DiffServ Pricing Games in Multi-class Queueing Network Models

Transcription

1 DffServ Prcng Games n Mult-class Queueng Networ Models Parjat Dube & Rahul Jan Abstract Introducton of dfferentated servces on the Internet has faled prmarly due to many economc mpedments We focus on the provder competton aspect, and develop a mult-class queueng networ game framewor to study t Each networ servce provder s modeled as a sngle-server mult-class queue Provders post prces for varous servce classes Traffc s elastc and there are multple types of t, each traffc-type s senstve to a dfferent degree to Qualty of Servce QoS Arrvng users choose a provder and a class for servce We study the prcng and servce competton between the provders n a game-theoretc settng We provde suffcent condtons for the exstence of Nash equlbrum n the Bertrand prcng game between the mult-class queueng servce provders We also characterze the neffcency prce of anarchy due to strategc DffServ prcng Index Terms DffServ, Queueng networs, Bertrand game, Nash equlbrum, Prce of Anarchy I INTRODUCTION It s well-nown that ntroducton of servce dfferentaton n networs has faled not due to nadequaces of technologcal solutons eg, DffServ but prmarly due to economc mpedments At the same tme, ths became part of the networ neutralty debate wheren networ servce provders NSPs eg, AT&T, Verzon, etc want to ntroduce servce dfferentaton through prce dfferentaton whle content provders such as Google, Yahoo, etc are opposed to such an arrangement Both sdes have made varous arguments for and aganst networ neutralty and servce dfferentaton whose veracty has been hard to judge Motvated by ths, we propose a smple queueng model whch can be useful n studyng some of these questons We cauton that the larger polcy ssue s a lot more complcated and we cannot hope to have all the answers by studyng smple models Nevertheless, the framewor can be used to address some basc questons: Is servce dfferentaton better for networ utlty maxmzaton? How much s lost n networ utlzaton due to competton between provders when they offer dfferentated servces? Parjat Dube s wth the IBM TJ Watson Research, Hawthorne, NY 0532 Emal: pdube@usbmcom Rahul Jan s wth the Unversty of Southern Calforna, Los Angeles, CA Emal: rahuljan@uscedu Hs research s supported by a James H Zumberge Faculty Research and Innovaton award, the NSF NetSE grant IIS and the NSF CAREER award CNS We gnore the nternal topology of a networ and thn of each autonomous networ of an NSP as a node represented by ts Norton-equvalent queue There are N provders each of whom operates a GI/G/-PS queue wth two classes wth processor-sharng between them we restrct to queues wth convex delay functons There are two types of traffc, type traffc s more delay-senstve than type 2 Users of each type arrve accordng to a renewal process eg, Posson Provders could charge a hgher servce prce for class than for class 2 Upon arrval, each user then has to decde whch provder/queue to jon type users only go to class If t jons a partcular queue, t has to pay a servce prce to the operator Each user s also delay senstve and hs servce utlty s a non-ncreasng functon of the delay t suffers Thus, he jons a queue that offers the maxmum net utlty, e, servce utlty mnus delay cost and prce We wll assume that the queues are partally observable n the sense that expected delay metrcs are avalable to the user upon hs arrval but not the nstantaneous queue lengths Each provder s objectve on the other hand s to maxmze hs total payoff The players are compettve and strategc The objectve of each provder s to pc servce prces for the two classes as well as the processor-sharng rato that maxmzes hs total payoff tang nto account the competton provded by the other provders The queston at hand s what nd of outcomes may one expect from the competton between the provders? And are the outcomes optmal or effcent n some sense? There has been a plethora of wor on networ prcng, most of t concerned wth usng prcng models for desgn and analyss of networ flow control There s a huge lterature on economc models of networs as well see [3] ncludng prcng of Internet servces [6], and n partcular, DffServ servces as well [26], [7], [8], [], [2], [25], [24] Our focus, however, s dfferent We as what mpact compettve and strategc DffServ prcng has on networ utlty maxmzaton? Snce QoS s an mportant metrc of networ utlty, we explctly ntroduce a queueng networ model We beleve such a mult-class queueng networ game model s new n the study of DffServ We provde a bref survey of related wor

2 Prcng as a means to queueng stablty was frst studed n [22] wth Posson arrvals to an exponental server queue [6] In [7], a schedulng polcy was derved for a mult-class queue that maxmzes the net utlty Mendelson and Whang [20] ntroduced a stylzed model for a sngle queung servce provder that has been wdely used They ntroduced an ncentvecompatble prorty prcng rule for the M/M/ queue whch maxmzes the socal welfare In [0], a cost sharng perspectve s taen on sharng the total delay cost, and the Aumann-Shapley mechansm s used to determne ndvdual user payments Whle sngle server models have been much analyzed [8], multple queueng servce provder models are not so well understood The earlest wor on ths s [5] whle a varaton of the Mendelson-Whang model for two dentcal servers wth unobservable queues was consdered n [4] Both of these establshed the suboptmalty of equlbrum flows under varous settngs - a Bertrand prcng game and a Cournot capacty game More recently, [9] consdered the Bertrand game between multple NSPs wth lnear affne delay functons a very rough approxmaton for delay n queueng networs and showed the game to have a prce of anarchy PoA of /325 Ths result was sharpened by [23], [2] to 2/3 for general concave delay functons Whle a bound on PoA was obtaned, t was conjectured that a Nash equlbrum may not exst n general In [5], [2], suffcent condtons for exstence and unqueness of Nash equlbrum are gven All these wors focused on sngle class queues only Our recent wor [] does study mult-class queues but there s only one type of traffc Earler n [5], we have addressed exstence of equlbrum n sngle-class queueng games where provders can choose both prces as well as servce capactes In [4], the authors consdered the case of nelastc traffc each arrvng user must choose among two M/G/ queues cannot bal There, a Stacelberg game was consdered and t was shown that an equlbrum does not exst, but an oscllatory prce wars behavor around a lmt pont was notced The focus n ths paper s on understandng DffServ prcng n a mult-provder envronment We ntroduce a mult-class queueng networ wth elastc traffc of multple types Provders compete on settng prces for varous classes We gve suffcent condtons for exstence of Nash equlbrum and show that the prce of anarchy wth DffServ classes n networs s the same as n the absence of DffServ [2], [23] Ths smple model seems to suggest that no worst-case welfare gans may be made by ntroducton of servce dfferentaton II MODEL AND PRELIMINARIES We consder N provders each of whom offers a queued servce to customers Each provder operates a queue wth two classes and processor sharng between them, and convex delay functons for each class There are two types of traffc wth traffc wth dfferent senstvty to delay Traffc of type always goes nto class of the queues, traffc of type 2 always goes nto class 2 of the queues As llustrated n Fgure, let x be the traffc flow of type nto the th queue n the th class Provder has servce capacty y and he splts hs servce capacty as α y for class and ᾱ y for class 2 by dong processorsharng where ᾱ = α Servce qualty metrcs are delay functons d x ; α for class where delays at α = 0, d x ; 0 wll be assumed to be nfnte A hgher value of d mples a degradaton n servce qualty If the delay suffered by a unt flow of type s d, then the perceved delay cost per unt s θ d Provders charge prces p = p, p 2 per unt flow Fg DffServ prcng game between mult-class queued servce provders We assume that each arrvng user s nfntesmal and non-strategc e, prce-taer and ther preferences can be aggregated so that the aggregate utlty derved by all users of type s V x Note for prces p and delays d, the total demand or flow of type can be gven by z = V mn {p + θ d } The p + θ d term shall be called the full prce of queue and class We wll tae the net socal welfare to be Sx, α := V x, θ x d x ; α We wll call a traffc vector x to be socally effcent f t maxmzes the socal welfare maxmzaton problem max x,α Sx, α, where α [0, ] and x 0,, We wll mae the followng assumptons Assumptons A V x s strctly ncreasng and concave n x = x A2 d x ; α are strctly ncreasng and convex n x for gven α A3 V C 3, and d C 2, e, V and d are contnuous

3 A4 V α y θ d 0, α,,, wth α, > 0 Note that for two players α = α and ᾱ = α 2 These assumptons are justfed n [5] We frst characterze the rate vector to be socally optmal Lemma : Under Assumptons A-A2, a socally effcent rate vector x exsts and s characterzed by:,, V j x j x θ d x θ d x 0 wth equalty when x > 0 A user of type sends the margnal traffc eg a pacet n the communcaton networ to queue Q n class f arg mn j p j +θ d j x j ; α j We assume that each customer type sends traffc at some rate x beng determned by the prces and the delay cost t faces Thus, a natural outcome of such decson-mang by each user s the followng Defnton Wardrop equlbrum: For a prce vector p, and processor-sharng rato vector α, a flow vector x p, α s a Wardrop equlbrum f and only f t satsfes the followng,, V x p θ d x ; α 0 2 wth equalty when x > 0 We wll assume that nstantaneous queue lengths are not observable but the expected delays n each class for each queue s avalable to all arrvng users We wll assume that the queues have nfnte buffers for every class For provder, f total traffc s x wth prces p, then hs net payoff s Π p, α ; p, α = p x p, α, p,, α Thus, gven prces of other provders, he must pc a prce vector p that maxmzes hs total payoff A natural queston now s does there exst an equlbrum where provders announce expected delay guarantees, and users then choose traffc rates and the system equlbrates through some prces and traffc rates such that the full prce s the same for all queues n all classes, delays guarantees offered are met, and all provders are satsfed at these prces? When players are non-strategc, then the exstence of compettve equlbrum was establshed n [27] When players are strategc however, we must loo at Nash equlbrum Defnton 2 Nash equlbrum: A prce vector and processor-sharng rato p, α s a Nash equlbrum f the correspondng Wardrop equlbrum s x p, α, and for each, gven p, α, p, α arg max p,α p x p, α, p, α 3 III DIFFSERV PRICING GAME When the provders are strategc, they try to antcpate the actons of the other players provders and t may not be possble to acheve socally optmal allocatons The provders strategze by pcng prces Capacty avalable to them wll be assumed fxed We study equlbra n such a prcng game also called a Bertrand game between queueng servce provders In prevous wor, we consdered sngle-class queues, and establshed exstence of Nash equlbrum [5] The prce of anarchy n that settng had already been establshed n [23], [2] We also consdered the settng where provders choose capacty and prces at the same tme [5] Here we consder a model more approprate to study DffServ There are multple traffc types Each queue also has multple classes wth processor-sharng between the classes For smplcty, we present results for only two traffc classes delay-senstve and best-effort and two correspondng classes n each queue but these can be extended to a general settng at the cost of more complcated analyss The allocatons for a gven prce vector p and processor-sharng rato α s a Wardrop equlbrum, and from 2 we have, V j x j θ d x ; α p 0, 4 wth equalty when x > 0 Furthermore, the prces p and processor-sharng rato α whch maxmze the aggregate payoff of provder, Π = p x, must satsfy, x + p x p = 0, p x α + p 2 x 2 α = 0 5 Note that we also must establsh the jont concavty of Π n p, α n terms of condtons on second order dervatves Let x, p, α satsfy 4 and 5 Then, p, α s a Nash equlbrum We denote by U = p x, the payoff of provder at ths Nash equlbrum Lemma 2: Under Assumptons A-A4, f p, α s a Nash equlbrum, then there exsts a Nash equlbrum p, α wth equalty n 4 Proof: Let x = x p, α be a WE If x > 0, defne p = p If x = 0 and there s an nequalty n 4, defne p = V j x j θ d 0; α Note ths s always non-negatve by Assumpton A4 Wth prce p, the allocaton s stll x, mplyng that the correspondng payoff Ũ = p x s optmal and equal to U for all Further x, p also solves 4

4 wth equalty Thus, p, α s a Nash equlbrum Remar Note that we dd not use convexty of delay functons assumed n Assumpton A2 Remar 2 In subsequent dscusson, wthout loss of generalty, we absorb θ nto the delay functon, d The mplcaton of ths lemma s that f a Nash equlbrum exsts, then we can loo for a Nash equlbrum of type p for whch we have equalty for every n 4 We can establsh exstence of such a Nash equlbrum under certan condtons Theorem : If Π s jontly concave n p, p 2, α for every then under Assumptons A-A4, there exsts a Nash equlbrum n the DffServ Bertrand game The proof s relegated to the appendx A Concavty of the payoff functon We next establsh the jont concavty of revenue of provder n p, p 2, α To show ths we need to establsh negatve defnteness/sem-defnteness of the Hessan for provder revenue usng the followng nown result Lemma 3: A n n symmetrc matrx s negatve defnte ff the leadng prncple mnors are < 0 f s odd and > 0 f s even negatve semdefnte ff all the th order prncple mnors of are 0 f s odd and 0 f s even The Hessan for provder s optmzaton s gven by H Π = Π,p 2 Π,p,p 2 Π,p,α Π,p,p 2 Π,p 2 2 Π,p2,α Π,p,α Π,p2,α Π,α 2 6 By repeated partal dfferentaton of Π, we get: Π,p 2 = 2x,p + p x,p 2 ; Π,p,p j = 0, j ; Π,α 2 = p x,α 2 + p 2 x 2,ᾱ 2; Π,p,α = x,α + p x,p,α ; Π,p2,α = x 2,ᾱ p 2 x 2,p2,ᾱ Defne H = Π,p 2 2 Π,α 2 Π 2,p 2,α, H 2 = Π,p 2 Π,α 2 Π 2,p,α, H 3 = Π,p 2 Π,p 2 2 Thus we have Proposton : If Π s margnally concave n p, =, 2 and α, and dervatves of the proft functon satsfy: Π,p 2 Π,α 2 Π 2,p,α =, 2 7 Π,p 2 H Π 2,p,α Π,p then Π s jontly concave n p, p 2, α Proof: Concavty of Π n p and α ensures that the frst order leadng prncpal mnors of H Π are negatve The three second order prncpal mnors of H Π are postve, H and H 2 postve by 7 and H 3 by concavty of Π n p 8 maes the thrd order prncpal dagonal of H Π, e, the determnant of H, negatve Thus from Lemma 3, we have the jont concavty of Π Furthermore, f Π s jontly concave n p, p 2, α, then the provder optmzaton problem s convex, and an optmum always exsts B Specal Case: Lnear Utlty We consder the case wth lnear utltes We establsh suffcent condtons for the concavty of Π n p and α for ths case Let α = α and α 2 = α Lemma 4: Under Assumptons A-A4, f d s strctly decreasng n α then x s strctly decreasng n p and strctly ncreasng n α for = 2 Proof: Wthout loss of generalty, we can tae = From 26 n Appendx we have x,p = From Assumpton A and A2 we,,x have,, > 0 and d,x > 0 Thus x,p < 0 From 27 we have x,α =,,α Snce d,x > 0, d 2,α < 0, we have x,α > 0 Smlarly we can establsh x 2,α < 0 We next use Lemma 4 to establsh the followng Proposton 2: Under Assumptons A-A4, f V s lnear and d s ncreasng n α and jontly convex n x, α then x and Π are margnally concave n p and α Proof: Wlog we tae = For lnear V, wth V = 0, we have, = = By tang dervatve wth p x 2,p n 26 we get x,p 2 = d,x 2 d,x Thus x,p 2 0 as d,x 2 0 convexty and d,x > 0 Smlarly we can establsh x 2,p2 s concave n p 2 By tang dervatve wth α of 27 we get: d,x x,α 2 + d,x,α x,α + d,α 2 = 0 From Prop 4 we have x,α > 0 Jont convexty of d mples d,α 2 0 and d 2,α d,x 2 d,α 2 0 Smlarly we can establsh concavty of x 2 n p 2 and α Then the concavty of Π n p and α follows from expressons for Π,p 2 and Π,α 2 obtaned before Prop Remar 3: Observe that M/M/ type delay functons are jontly convex n flow and capacty and can be canddate delays for Prop 2 We use Prop 2 to establsh jont concavty of Π n the next theorem whch usng Prop establshes exstence of Nash equlbrum Theorem 2: Under the assumptons of Prop 2, f V and dervatves of d also satsfy:,α = d V d,x,α d,x then Π s jontly concave n p, p 2, α, 9

5 Proof: Snce Π s concave n p and α from Prop 2, to establsh jont concavty of Π 7 and8 needs to be satsfed Observe that f Π,p,α = 0, then 7 s satsfed as Π s concave n p and α Also 8 s satsfed as H 0 For Π,p,α = 0 we need to show: x,α + p x,p,α = 0, =, 2 0 Tang dervatve of 26 wth respect to α we get, wth lnear utltes: d,x x,p,α + d,x,α x,p = 0 Substtutng x,p,α from n 0 we get: x,α = p d,x,α d,x x,p 2 Substtutng expressons for x,α from 27, x,p from 26 n 2 and wth p = V d Wardrop equlbrum condton we get 9 IV PRICE OF ANARCHY OF THE DIFFSERV GAME An mportant queston now s what s lost due to compettve, strategc prcng n DffServ networs as compared to the socal-welfare maxmzng optmum A relevant metrc s the prce of anarchy PoA whch we now defne Gven a socal welfare functon Sxp for strategy vector p, we defne the Prce of Anarchy as η = mn p s NE Sxp, α Sx In ths secton, we compute the PoA of the DffServ prcng game and compare t to the PoA of the sngle-class queueng game We denote partal dervatves, d,x by d We frst fnd the prces at Nash equlbrum and socal optmum Proposton 3: At Nash equlbrum, the prces are gven by p = d x + δ x,, =, 2, 3 where, δ = j d j V, and flows x satsfy x /x 2 = d 2,ᾱ /d,α The socally optmal prces are gven by p = x d,, =, 2 Proof: We can wrte a provder s optmzaton problem as maxmzng hs payoff subject to hs prces and all flows satsfyng the Wardrop and Nash equlbrum condtons Thus, max p,x 0,0 α p x + p 2 x 2 4 st p + d = V, p j + d j = p + d, j, j Wrtng the KKT condtons for the above, we get for =, 2: x + λ j λ j = 0, 5 p + λ d V j λ j d = 0, 6 λ V + λ j d j = 0, j, 7 From 5, 6 we have p = x d +λ V Further, x from 5 and 7, we get λ = P, whch j V d j mples the desred result Dfferentatng Sx = V x, d x, wrt x we get V d x d = 0, thus p = x d Usng the result above, we can now obtan a lower bound on the PoA of the DffServ game Surprsngly, t s the same as for the sngle-class queueng game Theorem 3: For the prce and capacty game wth affne delays, under Assumptons A-A4, f there exsts a pure strategy Nash equlbrum, then the prce of anarchy s at least 2/3 The proof s n the appendx V CONCLUSION One of the fundamental challenges for future Internet archtecture evoluton s ntroducton of Qualty-of Servce QoS Ths s not because current Internet archtecture cannot support QoS but because of economc mpedments In ths paper, we ntroduce a smple multclass queueng networ model that can be used to study such ssues In such a queueng game model we have establshed the followng: we gave suffcent condtons for exstence of Nash equlbrum wth processor-sharng among classes, and we showed that the prce of anarchy of the prcng game s 2/3 Ths happens to be the same as n a sngle-class settng Ths seems to suggest that as far as worst-case equlbrum socal welfare s concerned, DffServ wll not result n an mprovement, though there may be many other reasons for ts adopton These results are prelmnary, and based on a smple model Thus, we advse cauton on the concluson untl the same result s acheved n more vared models In future wor, we wll extend ths framewor to more general queueng networs than just parallel queues whch are more relevant because mplementaton of DffServ requres QoS SLAs between peerng networ provders We wll also study the game when mult-class provders can choose both prces as well as capacty

6 APPENDIX Proof: Theorem From the frst order optmzaton condtons for Π, we get: x + p x,p = 0 x 2 + p 2 x 2,p2 = 0 8 p x,α p 2 x 2,ᾱ = 0 where and ᾱ = α Usng Lemma 2, we now study Nash equlbrum whch satsfy the Wardrop equlbrum condtons n equaton 4 wth equalty From equaton 4 we have: V = p + d x, α, 9 V 2 = p 2 + d 2 x 2, ᾱ 20 By tang partal dervatves n equaton 9 wrt p and α and n 20 wrt p 2, we get { V + dj,xj x x j,p = j,p, j = 2 d j,xj x j,p, j V 2 V V 2 j { + dj2,x2j x 2j,p2 = x 2j,p, j = 2 d j2,x2j x 2j,p2, j j { dj,xj x j,α = x j,α + d j,α, j = d j,xj x j,α, j j { dj2,x2j x x 2j,ᾱ = 2j,ᾱ + d j2,ᾱ, j = d j2,x2j x 2j,ᾱ, j j Defne X p = x j,p ; j =,, n, =, 2, X α = x j,α ; j =,, n and Xᾱ = x 2j,ᾱ ; j =,, n Then we have from equatons 2-24 wth b p a vector wth all zeros except at the th entry, b α a vector wth all zeros except d,α at the th entry and bᾱ a vector wth all zeros except d 2,ᾱ at the th entry, X p = D V U b p, =, 2 X α = D V U b α 25 Xᾱ = D 2 V 2 U bᾱ To show the exstence of X p, X α and Xᾱ we need to show that D V U s nvertble We state a result on the nverse of the sum of matrces Lemma 5 [3]: Let B be a nonsngular matrx and u and v be column and row vectors respectvely, then {B + buv } = B b + bv B u B uv B Usng ths Lemma, we can wrte: D V U = [ D + V V N D j= d j,x j N j= d j,x j I D Defne = V and, = V N j=,j d j,x j Wth d j,xj > 0 and V 0 by assumpton we have > 0 Thus we need to establsh the exstence of D whch follows as D s a dagonal matrx wth all postve entres as d j,xj > 0, t s nvertble Snce D, =, 2 exsts we have exstence of X p, X α and Xᾱ from equatons 25 Wthout loss of generalty we tae = Specfcally, we have: X p = X α = Xᾱ =,,x 2,x 2,x,α, ] 26 V 2,x 2,α 2,x 2 2,ᾱ 2, 2 2,x 2 V 22,x 22 2,ᾱ Wrtng the frst order condtons from equatons 8 n matrx form wth X = x, =,, n and P = p, =, n, we have X + X p P = 0, =, 2 29 X α P Xᾱ P 2 = 0, 30 where X p = dag,,x, 2,x 2,x, and X α s dag,,α, V 2,x 2,α, and Xᾱ dag s 2,x 2 2, 2,ᾱ 2, 2,x 2 2,ᾱ V From equatons 9, 20 and 29 we get 22,x 22 2, X = X p V I d, 3 and from equatons 9, 20 and 30 we have 0 = X α V I d Xᾱ V 2I d 2, 32

7 where d = d, =, N From equatons 3 and 32 we have, wth Y = α, =,, N, X = G X, X 2, Y, 33 X 2 = G 2 X, X 2, Y, 34 0 = G 3 X, X 2, Y 35 wth G X, X 2, Y : R 3n R, for =, 2, 3, G X, X 2, Y := X p V I d, G 2 X, X 2, Y := X p V I d and G 3 X, X 2, Y := X α V I d Xᾱ V 2I d 2 Frst note that snce V s concave, > 0 snce the delay functons are strctly ncreasng From the defnton of G, X p, X α, Xᾱ, and the fact that d and V are contnuously dfferentable by assumpton, we can conclude that G, =, 2 are contnuously dfferentable Snce G 3 s contnuously dfferentable, by mplct functon theorem [9] nvertblty of the Jacoban of G 3 X, X 2, Y n general, can be establshed, there exsts an F : R n R n st {X, X 2, F X, X 2 } = {X, X 2, Y : G 3 X, X 2, Y = 0} whch mples we have Y = F X, X 2 36 st X, X 2, Y U V W where U, V and W are open sets Then, there exst closed and bounded sets Ū, V and W st, Ū U, V V and W W Snce G, G 2 and F are contnuous functons and X, X 2, Y Ū, V, W whch are compact, then by Brouwer s fxed pont theorem there exsts an X, X 2, Y whch satsfes equatons 33, 34 and 36 Then, from equaton 9 and 20 we get P Ths establshes the exstence of a fxed pont X, X 2, Y, P Proof: Theorem 3 As proved n [23], [2], the prce of anarchy for general concave utlty and convex delay functons s lower bounded by prce of anarchy for lnear-truncted utlty functons, and affne delay functons We thus wor wth lnear affne delay functons Let S t denote the socal welfare n the game wth lnear truncated utlty functon, St the socal welfare at Nash equlbrum and p t be the prce and αt the processor sharng rato vector at Nash equlbrum Lemma 6 [2]: Under assumpton on contnuty and dfferentablty of V and d,, we have S t p t, αt Sp, α mn p t,α t St mn p,α S For the PoA analyss, we wor wth convex delay functons Let V be the Y-ntercept of the tangent to the dsutlty curve at V Fgure 2 characterzes the Nash Fg 2 Provders wth both convex and lnear delays V s dependent on the processor sharng rato α at Nash equlbrum equlbrum flow At Nash equlbrum, the flows x and prces p of any provder satsfy V + V x = d + p d x + p 37 where V = V α y, where α y s the servce rate of class of provder If d s affne we have equalty n 37 for d of the form a x From 37 and 3 we have V + V x = 2d x + δ x 38 Wrtng 38 n matrx form: V V M + 2B + X, 39 where, B and are order N dagonal matrces wth d d 2 0 B =, 0 0 d N δ δ 2 0 =, 0 0 δ N M s a order N square matrx of all ones Smlarly for socal optmum flow X we have V 2B V MX, 40 where B s order N dagonal matrx of d whch are delay dervatves evaluated at socal optmum For lnear

8 dsutlty not truncated the socal welfare at Nash, S, s gven by S = X T B 2 V M + X 4 For lnear truncated dsutlty the socal welfare at Nash, S t s gven by S t = X T B + X 42 Let Sl be the socal welfare at the socal optmum confguraton for a lnear dsutlty S t = S l + = V 2 XT MX 43 X T B 2 V MX + V 2 XT MX From 42 and 43 we can wrte, 3S t 2St = 44 X T 3B + 3 V MX V T + D T 2B V M V + D = X T 2B B + X X T 2B B + 2B V M 2B B + X X T V M + 2B + E E T X + 2B V M 2B B + X + E For affne delays C = D = E = 0 and B = B Thus 44 reduces to 3S t 2S t = XT [ B + 2B V M ] X Snce X 0, we need to show that all entres n the matrx nsded the square bracet n 44 are postve so that LHS s postve Ths can be establshed n the same way as n [2] REFERENCES [] E Altman, D Barman, R El-Azouz, D Ros, and B Tuffn, Prcng Dfferentated Servces: a game theoretc approach, Procof IFIP Networng, p43044, May 2004 [2] D Acemoglu and A Ozdagler, Competton and effcency n congested marets, Mathematcs of Operatons Research, 32:-3, 2007 [3] C Courcoubets and R Weber, Prcng Communcaton Networs: Economcs, Technology and Modellng, Wley Interscence, 2003 [4] P Dube, C Touat and L Wynter, Capacty Plannng, Qualty of Servce and Prce Wars, ACM Sgmetrcs Performance Evaluaton Revew, 35:3, 3-33, 2007 [5] P Dube and R Jan, N-player Bertrand and Cournot queueng games: Exstence of equlbrum, Proc Allerton Conference, 2008 [6] N Edelson and D Hldebrand, Congeston tolls for Posson queueng processes, Econometrca, 43:8-92, 975 [7] JM Harrson, Dynamc schedulng of a mult-class queue: Dscount optmalty, Operatons Research 232: , 975 [8] R Hassn and M Havv, To Queue or Not to Queue, Kluwer Academc Publshers, 2003 [9] A Hayrapteyan, E Tardos and T Wexler, A networ prcng game for selfsh traffc, ACM SIGACTS-SIGOPS Symposum on Prncples of Dstrbuted Computng PODC, 2005 [0] M Havv, The Aumann-Shapley prce mechansm for allocatng congeston costs, Operatons Research Letters 29:2-25, 200 [] R Jan and P Dube, Queueng game models for dfferentated servces, Proc Int Conf on Game Theory n Networs GameNets, 2009 [2] R Johar, G Wentraub and B Van Roy, Investment and Maret Structure n Industres wth Congeston, Operatons Research, 2009 [3] K S Mller, On the Inverse of the Sum of Matrces, Mathematcs Magazne, Vol 54, No 2 Mar 98, pp [4] C Loch, Prcng n marets sensttve to delay, PhD Dssertaton, Stanford Unversty, 99 [5] I Lus, On partal equlbrum n a queung system wth two servers, Revew of Economc Studes 433:59-525, 976 [6] J Mace-Mason and H Varan, Prcng the Internet, n Publc Access to the Internet, B Kahn and J Keller, eds, pp , 995, MIT Press [7] C Maglaras and A Zeev, Prcng and desgn of dfferentated servces: Approxmate analyss and structural nsghts, Operatons Research, 532: , 2005 [8] P Marbach, Prcng Dfferentated Servces Networs: Bursty Traffc, Proc INFOCOM, 200 [9] A Mas-Colell, M Whnston and J Green, Mcroeconomc Theory, Oxford Unversty Press, 995 [20] H Mendelson and S Whang, Optmal ncentve-compatble prorty prcng for the M/M/ queue, Operatons Research, 385: , 990 [2] J Musaccho and S Wu, The prce of anarchy n a networ prcng game, Proc of 45th Annual Allerton Conference, September 2007 [22] P Naor, The regulaton of queue sze by levyng tolls, Econometrca, 37:5-24, 969 [23] A Ozdagler, Prce competton wth elastc traffc, Networs, 2006 [24] A Odlyzo, Pars Metro prcng for the Internet, Proceedngs of ACM conference on electronc commerce, pp 4047, 999 [25] A Orda and N Shmn, Incentve Prcng n Mult-Class Communcaton Networs, Proc IEEE INFOCOM, 997 [26] J Shu and P Varaya, Prcng Networ Servces, Proc IEEE INFOCOM, 2003 [27] D Stahl and A Whnston, A general economc equlbrum model of dstrbuted computng, n New drectons n computatonal economcs, eds W Cooper and A Whnston, Kluwer Acad Pub, pp75-89, 994