Stability and Benefits of Suboptimal Utility Maximization

Transcription

1 Stablty and Benefts of Suboptmal Utlty Maxmzaton Tan Lan, Xaojun Ln 2, Mung Chang, Ruby Lee Department of lectrcal ngneerng, Prnceton Unversty, NJ 08544, USA 2 School of lectrcal and Computer ngneerng, Purdue Unversty, IN 47907, USA Abstract Network utlty maxmzaton has been wdely used to model resource allocaton and network archtectures. But n practce often t cannot be solved optmally due to complexty reasons. Thus motvated, we address the followng two questons n ths paper: can suboptmal utlty maxmzaton mantan queue stablty? Can under-optmzaton of utlty objectve functon n fact benefts other network desgn objectves? We quantfy that a resource allocaton that s suboptmal wth respect to a utlty maxmzaton formulaton mantans the maxmum flowlevel stablty when the utlty gap s suffcently small and the nformaton delay s bounded, and can stll provde a guaranteed sze of stablty regon otherwse. Utlty-suboptmal rate allocaton can also enhance other network performance metrcs, e.g., t may reduce lnk saturaton. Quantfyng these ntutons, ths paper provdes a theoretcal support for turnng attenton from optmal but complex solutons of network optmzaton to those that are smple even though suboptmal. I. INTRODUCTION The framework of Network Utlty Maxmzaton NUM has been very extensvely studed over the last decade snce. Formulatng many resource allocaton problems as maxmzaton of an ncreasng and concave utlty functon over a convex constrant set, a large number of publcatons have developed teratve, dstrbuted algorthms that converge to the optmum. Achevng optmalty s clearly desrable for two reasons. Not only does ths attan the benchmark of the hghest value of network utlty, t also guarantees flow-level stochastc stablty. The number of flows vares over tme as they are randomly generated by users and served by the network. Ths system can be vewed as a queung system where the servce rate depends on the resource allocaton e.g., rate control polcy employed by the network. For convex NUM and under the assumptons of Posson arrvals, exponentallydstrbuted fles szes, and zero nformaton delay.e. perfect queue-length nformaton, t has been shown that for all rate allocaton polces maxmzng α-far utltes wth α > 0, flow-level stochastc stablty can be acheved f and only f the traffc ntensty les wthn the rate regon, see, e.g., 2, 3, 5, 4. In other words, rate regon n the α-far utlty maxmzaton problem s also the maxmum stablty regon under arrval and departure dynamcs. Utlty-optmalty and flow-level stablty are strong benefts of optmzng NUM. However, n practce t s often prohbtve Ths work has been n part supported by the Natonal Scence Foundaton through awards CCF , CNS , CNS , CCF , CNS , CNS , and ONR YIP award N An earler verson of ths paper has appeared n I Infocom to solve NUM optmally, due to computatonal complexty and nformaton delay. The mpact of suboptmal soluton s not well-studed n the exstng lterature. It s of practcal mportance to sharpen our understandng for two reasons: All rate-control algorthms requre a non-neglgble amount of executon tme before they can reach the optmal rate allocaton that maxmzes the system utlty. In fact, most dstrbuted rate-control algorthms cannot reach the optmal rate allocaton n any fnte number of teratons. The tme for the teratons to converge s often so long that the network states, e.g., the composton of the user populaton, change many tmes before convergence occurs. As a result, practcal rate allocaton algorthms are subject to a postve, and possbly random, tme delay. Further, snce we cannot afford to run the algorthms untl t converges, each nstance of the utlty maxmzaton problem can only be solved suboptmally. In wreless networks, the schedulng problem has an exponental computatonal complexty despte the fact that the rate regon of the system s convex. For example, when the feasble rate regon of a network s obtaned by tme-sharng among dfferent subsets of users, a nonconvex mult-user/lnk schedulng problem stll needs to be solved n order to fnd the exact rate regon acheved by tme-sharng 6. Such hgh computatonal complexty further ncreases the amount of tme that s needed to reach the optmal rate allocaton. In addton, many cross layer optmzaton algorthms that mplement rate schedulers as an nner loop requre the schedulng teratons to stop at some suboptmal pont,for example, due to tmescale separaton assumpton. There are many theoretcal studes that nvestgate the use of lowcomplexty and even dstrbuted schedulng algorthms. However, wth these lower-complexty schedulng algorthms the rate-allocaton algorthm wll ether take longer convergence tme 7, or wll not converge to the optmal rate-allocaton any more 6. In ether case, f we are lmted to a fnte number of teratons, suboptmalty n the rate-allocaton becomes the only realstc outcome. The gap between elegant theory and useful practce thus leads us to the followng queston: between optmalty and smplcty, whch one should we pck n solvng NUM? Drven by the practcal need for smple yet suboptmal solutons, we focus on suboptmal utlty maxmzaton, and then quantfy the effects of nformaton delay and utlty-gap on flow-level stablty, and on other mportant network performance metrcs,

2 2 such as lnk saturaton. In 8, the authors show that for a class of rate allocaton algorthms based on so-called dual solutons, the optmal stablty regon can be acheved even f the algorthm does not converge to the optmal rate allocaton at any tme. Smlar observatons have also been made n swtchng 9 and schedulng 0 problems. In ths paper, we take a dfferent approach. We characterze the capablty of a resource allocaton algorthm by two features: the gap between ts utlty and the optmal utlty; and 2 the tme delay of the queue-length nformaton. We study stablty as a functon of both the utlty gap and the nformaton delay. Our results apply to a class of general NUM formulatons, n whch the flow-level queueng models are not frst-order Markov, thus makng our proof technque of ndependent nterest to general flow-level queueng model. Intutvely, one would thnk that the maxmum stablty regon may be retaned f the utlty gap s small and the tme delay s bounded, whle only a reduced stablty regon can be acheved when the utlty gap becomes large. Ths s ndeed true. In Secton III, we show that when nformaton delay s unformly bounded by a constant and the rato of the utlty gap caused by a suboptmal rate allocaton polcy to the maxmum utlty approaches zero as queue length tends to nfnty, the maxmum stablty regon can be retaned. However, when the utlty gap s proportonal to the maxmum utlty, only a reduced stablty regon can be acheved. In ths case, we can stll provde a lower bound for the achevable stablty regon under rate allocaton polces satsfyng the nformaton delay and the utlty gap condtons. These results characterze the stablty of a broad class of suboptmal rate allocaton polces. When nformaton delay s bounded, snce suboptmal rate allocatons wth a small enough utlty gap are capable of achevng the maxmum stablty, we nvestgate the potental benefts of allowng such a utlty gap,.e., the upsde of underoptmzng utlty objectves. It s clear that by delberately under-optmzng a utlty, we can mprove network performance n other metrcs. What remans unclear s precsely how much mprovement we can possbly acheve by underoptmzng the utlty wth a gven allowable gap. We formulate the potental performance mprovement as a functon of utlty gap, and derve a frst-order approxmaton for the tradeoff curve based on local senstvty shadow prce analyss. Ths formulaton generalzes that n, whch focuses on how network performance can be affected by the choce of α- far utlty and assumes that optmalty always holds. Our result not only llustrates the potental benefts of underoptmzng a utlty, but also quanttatvely characterzes the tradeoff between sacrfcng utlty value and mprovng other network performance metrcs, e.g. lnk saturaton. Our analyss can be easly extended beyond the class of α-far utltes. The results n ths paper explore a new perspectve to look at suboptmal solutons of the utlty maxmzaton problem. We show that suboptmal rate-allocaton polces may not always be nferor n performance. More precsely, by underoptmzng a utlty and allowng a certan optmzaton gap, we can stll retan the maxmum flow-level stablty and obtan network performance mprovements n other metrcs. The rest of the paper s organzed as follows: In Secton II, we ntroduce the class of utlty functons consdered n ths paper and defne the utlty gap for suboptmal rate allocatons. Two stablty results are stated next. In Secton III.A, a suffcent condton on utlty gap and nformaton delay for achevng the maxmum flow-level stablty s provded. In Secton III.B, when the utlty gap s proportonal to the maxmum utlty, we show that the achevable stablty regon can be strctly smaller, and we further obtan a lower bound for all achevable stablty regons. In Secton IV, we analyze the tradeoff between the utlty gap and lnk saturaton. Results based on senstvty analyss are derved to measure the benefts of under-optmzng α-far utlty. Smulaton results are provded at the end of secton III and IV respectvely. For smoother flow of the man results, we collect all the proofs n the appendces. Throughout ths paper, we use the followng notatons: Vectors are denoted n small letters, e.g., x, wth ther th component denoted by x. Matrces are denoted by captalzed letters, e.g., A, wth A j denotng the {, j}th component. Vector nequaltes denoted by x y are consdered componentwse. The superscrpt T denotes the matrx transpose. PM s the probablty of an event M. We use R to denote a set of vectors and Ř for ts nteror. II. UTILITY MAXIMIZATION AND GAP Consder a communcaton network shared by a set of data flows, whch belong to N dstnct flow classes. We refer to the vector x = x,..., x N as the network state, where x denotes the number of flows of class that reman n the system. The problem of rate allocaton s to determne the total rate allocated to class- flows at state x, denoted by φ x. Rate φ x s equally shared by all class- flows, each assgned rate φ x/x. We refer to the vector φx = φ x,..., φ N x as the rate allocaton at state x. Let R R N + be a set of all possble rate allocaton vectors. Rate allocaton φx s restrcted by φx R, whch means that the network can support the rate vector φx. In ths paper, we only requre the set R to be convex, compact, and coordnate-convex, whch holds n many settngs, e.g., 3, 6. Varous network rate control polces can be derved as solvng some utlty maxmzaton problem wth dfferent utlty functons: φ opt x = argmax φ R :x x U φ x where U s a utlty functon for flow class. In ths paper, we assume that the functon U s contnuous and twce dfferentable on 0, +. In addton, we assume the utlty functons satsfy the followng condtons: a U z 0 z and U 0 = 0, or U z 0 z,. b U z s concave and monotoncally ncreasng. c lm z 0 U z =,. d There exsts s, s.t. zu z U z s, z,. Assumptons a and b are commonly used n the lterature 4. Assumpton c can be nterpreted as one that prevents A set R s coordnate convex when the followng s true: If x R, then y R for all y component-wsely less than x.

3 3 starvaton, snce t mples that slope of the utlty functon ncreases to nfnty as the rate of the flow class approaches zero. Condton d requres that the utlty functon does not have sharp changes. One example of such utlty functons satsfyng assumptons a-d s a class of the well-used α-far utlty functons 2, defned by { z α U z = α, α > 0 and α 2 log z, α = where α s a postve constant. It s easy to verfy that the assumptons a-d are satsfed wth s = α. Parameter α 0 models the level of farness, whch ncludes several specal cases such as proportonal farness and max-mn farness. For example, maxmzng the total utlty corresponds to maxmzng weghted throughput as α 0, weghted proportonal farness as α =, mnmum potental delay as α = 2 and max-mn farness as α 3. It may be mpractcal to solve the rate allocaton problem optmally for all network states. In ths paper, we consder a more general scenaro where rate allocatons are not optmal and thus could possbly reduce network performance. Ths work s motvated by the followng two ssues n practcal networks: Frst, all practcal rate allocaton polces are subject to a postve delay due to the tme requrement for gatherng network nformaton and for algorthm convergence. In other words, the practcal rate allocaton vector φˆx can at best correspond to the optmal rate allocaton φ opt ˆx for some vector ˆx, where ˆx x s the network state observed by a practcal rate allocaton polcy. Second, due to computatonal overhead or requrement for dstrbuted computaton, even gven the network state ˆx, a practcal rate allocaton polcy may stll not be able to solve the NUM problem optmally. In other words, there may exst a utlty gap due to the suboptmalty of the rate allocaton polcy. We quantfy the suboptmalty of a practcal rate allocaton φˆx wth respect to state ˆx by a utlty gap as follows ˆx = :ˆx ˆx U φopt, ˆx ˆx φ ˆx ˆx U 3 ˆx The gap ˆx measures the dfference between suboptmal rate allocatons and the optmal allocaton, caused only by the mperfect computaton of the rate allocaton algorthm. Gven certan condtons on the utlty gap ˆx and a model for the observed network state ˆx, n secton III we wll characterze the stablty regon of networks wth an arbtrary suboptmal rate allocaton polcy. In secton IV, we wll formulate and analyze the tradeoff between utlty gap and two network performance metrcs. III. INFORMATION DLAY AND STABILITY Consder a network where class- flows arrve as a Posson process of ntensty λ 0 and have..d. exponental fle szes of mean /µ. A flow s consdered to have left the network when ts fle transfer s completed. Let ρ = λ /µ be the traffc ntensty of class- flows. We formulate a stochastc process of the network state, denoted by xt. To model the observed network state ˆxt at tme t, we ntroduce an nformaton delay process τt, such that the observed state of class- flows s gven by ˆx t = x t τ t, for an nformaton delay τ t. Snce network nformaton may arrve out of order, nformaton delay τ t s not necessarly ncreasng over tme t. Therefore, the rate allocaton at tme t s gven by φˆxt = φx t τ t. 4 We say that flow-level stablty holds under rate allocatons φˆxt f there exsts a postve non-decayng functon f wth lm z fz =, such that the resultng queue-length process xt satsfes lm sup T T T 0 N fx t dt <. 5 = The stablty condton n 5 s usually refer to as stablty n the mean 0, 5. If we further assume that the rate allocaton polcy and the nformaton delay make the queueng system an aperodc Markov chan wth a sngle communcatng class whch s the case where there s no nformaton delay and utlty gap, then the stablty n the mean property n 5 further mples that the Markov chan s postve recurrent 5. Remark : For ease of exposton, n the above model we have assumed that the actual rate allocaton s a gven functon φ of the observed network states ˆx. The result of ths paper could also be extended to the case when ths functon φ s replaced by a functon φ t that also vares wth tme. In that case, ˆx n 3 can be defned as an upper bound on the utlty gap caused by the functons φ t over all tme t. When there s no utlty gap or nformaton delay, flowlevel stablty has been studed n 2, 3, 4, 5 usng frstorder Markov models. If the feasble rate regon R s compact and convex, and the optmal rate allocaton φ opt xt that maxmzes problem wth an α-far utlty s mplemented at each tme t, t s proven that such rate allocaton acheves the maxmum stablty regon.e. the nteror of the feasble rate regon R. In other words, a suffcent and necessary condton for stablty s that the traffc ntensty vector must belong to the nteror of the feasble rate regon: ρ Ř. However, due to the nformaton delay, the rate allocaton φˆxt depends on prevous network states at tme t τ t, for =,..., N. Thus, the usual method of flow-level stablty analyss n 2, 3, 4, whch requres a frst-order Markov model of the queue-length process, s nadequate. In ths paper, we consder non-optmal rate allocaton polcy and prove stablty by evaluatng an expected Lyapunov drft. Let h > 0 be a small tme nterval. The evoluton of the th queue s descrbed by the followng equaton: x t + h = x t + a t, h d t, h, 6 where a t, h s the number of flows arrvng to flow class durng tme t to t + h and d t, h s the number of departng flows. Due to utlty gap, the departure rate now depends on the sub-optmal rate allocaton φˆxt, gven an observed state ˆxt = x t τ t. In ths secton, we wll derve a suffcent condton for achevng maxmum stablty. When the condton s not satsfed, we prove that the achevable stablty regon may be strctly smaller than the feasble rate

4 4 regon. The man results are stated n Theorems and 2. A. A Suffcent Condton for Maxmum Stablty Theorem : For an arbtrary suboptmal rate allocaton φˆx, f the nformaton delay τt s unformly bounded by a constant Ω > 0 and the order of the utlty gap whch s non-negatve due to the defnton n 3 caused by the mperfectness of rate allocaton algorthm s less than the order of the optmal utlty when the number of actve flows grows large,.e., ˆx lm sup max ˆx :ˆx ˆx U φopt,ˆx ˆx = 0, 7 then the network s stable f the traffc condton ρ Ř s satsfed,.e., the maxmum stablty regon can be obtaned. There are two key dffcultes n the proof. Frst, to account for the utlty gap, we need to derve a relatonshp between the traffc ntensty and the sub-optmal rate allocaton φˆxt, whch s of a stronger form than those used n 3, 4 see Lemma 4 n Appendx B. Second, the nformaton delay leads to a further gap between φˆxt and φxt. We need to carefully bound the effect of ths gap, especally when the dfference between ˆxt and xt s large see the detaled comment after Lemma 5 n Appendx B. We refer the readers to Appendx B for the detaled proof of Theorem. Remark 2: Theorem establshes a suffcent condton for achevng maxmum stablty: It shows that t s not necessary to solve the optmal soluton to the utlty optmzaton problem and to requre perfect nformaton on the network states xt. The condton 7 could be used to construct stoppng rules for desgnng sub-optmal rate-control polces that can acheve maxmum stablty. We note that such a stoppng rule could be desgned wthout knowledge of the optmal rate-allocaton φ opt. For example, consder a rate controller based on a dual algorthm for solvng the NUM problem. Note that the optmal utlty s upper bounded by the objectve value of the dual problem, whch can be easly calculated from the current dual prces. Smlarly, the current prmal varables can be used to generate a feasble rate-allocaton and to compute the achevable utlty values. Therefore, f the controller stops whenever the dfference between the dual objectve value and the achevable utlty values s smaller than a threshold Γ, then we can show that the resultng utlty gap between the optmal utlty and the achevable utlty value s also bounded by ˆxt Γ. Condton 7 s then satsfed, snce: ˆx lm sup max ˆx :ˆx ˆx U φopt,ˆx ˆx Γ lm sup max ˆx :ˆx ˆx U φopt,ˆx ˆx = 0. Hence, the maxmum stablty regon can be acheved by ths stoppng rule accordng to Theorem. Although the above example assumes a centralzed controller to check the stoppng rule, we envson that dstrbuted versons of the stoppng rule are also possble, whch we leave for future work. The condton n 7 s useful because t provdes the gudelne for desgnng such stoppng rules for general networks. B. A Lower Bound on the Achevable Stablty Regon When the condton 7 n Theorem s not satsfed and the utlty gap s on the same order as that of the optmal utlty, the achevable stablty regon could be smaller than the feasble rate regon, even f the delay τt s zero. Proposton : There exsts a suboptmal rate allocaton φˆx such that the utlty gap s on the same order as the order of the optmal utlty and the nformaton delay s zero,.e., for some constant η 0,, ˆx lm sup max ˆx :ˆx ˆx tu φopt,ˆx ˆx η, 8 but the achevable stablty regon s strctly smaller than R, even f the rate vector φˆx s Pareto-optmal.e. φˆx les on the boundary of the feasble rate regon. Proposton mples that f the utlty gap s large, there exsts a suboptmal rate allocaton polcy whose achevable stablty regon s strctly smaller than the nteror of the feasble rate regon R, regardless of the nformaton delay. Rased from ths example, a challenge s to answer the queston: what s the mnmum stablty regon that a suboptmal rate allocaton polcy can acheve gven that condton 8 s satsfed? In the next theorem, we show that η s R s a lower bound of all achevable stablty regons, f the rato of the utlty gap and the optmal utlty s asymptotcally bounded by a constant η < as the number of actve flows grows large. Ths lower bound s tght n the sense that there exsts a suboptmal rate allocaton polcy whose stablty regon s exactly η s R. Theorem 2: For an arbtrary suboptmal rate allocaton φˆx, f the nformaton delay s unformly bounded by a constant and the order of the utlty gap s the same as that of the optmal utlty,.e., ˆx lm sup max ˆx N = ˆx U φopt,ˆx ˆx η, 9 then the achevable stablty regon s lower bounded by η s Ř, where s s the parameter defned n Assumpton d n Secton II. There also exsts a suboptmal rate allocaton polcy satsfyng 9 whose stablty regon s exactly η s Ř,.e., the lower bound s tght. Accordng to Lemma, f the utlty functon satsfes Assumptons a-d and s postve, then we have Ua a s b Ub,.e. the utlty functon U shows a polynomal growth rate wth exponent s. Ths mples that when

5 5 the order of the utlty gap s the same as that of the optmal utlty, the dependence of the performance bound on s s due to the fact that utlty functon has a polynomal growth rate wth the exponent s. Hence, when the rato between the utlty gap and the optmal utlty s η, the dfference n the rate allocaton can be on the order of η s. Remark 3: Theorem 2 provdes a lower bound for achevable stablty regons. Of course, under condton 9, there mght stll exst certan suboptmal rate allocaton polces that are capable of achevng the maxmum stablty. However, the lower bound n Theorem 2 s tght n the sense that there exsts a suboptmal rate allocaton polcy wth zero nformaton delay and ts stablty regon s exactly η s R. Proposton and Theorem 2 together characterze the stablty of a broad class of suboptmal rate allocaton polces. of the optmal rate allocaton polcy, when utlty gap and nformaton delay decrease. Average Queue Length =0,τ 0 =0 0 =5,τ 0 =0 0 =0,τ 0 =2 0 =5,τ 0 =2 C. Numercal xamples Traffc Intensty ρ 0 Fg. 2. Ths fgure plots the average total queue-length of the rng network for four dfferent rate allocaton polces. It s shown that the three suboptmal rate allocaton polces wth constant utlty gap and nformaton delay stll stablze the queue for traffc ntensty ρ 0 <, although ther delay performances 3 measured by the average queue length are worse compared to that of the optmal rate allocaton polcy wth 0, τ 0 = 0, 0. Fg.. A rng network wth ten users and ten flow classes. Consder a rng network wth N = 0 flow classes and L = 0 unt-capacty lnks as shown n Fg.. Flow class s ntated by user and contans x t 0 actve flows at tme t. Let R be the shortest-dstance routng matrx for ths rng network. For an α-far utlty functon wth α =.e. a logarthmc utlty, we compute the optmal rate allocaton φ opt xt for each tme t and then perturb t randomly to construct a set of suboptmal rate allocatons, such that the resultng nformaton delay and utlty gap are constants: τ 0 = τ t and 0 = ˆxt, t. 0 Accordng to Remark, snce both utlty gap and nformaton delay are constants, the suboptmal rate allocaton polcy φ constructed above wll acheve the maxmum stablty regon that s equal to the nteror of the feasble rate regon R = {φ R N + : Rφ, φ 0}, where R s L N routng matrx: R l = f class- flows use lnk l, and R l = 0 otherwse. Therefore, ths rate regon s a polytope determned by a set of lnear lnk-capacty constrants.. Fgure 2 llustrates flow-level stablty of the network under dfferent suboptmal rate allocaton polces for 0, τ 0 = {0, 0, 5, 0, 0, 2, 5, 2} respectvely, by plottng the average total queue length vs. traffc load. In ths smulaton, we assume that the flow arrval rates for all flow classes are equal,.e. ρ = ρ 0 for =,...,0. For ρ 0 0, 3, we have ρ = ρ 0 R, whch mples that the expected queue-length should reman fnte. Fgure 2 also shows that the average queue-length of a suboptmal polcy φˆxt approaches that IV. UTILITY GAP AND NTWORK PRFORMANC Secton III showed that a suboptmal rate allocaton polcy that under-optmzes a certan utlty stll acheves the maxmum stablty regon. Snce each utlty functon s desgnated to capture one partcular network objectve, allowng a nonzero utlty gap or, equvalently, under-optmzng the utlty gves us freedom to potentally mprove other network performance objectves, such as the maxmum lnk saturaton dscussed n ths secton. More precsely, there exsts a tradeoff between the utlty gap and the maxmum network performance mprovement we can potentally acheve. In ths secton we frst provde a formulaton of ths tradeoff. Then we develop an approxmaton of the tradeoff curve based on local senstvty analyss. To obtan closed-form solutons, we focus on α-far utlty functons n ths secton, although our result can be extended to all concave utlty functons. Our approach s dfferent from, whch s restrcted to a throughputfarness tradeoff for optmal solutons only. In contrast, n ths secton, we addresses the followng queston pertanng to suboptmalty: by under-optmzng an utlty wth gap, what s the maxmum performance mprovement we can possbly acheve? A. Model and Analyss We focus on the followng model for wrelne networks, whch s an mportant specal case of the model descrbed n secton III. Consder a network of L lnks, ndexed by l, each wth a fnte lnk capacty c l. Then the feasble rate regons are defned by R = {φ : Rφ c, φ 0}, where c s the vector of lnk capactes and R s the L N routng matrx: R l = f class- flows use lnk l, and R l = 0 otherwse. At

6 6 each state x, the optmal rate allocaton s obtaned by solvng problem wth α-far utlty,.e., max φ N = x α φ α α s.t. Rφ c, φ 0 Let φ opt be the optmal rate allocaton that solves the maxmzaton problem. We say a rate allocaton φ underoptmzes the α-far utlty by a gap f = U opt N = x α φ α α 2 where U opt = N = xα φ α opt, / α s the optmal utlty acheved by rate allocaton φ opt. Snce the α-far utlty s desgnated for achevng farness, under-optmzng the α- far utlty wth a gap relaxes the maxmzaton problem. Thus, t gves freedom to system desgners to potentally mprove other network performance objectves, such as maxmum lnk saturaton. However, t s unclear how much performance mprovement we can acheve by underoptmzng the α-far utlty wth a gven allowable gap. For example, f we prepare to sacrfce 5% of the utlty, how much s the reducton of lnk saturaton we could expect n return? We formulate ths type of tradeoff functons and provde a local senstvty analyss based on examnng the Karush- Kuhn-Tucker KKT condtons at the optmum allocaton. We consder the maxmum lnk saturaton as a network performance metrc, defned by Z = max R lφ. 3 l L c l By under-optmzng the utlty, t s possble to reduce the maxmum lnk saturaton and then balance the network traffc over all lnks, an mportant goal n the operaton of large networks by the Internet Servce Provders. Moreover, reducng Z could potentally mnmze the occurrence of bottleneck lnks n the network, reduce packet delay, and also make the network more robust to lnk capacty fluctuaton and traffc bursts. To characterze the optmal tradeoff between the utlty gap and the maxmum lnk saturaton, we compute the mnmum Z that can be acheve by under-optmzng the α-far utlty wth a desgnated gap. Ths tradeoff functon Z can be formulated as follows Z = mn max φ l L subject to Rφ c, φ 0 N = x α R lφ c l 4 φ α α U opt Remark 4: For the tradeoff functon defned by 4, t s easy to see that ncreasng utlty gap relaxes the constrant set of the optmzaton problem, and leads to a smaller optmal objectve value. Thus, the maxmum lnk saturaton Z s a monotoncally decreasng functon of the utlty gap. Furthermore, t s easy to verfy that the optmzaton problem 4 s convex. Now we conduct a local senstvty analyss for the saturaton-utlty tradeoff defned by optmzaton problem 4. Agan, we make the assumpton that the actve constrant set n the problem 4 s unchanged when the utlty gap s perturbed locally. The man result s summarzed n the next theorem. We denote Z 0 as the lnk saturaton for = 0. Theorem 3: When the utlty gap s small, the saturatonutlty tradeoff functon can be approxmated usng ts frst order expanson: dz Z Z 0 = + o. 5 d =0 The frst order dervatve shadow prce of the saturatonutlty tradeoff functon s gven by dz = d =0 c T RD R T c, 6 { } where D = α dag x α φ α opt,,...,xα N φ α opt,n s a dagonal matrx. B. Numercal xamples In ths subsecton, we plot the tradeoff curve and ts frstorder approxmatons obtaned n Theorem 3 for the rng network descrbed n Secton III.C. Snce all lnks have unt capacty, the feasble rate regon s gven by R = {φ : Rφ, φ 0}, where R s the routng matrx for the rng network. Let x denote the number of actve flows for source. We can solve the convex optmzaton problem 4 for varyng utlty gap to obtan the exact tradeoff curve Z, whch s plotted n Fgure 3 usng sold lnes. We assume that the number of actve flows s x = 0,. A proportonal farness utlty functon correspondng to α = s consdered. When the utlty gap s small, the saturaton-utlty tradeoff curves can be approxmated by ts frst order expansons gven by 5. Usng the closed-form soluton n Theorem 3, we compute the frst order gradent dz d = Thus =0 the tradeoff curve can be approxmated by T Z Fgure 3 shows that the saturaton-utlty tradeoff defned n 4 can be well approxmated by ts frst order expanson, gven by the closed-form expresson n Theorem 3. Ths tradeoff curve allows us to predct how much performance mprovement we can possbly acheve by under-optmzng the utlty wth a desgnated small utlty gap. For example, f we under-optmze the utlty by %,.e. = % U opt = 0.32, t s clear from equaton 7 that a lnk saturaton reducton of 0.3% Z Z 0 = could be expected n return. Ths result not only llustrates the potental benefts of under-optmzng an α-far utlty, but also quanttatvely characterzes the tradeoff between sacrfcng utlty value and achevng network performance mprovement. Whether ths partcular tradeoff s worth makng or not depends on operator s preference, but t s mportant to provde the choces of tradeoff through results lke those n ths secton.

7 7 Fg. 3. Lnk Saturaton Z Tradeoff Curve Z Lnear Approxmaton Utlty Gap saturaton-utlty tradeoff curve and ts frst order approxmaton. V. CONCLUDING RMARKS Suboptmal resource allocaton wth a utlty gap s smply an nevtable phenomenon n real networkng. Fortunately, t may stll be able to mantan stablty regon and even enhance other network performance metrcs. Intuton on stablty and utlty-versus-saturaton tradeoff are quantfed wth closedform expressons n ths paper. There are open questons to be addressed n studyng the mpact of suboptmal solutons to network optmzaton, e.g., n characterzng degradaton of farness due to utlty gap and n conductng a global senstvty analyss, before we fully understand how bad suboptmal rate allocaton s. APPNDIX: PROOFS From Assumptons a-d n Secton II, n Appendx A we frst prove two lemmas, whch are useful throughout the rest of the proofs. Appendx B gves the proof of stablty n Theorem. It makes use of Lemma 3 whose proof s omtted due to space lmtaton and can be found n 3 and Lemmas 4-6 whch are proven n Appendces C-, respectvely. The proof of Theorem 2 n Appendx G s very smlar to that of Theorem, except that Lemma 4 s replaced by Lemma 7 under a dfferent utlty gap condton. Fnally, the saturatonutlty tradeoff s proven n Appendx H. A. Useful Lemmas. We frst collect three useful propertes of the utlty functon. Lemma : If U s a utlty functon satsfyng Assumptons a-d, then we have U a a s b U b for all a b > 0, and U a a s b U b for all b a > 0. Further, f the utlty functon s negatve, then Ua a s b Ub for all a b > 0. Otherwse, f the utlty functon s postve, Ua a s b Ub for all a b > 0. Proof: To prove part a, from Assumpton d, we have U z s U z z. Choose a b > 0 and ntegrate both sde of the nequalty from b to a. We obtan U a a s b U b. By swtchng a and b, we can prove part b. To show part c, f the utlty functon s negatve.e. case 2 n Assumpton a, we fx b n the nequalty n part a and ntegrate t from a = b to a = +,.e. b s U b + b ysdy U Ub Ub 8 where U exsts because the utlty functon s monotoncally ncreasng and upper bounded by zero as n Assumpton a. Ths mples that the ntegraton on the left hand sde also exsts and thus s >. We can derve U b Ub s b. Integratng t agan from a to b a, we obtan Ua a s b Ub, whch s the desred result. Smlarly, when the utlty functon s postve, we consder the ntegral of U a a s b U b from b = 0 to b = a and derve the result n Lemma. Lemma 2: For C 0, there exst a constant K c such that + C s z + K cc z holds for all z, and C s z KcC z holds for all z C. Proof: The proof s straghtforward by comparng the frst order dervatves wth respect to z. B. Proof of Theorem. Proof: We frst sketch the man steps of the proof. To prove stablty, we defne a Lyapnov functon V xt and analyze ts expectaton Wt = V xt as a functon of tme 2. We frst derve an expresson for Ẇt, the drft of the expected value of the Lyapunov functon. Here we need to use the Domnated Convergence Theorem n order to exchange the order of a lmt and an expectaton Lemma 3. Then, usng Lemmas 4-6, we upper bound the drft Ẇt by a negatve functon of the network state xt, plus some postve constants. Fnally, ntegratng the drft Ẇt and ts upper bound from tme t = 0 to t = T establshes the stablty condton n 5 and completes the proof. For ease of presentaton, n ths secton we present the man flow of the proof wth statements of Lemmas 3-6. The detaled proofs are summarzed n Appendces C-F. We remnd the readers that we wll use the Lemmas n Appendx A. Consder the followng Lyapunov functon N N V xt = V x t = = x t = n= µ U cρ, 9 n where c > 0 s a constant defned later n the proof. Let F t = σ {xu, u t} denote the σ-feld generated by the hstory up to tme t. For h > 0, we derve Wt + h Wt lm 20 h 0 h V xt + h F t V xt = lm h 0 h N + Px t + h x t = n F t = lm h 0 hµ = n= X t V x t + n V x t. 2 A smlar problem wth feedback delay s treated n 0 although the model there does not nvolve any flow-level dynamcs.

8 8 In order to move the lmt as h 0 nsde the expectaton and the summaton on the rght hand sde of 20, we wll use the Domnated Convergence Theorem and the followng Lemma. Lemma 3: There exsts an ntegrable functon gx t such that, for all 0 < h < and m 0, + n= X t Px t + h x t = n F t 2 hµ V x t + n V x t gx t Ths lemma provdes the bound needed for the Domnated Convergence Theorem to hold. Therefore, we can move the lmt nsde the expectaton on the rght hand sde of 20. Due to the orderlness property of Posson process whch mples that arrvals do not occur smultaneously, we can take the lmt as h and narrow down the condtonal probablty terms n 20 to get Ẇt = N λ V x t + V x t = N φ tµ V x t V x t. = Usng ρ = λ /µ and pluggng n the Lyapunov n 9 functon nto above, we obtan N Ẇt = ρ U cρ 22 x = t + N φ ˆxtU cρ {xt } 23. x t :ˆx t = Derve a bound for 23. In order to bound Wt by a negatve functon of xt, we frst derve a bound for the second summaton 23 and replace the rate allocaton φˆxt by a functon of traffc ntensty ρ. The resultng bound wll then have a form smlar to 22 so that we can compare the dfference. We make use of the followng lemma: Lemma 4: For any traffc ntensty ρ Ř and constant C > 0, f the suboptmal rate allocaton φˆxt satsfes the utlty gap condton n 7, there exst postve constants γ > 0 and ɛ > 0 such that, for all r < C and for any network state satsfyng max ˆx t > γ, the followng nequalty holds: ρ + ɛ 3 U cρ φ ˆxtU cρ ˆx t ˆx t + r where c = + ɛ 4 s a constant. We choose r = xt ˆxt n 24. Note that n order to use Lemma 4 to bound 23, we need the network state to satsfy max ˆx t > γ and the quantty xt ˆxt to be bounded by C. Toward ths end, we defne the event t = { xt u xt Ω < C/2, u 0, Ω},.e., t s the event that the maxmum change of network state wthn tme t Ω to t s bounded by C/2. Snce the nformaton delay s bounded by τt < Ω, event t mples the followng: xt ˆxt xt xt Ω + ˆxt xt Ω C Therefore, we can bound 23 by N φ ˆxtU cρ x t = A :ˆx t A + ɛ 3 φ ˆxtU {xt } cρ x t Derve a bound for 22. We next provde a correspondng bound for 22 to compare wth 27. Note that regon G s 0, 24 compact. As part of the proof of Lemma 5, A 2 s chosen n 49 such that: N ρ U cρ {xt G} A 2 <. 28 x t + :ˆx t ρ U cρ ˆx t 25 t {max ˆx t>γ} t {max ˆx t>γ} Note that ˆx t C f x t = 0 under event t. To change the summaton from { : x t } to { : ˆx t } n the frst step above, we have added A = N = ψu cρ. 26 C where we choose φ ˆxt ψ snce the feasble rate regon R s compact. The last step of 25 follows from Lemma 4. In order to compare the dfference wth 22, we stll need to replace the ˆxt on the rght hand sde by xt. Let G = {xt : max x t ξ} be a bounded regon wth ξ = γ+2c. We can prove the followng result: Lemma 5: There exsts A 2, A 3 > 0 such that the rght hand sde of 25 can be further bounded by + ɛ 2 ρ U cρ t {max ˆx ˆx t t>γ} 27 :ˆx t A 2 + 3A 3 ρ U cρ {xt G x t c }. :x t The ntuton behnd Lemma 5 s that: when t occurs, the absolute dfference between xt and ˆxt s bounded by C. If n addton the network state s large, then the relatve dfference between xt and ˆxt wll be small, and hence the correspondng values of U wll be close to each other. Provng Lemma 4 turns out to be non-trval. The challenge s that ˆxt and t n 27 are dependent. To handle ths dffculty, we consder the σ-feld F t Ω = σ {xu, u t Ω}. If we ntroduce the network state x Ω t = xt Ω as an auxlary varable, then ˆxt and t can be bounded wth respect to x Ω t, separately. For detals, please refer to Appendx D. = Applyng A 2 to 22, we have N ρ U cρ x t + = 29

9 A + :x t A + A 2 + ρ U cρ :x t ρ U x t + cρ x t + {xt Gc } where A s added to change the summaton to { : x t }. To further bound 29, we make use of Lemma b, wth a = cρ /xt and b = cρ /xt + : :x t :x t :x t ρ U cρ x t + ρ + x t ρ + K x t {xt G c } s U cρ x t U cρ x t {xt G c } {xt G c } 30 where K n the last step s the constant defned n Lemma 2. Fnally, we use the followng result: Lemma 6: For any K, C, ɛ > 0 there exsts ξ such that, for any max ˆx t > ξ, we have K C ρ ˆx t ɛ U cρ 0. 3 Such ɛ > 0 and δ > 0 exst due to the traffc condton ρ 2 ˆx t Ř. :ˆx t Let δ 0 = δ s. We then have Combnng Lemma 6 wth 29 and 30, we derve the 0 φ ˆxt u ˆx tu δˆx tu followng upper bound for 22: ˆx t ˆx t :ˆx t N ρ U cρ 32 φ ˆxt δ0 u ˆx tu ˆx tu x = t + ˆx t ˆx t :ˆx t A + A 2 + ρ + ɛ U cρ {xt G 2 x t c } φ ˆxt δ 0 u U δ0 u ˆx t :x t :ˆx t Prove stablty. Let A 0 = 2A +3A 2 +2A 3. Replacng 22 = φ ˆxt + ɛ 4 + ɛ 4 ρ ρ U 38 ˆx t :ˆx and 23 by ther upper bounds, we can bound the expected t drft Ẇt by: Ẇt A 0 ɛ 2 :x t A 0 + A 2 ɛ 2 ρ U cρ x t :x t ρ U {xt G c } cρ x t. 33 where nequalty 28 s used n the last step and a constant A 2 s added. Rearrangng the terms and ntegratng 33 from t = 0 to t = T, we obtan lm sup T T T t=0 = lmsup T :x t ρ U 2W0 2WT Tɛ cρ dt x t + 2A 0 + 2A 2 ɛ 2A 0 + 2A 2 34 ɛ where the last step uses the fact that WT 0 s postve. Snce functon U s a non-negatve and non-decreasng, and lm U z /z =, equaton 34 mples the stablty of the network, as clamed n the stablty defnton 5. C. Proof of Lemma 4 Proof: Accordng to the utlty gap condton n 7, we can conclude that for any δ 0, there exsts a postve γ such that, for ˆxt satsfyng max ˆx t > γ, we have ˆxt δ ˆx tu φ opt,ˆxt ˆx t. 35 :ˆx t To remove the absolute value on the rght hand sde of 35, we frst assume that the utlty functon s non-negatve. Pluggng the expresson of utlty gap ˆxt nto 35, we have 0 :ˆx t ˆx tu φ ˆxt ˆx t δˆx tu φopt, ˆx t ˆx t 9 36 Snce φ opt, ˆx t s the optmal rate allocaton for the NUM problem at state ˆx t, no rate vector u R can acheve a hgher utlty value. Choose ɛ > 0 and δ > 0 such that: u + ɛ 4 δ s ρ R. 37 where the second step uses Lemma wth a = u /ˆx t and b = δ 0 u /ˆx t, the thrd step uses the concavty of the utlty functon U, and the last step s from 37. Note that a smlar expresson for 38 can also be shown f the utlty functon s negatve. Specfcally, When the utlty value n 35 s negatve, usng the same proof technque and choosng δ 0 = + δ s, we can show that the nequalty 38 s also satsfed. Refer to 3 for the proof. Note that 38 s almost the same as 24, except for a constant r n the denomnator and an addtonal + ɛ factor. Let c = + ɛ 4. For r < C, we have φ ˆxtU cρ ˆx t + r :ˆx t φ ˆxtU cρ ˆx t + r :ˆx t C φ ˆxtU cρ ˆx t C :ˆx t C φ ˆxt C s U ˆx t :ˆx t C cρ ˆx t 39

10 0 :ˆx t C :ˆx t C φ ˆxt K cc ˆx t U cρ ˆx t + ɛ 4 ρ K ccφ ˆxt ˆx t U cρ ˆx t In the frst step of 39 some negatve terms are dropped. The second step makes use of the monotoncty of U. The thrd step uses Lemma. The fourth step uses Lemma 2. The last step s from 38. Snce φ ˆxt ψ, usng the proof n Lemma 6, we can show that for large enough γ and max ˆx t > γ, + ɛ 3 ɛρ K ccψ U cρ ˆx t ˆx t :ˆx C Applyng 40 to 39, we derve + ɛ 3 ρ U cρ ˆx t :ˆxt φ ˆxtU whch completes the proof of Lemma 4. D. Proof of Lemma 5 :x Ω t cρ 0, ˆx t + r Proof: To prove 27, we use network state x Ω t = xt Ω at tme t Ω as an auxlary varable, and bound both sdes of 27 wth respect to x Ω t, respectvely. Bound the left hand sde of 27. Defne the event Xt Ω = {max x Ω t > ξ C},.e. t s the event that the maxmum queue length at tme t Ω s greater than ξ C. We start wth ρ U cρ x Ω t t A 3 + :x Ω t :ˆx t ρ U cρ x Ω t t 4 where { a constant A 3 s added to change the summaton from : x Ω t } to { : ˆx }. Such A 3 satsfes ρ U cρ x Ω t {ˆxt=0} :x Ω t = :x Ω t ρ x Ω t s U ρ ˆx t x Ω t s U cρ {ˆxt=0} cρ {ˆxt=0} A 3 < +, 42 where A 3 n the last step exsts because ˆx t x Ω t can be bounded by a Posson random varable wth mean ωω. The second step uses Lemma. To proceed from 4, we have ρ U cρ t :ˆx t :ˆx t :ˆx t ρ U x Ω t cρ t ˆx t + C X Ωt s U ρ + C ˆx t cρ t ˆx t :ˆx t :ˆx t ρ + K cc ˆx t ρ + ɛ U 2 U cρ ˆx t cρ ˆx t t t 43 The frst step uses the monotoncty of U and the condton x Ω t ˆx t < C under event t. The second step above uses Lemma. The thrd step s due to + C s z + K cc z n Lemma 2. The last step follows from the proof of Lemma 6. Applyng the result to 4, we derve an upper bound for the left hand sde of 27 as follows: ρ U cρ x Ω :x Ω t t t A 3 44 ρ + ɛ 2 U cρ t :ˆx t :ˆx t ρ + ɛ 2 U ˆx t cρ ˆx t t {max ˆx t γ} where the last step uses ξ = γ + 2C and { t Xt Ω x Ω t ˆx t < C } 2, max x Ω t > ξ C { } max ˆx t > γ. 45 Derve a bound for the rght hand sde of 27. Defne the event X t = {xt G c } = {max x t > ξ},.e. t s the event that the maxmum queue length at tme t s greater than ξ. Usng the argument for 4 and 43, we frst change the summaton on the rght hand sde of 27 from { : x t } to { : x Ω t } by addng A 3 whch s defned n 42: ρ U cρ Xt :x t A 3 + :x Ω t x t ρ U cρ Xt x t 46 To bound 46, we defne the event L = {x t > x Ω t}, and evaluate 46 over three events: t c L, t c Lc, and t, separately. For t c L, note that t c L mples that the ncrease of xt n the nterval t Ω to t s larger than C. The number of arrvals n ths nterval must also be larger than C. Snce the arrval process s Posson, when C s large, we can bound the probablty of ths event by a small number. Hence, the contrbuton to the expectaton n 46 wll be small. Specfcally, let a t be the arrval wthn tme t Ω to t. We have ρ U cρ Xt ct L x t :x Ω t ρ U cρ ct L x t :x Ω t :x Ω t ρ x t x Ω t s U cρ x Ω t ct L

11 :x Ω t :x Ω t :x Ω t ρ + a s t x Ω t U cρ x Ω t ct L ρ + a t s U cρ x Ω t ct L ρ U cρ x Ω t K 47 where the second step uses Lemma, and the thrd step uses the monotoncty of U and x t x Ω t a t snce a t s the arrval wthn tme t Ω to t, and the quantty K n the last step s chosen as, K = + a t s c t F t Ω. 48 The condtonal expectaton s taken over the σ-feld F t Ω = σ {xu, u t Ω}, generated by the hstory up to tme t Ω. Gven F t Ω, a t s Posson-dstrbuted wth mean λ Ω. Therefore, usng + a t s <, we can use the Domnant Convergence Theorem to show lm K = lm + a t s t Ft Ω C C = + a t s lm t F t Ω = 0. C Ths mples that we can make C > 0 to be suffcently large, so that K ɛ 4. Applyng ths result to 47, we conclude ρ U cρ x Ω :x Ω t t K ɛ 4 ρ U cρ x Ω t :x Ω t ɛ 4 A 2 + :x Ω t :x Ω t ɛ 4 ρ U cρ x Ω t cρ 49 where A 2 defned n 49 bounds :x Ω t ρ U x Ω t A 2, over the compact regon Xt Ω c = {max x Ω t ξ C}. Smlarly, we evaluate 46 over t c Lc : ρ U cρ Xt ct L c x t :x Ω t ρ U cρ x Ω :x Ω t t Xt ct L c ρ U cρ x Ω :x Ω t t ct ρ U cρ x Ω :x Ω t t c t F t Ω ρ U cρ x Ω :x Ω t t K ɛ 4 A 2 + ɛ 4 ρ U cρ x Ω t X 50 Ωt The frst step uses the monotoncty of U and x t x Ω t under Lc. The fourth step s due to t c Ft Ω K accordng to 48. The last step uses the nequalty 49. Fnally, we evaluate 46 over t. Note that when t occurs, the dfference between xt and x Ω t s small. We wll use ths dea to replace the xt n the rght-hand sde of 46 by x Ω t. Recognzng { t X t x Ω t x t < C } 2, max x t > ξ we obtan :x Ω t { :x Ω t :x Ω t :x Ω t :x Ω t max x Ω t > ξ C } = X Ω t, 5 ρ U cρ Xt t x t ρ U cρ x t t ρ U cρ x Ω t + C t ρ + C x Ω t ρ + ɛ U 2 s U cρ x Ω t cρ x Ω t t t 52 where the second step uses the monotoncty of U, and the thrd step uses Lemma, and the fourth step s from Lemma 2 and 6 smlar to the argument n 43. Note that c t L + c t L c + t =. Therefore, puttng 49, 50, and 52 together, and pluggng the result nto 46, we derve :x t :x Ω t :x Ω t ρ U cρ x t ρ U ρ U cρ x t cρ x Ω t Xt A 3 ɛ 2 A 2 53 Xt ɛ 2 A 2 t + ɛ 2 t + ɛ 2 We stll need to replace the last constant ɛ/2 by a functon of t +. Toward ths end, we can show the followng the nequalty, whch s obtaned n a smlar way as the thrd and the last lnes of 50: ρ U cρ x Ω t ct :x Ω t ɛ 4 A 2 + ɛ 4 We add :x Ω t ρ U :x Ω t cρ x Ω t ρ U cρ x Ω t 54 X Ω t t to both sdes of

12 2 54 and rearrange the terms: ɛ ρ U cρ 4 x Ω :x Ω t t 55 ɛ 4 A 2 + ρ U cρ x Ω t t :x t :x Ω t Snce ɛ 4 < 2 for ɛ <, we replace ɛ 4 by 2 on the left hand sde of 55, and plug the result nto the last lne of 53 to obtan ρ U cρ Xt A 3 ɛa 2 x t :x Ω t ρ U cρ x Ω t t + 32 ɛ,56 whch establsh a lower bound for the rght hand sde of 27. Prove 27. To show nequalty 27, we combne the two bounds 44 and 56 to get :x t :ˆx t ρ U cρ x t {xt Gc } ɛa ɛ 2 A ɛ 3ɛ ρ U cρ t {max ˆx ˆx t t>γ} 57 Fnally, note that ɛ <, 3 2 ɛ < 2, and + ɛ 3ɛ < +ɛ2. Inequalty 27 s mmedate from 57.. Proof of Lemma 6. that Proof: We frst show that there exsts a constant A 4, such :ˆx t K C ρ ˆx t ɛ U cρ A ˆx t Snce the left hand sde of 58 s non-negatve only f ˆx t 4K C/ɛ, we have :ˆx t K C ρ ˆx t ɛ U cρ 4 ˆx t max ˆx t 4K C ɛ ρ K C U ρ K C ˆx t U cρ ˆx t A 4 59 cρ ɛ 4K C where negatve terms are dropped n the frst step, and the second step uses the monotoncty of U and ˆx t. Pluggng ths nto the left hand sde of 3, we have cρ :ˆx t A 4 ρ K C ˆx t ɛ 2 :ˆx t ρ ɛ 4 U U cρ ˆx t ˆx t 60 Accordng to Assumpton c n Secton II, for ξ and max ˆx t > ξ, we have 60. Therefore, there exsts large enough ξ, such that F. Proof of Proposton. Proof: To prove that the network s unstable when the nformaton delay s zero and the utlty gap s on the same order as that of the optmal utlty 8, we construct a counterexample, n whch the expected total queue-length grows unbounded as tme t ncreases. Consder a network wth two classes of flows and a feasble rate regon whch s convex and compact, e.g., an ellpse, depcted n Fgure 4. For an Fg. 4. The feasble rate regon under consderaton. A C α-far utlty wth α = /2, let φ opt xt denote the optmal rate allocaton for state xt at tme t. We defne a suboptmal rate allocaton by φ opt xt, f φ opt xt does not le on ÂB φˆxt = φ A, otherwse, f x t > x 2 t φ B, otherwse, f x t x 2 t where ÂB denotes the boundary of the rate regon between ponts A and B, and φ A and φ B are the optmal rate vectors at ponts A and B respectvely. To prove Proposton, we notce that xt = 0 for all φ opt xt ÂB. It can be shown that the utlty gap s on the same order as the optmal utlty,.e., lm sup max x t N = lmnf max x t xt B = x tu φopt,t 2 x t x tφ t + x 2 tφ 2 t x tφ opt, t + x 2 tφ opt,2 t x tφ B, + x 2 tφ A,2 lmnf max x t x tφ A, + x 2 tφ B,2 φ B, φ A,2 mn, φ A, φ B,2 where the second step holds because φ B, φ t φ A, and φ A,2 φ 2 t φ B,2 for any rate vector φˆxt that les on ÂB. Hence, the rate allocaton polcy φˆxt satsfes condton 8 as clamed. Next, we choose a pont C as the mddle pont of lne AB and show that for small enough ɛ > 0, the network s unstable under traffc ntensty ρ = +ɛφ C Ř. Consder a Lyapunov functon defned by the weghted sum queue-length V x = µ w x + µ 2 w 2x 2 6 where w = φ B,2 φ A,2 and w 2 = φ A, φ B, are two postve constants. Then, we can formulate the expected Lyapunov

13 3 functon Wt = V xt and derve an expresson for the drft Ẇt usng the same manner as n 22 and 23. In the followng, we show that the expected drft s strctly above zero for traffc ntensty ρ = + ɛφ C R wth a small enough ɛ > 0,.e. Ẇt = 2 = λ µ w w φ ˆxt = ɛw φ C, + w 2 φ C,2 + w φ C, φ t +w 2 φ C,2 φ 2 t ɛw φ C, + w 2 φ C,2 > 0 where the thrd nequalty holds snce the suboptmal rate allocaton φˆxt always les below the straght lne AB, whose slope s w w 2. Thus, for the choce of Lyapunov functon 6 and the traffc ntensty ρ = + ɛφ C R, the expected drft Ẇt s strctly above zero by a constant ɛw φ C, +w 2 φ C,2. Ths mples that the network s unstable, snce lm Wt = as t. t G. Proof of Theorem 2. Proof: The proof s almost the same as that of Theorem, except that we need to prove a result smlar to Lemma 3, but wth traffc ntensty ρ η s Ř. For the Lyapunov functon V x = N x = n= U cρ n /µ, we derve the same drft Ẇt as n 22 and 23. To bound Ẇ under the new utlty gap condton 9, we prove the followng Lemma for traffc ntensty ρ η s Ř. Lemma 7: Let c = + ɛ 4. For any traffc ntensty ρ η s Ř and constant C > 0. If the suboptmal rate allocaton φˆxt satsfes the utlty gap condton n 9, there exst postve constants γ > 0 and ɛ > 0 such that, for all r < C and for any network state satsfyng max ˆx t > γ, the followng nequalty holds: :ˆx ρ t + ɛ 3 U cρ ˆx t φ ˆxtU cρ ˆx t+r Proof: From the utlty gap condton n 9, we conclude that for any δ 0, there exsts a postve γ such that, for ˆxt satsfyng max ˆx t > γ, we have ˆxt η + δ ˆx tu φ opt,ˆxt ˆx t.63 :ˆx t We frst assume that the utlty functon s non-negatve. Pluggng the expresson of utlty gap ˆxt nto 63, we obtan 0 ˆx tu φ ˆxt ˆx t :ˆx t + δηˆx tu φ opt,ˆx t 64 ˆx t Snce φ opt, ˆx t maxmzes the NUM problem at state ˆx t. We can replace φ opt, ˆx t above by the followng choce of rate vector: u = + ɛ 4 + δη s ρ R. 65 Such ɛ > 0 and δ > 0 exst due to the traffc condton ρ η s Ř. Let δ0 = + δη s. Smlar to 38 n the proof of Lemma 3, we have 0 ˆx tu φ ˆxt ˆx t :ˆx t + δηˆx tu u ˆx t ˆx tu φ ˆxt ˆx tu δ 0u ˆx t ˆx t φ ˆxt δ 0 u U cρ ˆx t 66 :ˆx t :ˆx t The frst step uses Lemma, and the last step uses the concavty of the utlty functon U. Note that 66 s exactly the same as 38 n the proof of Lemma 3. Therefore, re rest of the proof below 38 can be coped drectly to prove Lemma 5 for traffc ntensty ρ η s Ř. xcept for Lemma 5, other dervatons n the proof of Theorem can be appled drectly. To summarze, we conclude T lm sup λ U cρ dt T T t=0 x t :x t 2A 0 + 2A 2 ɛ Because functon U s a non-negatve and non-decreasng functon and lm U z =, the last equaton above z mples the stablty of the network under the traffc ntensty ρ η s R. H. Proof of Theorem 3. Proof: We frst notce that optmzaton problem 4 can be rewrtten as Z = mn φ Z 67 s.t. Rφ Zc, φ 0 N x α φ α α U opt = Its Lagrangan s then gven by Lφ, Z, p, q = Z + p T Rφ Zc + qu opt V φ At the optmal pont of 67, the KKT condtons for optmalty are gven by Rφ = Zc, V φ = U opt 68 N R T p q φ V φ φ φ = 0, p T c = 69 = From the mplct functon theorem, varables φ, Z, p and q can be vewed as mplct functons of, whch s unquely defned by the KKT condtons 68 and 69. We defne a