Network Architectures and Algorithms

Transcription

1 Network Archtectures and Algorthms R. Srkant ECE and CSL Unversty of Illnos Le Yng ECE Iowa State Unversty June 3, 2011

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3 Contents 1 Mathematcs of Internet Archtecture Mathematcal Background: Convex Optmzaton Convex Sets and Convex Functons Convex Optmzaton Resource Allocaton as Utlty Maxmzaton Notons of Farness Mathematcal Background: Stablty of Dynamcal Systems Dstrbuted Algorthms: Prmal Soluton Prce Functons and Congeston Feedback Dstrbuted Algorthms: Dual Soluton Relatonshp to TCP Protocols TCP-Reno TCP-Vegas: A Delay Based Algorthm Lnks: Statstcal Multplexng and Queues Mathematcal Background: The Chernoff Bound Statstcal Multplexng and Packet Bufferng Queue Overflow Mathematcal Background: Dscrete-tme Markov Chans Delay and Packet Loss Analyss n Queues Lttle s Law The Geo/Geo/1 Queue The Geo/Geo/1/B Queue The Dscrete-Tme G/G/1 Queue Schedulng n Packet Swtches Swtch Archtectures and Crossbar Swtches Head-of-Lne (HOL) Blockng and Vrtual Output Queues Capacty Regon and MaxWeght Schedulng Schedulng n Wreless Networks Channel-Aware Schedulng n Cellular Networks The MaxWeght Algorthm for the Cellular Downlnk MaxWeght Schedulng Ad Hoc P2P Wreless Networks General MaxWeght Algorthms

4 4 CONTENTS 4.5 Q-CSMA: A Dstrbuted Algorthm for Ad Hoc P2P Networks The Idea behnd Q-CSMA Q-CSMA Back to Network Utlty Maxmzaton A Jont Formulaton of the Transport, Network and MAC Problems Stablty and Convergence: An Example for Cellular Networks Ad Hoc P2P Wreless Networks Internet versus Wreless Formulatons: An Example

5 Chapter 1 Mathematcs of Internet Archtecture 1.1 Mathematcal Background: Convex Optmzaton In ths secton, we present some basc results from convex optmzaton whch we wll fnd useful n the rest of the chapter. Often, the results wll be presented wthout proofs, but some concepts wll be llustrated wth fgures to provde an ntutve feel for the results Convex Sets and Convex Functons We frst ntroduce the basc concepts from optmzaton theory, ncludng the defntons of convex sets and convex functons. Defnton (Convex Set) A set S R n s convex f αx + (1 α)y S whenever x, y S and α [0, 1]. Snce αx + (1 α)y, for α [0, 1], descrbes the lne segment between x and y, a convex set can be pctorally depcted as n Fgure 1.1: Gven any two ponts x, y S, the lne segment between x and y les entrely n S. Fgure 1.1: A convex set S R 2 Defnton (Convex Hull) The convex hull of set S, denoted by Co(S) s the smallest convex set that contans S. See Fgure 1.2 for an example. 5

6 6 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Fgure 1.2: The sold lne forms the boundary of the convex hull of the shaded set. Defnton (Convex Functon) A functon f(x) : S R n R s a convex functon f S s a convex set and the followng nequalty holds for any x, y S and α [0, 1] : f(αx + (1 α)y) αf(x) + (1 α)f(y). f(x) s strctly convex f the nequalty above s strct for all α (0, 1) and x y. Pctorally, f(x) looks lke a bowl as shown n Fgure 1.3. The lne segment connectng the two ponts and les "above" the plot of Fgure 1.3: Pctoral descrpton of a convex functon n R 2 Defnton (Concave Functon) A functon f(x) : S R n R s a concave functon (strctly concave) f f s a convex (strctly convex) functon. Pctorally, f(x) looks lke an nverted bowl as shown n 1.4. Defnton (Affne Functon) A functon f(x) : R n R m s an affne functon f t s a sum of a lnear functon and a constant,.e., there exst α, a R such that f(x) = αx + a.

7 1.1. MATHEMATICAL BACKGROUND: CONVEX OPTIMIZATION 7 The lne segment connectng the two ponts and les "below" the plot of Fgure 1.4: Pctoral descrpton of a concave functon n R 2 The convexty of a functon may be hard to verfy from the defnton gven above. Therefore, next we present several condtons that can be used to verfy the convexty of a functon. The proofs are omtted here, and can be found n most textbooks on convex analyss or convex optmzaton. Result (Frst Order Condton I) Let f : S R R be a functon defned over a convex set S. If f s dfferentable and the dervatve f (x) s non-decreasng (ncreasng) n S, then f(x) s convex (strctly convex over S). Result (Frst Order Condton II) Let f : S R n R be a dfferentable functon defned over a convex set S. Then f s a convex functon f and only f f(y) f(x) + f(x)(y x) x, y S, (1.1) where f(x) = ( f (x), f (x),, f ) (x) x 1 x 2 x n and x s the th component of vector x. Pctorally, f x s one-dmensonal, ths condton mples that the tangent of the functon at any pont les below the functon as shown n Fgure 1.5. f(x) s strctly convex f the nequalty above s strct for any x y. Result (Second Order Condton) Let f : S R n R be a twce dfferentable functon defned over the convex set S. Then, f s a convex (strctly convex) functon f the Hessan matrx H wth H j = 2 f (x) x x j s postve semdefnte (postve defnte).

8 8 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Fgure 1.5: Pctoral descrpton of nequalty (1.1) n one-dmensonal space Result (Strct Separaton Theorem) Let S R n be a convex set and x be a pont that s not contaned n S. Then there exsts a vector β R n, β 0, and constant δ > 0 such that n n β y β x δ holds for any y S Convex Optmzaton We frst consder the followng unconstraned optmzaton problem: =1 =1 and present some mportant results wthout proofs. max f(x), (1.2) x S Defnton (Local Maxmzer and Global Maxmzer) For any functon f(x) over S R n, x s sad to be a local maxmzer or local optmal pont f there exsts an ɛ > 0 such that f(x + δx) f(x ) for any δx ɛ, where can be any norm. x s sad to be a global maxmzer or global optmal pont f f(x) f(x ) for any x S. When not specfed, maxmzer refers to global maxmzer n ths book. Result If f(x) s a contnuous functon over a compact set S (.e., S s closed and bounded f S R n ), then f(x) acheves a maxmum over ths set,.e., max x S f(x) exsts. Result If f(x) s dfferentable, then any local maxmzer x n the nteror of S R n satsfes f(x ) = 0. (1.3) If f(x) s a concave functon over S, condton (1.3) s also suffcent for x to be a local maxmzer.

9 1.1. MATHEMATICAL BACKGROUND: CONVEX OPTIMIZATION 9 Result If f(x) s concave, then a local maxmzer s also a global maxmzer. In general, multple global maxmzers may exst. If f(x) s strctly concave, then the global maxmzer x s unque. Result Results and hold for convex functons f the max n the optmzaton problem (1.2) s replaced by mn, and maxmzer s replaced by mnmzer n Results and Result If f(x) s a dfferentable functon over set S and x s a maxmzer of the functon, then f(x )δx 0 for any feasble drecton δx,.e., for any δx such that x + δx S. Further f f(x) s a concave functon, then x s a maxmzer f and only f f(x )δx 0 for any δx such that x + δx S. Next, we consder an optmzaton problem wth equalty and nequalty constrants as follows: max x S f(x) (1.4) subject to h (x) 0, = 1, 2,..., I (1.5) g j (x) = 0, j = 1, 2,..., J. (1.6) A vector x s sad to be feasble f x S, h (x) 0 for all, and g j (x) = 0 for all j. Whle (1.5) and (1.6) are nequalty and equalty constrants, respectvely, the set S n the above problem captures any other constrants that are not n equalty or nequalty form. A key concept that we wll explot later n the chapter s called Lagrangan dualty. Dualty refers to the fact that the above maxmzaton problem, also called the prmal problem, s closely related to an assocated problem called the dual problem. Gven the constraned optmzaton problem n (1.4)-(1.6), the Lagrangan of ths optmzaton problem s defned to be L(x, λ, µ) = f(x) I λ h (x) + The constants λ 0 and µ j are called Lagrange multplers. defned to be D(λ, µ) = sup L(x, λ, µ). x S =1 J µ j g j (x), λ 0. j=1 The Lagrangan dual functon s Let f be the maxmum of the optmzaton problem (1.4),.e., f = max x S f(x). Then, we have the followng theorem. Theorem D(λ, µ) s a convex functon and D(λ, µ) f.

10 10 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Proof The convexty comes from a known fact that the pontwse supremum of affne functons s convex (see Fgure 1.6). To prove the bound, note that h (x) 0 and g j (x) = 0 for any feasble x, so the followng nequalty holds for any feasble x, Ths nequalty further mples that L(x, λ, µ) f(x). sup L(x, λ, µ) sup f(x) = f. x S h(x) 0 x S h(x) 0 g(x)=0 g(x)=0 Snce removng some constrants of a maxmzaton problem can only result n a larger maxmum value, we obtan sup L(x, λ, µ) sup L(x, λ, µ). x S x S h(x) 0 g(x)=0 Therefore, we conclude that D(λ, µ) = sup L(x, λ, µ) f. x S Fgure 1.6: The sold-lne s the pontwse supremum of the four dashed-lnes, and s convex. Theorem states that the dual functon s an upper bound on the maxmum of the optmzaton problem (1.4)-(1.6). We can optmze over λ and µ to obtan the best upper bound, whch yelds the followng mnmzaton problem, called the Lagrange dual problem: nf D(λ, µ). λ 0,µ

11 1.1. MATHEMATICAL BACKGROUND: CONVEX OPTIMIZATION 11 Let d be the mnmum of the dual problem,.e., d = nf λ 0,µ D(λ, µ). The dfference between d and f s called the dualty gap. For some problems, the dualty gap s zero. We say strong dualty holds f d = f. If strong dualty holds, then one can solve ether the prmal problem or the dual problem to obtan f. Ths s often helpful snce sometmes one of the problems s easer to solve than the other. A smple yet frequently used condton to check strong dualty s Slater s condton, whch s gven below. Theorem (Slater s condton) Consder the constraned optmzaton problem defned by (1.4)-(1.6). Strong dualty holds f the followng condtons are true: f(x) s a concave functon and h (x) are convex functons. g j (x) are affne functons. There exsts an x that belongs to the relatve nteror 1 of S such that h (x) < 0 for all and g j (x) = 0 for all j. As mentoned earler, when strong dualty holds, we have a choce of solvng the orgnal optmzaton n one of two ways: ether solve the prmal problem drectly or solve the dual problem. Later n ths chapter, we wll see that resource allocaton problems n communcaton networks can be posed as convex optmzaton problems, and we can use ether the prmal or the dual formulatons to solve the resource allocaton problem. We now present a result whch can be used to solve convex optmzaton problems. Theorem (Karush-Kuhn-Tucker (KKT) Condtons) Consder the constraned optmzaton problem defned n (1.4)-(1.6), where f s a concave functon, h ( = 1,..., I) are convex functons and g j (j = 1,..., J) are affne functons. Let x be a feasble pont,.e., a pont that satsfes all the constrants. Suppose there exst constants λ 0 and µ j such that f x k (x ) λ h x k (x ) + j µ j g j x k (x ) = 0, k, (1.7) λ h (x ) = 0,, (1.8) then x s a global maxmzer of the constraned optmzaton problem, (λ, µ ) s a global mnmzer of the Lagrange dual problem, and strong dualty holds. If f s strctly concave, then x s also the unque global maxmzer. The KKT condtons (1.7)-(1.8) can be nterpreted as follows. Consder the Lagrangan L(x, λ, µ) = f(x) λ h (x) + j µ j g j (x). 1 For convex set S, a relatve nteror s a pont x such that for any y S there exst z S and 0 < λ < 1 such that x = λy + (1 λ)z.

12 12 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Condton (1.7) s the frst-order necessary condton for the maxmzaton problem max x S L(x, λ, µ ). When strong dualty holds, we have f(x ) = f(x ) λ h (x ) + j µ jg j (x ), whch results n condton (1.8) snce g j (x ) = 0 j, and λ 0 and h (x ) 0. We remark that condton (1.8) s called complementary slackness. 1.2 Resource Allocaton as Utlty Maxmzaton The Internet s a shared resource, shared by many mllons of users, who are connected by a huge network consstng of many, many routers and lnks. The capacty of the lnks must be splt n some far manner among the users. To apprecate the dffculty n defnng what farness means, let us consder an every day example. Suppose that one has a loaf of bread whch has to be dvded among three people. Almost everyone wll agree that the far allocaton s to dvde the loaf nto three equal parts and gve one pece to each person. Whle ths seems obvous, consder a slght varant of the stuaton where one of the people s a two-year old chld and the other two are football players. Then, an equal dvson does not seem approprate: the chld cannot possbly consume the thrd allocated to her, so a dfferent dvson based on ther needs may appear to be more approprate. The stuaton gets more complcated when there s more than one resource to be dvded among the three people. Suppose that there are two loafs of bread, one wheat and one rye, then a far dvson has to take nto account the preferences of the ndvduals for the dfferent types of bread. Economsts solve such problems by assocatng a so-called utlty functon wth each ndvdual, and fndng an allocaton that maxmzes the net utlty of the ndvduals. We now formally descrbe and model the resource allocaton problem n the Internet. Suppose we have a network wth a set of traffc sources S and a set of lnks L. Each lnk l L has a fnte fxed capacty c l. Each source n S s assocated wth a fxed route r L along whch t transmts at some rate x r to a fxed destnaton. In our model, a route s smply a collecton of lnks connectng a source to ts destnaton. In fact, the order of the lnks n the route s rrelevant for our mathematcal model. Note that we can use the ndex r to ndcate both a route and the source that sends traffc along that route and we wll follow ths notaton. Also snce multple sources could use the same set of lnks as ther routes, there could be multple ndces r whch denote the same subset of L. The utlty that the source obtans from transmttng data on route r at rate x r s denoted by U r (x r ). We assume that the utlty functon s contnuously dfferentable, non-decreasng and strctly concave. The concavty assumpton follows from the dmnshng returns dea a person downloadng a fle would apprecate the effect of a rate ncrease from 1 kbps to 100 kbps much more than an ncrease from 1 Mbps to 1.1 Mbps although the ncrease s the same n both cases. The goal of resource allocaton s to solve the followng optmzaton problem, called Network Utlty Maxmzaton (NUM): max x r U r (x r ) (1.9) r S

13 1.2. RESOURCE ALLOCATION AS UTILITY MAXIMIZATION 13 subject to the constrants x r c l, l L, (1.10) r:l r x r 0, r S. (1.11) The above nequaltes state that the capacty constrants of the lnks cannot be volated and that each source must be allocated a non-negatve rate of transmsson. The utlty maxmzaton problem has a unque soluton snce a strctly concave functon has a unque maxmzer over a closed and bounded set. In addton, the constrant set for the utlty maxmzaton problem s convex whch allows us to use the method of Lagrange multplers and the Karush-Kuhn-Tucker (KKT) theorem to solve the optmal soluton. We consder an example of such a maxmzaton problem n a small network and show how one can solve the problem usng the above method of Lagrange multplers. Example 1 Consder the network n Fgure 1.7 n whch three sources compete for resources n the core of the network. Lnks L 1, L 3 and L 5 have a capacty of 2 unts per second, whle lnks L 2 and L 4 have capacty 1 unt per second. There are three flows and denote ther data rates by x 0, x 1 and x 2. Fgure 1.7: Example llustratng network resource allocaton. We assume that lnks L 1, L 3 and L 5 have capacty 2, whle L 2 and L 4 have capacty 1. The access lnks of the sources are assumed to have nfnte capacty. There are three flows n the system. In our problem, lnks L 3 and L 4 are not used, whle L 5 does not constran source S 2. Assumng log utlty functons, the resource allocaton problem s gven by max x 2 log x r (1.12) r=0

14 14 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE wth constrants x 0 + x 1 1, x 0 + x 2 2, x 0, where x s the vector consstng of x 0, x 1 and x 2. Now, snce log x as x 0, the optmal resource allocaton wll not yeld a zero rate for any source, even f we remove the non-negatvty constrants. So the last constrant above s not actve. We defne p 1 and p 2 to be the Lagrange multplers correspondng to the capacty constrants on lnks L 1 and L 2, respectvely, and let p denote the vector of Lagrange multplers. Then, the Lagrangan s gven by Settng L x r L(x, p) = log x 0 + log x 1 + log x 2 p 1 (x 0 + x 1 ) p 2 (x 0 + x 2 ). = 0 for each r gves x 0 = 1 p 1 + p 2, x 1 = 1 p 1, x 2 = 1 p 2. (1.13) Lettng x 0 + x 1 = 2 and x 0 + x 2 = 1 (snce we can always ncrease x 1 or x 2 untl ths s true) yelds 3 p 1 = = 0.634, p 2 = 3 = Note that (1.8) s automatcally satsfed snce x 0 +x 1 = 2 and x 0 +x 2 = 1, and (1.7) s satsfed due to equaltes (1.13). Therefore, the values of the Lagrange multplers actually are the mnmzers of the dual functon. Hence, we have the fnal optmal allocaton x 0 = = 0.422, x 1 = A few facts are noteworthy n ths smple network scenaro: = 1.577, x 2 = 1 3 = Note that x 1 = 1/p 1, and t does not depend on p 2 explctly. Smlarly, x 2 does not depend on p 1 explctly. In general, we wll see later that the optmal transmsson rate for source r s only determned by the Lagrange multplers on ts route. We wll also see that ths feature s extremely useful n desgnng decentralzed algorthms to reach the optmal soluton. The value of x r s nversely proportonal to the sum of the Lagrange multplers on ts route. We wll see later that, n general, x r s a decreasng functon of the Lagrange multplers. Thus, the Lagrange multpler assocated wth a lnk can be thought of the prce for usng that lnk and the prce of a route can be thought of as the sum of the prces of ts lnks. If the prce of a route ncreases, then the transmsson rate of a source usng that route decreases. In the above example, t was easy to solve the Lagrangan formulaton of the problem snce the network was small. In the Internet whch conssts of thousands of lnks and possbly mllons of users, such an approach s not possble. In the next secton, we wll see that there are dstrbuted solutons to the optmzaton problem whch are easy to mplement n the Internet.

15 1.2. RESOURCE ALLOCATION AS UTILITY MAXIMIZATION Notons of Farness In our dscusson of the network utlzaton maxmzaton, we have assocated a utlty functon wth each user. The utlty functon can be vewed as a measure of satsfacton of the user when t gets a certan data rate from the network. A dfferent pont of vew s that a utlty functon s assgned to each user n the network by a servce provder wth the goal of achevng a certan type of resource allocaton. For example, suppose U(x r ) = log x r, for all users r. It s a well-known property of concave functons that f(x )(x x ) 0, (1.14) where x s the maxmzer of f(x). So the optmal rates whch solve the network utlty maxmzaton problem, {x r}, satsfy x r x r x 0, r r where {x r } s any other set of feasble rates. For log utlty functons, ths property states that, under any other allocaton, the sum of proportonal changes n the users utltes wll be nonpostve. Thus, f some User A s rate ncreases, then there wll be at least one other user whose rate wll decrease and further, the proporton by whch t decreases wll be larger than the proporton by whch the rate ncreases for User A. Therefore, such an allocaton s called proportonally far. If the utltes are chosen such that U r (x r ) = w r log x r, where w r 0 s some weght, then the resultng allocaton s sad to be weghted proportonally far. Another wdely used farness crteron n communcaton networks s called max-mn farness. An allocaton {x r} s called max-mn far f t satsfes the followng property: f there s any other allocaton {x r } such a user s s rate ncreases,.e., x s > x s, then there has to be another user u wth the property x u < x u and x u x s. In other words, f we attempt to ncrease the rate for one user, then the rate for a less-fortunate user wll suffer. The defnton of max-mn farness mples that mn x r r mn x r, r for any other allocaton {x r }. To see why ths s true, suppose that there exsts an allocaton such that mn x r r < mn x r. (1.15) r Ths mples that, for any s such that mn r x r = x s, the followng holds: x s < x s. Otherwse, our assumpton (1.15) cannot hold. However, ths mples that f we swtch the allocaton from {x r} to {x r }, then we have ncreased the allocaton for s wthout affectng a less-fortunate user (snce there s no less-fortunate user than s under {x r}). Thus, the max-mn far resource allocaton attempts to frst satsfy the needs of the user who gets the least amount of resources from the network. In fact, ths property contnues to hold f we remove all the users whose rates are the smallest under max-mn far allocaton, reduce the lnk capactes by the amounts used by these users and consder the resource allocaton for the rest of the users. The same argument as above apples. Thus, max-mn s a very egaltaran noton of farness.

16 16 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Yet another form of farness that has been dscussed n the lterature s called mnmum potental delay farness. Under ths form of farness, user r s assocated wth the utlty functon 1/x r. The goal of maxmzng the sum of the user utltes s equvalent to mnmzng r 1/x r. The term 1/x r can be nterpreted as follows: suppose user r needs to transfer a fle of unt sze. Then, 1/x r s the delay n assocated wth completng ths fle transfer snce the delay s smply the fle sze dvded by the rate allocated to user r. Hence, name mnmum potental delay farness. All of the above notons of farness can be captured by usng utlty functons of the form U r (x r ) = x1 α r 1 α, (1.16) for some α > 0. Resource allocaton usng the above utlty functon s called α-far. Dfferent values of α yeld dfferent deas of farness. Frst consder α = 2. Ths mmedately yelds mnmum potental delay farness. Next, consder the case α = 1. Clearly, the utlty functon s not welldefned at ths pont. But t s nstructve to consder the lmt α 1. Notce that maxmzng the sum of x1 α r 1 α yelds the same optmum as maxmzng the sum of Now, by applyng L Hosptal s rule, we get lm α 1 x 1 α r 1 1 α. x 1 α r 1 1 α = log x r, thus yeldng proportonal farness n the lmt as α 1. Next, we argue that the lmt α gves max-mn farness. Let x r(α) be the α-far allocaton. Assume that x r(α) x r as α and x 1 < x 2 <... < x n. Let ɛ be the mnmum dfference of {x r},.e., ɛ = mn r (x r+1 x r). Then when α s suffcently large, we have x r(α) x r ɛ/4, whch mples that x 1 (α) < x 2 (α) <... < x n(α). Now by the property of concave functons mentoned earler (nequalty (1.14)), x r x r(α) 0. (α) r Consderng an arbtrary flow s, the above expresson can be rewrtten as s (x r x r(α)) x α s (α) n x α r (α) + (x s x s(α)) + (x x (α)) x α s (α) x α (α) 0. r=1 Snce x r(α) x r ɛ/4, we further have x α r s (x r x r(α)) x α s (α) x α (α) + (x s x s(α)) r=1 r =s+1 n =s+1 x x (α) (x s + ɛ/4) α (x ɛ/4)α 0. Note that x ɛ/4 (x s + ɛ/4) ɛ/2 for any > s, so by ncreasng α, the thrd term n the above expresson wll become neglgble. Thus, f x s > x s(α), then the allocaton for at least one user whose rate satsfes x r(α) < x s(α) wll decrease. The argument can be made rgorous and extended to the case x r = x s for some r and s. Therefore as α, the α-far allocaton approaches max-mn farness.

17 1.3. MATHEMATICAL BACKGROUND: STABILITY OF DYNAMICAL SYSTEMS Mathematcal Background: Stablty of Dynamcal Systems Consder a dynamcal system defned by the followng dfferental equaton ẋ = f(x), f : R n R n, (1.17) where ẋ s the dervatve of x wth respect to the tme t. The tme varable t has been omtted n most of places when no confuson s caused. Assume that x(0) s gven. Throughout we wll assume that f s a contnuous functon and that t also satsfes other approprate condtons to ensure that the dfferental equaton has a unque soluton x(t), for t 0. A pont x e R n s sad to be the equlbrum pont of the dynamcal system f f(x e ) = 0. We assume that x e = 0 s the unque equlbrum pont of ths dynamcal system. Defnton (Globally, asymptotcally stable) x e = 0 s sad to be a globally asymptotcally stable equlbrum pont f lm t x(t) = 0 for any x(0) R n. We frst ntroduce the Lyapunov boundedness theorem. Theorem (Lyapunov boundedness theorem) Let V : R n R be a dfferentable functon wth the followng property: V (x) as x. (1.18) Denote by V (x) the dervatve of V (x) wth respect to t,.e., V (x) = V (x)ẋ = V (x)f(x). If V (x) 0 for all x, then there exsts a constant B > 0 such that x(t) B for all t. Proof At any tme T, we have T V (x(t )) = V (x(0)) + 0 V (x(t)) dt V (x(0)). Note that condton (1.18) mples that {x : V (x) c} s a bounded set for any c. Lettng c = V (x(0)), the theorem follows. Theorem (Lyapunov global asymptotc stablty theorem) If n addton to the condtons n the prevous theorem, we assume that V (x) s contnuously dfferentable and also satsfes the followng condtons: (1) V (x) 0 x and V (x) = 0 f and only f x = 0. (2) V (x) < 0 for any x 0 and V (0) = 0. Then, the equlbrum pont x e = 0 s globally, asymptotcally stable.

18 18 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Proof We prove ths theorem by contradcton. Suppose x(t) does not converge to the equlbrum pont 0 as t. Note that V (x(t)) s non-ncreasng because ts dervatve wth respect to t s non-postve ( V (x) 0) for any x. Snce V (x(t)) decreases as a functon of t and s lower bounded (snce V (x) 0 x), t converges as t. Suppose that V (x(t)) converges to, say, ɛ > 0. Defne the set C {x : ɛ V (x) V (x(0))}. The set C s bounded snce V (x) as x and t s closed snce V (x) s a contnuous functon of x. Thus, C s a compact set. Let a = sup V (x) x C where a > 0 s fnte because V (x) s contnuous n x and C s a compact set. Now we wrte V (x(t)) as t V (x(t)) = V (x(0)) + V (x(s)) ds 0 V (x(0)) at, whch mples that V (x(t)) = 0, t V (x(0)), a and x(t) = 0, t V (x(0)). a Ths contradcts wth the assumpton that x(t) does not converge to 0. The Lyapunov global asymptotc stablty theorem requres that V (x) 0 for any x 0. In the case V (x) = 0 for some x 0, global asymptotc stablty can be studed usng Lasalle s nvarance prncple. The proof of the theorem s omtted n ths book Theorem (Lasalle s nvarance prncple) Replace condton (2) of the prevous theorem by V (x) 0 x, and suppose that the only trajectory x(t) that satsfes ẋ(t) = f(x(t)) and V (x(t)) = 0, t s x(t) = 0 t. Then x = 0 s globally, asymptotcally stable.

19 1.4. DISTRIBUTED ALGORITHMS: PRIMAL SOLUTION Dstrbuted Algorthms: Prmal Soluton In the prevous secton, we formulated an optmzaton problem, the soluton of whch provded far resource allocaton. However, the technque used to solve the optmzaton problem n Example 1 assumed that we had complete knowledge of the topology and routes. Clearly ths s nfeasble n a gant network such as the Internet. In ths secton and the next, we wll study dstrbuted algorthms whch only requre lmted nformaton exchange among the sources and the network for mplementaton. The approach n ths secton s called the prmal soluton. We frst relax the capacty constrants: nstead of requrng that the total arrval rate at each lnk s less than the capacty, we assume that there s a cost for sendng data at a certan rate over a lnk. Then, nstead of havng these resource constrants n the optmzaton problem, we subtract the cost from the total utlty of the sources n the network as follows: W (x) = U r (x r ) ( ) B l x s, (1.19) r S l L s:l s where x s the vector of rates of all sources and B l (.) s the cost of sendng data on lnk l: t can be nterpreted as ether a barrer functon assocated wth lnk l whch ncreases to nfnty when the arrval rate on lnk l approaches the lnk capacty c l or a penalty functon whch penalzes the arrval rate for exceedng the lnk capacty. By approprate choce of the functon B l, one can solve the exact utlty optmzaton problem posed n the prevous secton; for example, choose B l (y) to be zero f y c l and equal to f y > c l. However, such a soluton may not be desrable or requred. For example, the desgn prncple may be such that one requres the delays on all lnks to be small. Whle t s not apparent n the determnstc formulaton here, later n the book we wll see that even when the arrval rate on a lnk s less than ts capacty, due to randomness n the arrval process, packets n the network wll experence delay or packet loss. The functon B l (.) may thus be used to represent average delay, packet loss rate, etc. Thus, W (x) represents a tradeoff: large values of x r ncrease utlty, but result n packets ncurrng excessve delays or other mparments at the lnk. We frst assume that B l s a convex functon so that the functon (1.19) s a strctly concave functon.. Further, assume that B l s contnuously dfferentable. Then, we can equvalently requre that B l ( s:l s x s ) = s:l s xs 0 f l (y)dy, (1.20) where f l ( ) s an ncreasng, contnuous functon. We call f l (y) the congeston prce functon, or smply the prce functon, assocated wth lnk l, snce t assocates a prce wth the level of congeston on the lnk. It s straghtforward to see that B l defned n the above fashon s convex, snce ntegratng an ncreasng functon results n a convex functon (see Result 1.1.1). We wll assume that U r and f l are such that the maxmzaton of (1.19) results n a soluton wth x r > 0 r S. Now, the condton that must be satsfed by the maxmzer of (1.19) s obtaned by dfferentaton and s gven by U r(x r ) l:l r ( ) f l x s = 0, r S. (1.21) s:l s

20 20 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE We now requre a dstrbuted algorthm that would drve x towards the soluton of (1.21). A natural canddate for such an algorthm s the so-called gradent ascent algorthm from optmzaton theory. The dea here s that f we want to maxmze a functon of the from g(x), then we progressvely change x so that g(x(t + δ)) > g(x(t)). We do ths by fndng the drecton n whch a change n x produces the greatest ncrease n g(x). Ths drecton s gven by the gradent of g(x) wth regard to x. In one dmenson, we merely choose the update algorthm for x as x(t + δ) = x(t) + k(t) dg(x) dx δ, where k(t) s a scalng parameter whch controls the amount of change n the drecton of the gradent, or lettng δ 0 ẋ = k(t) dg(x) dx. (1.22) Let us try to desgn a smlar algorthm for the network utlty maxmzaton problem. Consder the algorthm ( ẋ r = k r (x r ) U r(x r ) ( )) f l x s. (1.23) l:l r s:l s We have obtaned the above by dfferentatng (1.19) wth respect to x r to fnd the drecton of ascent, and used t along wth a scalng functon k r ( ) to construct an algorthm of the form shown n (1.22). The scalng functon k r ( ) must be chosen such that the equlbrum of the dfferental equaton s the same as the one obtaned from (1.21). Thus, the equlbrum pont of the dfferental equaton s the same as the soluton to the resource allocaton problem. The controller s called a prmal algorthm snce t arses from the prmal formulaton of the utlty maxmzaton problem. Note that the prmal algorthm has many ntutve propertes that one would expect from a resource allocaton/congeston control algorthm. When the route prce q r = l:l r f l( s:l s x s) s large, then the congeston controller decreases ts transmsson rate. Further, f x r s large, then U (x r ) s small (snce U r (x r ) s concave) and thus the rate of ncrease s small as one would expect from a resource allocaton algorthm whch attempts to maxmze the sum of the user utltes. We must now answer two questons regardng the performance of the prmal congeston control algorthm: What nformaton s requred at each source n order to mplement the algorthm? Does the algorthm actually converge to the desred equlbrum pont? Below we consder the answer to the frst queston and develop a framework for answerng the second. The precse answer to the convergence queston wll be presented n the next subsecton. The frst queston s easly answered by studyng (1.23). It s clear that all that the source r needs to know n order to reach the optmal soluton s the sum of the prces of each lnk on ts route. How would the source be apprased of the lnk prces? The answer s to use a feedback mechansm each packet generated by the source collects the prce of each lnk that t traverses. When the destnaton receves the packet, t sends ths prce nformaton n a small packet (called the acknowledgment packet or ack packet) that t sends back to the source.

21 1.4. DISTRIBUTED ALGORITHMS: PRIMAL SOLUTION 21 To vsualze ths feedback control system, we ntroduce a matrx R whch s called the routng matrx of the network. The (l, r) element of ths matrx s gven by Let us defne { 1 f route r uses lnk l R lr = 0 else y l = s:l s x s, (1.24) whch s the load on lnk l. Usng the elements of the routng matrx, y l can also be wrtten as y l = s R ls x s. Lettng y be the vector of all y l (l L), we have Let p l (t) denote the prce of lnk l at tme t,.e., y = Rx. (1.25) p l (t) = f l ( s:l s x s (t) ) = f l (y l (t)). (1.26) Then the prce of a route s just the sum of lnk prces p l of all the lnks n the route. So we defne the prce of route r to be q r = l:l r p l. (1.27) Also let p be the vector of all lnk prces and q be the vector of all route prces. We thus have q = R T p (1.28) The relatonshps derved above can be made clear usng the block dagram n Fgure 1.8. We show that the prmal controller of (1.23) s globally asymptotcally stable by usng the Lyapunov functon dea descrbed n Secton 1.3. Recall that W (x) s a strctly concave functon. Let ˆx be ts unque maxmzer. Then, W (ˆx) W (x) s non-negatve and s equal to zero only at x = ˆx. Thus, W (ˆx) W (x) s a natural canddate Lyapunov functon for the system (1.23). We use ths Lyapunov functon n the followng theorem. Theorem Consder a network n whch all sources follow the prmal control algorthm (1.23). Assume that the functons U r ( ), k r ( ) and f l ( ) are such that V (x) = W (ˆx) W (x) s such that V (x) as x, ˆx > 0 for all, and W (x) s as defned n (1.19). Then, the controller n (1.23) s globally asymptotcally stable and the equlbrum value maxmzes (1.19).

22 22 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE Fgure 1.8: A block dagram vew of the congeston control algorthm. The controller at the source uses congeston feedback from the lnk to perform ts acton. Proof Dfferentatng V (.), we get V = r S V ẋ r = k r (x r ) ( U ) 2 x r(x r ) q r < 0, x ˆx, (1.29) r r S and V = 0 f x = ˆx. Thus, all the condtons of the Lyapunov theorem are satsfed and we have proved that the system state converges to ˆx, startng from any ntal condton. In the proof of the above theorem, we have assumed that the utlty, prce and scalng functons are such that W (x) has some desred propertes. It s very easy to fnd functons that satsfy these propertes. For example, f U r (x r ) = w r log(x r ), and k r (x r ) = x r, then the prmal congeston control algorthm for source r becomes ẋ r = w r x r f l (y l ), and thus the unque equlbrum pont s w r /x r = l:l r f l(y l ). If f l (.) s any polynomal functon, then W (x) goes to as x and thus, V (x) as x. l:l r Prce Functons and Congeston Feedback We had earler argued that collectng the prce nformaton from the network s smple. If there s a feld n the packet header to store prce nformaton, then each lnk on the route of a packet smply adds ts prce to ths feld, whch s then echoed back to the source by the recever n the acknowledgment packet. However, packet headers n the Internet are already crowded wth a lot of other nformaton, so Internet practtoners do not lke to add many bts n the packet header to collect congeston nformaton. Let us consder the extreme case where there s only one bt avalable n the packet header to collect congeston nformaton. How could we use ths bt to collect the prce of route? Suppose that each packet s marked wth probablty 1 e p l when the

23 1.5. DISTRIBUTED ALGORITHMS: DUAL SOLUTION 23 packet passes through lnk l. Markng smply means that a bt n the packet header s flpped from a 0 to a 1 to ndcate congeston. Then, along a route r, a packet s marked wth probablty 1 e l:l r p l. If the acknowledgment for each packet contans one bt of nformaton to ndcate f a packet s marked or not, then by computng the fracton of marked packets, the source can compute the route prce l:l r p l. The assumpton here s that x r s change slowly so that each p l remans roughly constant over many packets. Thus, one can estmate p l reasonably accurately. Another prce functon of nterest s found by consderng packet droppng nstead of packet markng. If packets are dropped due to the fact that a lnk buffer s full when a packet arrves at the lnk, then such a droppng mechansm s called a Droptal scheme. ( A ) crude approxmaton to the drop probablty (also known as packet loss rate) at lnk l s yl c +, l whch s non-zero only f y l = r:l r x r s larger than c l. When packets are dropped at a lnk for source r, then the arrval rate from source r at the next lnk on the route would be smaller due to the fact that dropped packets cannot arrve at the next lnk. Thus, the arrval rate s thnned as we traverse the route. However, ths s very dffcult to model n our optmzaton framework. Therefore, we assume that the drop probabltes are small so that arrval rate of packets from a gven source s approxmately the same at all lnks on ts route. Further, the end-to-end drop probablty on a route can be approxmated by the sum of the drop probabltes on the lnks along the route f the drop probablty at each lnk s small. Thus, the optmzaton formulaton approxmates realty under these assumptons. 1.5 Dstrbuted Algorthms: Dual Soluton In ths secton we consder another dstrbuted algorthm based on the dual formulaton of the utlty maxmzaton problem. Consder the resource allocaton problem that we would lke to solve max U r (x r ) (1.30) x r r S y l subject to the constrants x r c l, l L, (1.31) r:l r x r 0, r S. (1.32) The Lagrange dual of the above problem s obtaned by ncorporatng the constrants nto the maxmzaton by means of Lagrange multplers as follows: D(p) = max U r (x r ) ( ) p l x s c l (1.33) {x r 0} r l s:l s Here the p l s are the Lagrange multplers that we saw n Secton 1.1. The dual problem may then be stated as mn p 0 D(p).

24 24 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE As n the case of the prmal problem, we would lke to desgn an algorthm whch ensures that all the source rates to converge to the optmal soluton. Notce that n ths case we are lookng for a gradent descent (rather than a gradent ascent that we saw n the prmal formulaton), snce we would lke to mnmze D(p). To fnd the drecton of the gradent, we need to know D p l. We frst observe that n order to acheve the maxmum n (1.33), x r must satsfy U r(x r ) = q r, (1.34) where, as usual, q r = l:l r p l, s the prce of a partcular route r. Note that we have assumed that x r > 0 n wrtng down (1.34). Ths would be true, for example, f the utlty functon s an α-utlty functon wth α > 0. Now, Thus, usng (1.34), we have D p l = r = r = r = r U r(x r ) x r p l (y l c l ) k U r(x r ) x r p l (y l c l ) k U r(x r ) x r p l (y l c l ) r U r(x r ) x r p l (y l c l ) r y k p k p l p k x r p l r:k r k:k r x r p l q r. x r p l D p l = (y l c l ). (1.35) Recallng that, to mnmze D(p), we have to descend down the gradent, from (1.34) and (1.35), we have the followng dual control algorthm: where h l > 0 s a constant and (g(x)) + y denotes { (g(x)) + g(x), y > 0, y = max(g(x), 0), y = 0. x r = U r 1 (qr ) and (1.36) ṗ l = h l (y l c l ) + p l, (1.37) We use ths modfcaton to ensure that p l never goes negatve snce we know from the KKT condtons that the optmal prce s non-negatve. Note that, f h l = 1, the prce update above has the same dynamcs as the dynamcs of the queue at lnk l. The prce ncreases when the arrval rate s larger than the capacty and decreases when the arrval rate s less than the capacty. Moreover, the prce can never become negatve. These are exactly the same dynamcs that govern the queue sze at lnk l. Thus, one does not even have to explctly keep track of the prce n the dual formulaton; the queue length naturally provdes ths nformaton. The stablty of ths algorthm follows n a manner smlar to the prmal algorthm by consderng D(p) as the Lyapunov functon snce the dual algorthm s smply a gradent algorthm for fndng the mnmum of D(p). In the next secton, we wll dscuss practcal TCP protocols based on the prmal and dual formulatons. When we dscuss these protocols, we wll see that the prce functons and congeston control mechansms obtaned from the two formulatons have dfferent nterpretatons. p k

25 1.6. RELATIONSHIP TO TCP PROTOCOLS Relatonshp to TCP Protocols In ths secton, we explore the relatonshp between the algorthms dscussed n the prevous sectons and the protocols used n the Internet today. It s mportant to note that Internet congeston control protocols were not desgned usng the optmzaton formulaton of the resource allocaton problem that we have seen n the prevous two sectons. The predomnant concern whle desgnng these protocols was to mnmze the rsk of congeston collapse,.e., large-scale buffer overflows, and hence they tended to be rather conservatve n ther behavor. Even though the current Internet protocols were not desgned wth clearly-defned farness and stablty deas n mnd, they bear a strong resemblance to the deas of far resource allocaton that we have dscussed so far. In fact, the utlty maxmzaton methods presented earler provde a sold framework for understandng the operaton of these congeston control algorthms. Further, gong forward, the utlty maxmzaton approach seems lke a natural canddate framework used to modfy exstng protocols to adapt to the evoluton of the Internet as t contnues to grow faster. As mentoned n the frst chapter, the congeston control algorthms used n today s Internet are based on wndow flow control. The dea s that each user mantans a number called a wndow sze, whch s the number of unacknowledged packets that t s allowed to send nto the network. Any new packet can be sent only when an acknowledgment for one of the prevous sent packets s receved by the sender as shown n Fgure 1.9. Fgure 1.9: Wndow Flow Control. The wndow sze s set to be 5, so at most fve unacknowledged packets are allowed. After an addtonal acknowledgement s receved, one more packet can be sent out. The wndow sze W s closely related to the data rate x of the flow as explaned below. The amount of tme that elapses between the sendng of a packet and the recepton of feedback from the destnaton s called the Round-Trp Tme (RTT). We denote the RTT by T. Assume the lnk speeds are very fast so that the tme that t takes to process a packet at a lnk s neglgble compared to the RTT. Suppose that the wndow sze s W and as a result, the source send W packets nto the network. Then, snce the processng tme s neglgble, the acks for these packets wll arrve roughly at the same tme at the source (after one RTT tme). Thus, the average rate of transmsson x s just the wndow sze dvded by T,.e., x = W/T. Clearly ths model s very crude, but t works surprsngly well n practce. Because of ths relaton between wndow sze and data rate, the data rate of a flow can be

26 26 CHAPTER 1. MATHEMATICS OF INTERNET ARCHITECTURE controlled by adaptng the wndow sze. Transmsson Control Protocol (TCP) s the protocol that determnes the ncrease/decrease of the wndow sze n today s Internet. There are several dfferent flavors of TCP congeston control, each of whch operates somewhat dfferently. But all versons of TCP are wndow-based protocols. TCP adapts the wndow sze n response to congeston nformaton. The wndow sze s ncreased f the sender determnes that there s excess capacty present n the route, and decreased f the sender determnes that the current number of n-flght packets exceeds the capacty of the route. A decson on whether to send a new packet, and whether the wndow s to be ncreased or decreased, s taken upon recepton of the acknowledgement packet. Ths means that the decson-makng process has no perodcty that s decded by a clock of fxed frequency. TCP s therefore called self-clockng. Dfferent versons of TCP use dfferent algorthms to determne when and how to ncrease/decrease the wndow szes. We dscuss these versons of TCP next TCP-Reno The most commonly used TCP versons used for congeston control n the Internet today are Reno and NewReno. Both of them are updates of TCP-Tahoe, whch was ntroduced n Although they vary sgnfcantly n many regards, the basc approach to congeston control s smlar. The dea s to use successful recepton packets as an ndcaton of avalable capacty and dropped packets as an ndcaton of congeston. We consder a smplfed model for the purpose of exposton. Each tme the destnaton receves a packet, t sends an acknowledgement (also called ack) askng for the next packet n sequence. For example, when packet 1 s receved, the acknowledgement takes the form of a request for packet 2. If, nstead of the expected packet 2, the destnaton receves packet 3, the acknowledgement stll requests packet 2. Recepton of three duplcate acknowledgments or dupacks (.e., four successve dentcal acks) s taken as an ndcaton that packet 2 has been lost due to congeston. The source then proceeds to cut down the wndow sze and also to re-transmt lost packets. In case the source does not receve any acknowledgements for a pre-determned tme, t assumes that all ts packets n flght have been lost and tmes out. When a non-duplcate acknowledgment s receved, the protocol ncreases ts wndow sze. The amount by whch the wndow sze s ncreased depends upon the TCP transmsson phase. TCP operates n two dstnct phases. When fle transfer begns, the wndow sze s 1, but the source rapdly ncreases ts transmsson wndow sze so as to reach the avalable capacty quckly. Let us denote the wndow sze by W. The algorthm ncreases the wndow sze by 1 each tme an acknowledgement s receved,.e., W W + 1. Ths s called the slow-start phase. Snce we have assumed that packet processng tme at a lnk s small compared to the RTT, the number of acknowledgements receved by a source n one RTT would be approxmately equal to the wndow sze. If we ncrease the wndow sze by one for each successful packet transmsson, ths also means that (f all transmssons are successful) the wndow would roughly double n each RTT, so we have an exponental ncrease n rate as tme proceeds. Slow-start refers to the fact that the wndow sze s stll small n ths phase, but the rate at whch the wndow ncreases s qute rapd. When the wndow sze ether hts a threshold, called the slow-start threshold or ssthresh or f a packet loss s detected (mmedately leadng to a halvng of wndow sze), the algorthm shfts to a more conservatve wndow-ncrease algorthm called the congeston avodance phase. When n the congeston-avodance phase, the algorthm ncreases the wndow sze by 1/W every tme feedback of a successful packet transmsson s receved, so we now have W W + 1/W. Thus, n

27 1.6. RELATIONSHIP TO TCP PROTOCOLS 27 each RTT, the wndow ncreases by one packet,.e., a lnear ncrease n rate as a functon of tme. When a packet loss s detected by the recept of three dupacks, the slow-start threshold (ssthresh) s set to W/2 and TCP Reno cuts ts wndow sze by half,.e., W W/2. Protocols of ths sort where the ncrement s by a constant amount, but the decrement s by a multplcatve factor are called addtve-ncrease multplcatve-decrease (AIMD) protocols. When packet loss s detected by a tme-out, the wndow sze s reset to 1 and TCP enters the slow-start phase. We llustrate the operaton of TCP-Reno n Fgure Fgure 1.10: Operaton of TCP-Reno. The wndow sze frst exponentally ncreases durng the slow start phase untl reachng ssthresh (=12). Then TCP-Reno enters congeston avodance phase and the wndow sze lnearly ncreases. When a loss s detected by recevng three dupacks at the 9 th RTT, the ssthresh s set to be 8, and the wndow sze s set to 8 and ncreases lnearly. When a tme-out occurs at the 16 th RTT, the ssthresh s set to be 7. The wndow sze s set to be 1, and TCP-Reno enters the slow-start phase. Now, the slow-start phase of a flow s relatvely nsgnfcant f the flow conssts of a large number of packets. So we wll consder only the congeston-avodance phase. Let us call the congeston wndow at tme t as W (t). Ths means that the number of packets n-flght s W (t). The tme taken by each of these packets to reach the destnaton, and for the correspondng acknowledgement to be receved s T. The RTT s a combnaton of propagaton delay and queueng delay, but we gnore the fluctuatons n queueng delay and assume that the RTT s a constant. Let us now wrte down TCP Reno s behavor n terms of the dfferental equaton models. Consder a flow r. As defned above, let W r (t) denote the wndow sze and T r ts RTT. Earler we used the notaton q r (t) to denote the prce of a route r. TCP uses packet loss probablty as the prce of a route. So we use the same notaton q r (t) to denote the packet loss probablty under TCP. We can model the congeston avodance phase of TCP-Reno as Ẇ r (t) = x r(t T r )(1 q r (t)) W r (t) The above equaton can be derved as follows: βx r (t T r )q r (t)w r (t). (1.38) The rate at whch the source obtans acknowledgements s x r (t T r )(1 q r (t)). Snce each acknowledgement leads to an ncrease by 1/W r (t), the rate at whch the wndow sze ncreases s gven by the frst term on the rght sde.