Convergence Complexity of Optimistic Rate-Based Flow-Control Algorithms*

Transcription

1 Journal of Algorthms 30, Ž Artcle ID agm , avalable onlne at on Convergence Complexty of Optmstc Rate-Based Flow-Control Algorthms* Yehuda Afek, Yshay Mansour, and Zv Ostfeld Computer Scence Department, Tel-A Unersty, Tel-A 69978, Israel E-mal: or Receved March 17, 1996; revsed January 9, 1998 Ths paper studes basc propertes of rate-based flow-control algorthms and of the max-mn farness crtera. For the algorthms we suggest a new approach for ther modelng and analyss, whch may be consdered more optmstc and realstc than tradtonal approaches. Three varatons of the approach are presented, and ther rate of convergence to the optmal max-mn farness soluton s analyzed. In addton, we ntroduce and analyze approxmate rate-based flow-control algorthms. We show that under certan condtons the approxmate algorthms may converge faster. However, we show that the resultng flows may be substantally dfferent from the flows dctated by the max-mn farness. We further demonstrate that the max-mn farness soluton can be very senstve to small changes,.e., there are confguratons n whch an addton or deleton of a sesson Ž n 2 wth rate may change the allocaton of another sesson by 2., but by no more than OŽ 2 n.. Ths mples that the max-mn farness crtera may provde a bad estmate of how far a set of flow allocatons s from the optmal allocaton Academc Press 1. INTRODUCTION The ATM Forum on Traffc Management has adopted rate-based flow control Cha94, Jaf81, Hay81, Mos84, GB84, Gaf82, BG87as the bass for flow control n ts networks for ABR Ž Avalable Bt Rate. traffc Žsee BF95, for example.. The maor arguments for rate-based flow control are the smplcty and modest hardware requrements per vrtual crcut, compared wth those of the credt-based scheme KM95. Conceptually, the rate-based flow control adusts transmsson rates of dfferent sessons Ž vrtual crcuts. n the network n an end-to-end manner, amng to * A prelmnary verson of ths paper was presented n the 28th Annual ACM Symposum on Theory of Computng, STOC, Phladelpha, PA, 1996, $30.00 Copyrght 1999 by Academc Press All rghts of reproducton n any form reserved. 106

2 COMPLEXITY OF RATE-BASED FLOW CONTROL 107 acheve a good utlzaton of the network whle mantanng farness between the sessons. Most mplementatons of rate-based flow control work as follows. A control message Ž Resource Management cell, RM. loops around the path of each sesson. 1 On ts way the control message calculates the mnmum share the sesson may take from the excess capacty of the lnks along the path. Roughly, the share a sesson may take from a lnk s the excess capacty of the lnk dvded by the number of sessons that may stll ncrease ther transmsson rate and share the excess. The rate of the sesson s then ncreased by ths mnmum Žsee Cha94.. A key pont n these algorthms s the computaton n each step of how much each sesson may be ncreased. Ths computaton essentally determnes how the network resources Ž lnk capactes. are dvded among the dfferent vrtual crcuts. The basc prncple gudng ths computaton s the desre to share the lnk capactes n a far way among the dfferent sessons. The max-mn farness crteron BG87, Jaf81, GB84, Gaf82, Hay81, Mos84, Cha94 s wdely accepted as the theoretcal crteron gudng ths computaton. A set of sessons s sad to be n the state of max-mn farness f t s mpossble to nfntesmally ncrease the rate of any sesson wthout decreasng the rate of some sessons whose rate s equal or smaller. It can be shown that there s exactly one unque set of flow values that acheves the max-mn farness for a set of sessons wth requrements n a gven network. The max-mn crteron s consdered far because no sesson gets a larger flow on the account of a sesson wth a smaller or equal flow. 2 Gven a network wth lnk capactes and a set of sessons wth transmsson requrements, there s a smple teratve algorthm for computng the set of transmsson rates that correspond to the max-mn farness rates BG87 : In each teraton fx the rates of sessons that use the most congested lnk Ž.e., the lnk that restrcts a mnmum rate for sessons that have not yet been fxed wth ther fnal rate.. In the next teraton the rates that have been fxed are subtracted from the lnk capactes, and the process s terated. ŽThus, we are maxmzng the mnmum sesson and contnung recursvely on the remanng sessons, after updatng the capactes.. Whle ths algorthm seem to be effcent, t s mpractcal, snce n each teraton the new globally most congested lnk has to be selected. Most theoretcal rate-based flow-control algorthms that guarantee convergence to the state of max-mn farness adust the rates of sessons n a 1 Note that accordng to the ATM concept each sesson has a fxed path. Hence, the flow control mechansms addressed n ths paper have nothng to do wth the ssues of routng. 2 The problem of fndng the max-mn farness can be expressed as a set of lnear nequaltes, as presented n Hay81.

3 108 AFEK, MANSOUR, AND OSTFELD conservatve way. An algorthm s consdered to be conservatve f t never overshoots. That s, gven a fxed set of sessons, t converges to the max-mn rates wthout ever assgnng a sesson a rate that s larger than ts fnal rate. In other words, the rates that are assgned durng the computaton may only ncrease, untl the fnal value s reached BG87, GB84, Cha94. One problem wth most of the conservatve algorthms s that they do not nclude a decrease operaton that reduces the rate of sessons Žwth the excepton of Cha94.. Ths does not model the stuaton of real networks that have to be able to decrease the rate of some sessons Že.g., after a new sesson has been added.. Furthermore, the network has some current set of rates, whch should be adusted, and t would be preferable to perform an ncremental update rather than startng each tme from scratch. In ths paper we examne a more aggressve approach, whch we term optmstc. Accordng to ths approach, durng the computaton, sessons may be gven a rate that s larger than ther fnal rate. Hence, durng the convergence process rates may go up and down. Ths approach s closer to the proposed rate-based approach of the ATM Forum Rob94, ST94, JKV94. We choose a smple abstracton for the update operatons. The update operaton s performed atomcally on a sesson. Ths means that an update operaton s not performed untl the prevous one has completed. Another assumpton s that the swtch has full knowledge of the current rates of all of the sessons crossng t. Ths model s presented for the purpose of analyss and s not expected to be mplemented. However, we beleve that ths model captures the essental propertes of the problem as faced by practcal dstrbuted rate-based algorthms, such as Rob94, ST94, JKV94. The optmstc approach s close to the approach used by Roberts algorthm ŽRob94.. However, snce n our model the update operaton s atomc and the swtch knows the rates of all sessons that cross t, the sum of the rates of the sessons that use a lnk may never exceed the lnk capacty. In Roberts algorthm the swtch has only partal knowledge about the rate of the sessons that cross t, and snce lnk updates are not atomc an output lnk mght be overloaded. Our model s somewhat smlar to that of Charny Cha94, where the swtch knows the rates of all sessons that cross t. However, Charny uses a conservatve approach and does not assume atomc updates. Furthermore, the convergence complexty n ŽCha94. s measured n tme, whle n ths paper the complexty measure s abstracted by the number of update operatons. Each of the algorthms n ths paper s consdered as a sequence of atomc operatons called update operatons that are appled on the

4 COMPLEXITY OF RATE-BASED FLOW CONTROL 109 network. Each update operaton ncreases the rate of one sesson and decreases the rate of a subset of sessons that share an edge wth t. An update operaton s performed on sesson S only f ts rate may be ncreased wthout decreasng the rate of any other sesson, below the new fnal rate of S. The magntude of ncrease s the maxmum that satsfes the above condton. The man dfference between the dfferent algorthms s the scheduler that n each network state decdes whch sesson to ncrease next. For example, f a lnk wth capacty 19 s shared by fve sessons wth rates 1, 2, 3, 6, and 7, and the update operaton s appled to the second sesson Ž whose rate s 2., then the resultng rates are 1, 5, 3, 5, and 5. Two parameters dstngush between the dfferent algorthms consdered here. One s the scheduler that n each state selects the next sesson to be consdered for an ncrease, and the second s the rule by whch the algorthm decdes whether to perform the update. The complexty measure used n analyzng the dfferent algorthms s the total number of update operatons appled untl the algorthm quesces Ž stops to change rates., whch we call the convergence complexty. For the frst mechansm, the scheduler, we mark sessons that have reached ther fnal rate as done. Sessons that have not yet reached ther fnal rate are consdered acte. We consder and analyze three dfferent schedulers: Ž. 1 a global mn scheduler n whch the next update operaton s performed usng an acte sesson wth the global smallest rate, Ž. 2 a local mn scheduler n whch the next update operaton s performed usng an acte sesson whose rate s smallest among all of the acte sessons that share a lnk wth t, and Ž. 3 an arbtrary scheduler that arbtrarly selects the next sesson to be ncreased. Note that the local mn s more approprate for a dstrbuted envronment than the global mn, and the arbtrary s even more dstrbuted than both. In dscussng the algorthms n ths order we go from the easer Ž to analyze. to the more dffcult. For the update decson we consder two varants: Ž. 1 a selected sesson that s ncreased as long as ts rate may be ncreased, and Ž. 2 approxmate algorthms, n whch the rate of a sesson s ncreased only f the ncrease s by more than. The approxmate algorthms reach quescence when no sesson may be ncreased by more than. The man results presented n ths paper are as follows: 1. In both the global and the local mnmum algorthms Žn 2. update operatons are both necessary and suffcent to reach the max-mn farness state, where n s the total number of sessons n the network Ž.e., t s suffcent for all scenaros, and there are scenaros for whch t s necessary..

5 110 AFEK, MANSOUR, AND OSTFELD 2. In the arbtrary algorthm OŽ n. update operatons are shown to be suffcent to reach the max-mn farness state Žwhere Ž ' An example s gven n whch the adversary may force Ž2 n 2. update operatons. 3. Any approxmate algorthm Žany algorthm that refrans from updates that are smaller than or equal to. quesces at a state n whch the rate of every sesson may be smaller or larger than ts rate n the optmal max-mn farness allocaton by at most OŽ 2 n.. 4. An example of a confguraton n whch each sesson cannot be Ž n 2 ncreased by more than, yet some sesson s 2. away from ts allocaton n the max-mn vector, s provded. Ths s a lower bound on the possble gap between the rates computed by any approxmate algorthm and the rates n the max-mn farness allocaton. Although ths property s exhbted only under pecular condtons, t s shared by any algorthm that refrans from small updates. Furthermore, ths lower-bound phenomenon s possble also f the rate of only one sesson can be ncreased by,.e., f the rate of no other sesson may be ncreased. However, ths phenomenon does not ndcate a dsadvantage of the approxmate algorthms, snce n partcular, the removal or addton of a sesson wth a small rate to the network could have the same effect. Hence, ths example manly suggests that the state of max-mn farness may be unstable n the sense that small fluctuatons n the rate of one sesson may cause dramatc changes n the rate of other sessons. 5. If the number of sessons that share an edge s bounded by a constant, then the approxmate algorthm under ether the global mnmum scheduler or the enhanced local mnmum scheduler Ža slght modf-. Ž Ž 2 caton of the local mnmum scheduler reaches quescence n O mn n, n log Ž W... update operatons, where W s at most the maxmum capacty of an edge n the network Ža tghter bound on W s gven n Secton 7.. Note that these mproved bounds for the approxmate algorthms hold for the schedulers that are based upon a mnmum concept Ž ether local or global.. For arbtrary schedulers, t can be shown that the example that gves the lower bound of Ž2 n 2. on the number of steps stll holds regardless of the value of. The results mentoned n 3 and 4 answer an open queston rased by Charny Cha94 about the convergence of rate-based flow-control algorthms f the rates are restrcted to dscrete values. Notce that the goal of ths paper s to analyze the behavor of flow control n ATM networks. Hence, we do not consder specfc mplementatons of the above models Ž n complance wth the ATM standards. and are satsfed wth the clam that the model reflects the behavor of ATM networks. The results presented n ths paper provde a deeper understand-

6 COMPLEXITY OF RATE-BASED FLOW CONTROL 111 ng of the behavor of far rate-based flow-control algorthms. One concluson s that t s benefcal to update the sessons wth the lower rates before updatng the sessons wth the larger rates. The lower bound examples pont to the hopefully rare cases that one should watch out for, n whch ether the convergence to the max-mn state s very slow, or the possblty n whch a small gap or a small change n the rate of one sesson may cause a large gap n the rates of other sessons. We assumed throughout ths work that every sesson s greedy and would lke to consume as much rate as allowed by the network. In case there s a permanent self-restrcton of a source Žpeak cell rate n terms of ATM., t s done by addng a sngle edge that connects the source to a swtch, n whch the capacty s the restrcted bandwdth. The man ssue addressed here s the convergence complexty,.e., the number of update operatons necessary to reach the fnal state. Therefore, we assume that sessons do not change ther requrements durng the executon of the algorthm and all sesson rates are ntally zero. However, none of the algorthms and bounds dscussed need an all-zero ntal state to correctly operate; they may start from any Ž legal. ntal set of flows. The model, general defntons, and notatons are gven n Secton 2. In Secton 3 basc propertes that are common to all of our algorthms are presented. The global mnmum algorthm s analyzed n Secton 4, the local mnmum algorthm s analyzed n Secton 5, and the arbtrary algorthm s analyzed n Secton 6. Approxmate algorthms are ntroduced and analyzed n Secton GENERAL DEFINITIONS AND NOTATIONS We model a network as a drected graph G Ž V, E., where each edge e E s a lnk n the network. Each edge e E has a certan capacty denoted by CapŽ e.. A set of n sessons S,...,S 4 1 n s lad out n the network, where sesson S s a set of lnks that consttutes a smple path n G between a source and a destnaton. The bandwdth allocated to sesson S s denoted by ratež S.. For every edge e E we defne Ž e. to be the set of sessons that use e,.e., Ž e. S e S 4. The capacty constrant requres that for every edge e E the sum of the rates of the sessons that share t s at most the edge capacty,.e., Ý ratež S. CapŽ e. S Ž e.. If all of the capac ty constrants are satsfed, then the allocaton s a feasble allocaton. A feasble allocaton s maxmal f no sesson can ncrease ts rate wthout decreasng the rate of any other sesson. A far allocaton s a feasble max-mn allocaton f to ncrease the rate of any sesson one needs to decrease the rate of sessons wth lower or equal rates.

7 112 AFEK, MANSOUR, AND OSTFELD We now gve a formal defnton for the term far flow Žsmlar defntons may be found n other works Cha94, Jaf81, Hay81, Mos84, GB84,. Ž 1 2 n. n Gaf82, BG87. A vector x x, x,..., x R s sad to be lexco- Ž 1 2 n. n graphcally greater than y y, y,..., y R f k,1kn such that x y, for k, and x k y k, n whch case we denote ths by x lex y. Ž 1 2 n. n For a vector x x, x,..., x R we defne sortž x. Ž 1 2 n. n x, x,..., x R to be the elements of x arranged n a nondecreasng order. Ž An allocaton vector x n whch x ratež S.. s called max-mn ector f t s a feasble allocaton and s lexcographcally maxmum along all feasble allocaton vectors, wth respect to the sort order Ž.e., for every feasble allocaton y, we have that sortž x. sortž y.. lex. Note that n the max-mn vector allocaton each sesson has an edge on whch t has a rate equal to the maxmum rate that passes through that edge. The goal of our algorthms s to compute a max-mn vector x, whch consttutes a far allocaton Algorthms Progress Our algorthms start at some ntal allocaton of rates and converge to a fnal rate allocaton. In the processes the algorthms mark the sessons that reached ther fnal rate as done, whle the other sessons are marked acte. We denote the set of acte sessons by A and ts complement, the done sessons, by D. When our algorthms start all of the sessons are acte e., D.. The algorthms quesce when all of the sessons are done Ž A.. Each executon of an algorthm s modeled as a sequence QoQo QoQ 1 where Q represents a state of the network at a specfc pont n tme and o s an update operaton. The procedure that selects the next sesson to be ncreased s called the scheduler. The acte and done markngs are used by the scheduler to decde whch sesson may be a canddate for an ncrease n the next operaton. However, ths markng s not used by the computaton of the new rate values. We denote by So Ž. the sesson that was selected for an ncrease n operaton o. The varables n state Q are denoted by superscrpt, e.g., rate S s the rate of sesson S n state Q. Each algorthm s descrbed as a sequence of operatons Žcalled update operatons. that modfy the rates of sessons n the network. In each update operaton o the rate of one sesson, S, s ncreased, and the rate of other sessons that share an edge wth S may be decreased. An update operaton o s performed on sesson S only f there s a way to ncrease ratež S. to a new value, ratež S. 1 ratež S., whle the

8 COMPLEXITY OF RATE-BASED FLOW CONTROL 113 rate of any other sesson Sl S s ether unaffected or decreased to a new value whch s greater or equal to. Gven a vector of rates x, the outputs of the update operaton on S wll result n a maxmal ncrease n the rate of S wthout volatng the above condton. The possble ncrease n ratež S. s,.e., f rate S s ncreased by o, then rate S 1 ratež S.. Ž A formal specfcaton of the update operaton s gven n Appendx A.. EXAMPLE. Consder x Ž 1, 2, 3, 4, 6., and assume that all of the sessons share a sngle edge e whose capacty s 16. Frst note that x s maxmal. Sesson S 5, wth rate 6, cannot be updated n ths confguraton. Sesson S4 can be updated, resultng n the followng rate allocaton Ž 1, 2, 3, 5, 5.. Sesson S 1, S 2, and S3 can also be updated, e.g., the result of updatng S s Ž 1, 4, 3, 4, The above example was for the smple case of a sngle edge. In general a sesson may have several edges, and t may be ncreased only f t can be ncreased on eery edge. For each edge e through whch sesson S passes, we defne Ž e,. to be the maxmum amount by whch ratež S. may be ncreased f edge e s the only edge constranng S Ž.e., as f e s the whole network.. The possble update, Ž., by defnton, s the mnmum over all of the ncreases possble by the dfferent edges of S,.e., mn Ž e,.4 e S. Note that any sesson that shares an edge wth the ncreased sesson and whose rate was larger than may be reduced n a far manner to a value that s larger than or equal to. For example, consder Fg. 1, where x Ž 40, 45, 10, 80. Ž note that x s not maxmal.. The result of an update operaton on S3 would be the rate allocaton Ž 35, 35, 35, 75.. For each edge e, N Ž e. s the number of acte sessons that use e n state Q,.e., N e A Ž e.. FIG. 1. An example for an update operaton.

9 114 AFEK, MANSOUR, AND OSTFELD We defne the allotted capacty of an edge e at state Q as the total capacty already allocated to done sessons,.e., Ý allotted e rate S. S D Ž e.4 Intutvely, the avalable capacty of an edge s the capacty that can be stll dvded between the acte sessons. Ths leads to the followng defnton o a far share of an edge n state Q : CapŽ e. allotted Ž e. FS Ž e.. N Ž e. 3. BASIC PROPERTIES OF BOTTLENECK ALGORITHMS In ths secton we defne bottleneck algorthms and prove several basc propertes on them. In an algorthm that computes the exact max-mn vector, an edge e s called a bottleneck edge n state Q f for every acte sesson S Ž e. A, FS Ž e. s the smallest far share among the edges along S BG87. DEFINITION 3.1. A general bottleneck algorthm s an algorthm that computes the exact max-mn vector for a set of sessons, and n any state Q n ts executon, a sesson S s marked done Ž.e., has reached ts fnal rate. only f there s a bottleneck edge e S n Q and ratež S. FS Ž e.. In some algorthms Že.g. BG87, Cha94. that compute the max-mn vector, when a sesson has reached ts fnal value e., marked done., then n addton to the above condton, all of the acte sessons that use a bottleneck edge have the same rates. ŽThat s, they share the edge avalable capacty n a far manner.. When all of the sessons satsfy ths condton e., marked done. the algorthm termnates. All of the algorthms presented n ths paper except for those n Secton 7 are called bottleneck algorthms. DEFINITION 3.2. A bottleneck algorthm s a general bottleneck algorthm that, n addton, uses the update operaton, as descrbed n Secton 2. The man pont that dstngushes our algorthms from the prevous ones s that n transent states, when the max-mn condton has not yet been reached, some sessons may have a rate that s larger than ther rate n the max-mn vector. In ths secton we present and prove some basc general lemmas concernng bottleneck algorthms. These basc lemmas are shared by the

10 COMPLEXITY OF RATE-BASED FLOW CONTROL 115 dfferent algorthms consdered and are used n the sequel for ther proof and analyss. The followng theorem shows the correctness of a bottleneck algorthm by statng that when t termnates e., all sessons are marked done., then ŽrateŽ S., ratež S.,...,rateŽ S.. s the max-mn vector. 1 2 n THEOREM 3.3 BG87. In a bottleneck algorthm, f A, then the rate allocaton s the max-mn ector. The next lemma shows that the far share of an edge s a monotoncally nondecreasng functon of tme. LEMMA 3.4. In any bottleneck algorthm the far share FSŽ e. of edge e may not decrease as the algorthm progresses; that s, for 0, f there are Ž 1. 1 acte sessons that use e n state Q.e., N e 0,then FS Ž e. 1 FS Ž e.. Proof. If n operaton o, a number of sessons that use edge e change ther rate, but none becomes done, then clearly the far share of e, FSŽ e. remans unaffected. The only way FSŽ e. s affected s f some sessons that use e become done n the last step. Ths ncreases allottedž e. and decreases Ne. If the rate of all of the done sessons equals FS Ž e., then t does not affect the Far Share. Otherwse, some of the done sessons must get less than FS Ž e. and some of the done sessons must get exactly FS Ž e.. Hence, the remanng actve sessons may get more, thus ncreasng the 1 1 Far Share. Formally: If N e N e, then by defnton, FS Ž e. 1 FS e. Otherwse, N Ž e. N Ž e. k, where N Ž e. k 1, and n 4 state Q there are k sessons, S, S,...,S A Ž e.4 wth edges 1 2 k e S, e S,...,e 1 2 k S such that for every l, where k l k ratež S. FS Že. FS Ž e. l. Then, by the defnton of Far Share, t l follows that CapŽ e. allotted 1 Ž e. 1 FS Ž e. 1 N Ž e. k Cap e allotted e Ýl1 rate S l N e k CapŽ e. allotted Ž e. k FS Ž e. FS Ž e.. N Ž e. k The followng lemma states that after ncreasng the rate of a sesson va an update operaton, ts rate s larger than or equal to the smallest Far Share along ts path.

11 116 AFEK, MANSOUR, AND OSTFELD LEMMA 3.5. Consder an operaton o n whch the rate of sesson SŽ o. S A s ncreased. Then n the succeedng state, Q 1, there s at least one edge e S such that 1. e s saturated Ž.e., Ý ratež S. 1 CapŽ e.. S Ž e. l. l 2. ratež S. 1 FS Ž e.. Proof. Let e be the edge accordng to whch the amount of ncrease was determned, that s, the edge that enabled the smallest ncrease n the rate of S Ž.e., Ž e,. as defned n Appendx A.. Followng Defnton A.1 and Defnton A.2, after the ncrease, e s saturated and ratež S. 1 s the maxmum rate among all sessons n Ž e., whch com- pletes the proof. DEFINITION 3.6. Let S A. We defne LB to be the smallest Far Share along the path of S n state Q, formally LB mn FS Ž e.4. e S Our goal now s to prove Theorem 3.10, n whch we state that f the rate of an arbtrary sesson S s adusted n state Q Ž ether ncreased or. decreased, then the rate of ths sesson wll never be smaller than LB. Ths s proved n two steps. Frst, n Lemma 3.7, we prove that f the rate of sesson S s ncreased n state Q or afterward, then n the subsequent state ts rate s larger than LB. Second, n Corollary 3.9, we show that no decrease n the rate of S subsequent to state Q may decrease t below LB. LEMMA 3.7. Consder state Q of a bottleneck algorthm and let e be an edge along the path of S A where FS e LB. Let l where SŽ o. l S. Then, ratež S. l1 FS Ž e.. l l Proof. Let e be an edge along the path of S such that FS e LB. Hence, ratež S. l1 FS l Ž e. FS Ž e. FS Ž e. Ž the frst nequalty fol- lows from Lemma 3.5, the second nequalty follows from Lemma 3.4, and the thrd nequalty follows from the fact that FS Ž e. s the mnmum Far Share among all edges of S n state Q.. The next lemma shows that f a sesson s rate s decreased, then t s stll above the Far Share of at least one of ts edges. The ntuton for ths lemma s that a sesson s decreased accordng to a constrant caused by an edge that s common to ths sesson and to the sesson that was chosen to be ncreased. Therefore, the updated rate of both the ncreased sesson and the decreased sesson wll be at least the Far Share of that edge. LEMMA 3.8. Consder state Q of an arbtrary bottleneck algorthm for computng the max-mn ector of rates. If ratež S. 1 ratež S. Ž.e., S s decreased n o., then ratež S. 1 LB.

12 COMPLEXITY OF RATE-BASED FLOW CONTROL 117 Proof. Let So Sl be an arbtrary sesson that s ncreased. Follow- ng Defnton A.2, one can see that ratež S. s decreased only f there exsts an edge e such that S, S 4 Ž e. l, e s saturated n state Q 1, and ratež S. 1 s maxmum n Q among all sessons of Ž e. 1. Hence, 1 rate S FS e LB. COROLLARY 3.9. Consder states Q and Ql of an arbtrary bottleneck algorthm for computng the max-mn ector of rates, where l. If ratež S. 1 ratež S. l e., S s decreased n o., then ratež S. l1 LB. Proof. LB. l Follows mmedately from Lemma 3.8 and the fact that LB l THEOREM Consder state Q of a bottleneck algorthm where ether So 1 S or rate S rate S. Then, for all l, we hae rate S l LB. 4. GLOBAL MIN SCHEDULER In ths secton we analyze a bottleneck algorthm n whch the scheduler selects n each state the globally smallest acte sesson for an ncrease. The followng rules are used n each state of the algorthm: Markng a sesson done: The smallest rate acte sesson n the network s marked done f t cannot be ncreased. Sesson selected for ncrease: In state Q, So S, such that ratež S. mn t A rate S t. Termnaton: The algorthm termnates when A. Ths technque s qute ntutve, and ndeed t turns out that the algorthm converges n Žn 2. update operatons. The upper bound s proved n Theorem 4.3, and the lower bound s proved n Theorem 4.4. Note that ths technque s used n BG87 for a dfferent model where an upper bound analogous to that n Theorem 4.3 s proved. Obseraton 4.1. The Global Mn algorthm s a bottleneck algorthm. Proof. We have to show compatblty to Defnton 3.2. Snce the algorthm conforms to the conventons of Secton 2, we ust have to show that t s a general bottleneck algorthm. Let S A be a sesson where ratež S. s global mnmum and cannot be ncreased. Snce rate S s the mnmum among all sessons n A, then n partcular, for every edge e S, rate S rate S for all sessons S A Ž e.4 m m, and hence ratež S. FS Ž e. ŽrateŽ S. FS Ž e. mples that the amount of flow n the edge exceeds ts capacty..

13 118 AFEK, MANSOUR, AND OSTFELD Snce ratež S. cannot be ncreased, then there s an edge e S where e, 0, whch mples that e s saturated, and for all Sm A Ž e.4, ratež S. rate S m. Hence, for all sessons S 4 m A e, rate S rate Sm FS Ž e.. Hence, also ratež S. m s the mnmum among all sessons n A, whch mples that for any edge e S, FS e rate S FS Ž e. m m. The followng lemma shows that f a sesson rate becomes a global mnmum n two dfferent states Ql and Q, then at least one acte sesson becomes done between those two states. LEMMA 4.2. In the Global Mn algorthm, f for l, So So l S, l Ž l then A A.e., A s a proper subset of A.. Proof. Let e S such that FS e LB. By Theorem 3.10, for all k l k, rate S FS e. Hence, f A A, then FS l Ž e. FS Ž e., whch mples that ratež S. l l FS e.if rate S l s global mnmum of all ses- l l sons n A, t s n partcular, smallest n all sessons of A Ž e., whch l mples that the rate of all sessons n A Ž e. s larger than or equal to FS l Ž e.. Followng the defnton of FS l Ž e., ths mght happen only f the l rate of all sessons n A Ž e. s exactly FS l Ž e.. However, n such a case l the rate of all of the sessons n A Ž e. Žand n partcular, the rate of sesson S. cannot be ncreased s state Q l, and they should be transferred from A to D before the executon of o l, completng the proof. THEOREM 4.3. The number of update operatons n any run of the Global Mn algorthm s OŽn 2.. Proof. Frst notce that as long as A, the algorthm progresses, snce n state Q every acte sesson n whch the rate s global mnmum can be ether ncreased or marked done. Followng Lemma 4.2, the sze of the actve set must decrease between every par of updates to the same sesson; ths mmedately mples an upper bound of n Ž n 1. Žn. Ž On.. The next theorem establshes a tght lower bound for ths algorthm. THEOREM 4.4. There s an nfnte famly of networks wth n sessons n whch the number of update operatons n a run of the Global Mn algorthm s Žn 2.. Proof. We construct recursvely a network denoted NetŽm, bottom, potental.. The network has 2m sessons, where bottom s the ntal rate for each of the sessons n the network Žand a lower bound for the rate t can acheve n the future., and bottom potental s an upper bound for the amount of flow Ž ether temporary or fnal. that any sesson can acheve n the future. Fgure 2 s an example of a network NetŽ 4, 0, 128..

14 COMPLEXITY OF RATE-BASED FLOW CONTROL 119 FIG. 2. An example of a network Net 4, 0, 128. The procedure that bulds that network s as follows: Procedure NetŽ m, bottom, potental.: 1. If m 1, then S1 and S2 share a common edge e such that CapŽ e. 2 bottom potental. 2. If m 1, then Ž. a S and S share a common edge e such that CapŽ. 2 m1 2m e 2 bottom potental2. Ž b. Construct confguraton NetŽm 1, bottom potental2, potental4.. Ž. c Each of the sessons S 1, S 2,...,S2Žm1. shares an edge e wth S where CapŽ e. 2 bottom potental. 2 m We buld the network by callng the procedure NetŽ m,0,p., wth a large constant P ŽP OŽ4 m... Note that the network s dentfed by the edges n whch the sessons cross each other, where there s no mportance to the order between those edges along a sesson path. Now we descrbe a bad scenaro that acheves the requred number of operatons. Note that n some states of the algorthm the global mnmum

15 120 AFEK, MANSOUR, AND OSTFELD among the acte sessons s not unque, and hence there s more than one acte sesson that can be selected for ncrease. Therefore, there are other scenaros besdes the one that s descrbed below, where some of the other scenaros may acheve a lower complexty. The scenaro of a run of the Global Mn Algorthm s then constructed by callng the followng recursve procedure ScenaroŽ m, 0,P. Ž where m 1. on the network NetŽ m,0,p.. See Appendx B for an example of ScenaroŽ 4, 0, Procedure ScenaroŽ m, bottom, potental.: In the begnnng of the procedure we assume that the acte sessons are S, S,...,S m, ther rate s bottom, and any update operaton can ncrease the rate of an acte sesson to be at most bottom potental. 1. If m 1, then apply step Ž. a followed by step Ž. b : Ž. a The rate of sesson S s ncreased to bottom potental. Ž 2 The edge that s common to S1 and S2 becomes saturated.. Ž b. The rates of sessons S1 and S2 are set to bottom potental2 Ž because of an ncrease n the rate of S. 1. Sessons S1 and S2 become done. 2. If m 1, then apply the followng steps: Ž. a The rate of sesson S2 m s updated to bottom potental2. Ž The edge that s common to S and S becomes saturated.. 2 m1 2m Ž b. The rate of each of the sessons S where 1, 2,..., 2m 2 s updated to be bottom potental2 Žtotal of 2m 2 updates, where n each update the edge that s common to S and S2 m becomes saturated.. Ž. c The rates of sessons S2 m1 and S2m are set to bottom potental4 Ž because of an ncrease n the rate of S. 2 m1. Sessons S2m1 and S become done. 2 m Ž d. The scenaro s recursvely extended by callng ScenaroŽm 1, bottom potental2, potental4. on the network NetŽm 1, bottom potental2, potental4.. Note that after sessons S2 m1 and S2m become done, the rate of all acte sessons s ndeed bottom potental2, and ther potental s the potental4 flow that was released by S 2 m. One can see that when m 1, there are two update operatons. Hence, for an arbtrary m 1, there are m teratons where for all, the th teraton s ScenaroŽ,,. requres 2 update operatons. Therefore, the n 2 total number of update operatons s Ý Ž 2. Žn Note that n most cases we can assume that ntally bottom 0 Žwhen we start from scratch..

16 COMPLEXITY OF RATE-BASED FLOW CONTROL LOCAL MIN SCHEDULER In ths algorthm an acte sesson s selected to be ncreased f ts rate s smaller than or equal to the rates of all of the acte sessons t shares an edge wth. In each state of the algorthm the followng rules are appled: Markng a sesson done: Sesson S A s marked done f the followng three condtons hold: 1. ratež S. cannot be ncreased. 2. ratež S. s smaller than or equal to the rates of all of the acte sessons t shares an edge wth e., S s a local mnmum.. 3. For every acte sesson S m such that rate Sm rate S and S Ž m shares an edge wth S, Sm s also a local mnmum.e., rate Sm s smaller or equal to the rates of all of the acte sessons Sr that share an edge wth S. m. Sesson selected for ncrease: In state Q, So S A such that S can be ncreased and ratež S. s smaller than or equal to the rates of all of the sessons t shares an edge wth. Termnaton: The algorthm termnates n state Q f A. Ths algorthm allows the rate of a sesson to be ncreased accordng to local consderatons. Notce that the local consderaton reflects only the decson about whch sesson should be ncreased Ž or marked done., and by local we mean that only edges along the sesson path need to be checked Žn a very long sesson that covers many lnks, ths may end up not beng very local.. The executon of an update operaton s dentcal n all of the schedulers Ž and mght nvolve a scan of several lnks.. Hence, t seems to be closer to real network behavor than the global mn scheduler. Here we show that ths algorthm has the same convergence complexty as the global mn algorthm,.e., Žn 2. update operatons. Obseraton 5.1. The Local Mn algorthm s a bottleneck algorthm. Proof. We have to show compatblty wth Defnton 3.2. Snce the algorthm conforms to the conventons of Secton 2, we ust have to show that t s a general bottleneck algorthm. One can see that an acte sesson S s marked done n state Q only f there s an edge e where 1. ratež S. FS Ž e. Ž snce t cannot be ncreased.. 2. For all acte sessons S A Ž e., ratež S. ratež S. Ž m m snce S s a local mnmum.. Hence, snce S cannot be ncreased, we have that for all of those sessons Ž ncludng S. ratež S. FS Ž e. m, whch mples that ratež S. FS Ž e.. m

17 122 AFEK, MANSOUR, AND OSTFELD 3. All acte sessons S A Ž e. Ž ncludng S. m are local mn- mum Ž snce S s marked done.. Hence, e s a bottleneck edge for all of those sessons. The followng lemma and ts proof are smlar to Lemma 4.2. It shows that f a sesson rate becomes a local mnmum n two dfferent states, then at least one acte sesson becomes done between those two states. LEMMA 5.2. In the Local Mn algorthm, f for l So So l S, then A l A. Proof. The proof of Lemma 4.2 apples verbatm. Hence, we have the followng theorem, whch s analogous to Theorem 4.3. THEOREM 5.3. The number of update operatons usng the Local Mn algorthm s OŽn 2.. As for the tght lower bound, we have the followng theorem: THEOREM 5.4. There s an nfnte famly of networks wth n sessons n whch the number of update operatons n a run of the Local Mn algorthm s Žn 2.. Proof. Snce the global mnmum s also a local mnmum, the constructon of Theorem 4.4 apples to the Local Mn algorthm as well. 6. ARBITRARY SCHEDULER In ths case, the sesson that s ncreased n each round of the algorthm s chosen arbtrarly from A. In each step of the algorthm the followng rules are appled: Markng a sesson done: Sesson S s marked done n state Q f there s an edge e S that s a bottleneck edge n state Q, and for every sesson S e A, ratež S. equals FS Ž e.. r r Sesson selected for ncrease: At each state the schedular selects an arbtrary sesson that may be ncreased and updates t. Termnaton: The algorthm termnates n state Q f A. Note the followng: 1. The arbtrary scheduler s a bottleneck algorthm. 2. The done Ž or the acte. markngs are used by ths algorthm only for termnaton detecton Ž they do not affect the decson of the scheduler..

18 COMPLEXITY OF RATE-BASED FLOW CONTROL We can slghtly change the defnton of the scheduler and allow t to select an arbtrary sesson. If the selected sesson cannot be ncreased, then we consder ths event as f nothng happens Žths operaton s not consdered as an update operaton.. Ths scheduler allows the rate of a sesson to be ncreased wthout any consderaton of other sessons. Hence, t seems to be closer to real network behavor than the schedulers presented prevously. ŽIn fact, the only maor dfference between ths scheduler and the behavor of real rate-based networks stems from the fact that here the update operatons are done sequentally;.e., there are no concurrent update operatons.. In ths secton we show that usng ths mechansm, the rate vector always converges to the max-mn vector after no more than OŽ n. update operatons, where Ž ' s the golden rato Žand n s the number of sessons.. An exponental lower bound of Ž2 n 2. update operatons s presented Ž n Theorem for an nfnte famly of networks. Frst we gve some defntons that are used n the sequel to obtan the requred results. DEFINITION 6.1. For k 1, we denote by FŽ k. the maxmum number of update operatons that may be performed n a network wth k sessons under the arbtrary scheduler. The results presented n the sequel are acheved usng recursve technques. Here we defne a prefx, and later we show how the prefx s used to form the recursve formula. DEFINITION 6.2. Let G be a network wth n sessons S, S,...,S n. For every state Q of the algorthm, prefxž. S, S,...,S 4 Ž where 1 2 k k n. s defned to be the set of sessons that were updated Žether ncreased or decreased. at least once n o, o,...,o Followng the defnton of the prefx, we descrbe a proecton of the network. Ths proecton tres to solate the envronment of the sessons that are updated durng the early operatons. DEFINITION 6.3. Let G Ž V, E. be a network wth n sessons S, S,...,S n n whch the arbtrary scheduler s mplemented. Let prefxž. S, S,...,S 4, and let S, S,...,S 4 be the rest of the 1 2 k k1 k2 n sessons n G. The proecton of G over, PROJ Ž G., s a new network G Ž V, E. that s defned as follows: 1. E S S S. 1 2 k 2. For every sesson S, where k 1, the value of rate S 0 n G s the same as n G.

19 124 AFEK, MANSOUR, AND OSTFELD 3. For every edge ee we set CapŽ e. CapŽ e. Ý S Ž e. k1 0 rate S, where e E s the analogue edge to e. Ž For each edge we subtract the total amount of flow of sessons for whch the rate has not been adusted.. The followng observaton bounds the length of a prefx as a functon of the number of sessons that t updates. Obseraton 6.4. Let G Ž V, E. be a network wth n sessons S, S,...,S n n whch the arbtrary scheduler s mplemented. Let Q be a state where prefxž. S, S,...,S 4. Then, FŽ k. Ž.e., the number 1 2 k of possble update operatons that nvolve no more than k sessons n an arbtrary network s less than or equal to FŽ k... The next defnton s related to the maxmum rate of a sesson. DEFINITION 6.5. For every sesson S we defne boundž S. to be an upper bound on ratež S. durng the algorthm executon,.e., for all 0 we have boundž S. ratež S.. The noton of boundž S. s used to have a more refned restrcton for the potental rate of sessons that are sharng a lnk wth sesson S,as shown n the sequel. The followng defnton extends the defnton of Far Share: DEFINITION 6.6. For every edge e, where N Ž e. 0, and for every sesson S A, PFS Ž e,., the Potental Far Share of lnk e n state Q s defned as follows: If S Ž e. and N Ž e. 1, then CapŽ e. bound S allotted Ž e. PFS Ž e,., N Ž e. 1 else PFS Ž e,. FS Ž e.. The followng defnton s an extenson for the term LB: DEFINITION 6.7. Let S A and let Sr S. Then, 4 MLB Ž r. mn PFS Ž e, r. es Ž MLB Ž r. s the smallest Potental Far Share wth respect to Sr along the path of S n state Q.. Snce the Potental Far Share s nondecreasng Žby arguments smlar to those n Lemma 3.4., then one can see that after the rate of sesson S s updated n state Q l, rate S l1 s greater than or equal to the values of MLB Ž r. for every S S and for every l Žby arguments smlar to r

20 COMPLEXITY OF RATE-BASED FLOW CONTROL 125. those n Lemma 3.7 and n Corollary 3.9. Hence, we have the followng extenson of Theorem 3.10: THEOREM 6.8. Consder state Q of a bottleneck algorthm where ether So 1 S or rate S rate S. Then, for all l and for all Sr S, ratež S. l MLB Ž r.. The next lemma bounds the number of operatons performed after the rate of all sessons have been updated at least once. LEMMA 6.9. Let G Ž V, E. be a network wth n sessons S, S,...,S n and let Q be a state of the algorthm where prefxž. contans all of the n sessons. Then there are at least two sessons S, S 4 r that are not updated n all operatons o Ž l l, such that l.e., for eery l, rate S rate S and ratež S. l ratež S.. r r. Proof. Let e E be an edge where FS 0 Ž e. s the smallest Far Share among all edges n Q. Snce Ž e. prefxž. 0, then for every sesson S Ž e. there s m, where ratež S. k k s updated n state Q m. Snce the Far Share s nondecreasng, then, by Theorem 3.10, we have that for every sesson S Ž e., and for every l, ratež S. l FS 0 Ž e. k k, whch mples that for every l, ratež S. l FS 0 Ž e.. If Ž e. k contans at least two sessons we are done. Else, Ž e. contans exactly one sesson Sr and rate S boundž r. CapŽ e. r. Let E E be a set of edges where e E ff e S. Let e E be an edge where PFS Ž e, r. r s the mn- mum among all edges n E. Clearly, for every sesson S Ž e. k S r, and for every l, we have that ratež S. l PFS 0 Ž e, r. MLB 0 Ž r. k k. However, snce ratež S. boundž r. r, then t mples that for every sesson Sk Ž e. S, and for every l, we have that ratež S. l MLB 0 Ž r. r k k.we take an arbtrary sesson n Ž e. S to be S, and we are done. The followng theorem proves an upper bound on the number of update operatons. THEOREM Let FŽ n. be the number of update operatons performed n the arbtrary scheduler n a network wth n sessons. Then FŽ n. OŽ n., where Ž 1 ' 5. 2 s the golden rato. Proof. Clearly, FŽ. 1 1 and FŽ To prove the theorem, we show that for every k 2, we have the nequalty FŽ k 1. FŽ k. 1 FŽ k 1.. r Any scenaro QoQo can be dvded nto three parts, such that Qo 0 0 Ql1o l1, Qo, l l and Ql1o l1..., where l s selected such that prefxž. l k Ž.e., no more than k sessons were updated durng the frst l operatons. and prefxž l 1. k 1 Ž.e., Q l s the frst tme that the k 1st sesson s updated..

21 126 AFEK, MANSOUR, AND OSTFELD By Observaton 6.4, the frst part may take no more than FŽ k. operatons. By Lemma 6.9, there are at least two sessons that are not updated after operaton ol s fnshed. Hence, n the thrd part we have a network wth no more than k 1 sessons that may be changed, whch mples Žby Observaton 6.4. that ths part may take at most FŽ k 1. addtonal update operatons. The followng theorem gves a lower bound on the convergence complexty of the algorthm. THEOREM For eery odd number n, there exsts a network wth n sessons, for whch there s a schedule that requres Ž2 n 2. update operatons. Proof. We construct a network denoted ExpNetŽm, potental, progress.. The network has 2m 1 sessons, where potental s an upper bound on the amount of flow Ž ether temporary or fnal. that any sesson can acheve, and progress s a parameter that s used to decrease potental. Fgure 3 s an example of a network ExpNetŽ 3, 100, 10.. We use an array Ž called pot. where for every sesson S, pot s an exact upper bound for the amount of flow that ths sesson can acheve n the future. We assume that potental 2 m progress. FIG. 3. An example of a network Exp Net 3, 100, 10.

22 COMPLEXITY OF RATE-BASED FLOW CONTROL 127 The procedure that bulds the network s as follows: Procedure Exp NetŽ m, potental, progress.: for 1to2m1do begn S uses solely an edge e where CapŽ e. potental potž. potental potental potental progress f s even then for 1to1do S and S share an edge e where CapŽ e. pot pot progress else s odd4 f 1 then S and S share an edge e where CapŽ e. pot potž 1. 1 progress end of for 4 Notce that the network s dentfed by the edges n whch the sessons cross each other, where there s no mportance to the order between these edges along a sesson path. Now we descrbe a bad scenaro that acheves the requred number of operatons. Notce that n some states of the algorthm there s more than one acte sesson that can be selected for ncrease. Therefore, there are other scenaros besdes the one that s descrbed below, where some of the other scenaros may acheve a lower complexty. The scenaro of a run of the Global Mn Algorthm s then constructed by callng the followng recursve procedure Exp ScenaroŽ m, pot. on the network ExpNet Ž m, potental, progress. Žwhere pot s the array that was computed by ExpNetŽ m, potental, progress. as descrbed above.. We assume that the scenaro starts from scratch such that for every sesson S, the ntal rate s less than potž.. Procedure Exp ScenaroŽ m, pot.: 1. If m 1, then the rate of sesson S s ncreased to potž.ž 1 1. The edge that s used solely by S1 becomes saturated.. 2. If m 1, then Ž. a Call ExpScenaroŽ m 1, pot.. Ž b. The rate of sesson S s ncreased to potž 2m 2. 2 m2. Ths causes the rates of all sessons S, where 2m 2, to be pot progress. Ž. c The rate of sesson S s ncreased to be potž 2m 1. 2 m1. Ths decreases the rate of sesson S to potž 2m 1. potž 2m 2. 2 m2

23 128 AFEK, MANSOUR, AND OSTFELD progress. Now for all sessons S, such that 2m 2, the rate s pot progress, and there s no restrcton mposed by S2 m2 that dsables ther rate from ncreasng to potž.. Hence, we can repeat step Ž a. as follows: Ž d. Call ExpScenaroŽ m 1, pot.. Let HŽ m. be the number of update operatons n ExpScenaroŽ m, pot.. One can see that when m 1 there s one update operaton Ž.e., HŽ and for an arbtrary m 1, HŽ m. HŽ m 1. 2 HŽ m 1. 2 HŽ m Therefore, the total number of update operatons s Ž2 m. Ž2 n 2.. In Appendx C we show ths scenaro for m 3, potental 100, and progress APPROXIMATIONS In ths secton we present algorthms n whch a sesson rate s ncreased only f the ncrease s by more than. The algorthm termnates when the rate of no sngle sesson can be ncreased by more than. More formally, we say that 1. For every,, ether ratež S. 1 rate S or rate S 1 ratež S.. 2. In Q fnal, the fnal state of the algorthm, for every sesson S, fnal. We call an allocaton that satsfes the second condton a -max-mn allocaton. The motvaton for studyng ths ssue s that n dynamc networks, the computaton of the exact max-mn vector may not be reached Ž because of the dynamc nature of the system., and hence we are wllng to pay the overhead requred by the update operatons only f the beneft s sgnfcant. Furthermore, an algorthm that converges fast to the new Qfnal but slowly to the standard max-mn vector may be consdered better than another algorthm n whch t takes more tme to reach the new Qfnal state, even f t converges faster to the standard max-mn vector. In Subsecton 7.1 we analyze the mplcatons of such an approxmaton. It turns out that every sesson S may dffer from ts optmal rate Ž the value of ratež S. n the max-mn vector. by at most OŽ 2 n.ž where n s the number of sessons.. We show an example n whch there s a feasble allocaton, such that the rate of only one sesson can be mproved by Ž and the rate of all of the other sessons cannot be ncreased.. Yet, n ths

24 COMPLEXITY OF RATE-BASED FLOW CONTROL 129 Ž n 2 example, there s some sesson that s 2. from ts value accordng to the max-mn vector. Note that ths potental bad property of the network cannot be overcome by any approxmaton algorthm for rate control. However, ths dfference between the two resultng rate vectors s not necessarly a negatve ndcaton. It manly suggests that the max-mn vector may be unstable n the sense that small fluctuatons may change t dramatcally. The proofs presented here show that the addton or deleton of a sesson wth a small rate from the max-mn vector may cause a dramatc Žexponen- tal n the number of sessons. change n the rate of some other sessons. Thus, the max-mn vector s not necessarly better than the vector computed by an approxmate algorthm Žt may suffer from large nstablty that may result from a modest dynamc update.. One of the mplcatons of these results s the settlement of an open queston rased by Charny Cha94 Žwhere she asks about the convergence of rate-based algorthms n the case where rates are restrcted to dscrete values.. In Subsecton 7.2 we show that the Approxmate Global Mn algorthm and the Approxmate Enhanced Local Mn algorthm Žto be descrbed n. Ž Ž 2 the sequel wth parameter reach quescence n at most O mn n, n log W.. h Ž h1. operatons, where h s the maxmum number of sessons that share a sngle edge Ž we assume h 2; otherwse t s trval., and W s defned as follows: 4 W max mn CapŽ e. S es Ž.e., for each sesson we choose the edge wth the mnmum capacty that t passes and W s the maxmum over all of these edges.. Hence, f h s bounded by a constant, then the number of operatons s no more than Ž Ž 2 O mn n, n log W... Ž Ž 2 Remark. A bound of O mn n, n log W.. h Ž h1. on the number of update operatons n ether the Approxmate Global Mn algorthm or the Approxmate Enhanced Local Mn algorthm Ž gven n Subsecton 7.2. s vald for the conservatve approach as well and can be shown by usng arguments smlar to those presented n ths secton The Dfference Between the -Max-Mn Allocaton and Max-Mn In ths subsecton we bound the dfference between the rate of a sesson n -max-mn allocaton and ts rate n the max-mn vector. We consder an arbtrary -max-mn allocaton and use the followng defntons: 1 2 n X x, x,..., x : The max-mn vector, where x s the rate of sesson S.