Reverse Engineering TCP/IP-like Networks using Delay-Sensitive Utility Functions

Transcription

1 Reverse Engneerng TCP/IP-ke Networks usng Deay-Senstve Utty Functons John Pongsajapan and Steven H. Low Engneerng & Apped Scence, Catech Abstract TCP/IP can be nterpreted as a dstrbuted prmadua agorthm to maxmze aggregate utty over source rates. It has recenty been shown that an equbrum of TCP/IP, f t exsts, maxmzes the same deay-nsenstve utty over both source rates and routes, provded pure congeston prces are used as nk costs n the shortest-path cacuaton of IP. In practce, however, pure dynamc routng s never used and nk costs are weghted sums of both statc as we as dynamc components. In ths paper, we ntroduce deay-senstve utty functons and dentfy a cass of utty functons that such a TCP/IP equbrum optmzes. We exhbt some counter-ntutve propertes that any cass of deay-senstve utty functons optmzed by TCP/IP necessary possess. We prove a suffcent condton for goba stabty of routng updates for genera networks. We construct exampe networks that defy conventona wsdom on the effect of nk cost parameters on network stabty and utty. I. INTRODUCTION AND SUMMARY Any TCP congeston contro agorthm can be nterpreted as carryng out a dstrbuted prma-dua agorthm over the Internet to maxmze aggregate utty, see e.g. [12], [13], [17] [21], [25] for uncast and [4], [10], [25] for mutcast. A of these works assume that routng s gven and fxed at the tmescae of nterest, and TCP, together wth actve queue management AQM, attempt to maxmze aggregate utty over source rates. The paper [26] studes cross-ayer utty maxmzaton at the tmescae of route changes, many for the speca case of pure dynamc routng. In ths paper, we extend the resuts of [26] n severa ways. As n [26], we focus on the stuaton where a snge mnmum-cost route shortest path s seected for each sourcedestnaton par Secton II. Ths modes IP routng n the current Internet wthn an Autonomous Systems usng common routng protocos such as OSPF [22] 1 or RIP [8]. For jont congeston contro and routng optmzaton usng mutpe paths, see, e.g., [2], [5] [7], [9], [11], [12], [14], [15], [23]. Routng s typcay updated at a much sower tmescae than TCP AQM. We mode ths by assumng that TCP and AQM converge nstanty to equbrum after each route update to produce source rates and congeston prces for that update perod. These congeston prces may represent deays or oss probabtes across network nks. They determne the next routng update n the case of dynamc routng. Thus TCP AQM/IP form a feedback system where routng nteracts wth congeston contro n an teratve process. We are nterested n the equbrum and stabty propertes of ths teratve process. 1 Even though OSPF mpements a shortest-path agorthm, t aows mutpe equa-cost paths to be utzed. Our mode gnores ths feature. To smpfy notaton, we w henceforth use TCP AQM/IP and TCP/IP nterchangeaby. We assume routng s chosen to mnmze the weghted sum ap + bτ of congeston prces p and propagaton deays τ aong the path. When b = 0 pure dynamc routng, [26] characterzes the exact condton under whch an equbrum of TCP/IP exsts, and proves that such an equbrum maxmzes deay-nsenstve utty over both rates and routes. In practce, however, pure dynamc routng s never used because of ts nstabty. Instead, both weghts a and b are typcay nonzero, a case for whch no resut s avaabe. To reverse engneer TCP/IP networks wth nonzero weghts a and b, we ntroduce n Secton III deay-senstve utty functons that depend on not ony source rates but aso propagaton deays. We dentfy a cass C of deay-senstve utty functons that s mpcty optmzed by TCP/IP. As for the b = 0 case, we characterze the exact condton under whch TCP/IP has an equbrum and prove that such an equbrum maxmze utty functons n C over both rates and routes. As the reatve weght a/b, the utty functons n C become deay-nsenstve and these resuts reduce to those proved n [26] for pure dynamc routng. We exhbt some counter-ntutve propertes of cass C utty functons, and prove that any other cass of utty functons that TCP/IP optmzes necessary possess some strange propertes. We prove n Secton IV that, for genera networks, f the weght a s sma enough, ony mnmum-propagaton-deay paths are seected. Ths mpes that f a source-destnaton pars have unque mnmum-propagaton-deay paths, then equbrum of TCP/IP exsts and s gobay asymptotcay stabe. It s often beeved that decreasng a heps ensure routng stabty. We prove that ths may not be the case f not a source-destnaton pars have unque mnmum-propagatondeay paths. Indeed, for a genera network, ts equbrum and stabty propertes are the same as a modfed network whose routng s based on pure congeston prces p, a network that s prone to routng nstabty. More surprsngy, there exsts networks where reducng the weght a can destabze an orgnay stabe equbrum. It s conjectured n [26] that there s generay an nevtabe tradeoff between utty maxmzaton and stabty n TCP/IP networks. In partcuar, as the weght a ncreases, the routng s conjectured to become more unstabe but the achevabe utty hgher. We show however how to construct a network that has any gven utty profe as a functon of the weght a X/07/$ IEEE 418

2 II. MODEL We use the same mode as n [26]. In genera, we use sma etters to denote vectors, e.g., x wth x as ts th component; capta etters to denote matrces, e.g., H, W, R, or constants, e.g., L, N, K ; and scrpt etters to denote sets of vectors or matrces, e.g., W s, W m, R s, R m. Superscrpt s used to denote vectors, matrces, or constants pertanng to source, e.g., y, w, H, K. A. Network A network s modeed as a set of L undrectona nks shared by a set of N source-destnaton pars, ndexed by we w aso refer to the par smpy as source. Each nk has a fnte capacty c > 0 and a deay τ > 0 across the nk,.e., t takes τ to process and propagate a packet from one end of the nk to the other, excudng queueng deay. Let c =c, =1,...,L and τ =τ, =1,...,L. There are K acycc paths for source represented by a L K 0-1 matrx H where Hj 1, f path j of source uses nk = 0, otherwse Let H be the set of a coumns of H that represents a the avaabe paths to under snge-path routng. Defne the L K matrx H as H = [H 1... H N ] where K := K. H defnes the topoogy of the network. Let w be a K 1 vector where the jth entry represents the fracton of s fow on ts jth path such that w j 0 j and 1 T w =1 where 1 s a vector of an approprate dmenson wth the vaue 1 n every entry. We requre wj 0, 1} for sngepath routng, and aow wj [0, 1] for mut-path routng. Coect the vectors w, =1,...,N,ntoaK N bockdagona matrx W.LetW s be the set of a such matrces correspondng to snge path routng defned as W W = dagw 1,...,w N 0, 1} K N, 1 T w =1} Defne the correspondng set W m for mut-path routng as: W W = dagw 1,...,w N [0, 1] K N, 1 T w =1} As mentoned above, H defnes the set of acycc paths avaabe to each source, and represents the network topoogy. W defnes how the sources oad baance across these paths. Ther product defnes a L N routng matrx R = HW that specfes the fracton of s fow at each nk. Thesetofa snge-path routng matrces s R s = R R = HW,W W s } 1 and the set of a mut-path routng matrces s R m = R R = HW,W W m } 2 The dfference between snge-path routng and mut-path routng s the nteger constrant on W and R. A snge-path routng matrx n R s s an 0-1 matrx: 1, f nk s n a path of source R = 0, otherwse A mut-path routng matrx n R m s one whose entres are n the range [0, 1]: > 0, f nk s n a path of source R = 0, otherwse The path of source s denoted by r =[R 1... R L ] T, the th coumn of the routng matrx R. B. TCP AQM/IP We consder the stuaton where TCP AQM operates at a faster tmescae than routng updates. We assume a snge path s seected for each source-destnaton par that mnmzes the sum of the nk costs n the path, for some approprate defnton of nk cost. In partcuar, traffc s not spt across mutpe paths from the source to the destnaton even f they are avaabe. Ths modes, e.g., IP routng wthn an Autonomous System. We focus on the tmescae of the route changes, and assume TCP AQM s stabe and converges nstanty to equbrum after a route change. As n [17], we w nterpret the equbra of varous TCP and AQM agorthms as soutons of a utty maxmzaton probem defned n [12]. Dfferent TCP agorthms sove the same prototypca probem 3 wth dfferent utty functons; see e.g. [17], [19], [25] for the utty functons for varous popuar TCP proposas. Specfcay, suppose each source has a utty functon U x,d whch depends on both ts tota transmsson rate x and the end-to-end propagaton deay d. Gven a routng matrx R, we assume L d := d R = R τ Hence the deay d depends ony on routng R and not on congeston n the path. The routng matrx R s n R s for snge-path routng and n R m for mut-path routng. Note that n the mut-path case, d s the traffc-weghted average of propagaton deays aong ts paths. We assume that utty functons are strcty concave for fxed d. The speca case where the utty functon U x =U x,d depends on ts rate x but not on the deay d s studed n [26]. Here, we focus on the deay-senstve case. Gven a routng matrx R, U x,d s a functon ony of rate x.letrt R s be the snge-path routng n perod t. GvenaRt, et the equbrum rates xt = xrt and prces pt =prt generated by TCP AQM n perod t, respectvey, be the optma soutons of the constraned maxmzaton probem max x 0 U x,d s. t. Rtx c 3 and ts Lagrangan dua mn max U x,d x R tp + p 0 x 0 c p 4 419

3 The prces p t, =1,...,L, are measures of congeston, such as queueng deays or oss probabtes [17], [19]. We assume that the nk costs n perod t are z t = ap t+bτ 5 where a 0, b 0, and τ > 0 are constants. Based on these costs, each source computes ts new route r t +1 H ndvduay that mnmzes the sum of nk cost n ts path: r t + 1 = arg mn r H z tr 6 Reca that τ n 5 are propagaton deays across nks. If p t represents the queueng deays at nks and a = b =1, then z t represent tota deays across nks. The protoco parameters a and b determne the responsveness of routng to network traffc: a =0corresponds to statc routng, b = 0 corresponds to purey dynamc routng, and the arger the rato of a/b, the more responsve routng s to network traffc. They determne whether an equbrum exsts, whether t s stabe, and the achevabe utty at equbrum. The paper [26] focuses on the case of b =0; we study the genera case here. An equvaent way to specfy the TCP AQM/IP system as a dynamca system, at the tmescae of route changes, s to repace 3 4 by ther optmaty condtons. The routng s updated accordng to combnng 5 and 6 r t + 1 = arg mn r H ap t+bτ r, for a 7 where pt and xt are gven by [ ] + U R tp t = x t,d for a 8 c f p R tx t t 0 for a 9 = c f p t > 0 xt 0, pt 0 10 Ths set of equatons descrbe how the routng Rt, rates xt, and prces pt evove. Note that xt and pt depend on Rt ony through 8 10, mpcty assumng that TCP AQM converges nstanty to an equbrum gven the new routng Rt. We say that R,x,p s an equbrum of TCP/IP f t s a fxed pont of 3 6, or equvaenty, 7 10,.e., startng from routng R and assocated x,p, the above teratons yed R,x,p n the subsequent perods. C. The jont optmzaton probem Defnton 1: A deay-senstve utty functon s a contnuousy dfferentabe functon Ux, d from [0, [0, to [,, that satsfes the foowng propertes: 1 fxed d>0, Ux, d s strcty concave n x. 2 d >0,x>0, Ux, d and U x, d are fnte. 3 x > 0, d 1,d 2 s.t. 0 < d 1 < d 2 : Ux, d 1 > Ux, d 2. 4 D >0, 0 <d<d, Xd > 0, x <Xd : x, d > 0. U Essentay, a deay-senstve utty functon s defned so that the source aways gans utty from reducng propagaton deay. If propagaton deay s too hgh, the source can choose not to transmt. Otherwse, for fxed deay, the source s utty ncreases wth transmsson rate, possby up to some mt. We assume a sources on the network have deay-senstve utty functons U x,d. We adapt the snge-path deay-nsenstve network optmzaton probem from [26] to a deay-senstve network optmzaton probem: N L max U x, R τ s.t. Rx c 11 R R s,x 0 =1 Its Lagrangan dua s: N L mn max max L U x,d x R p + c p p 0 x 0 r =1 H 12 where r s the th coumn of R wth r = R. Ths probem maxmzes utty over both rates and routes. Defne the Lagrangan [1]: N L L LR, x, p = U x,d x R p + c p =1 Then we can express the prma and dua probems respectvey as: V sp = max max R R s V sd = mn max p 0 mn x 0 p 0 max R R s x 0 LR, x, p LR, x, p If we aow sources to use mutpe paths, the correspondng probems are: V mp = max max R R m V md = mn max p 0 mn x 0 p 0 max R R m x 0 LR, x, p LR, x, p The TCP/IP dynamca system s descrbed by 3 6, or equvaenty, D. Revew: deay-nsenstve utty functons In ths subsecton, we consder the speca case where the utty functons U x,d =U x depend ony on rates x but not on propagaton deays d. There are three sub-cases: a>0,b=0; a =0,b>0; a>0,b>0. For the frst case where a>0,b=0n 5,.e., IP uses ony congeston prces p generated by TCP AQM as nk costs, t s shown n [26] that TCP/IP maxmzes aggregate utty over both rates and routng when an equbrum exsts. Theorem 1 [26]: Suppose a>0 and b =0n 5. Then: 1 An equbrum R,x,p of TCP/IP exsts f and ony f there s no duaty gap between 11 and In ths case, the equbrum R,x,p s a souton of 11 and 12. Moreover, n that case, there s no penaty n not spttng the traffc among mutpe paths. Theorem 2 [26]: V sp V sd = V mp = V md. 420

4 One of the open questons rased n [26] s the characterzaton of TCP/IP equbrum n the other two cases where b>0 n 5,.e., when IP uses propagaton deay, excusvey or not, as nk costs. It s shown n [24] that for any deaynsenstve utty functon Ux, there exsts a network wth sources usng ths utty functon, where TCP/IP equbrum exsts but does not sove 11 and 12. See [24] for expct constructon of such networks. We now show that TCP/IP turns out to maxmze a cass of deay-senstve utty functons when b>0. III. DELAY-SENSITIVE NETWORK OPTIMIZATION In ths secton, we consder the case where the utty functons U x,d depend both on rates x and on propagaton deays d. We dentfy a cass C of utty functons for whch TCP/IP, wth a > 0 and b = 1 n 5, does maxmze aggregate utty at equbrum, when equbrum exsts. We anayze the propertes of C, and then derve propertes that any cass of utty functons that TCP/IP mpcty maxmzes at equbrum must possess. We start wth the case where a =0or b =0n 5,.e., f a nks use ony the propagaton deays τ as nk costs, or f a nks use ony the congeston prces p generated by TCP AQM as nk costs. In ths case, t can be shown that for every deay-senstve utty functon Ux, d, there exsts a network wth sources usng ths utty functon, where TCP/IP equbrum exsts but does not sove 11 and 12. See [24] for expct constructon of such networks. Hence we consder the case where both a>0 and b>0. A. Reverse engneerng for nk cost ap + τ In ths subsecton, we assume that IP uses ap + τ as nk cost,.e., a>0 and b =1n 5. Consder the cass C of functons Ux, d that can be wrtten as: Ux, d =V x a 1 xd where V x s a contnuousy dfferentabe functon from [0, to [, so that V x s strcty concave ncreasng, and x >0, V x and V x are fnte. In [24] we show that functons n C are deay-senstve utty functons defned n Defnton 1. Every utty functon n C has two components: V x measures the beneft of throughput x; a 1 xd measures the penaty of deay d weghted by the throughput x. Note that a arger throughput x ncreases both the beneft of throughput and the penaty due to deay. The reatve mportance s determned by the reatve weght a on congeston prce n the nk cost used n routng decsons: the arger the weght a, the more mportant the throughput beneft and the ess mportant the deay penaty. In the mtng case as a, correspondng to pure dynamc routng.e., b = 0, the utty functon become deay-nsenstve and our resuts beow reduce to those estabshed n [26] for the b =0case. Specazng to utty functons n C, the optmzaton probem 11 reduces to: N L max V x a 1 x R τ s.t. Rx c R R s,x 0 =1 Consder the Lagrangan LR, x, p = N L L V x x R p + a 1 τ + p c and the dua probem Dp := max R Rs,x 0 LR, x, p: Dp = N max x 0 =1 [V x x mn r H L L R p + a 1 τ ] + p c The mnmzaton over R n the dua probem appears to nvove mnma-cost routng usng p + a 1 τ as route cost. But ths s the same as mnma-cost routng usng ap + τ as route cost. Ths suggests that TCP/IP mght sove the jont optmzaton probem wth utty functons n C. Ths s ndeed the case. Our frst man resuts, and ther proofs, are anaogous to Theorem 1 and Theorem 2 but for b>0 and deay-senstve utty functons. They say that the equbrum of TCP/IP, when exsts, soves the jont utty maxmzaton over routes and rates and ts dua probem. Moreover, n that case, there s no penaty n not spttng traffc among mutpe paths. Ther proofs are n [24]. Theorem 3: Suppose a utty functons are n C, a>0 and b =1. 1 An equbrum R,x,p of TCP/IP exsts f and ony f there s no duaty gap between 11 and In ths case, the equbrum R,x,p s a souton of 11 and 12. Theorem 4: Suppose utty functons are n C. Then V sp V sd = V mp = V md. B. Counter-ntutve propertes of cass C In ths subsecton, we exhbt some counter-ntutve propertes of the cass C of utty functons. Specfcay, we present exampe networks where, because the penaty term a 1 xd s proportona to throughput x, these utty functons can underutze nk capactes or avaabe network paths. The frst exampe ustrates a network equbrum whch s n the strct nteror of the feasbe set Rx c, contrary to what the tradtona TCP mode wth deay-nsenstve utty functons woud predct. Remark 1: Gven any utty functon n C, there exsts a network where TCP/IP underutzes nks. Proof: U x, d =V x a 1 d.ifv x a 1 d =0, then x s the rate that maxmzes Ux, d for fxed d, snce Ux, d s strcty concave for fxed d. Choose any c > 0 and set τ = av c. Note that τ > 0, snce V x s strcty ncreasng. Consder a network wth one nk, whose capacty s 2c. A fow whose path s just ths nk w have rate c at equbrum, snce U x, d = V c a 1 τ = V c V c =0. But ths eaves the nk underutzed, snce the nk has capacty 2c. The second exampe shows that extra paths that woud be utzed f the utty functons were deay-nsenstve may not 421

5 Src Fg. 1. Network 1 L1: Capacty: c 1, Deay: τ 1 L2: Capacty: c 2, Deay: τ 2 Dest be utzed by utty functons n C. It aso ustrates Theorem 3 and Theorem 4. Consder any Ux, d = V x a 1 xd n C. It can be shown that there exst τ > 0, c>0 so that U c, τ > 0. Consder Network 1 n Fgure 1 wth c 1 = c, τ 1 = τ,c 2 =,τ 2 = τ + c, τ. Suppose there s ony one fow, Fow 1, and t can choose between routes R1: L1 and R2: L2. Suppose Fow 1 s ntay on route R1. It acheves rate c wth propagaton deay τ and utty Uc, τ. Then R1 has route cost τ + c, τ and R2 has route cost τ +a U c, τ. The nta routng s an equbrum routng snce a fows are usng mnma cost routes. Theorem 3 then mpes that there s no duaty gap. Theorem 4 then mpes that V sp = V mp and there s no beneft n mut-path routng,.e., there s no beneft n utzng route R2. Ths seems counter-ntutve wth a deay-nsenstve networks, there s aways a beneft n utzng prevousy unutzed routes. Indeed, we can show drecty that utzng R2 ncreases the average propagaton deay experenced by the fow, whch turns out to be suboptma regardess of the amount of extra throughput. Remark 2: Gven any utty functons n C, t s suboptma to use route R2 n Network 1 wth c 1 = c, τ 1 = τ,c 2 =,τ 2 = τ + c, τ, even when the fow s aowed to dstrbute ts traffc over mutpe paths. Proof: Let kc specfy the throughput on nk L2, so that k k+1 specfes the fracton of traffc sent over nk L2 where the rest s sent over nk L1. Then the tota throughput s gven by kc + c, and the weghted average propagaton deay s k [ τ + c, τ] + τ k +1 We cam that for every k>0, wemusthave Uc, τ U kc + c, k [ τ + c, τ] + τ k +1 To verfy ths, pug n our utty functon: V x a 1 cτ U kc + c, k τ + c, τ + τ k +1 V kc + c a 1 c kav c+τ V V kc + c V c c kc whch s true snce V x s concave. C. Aternatve casses We have shown n the ast subsecton some counter-ntutve propertes of cass C utty functons. We now show that any cass of deay-senstve utty functons that TCP/IP optmzes, usng ap + τ propertes. Defne as nk cost, must possess some strange MU, d := m Uc, d, 13 c whch computes the maxmum possbe utty at each deay d for any deay-senstve utty functon Ux, d. The next resut says that any cass B of utty functons that TCP/IP optmzes ether contan functons that are not strcty ncreasng n throughput contrary to the tradtona deay-nsenstve utty functons of TCP modes, or are dscontnuous n throughput n the space of utty functons, or can be dscontnuous n deay for arge enough deay. Theorem 5: Suppose B s any cass of deay-senstve utty functons such that when TCP/IP equbrum exsts, the equbrum soves 11 and 12 wth utty functons n B. Then B must have at east one of the foowng three propertes: 1 Ux, d B,d > 0 so that Ux, d s not strcty ncreasng n x. 2 U 1 x, d B, ɛ >0, wehaveu 2 x, d :=U 1 x + ɛ, d s not n B. 3 Ux, d B, D>0such that fd :=MU, d s fnte and dscontnuous for a d>d. Proof: See [24]. In partcuar, t can be checked that cass C utty functons possess the frst two propertes. IV. STABILITY AND UTILITY OF ROUTING POLICIES In ths secton, we anayze the effects on stabty and utty of dynamc routng. A. Suffcent condton for stabty Denote the set of paths G H wth mnma propagaton deay for fow by: } G := r H : τ T r = mn τ T s s H Denote the set of paths F H wthout mnma propagaton deay for fow by: F := H G Defne qr, a functon that computes the equbrum congeston prce vector for a gven routng matrx R R s. We assume that t mpcty depends on an arbtrary, fxed network L, N, F, G, H, R s,k,u,τ,c: N qr :=arg mn max U x,d p T Rx + p T c p 0 x 0 In ths subsecton, we show that wth suffcenty sma a,a fows w choose ony mnma propagaton deay paths. For notatona smpcty, a of the foowng functons, emmas, 422

6 and theorems n ths subsecton mpcty depend on an arbtrary, fxed network L, N, F, G, H, R s,k,u,τ,c. Defne hx, y, a functon that w be used to smpfy notaton: x y f y>0 hx, y = f y 0 Defne a # as foows: max mn bm,r a # := mn mn m G r F R R s 0< N f F f F = where bm,r :=h τ T r m,qr T m r. It can be checked that a # s strcty postve. Theorem 6: Suppose a<a #. Then R R s, 0 < N, m G such that aqr+τ T m < aqr+τ T r, r F 14 In other words, a fows on the network w choose a path wth mnma propagaton deay n the next teraton. Proof: We manpuate the rght hand sde of 14: aqr+τ T m < aqr+τ T r aqr T m r <τ T r m But by nspectng the defnton of hx, y, ths nequaty hods f a<h τ T r m,qr T m r snce τ T r m > 0 f m G and r F. The forma part of the theorem s then easy to see. It mpes that for any current routng, and for every fow, a path wth mnma propagaton deay has strcty ower cost than any path wthout mnma propagaton deay. Therefore no fow w seect any path wthout mnma propagaton deay. Therefore every fow w seect a path wth mnma propagaton deay. Theorem 7: Suppose a source-destnaton pars on a network have unque mnmum-propagaton-deay paths. Then f a<a #, TCP/IP has an asymptotcay stabe equbrum. Proof: Each fow ony has one path wth mnma propagaton deay. Appyng Theorem 6, each fow w aways seect the same path, and t w aways do ths from any routng. Coroary 1: Suppose every path n a network has dfferent propagaton deay. Then f a<a #, TCP/IP has an asymptotcay stabe equbrum. Proof: If every path n the network has dfferent propagaton deay, every source-destnaton par on a network has a unque mnmum-propagaton-deay path, so the resut of Theorem 7 appes. B. Counter-ntutve propertes: stabty It s often beeved that decreasng the weght a on the traffc senstve component of nk cost can aways stabze dynamc routng; e.g., see [3], [16], [26]. In ths subsecton, we show that ths s generay not true. It s easy to see that not a networks can be stabzed by decreasng a. For nstance, consder Network 1 wth c 1 = c 2 = c>0, τ 1 = τ 2 = τ>0. Suppose there s one fow, Fow 1, that can choose between routes R1: L1 and R2: L2. Ifthas a deay-nsenstve utty functon Ux, then for a a > 0, ths network has no equbrum. The next two resuts are ess obvous. The frst says that f a s sma enough, then a network wth routng based on ap + τ behaves ke a modfed network wth routng based on p, whch seems prone to routng nstabty [26]. Theorem 8: Gve a network wth a < a #. Consder the modfed network obtaned by deetng a paths wthout mnma propagaton deay from the orgna network. Then the orgna network wth routng based on ap + d has the same equbrum and stabty propertes as the modfed network wth routng based on p. Proof: Consder the TCP/IP dynamca system on the modfed network when nk costs are p : pt T r t + 1 = mn r G ptt r, for a pt,xt = arg mn max N U x,d p T Rtx+p T c p 0 x 0, t : r t G Consder the TCP/IP dynamca system on the orgna network when nk costs are ap + τ : apt+τ T r t + 1 = mn r H apt+τ T r, for a. pt,xt = arg mn max N U x,d p T Rtx+p T c p 0 x 0, t : r t H The next emma, whose proof s n [24], mpes that these dynamca systems are equvaent. Snce they are equvaent, they share the same equbrum and stabty propertes. Lemma 1: Suppose we have some network, and routng pocy on ths network s such that a<a #. Then for any attanabe prce vectors p so that p = qr for some R R s, and for a 0 < N, f and ony f p T r = mn s G pt s, where r G 15 ap + τ T r = mn s H ap + τ T s, where r H 16 The second counter-ntutve resut says that t s possbe to destabze a network by decreasng the statc component a n nk cost. Theorem 9: Consder any deay-senstve or deaynsenstve utty functon Ux, d. There exsts a network wth sources usng ths utty functon, and constants a 3 >a 2 a 1 > 0 so that: 1 The network s stabe for a 0,a

7 2 The network s unstabe for a a 2,a 3. 3 The network s stabe for a a 3,. We w prove the theorem by exhbtng such a network. Suppose Ux, d s an arbtrary deay-senstve or deaynsenstve functon. It can be shown [24] that t s possbe to choose parameters c 1,c 2,τ 1,τ 2 that satsfy the foowng nequates: U c 1,τ 1 > U c 2,τ 2 > 0 17 c 2 >c 1 > 0 18 τ 2 >τ 1 > 0 19 Consder Network 1 wth those parameters. Suppose there are two fows. Suppose the possbe routes are R1: L1 and R2: L2. Suppose that Fow 1 s constraned to R1, and Fow 2 can choose between R1 and R2. Denote ths nstance of Network 1byNetwork2. Defne functon A 3 U, c 1,τ 1,c 2,τ 2, where U s a deaynsenstve or deay-senstve utty functon and the rest of the parameters are n R, as foows: τ 2 τ 1 A 3 U, c 1,τ 1,c 2,τ 2 := U c 1,τ 1 U c 2,τ 2 Lemma 2: Consder any deay-senstve or deay-nsenstve utty functon Ux, d. Suppose c 1,τ 1,c 2,τ 2 satsfy 17 through 19 wth Ux, d. Network 2 wth a sources usng Ux, d s stabe for a a>a 3 U, c 1,τ 1,c 2,τ 2. Proof: We show that routng converges from every possbe nta condton. Suppose Fow 2 s on route R1. Then Fow 1 and Fow 2 share L1 equay, and both acheve rate c1 2 wth propagaton deay τ 1 and utty U c1 2,τ 1. Then R1 has route cost c1 2,τ 1+τ 1 and R2 has route cost τ 2. It can be verfed that a>a 3 U, c 1,τ 1,c 2,τ 2 mpes that: c1 1 2,τ + τ 1 >τ 2 So R2 s a ower cost route than R1. To show asymptotc stabty, t s then suffcent to show that the routng where Fow 2 s on R2 s an equbrum. Suppose Fow 2 s on R2. Then Fow 1 acheves rate c 1 wth propagaton deay τ 1 and utty Uc 1,τ 1, and Fow 2 acheves rate c 2 wth propagaton deay τ 2 and utty Uc 2,τ 2. Then R1 has route cost c 1,τ 1 +τ 1 and R2 has route cost c 2,τ 2 +τ 2. Consder Fow 2 s route choce at the next routng teraton. It can be verfed that a>a 3 U, c 1,τ 1,c 2,τ 2 mpes that c 1,τ 1 +τ 1 > c 2,τ 2 +τ 2 So R2 s a ower cost route than R1. Therefore R2 s mnma cost, and so ths routng s an equbrum routng. Defne functon A 2 U, c 1,τ 1,c 2,τ 2, where U s a deaynsenstve or deay-senstve utty functon and the rest of the parameters are n R as foows: A 2 U, c 1,τ 1,c 2,τ 2 := τ 2 τ 1 U c1 2,τ 1 Lemma 3: Consder any deay-senstve or deay-nsenstve utty functon Ux, d. Suppose c 1,τ 1,c 2,τ 2 satsfy 17 through 19 wth Ux, d. Then A 2 U, c 1,τ 1,c 2,τ 2 < A 3 U, c 1,τ 1,c 2,τ 2. Proof: See [24]. Lemma 4: Consder any deay-senstve or deay-nsenstve utty functon Ux, d. Suppose c 1,τ 1,c 2,τ 2 satsfy 17 through 19 wth Ux, d. Network 2 wth a sources usng Ux, d s unstabe for a a satsfyng A 2 U, c 1,τ 1,c 2,τ 2 < a<a 3 U, c 1,τ 1,c 2,τ 2. Proof: Suppose Fow 2 s on R2. Then Fow 1 acheves rate c 1 wth propagaton deay τ 1 and utty Uc 1,τ 1, and Fow 2 acheves rate c 2 wth propagaton deay τ 2 and utty Uc 2,τ 2. Then R1 has route cost c 1,τ 1 +τ 1 and R2 has route cost c 2,τ 2 +τ 2. Consder Fow 2 s route decson at the next routng teraton. It can be verfed that a<a 3 U, c 1,τ 1,c 2,τ 2 mpes c 1,τ 1 +τ 1 < c 2,τ 2 +τ 2. So Fow 2 next chooses route R1. Then Fow 1 and Fow 2 share L1 equay, and both acheve rate c1 2 wth deay τ 1 and utty U c1 2,τ 1. Then R1 has route cost c1 2,τ 1+τ 1 and R2 has route cost τ 2. It can be verfed that a>a 2 U, c 1,τ 1,c 2,τ 2 mpes c 1 2,τ 1+τ 1 >τ 2. So Fow 2 next chooses route R2. Ths mpes that Fow 2 s routng oscates between R1 and R2, so the network s unstabe. Proof Theorem 9. We choose c 1,τ 1,c 2,τ 2 so that they satsfy 17 through 19. Then consder Network 2 wth a sources usng Ux, d. Set a 1 = a # for ths network. Theorem 6 mpes that for a<a 1, the network s stabe. We then set a 2 = A 2 U, c 1,τ 1,c 2,τ 2 and a 3 = A 3 U, c 1,τ 1,c 2,τ 2.The emmas n ths subsecton then estabsh the desred resut. C. Counter-ntutve propertes: utty The paper [26] anayzes the effects of ncreasng a on the tme-averaged aggregate utty for a rng network, and a randomy generated network. On the rng network, tmeaveraged aggregate utty approached the maxmum possbe tme-averaged aggregate utty for any a, as a ncreased. On the generated network, tme-averaged aggregate utty ncreased unt routng stabty set n, and then decreased. In ths subsecton, we show that the effects of ncreasng a on tme-averaged aggregate utty are network dependent. In partcuar, we show how to construct a network wth any gven utty profe as a functon of the weght a. For every deay-nsenstve Ux, there exst k + >k > k > 0 and g > 0 so that g = Uk + Uk = Uk Uk. Aso, for every z [ 1, 1], there exsts k [k,k + ] so that Uk Uk =zg, that s gven by kz :=U 1 Uk +zg. Ths s easy to see snce Ux s strcty monotone ncreasng. 424

8 Consder the foowng network Nj, z, parameterzed by j>0, and z [ 1, 1], whch s defned to be Network 1 wth parameters c 1 = k,τ 1 = ju k,c 2 = kz,τ 2 =2jU k, wth one fow, Fow 1, choosng between routes R1: L1 and R2: L2. Denote the tme-averaged utty of the network under routng pocy ap + d by T N,a. Lemma 5: For every j>0, z [ 1, 1], network Nj, z has the foowng propertes: 1 a 1 [0,j],a 2 [0,j]: T N,a 1 =TN,a 2 2 a 1 j,,a 2 j, : T N,a 1 =TN,a 2 3 a 1 0,j],a 2 j, : T N,a 2 =TN,a 1 + zg 2 Proof: Suppose current routng s R2. Then R1 has route cost ju k and R2 has route cost 2jU k +au kz. By nspecton, for every a 0, R1 has ess cost than R2. Therefore routng on the next teraton w be R1. Suppose current routng s R1. Then R1 has route cost ju k +au k and R2 has route cost 2jU k. By nspecton, f a j then R1 s a ower cost route than R2, and f a>j, then R2 s a strcty ower cost route than R1. Fow 1 s utty on route R1 between routng changes s Uk. Fow 1 s utty on route R2 between routng changes s Ukz. If a j, then the network s stabe wth Fow 1 on R1 and the aggregate utty s Uk.Ifa > j, then the network oscates between R1 and R2 and acheves tmeaveraged utty Ukz + Uk /2. The excess utty ganed by ncreasng a from ess than j to more than j s gven by: Ukz + Uk 2 Uk = Ukz Uk 2 = zg 2 Defnton 2: A utty-versus-a profe s a par of vectors x, y such that x = y > 0, : x > 0, and < x : x <x +1. Defnton 3: A network N matches a utty-versus-a profe x, y f there exsts λ>0 so that: If x > 1, 1 s.t. 1 < < x, a 1 s.t. x 1 < a 1 x, a 2 s.t. x <a 2 <x +1 : T N,a 2 T N,a 1 =λy. 2 a 1 s.t. 0 < a 1 x 1, a 2 s.t. x 1 < a 2 < x 2 : T N,a 2 T N,a 1 =λy 1. 3 a 1 s.t. x x 1 <a 1 x x, a 2 s.t. x x <a 2 < : T N,a 2 T N,a 1 =λy x. If x =1, a 1 s.t. 0 <a 1 x 1, a 2 s.t. x 1 <a 2 < : T N,a 2 T N,a 1 =λy 1. In other words, for a, the tme-averaged aggregate utty of the network ncreases by λy at a = x. Theorem 10: For every utty-versus-a profe x, y, there exsts a network wth sources usng deay-nsenstve utty functons that matches ths profe. Proof: Consder any utty-versus-a profe x, y. Defne the normazed y as y := max y 1 y. Construct the network N by takng the unon of networks N 1...N y where, N := Nx,y. The subnetworks are entrey dsjont n the unon network. It s easy to see that the network N matches profe x, y g wth λ = 2 max v. V. CONCLUSION In ths paper, we have attempted to reverse engneer TCP/IPke networks. We have dentfed a cass of deay-senstve utty functons that s mpcty optmzed by an equbrum of TCP/IP. We have characterzed ts equbrum and stabty propertes for genera networks, and exhbted severa counterntutve resuts. Many ssues are st open. Frst, we beeve C s the ony cass of utty functons that TCP/IP wth nk cost ap + d jonty optmze, but we have not been abe to prove ths. Second, the deay-nsenstve utty functons obtaned from reverse engneerng TCP agorthms n the terature, assumng routng s fxed, are a strcty ncreasng n throughput. Hence the deay-senstve utty functons obtaned from reverse engneerng TCP/IP shoud deay be strcty ncreasng n throughput when routng s hed fxed. However, ths s not the case wth cass C utty functons. Ths msmatch shoud be reconced. REFERENCES [1] D. Bertsekas. Nonnear Programmng. Athena Scentfc, [2] D. Bertsekas, E. Gafn, and R. G. Gaager. Second dervatve agorthms for mnmum deay dstrbuted routng n networks. IEEE Trans. on Communcatons, 328: , [3] Dmtr P. Bertsekas. Dynamc behavor of shortest path routng agorthms for communcaton networks. IEEE Transactons on Automatc Contro, pages 60 74, February [4] S. Deb and R. Srkant. Congeston contro for far resource aocaton n networks wth mutcast fows. In Proceedngs of the IEEE Conference on Decson and Contro, December [5] R. G. Gaager. A mnmum deay routng agorthm usng dstrbuted computaton. 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