not hold. Therefore, for max η, κ ρ, the best rce resonse functon s gven by R. Here η solves π,; η, η π, ;, η Proof of Prooston 6 R. Prooston 6 follows Prooston 5 drectly. Euatons 4 and 5 guarantee that two best resonse functons cross at κ,. Euaton 3 s obtaned by smultaneously solvng κ R and κ R. κ Proof of Lemma 5 κ κ For a gven c, we solve π,; η, η π, ;, Euaton 6, η R for η. Ths gves, corresondng to, α ϕ, η ; α,, 35

36 η,,,,,. α,, From the roof of Lemma 3, we know the denomnator n the above two exressons s ostve. It can be verfed, by substtutng n the exlct exressons of state robabltes, that the numerator of η s ostve. Proof of Prooston 7 The eulbrum rces are straghtforward followng the best rce resonse functons, dected n Fgure 6. Lemma 5 states that η ncreases as c s reduced. The threshold value c L can be found by settng η to c. 36