Quantifying Responsiveness of TCP Aggregates by Using Direct Sequence Spread Spectrum CDMA and Its Application in Congestion Control

Transcription

1 Quantfyng Responsveness of TCP Aggregates by Usng Drect Sequence Spread Spectrum CDMA and Its Applcaton n Congeston Control Mehd Kalantar Department of Electrcal and Computer Engneerng Unversty of Maryland, College Park, MD 2742 Phone: , Emal: mehkalan@eng.umd.edu Mark Shayman Department of Electrcal and Computer Engneerng Unversty of Maryland, College Park, MD 2742 Phone: , Emal: shayman@eng.umd.edu Abstract In ths paper, we ntroduce the noton of responsveness for TCP aggregates and defne t as the degree to whch a TCP aggregate reduces ts sendng rate to the network as a response to packet drops. We defne metrcs that descrbe the responsveness of TCP aggregates, and then we suggest two methods for determnng the values of these quanttes. The frst method s based on a test n whch we drop a few packets from the aggregate ntentonally and measure the resultng rate decrease of that aggregate. Ths knd of test s not robust to multple smultaneous tests performed at dfferent routers. Extensons are done to make the test robust to multple smultaneous tests by usng deas from the CDMA approach to multple access channels n communcaton theory. Then we use these methods to perform congeston control. A dstngushng feature of our congeston control scheme s that t mantans a degree of farness among dfferent aggregates. Keywords: Responsveness of TCP Aggregates, CDMA, Aggregate Perturbaton Method, and Congeston Control.

2 I. INTRODUCTION A key characterstc of TCP traffc s ts responsveness to packet drops. Ths means that the TCP aggregates reduce ther sendng rate as a result of experencng packet drops made by the network at the tme of congeston. The degree to whch a TCP aggregate reduces ts rate n response to packet drops depends on packet sze, round trp tme and the dstrbuton of wndow szes among the consttuent flows. The goal of ths paper s to ntroduce a technque for quantfyng the responsveness of a TCP aggregate to packet drops and applyng the measure of responsveness to obtan a far congeston control scheme. In ths scheme, when a router faces congeston and needs ts traffc arrval rate to be reduced, t tres to drop the packets of less responsve aggregates more aggressvely. One of the unque features of our work s that we try to apply the responsveness measure at the aggregate level, not at the flow level. In general an aggregate s a group of flows wth a common property that pass through the same router or swtchng devce at some pont n the network. An example of an aggregate can be all FTP flows that pass through a router, or all the traffc beng routed toward yahoo.com. Performng the responsveness test n the aggregate level has several advantages. Frst, t does not suffer from scalablty; many flows can be bundled together to form an aggregate and the responsveness test s done for the resultng aggregate. The second advantage of aggregate based testng s the fact that the majorty of the current traffc of Internet s composed of short-lved flows known as Internet mce. It s extremely hard to perform responsveness tests for such traffc n the flow level because flows last only for a few round trp tmes, and often they end before a router can keep track of them. However, f we put many such flows together, we wll get an aggregate that s composed of many flows that appear, survve for a few round trp tmes and dsappear. The aggregate composed of these flows has some statstcal propertes that can help us to defne a responsveness measure for t. In general terms, our approach to measure the responsveness of an aggregate s to perturb the arrval rate of the aggregate by ntentonally droppng a small number of packets, and observng the way the aggregate responds. A normal TCP aggregate shows a transent degradaton n ts rate as a result of nstantaneous packet drops, and we measure ths degradaton and use t as a responsveness measure. By dong ths perodcally, the responsveness of the aggregate can be determned. We have called ths approach the Aggregate Perturbaton Method (APM). The above approach has a drawback n a dstrbuted mplementaton. Each router should be able to apply perturbatons and use these perturbatons to determne the responsveness of the aggregates t observes. However, the flows n an aggregate may experence perturbatons at multple routers. In a dstrbuted mplementaton, n order to perform ts test, a router should not need to be aware of the perturbatons appled by other routers. Our approach to solvng ths problem s nspred by the drect sequence spread spectrum (CDMA) approach n multple access communcaton channels. Each router s assgned a droppng sgnature that specfes ts packet droppng rate as a functon of tme. Dfferent routers are assgned sgnatures that are orthogonal n a certan sense. Usng smulatons, we show that ths approach enables each ndvdual router to fnd the response coeffcent of the aggregates that pass through t wthout requrng any nformaton to be shared among routers. We have called ths approach CDMA based Aggregate Perturbaton Method (CAPM). As an applcaton of APM and CAPM, we use the response coeffcents to offer a far congeston control mechansm. In ths scheme the response coeffcents are taken nto account to assgn the drop probablty of dfferent aggregates at the tme of congeston; less responsve aggregates are penalzed more by havng a hgher probablty of drop. Therefore, dfferent aggregates wll show the same absolute value or percentage of rate decrease. Our smulaton results confrm the effectveness of ths scheme to keep farness among the aggregates. The remander of the paper s organzed as follows. In Secton II, we descrbe related work. In Secton III, we explan APM. In Secton IV, we ntroduce CAPM and descrbe how the use of CDMA-nspred orthogonal perturbng sgnatures enables multple routers to perform perturbatons wthout mutual nterference. In Secton V, we wll explan our approach to usng the response coeffcents of aggregates for the purpose of congeston control. In Secton VI, the results of the smulaton experments are presented whch confrm the effcacy of the proposed methods. II. RELATED WORK Many researchers have conducted studes to do dentfcaton and modellng of TCP traffc n the granularty of flow under steady state condtons. In [] the authors propose a method of testng a flow by comparng the steady state throughput of a TCP flow wth the theoretcal predcted value for conformng flows. If the response and model are smlar, the flow s called TCP conformng. The objectve of that study s to dentfy and penalze the nonconformng flows for congeston control purposes. The approach n [] descrbes how large sustaned ndvdual flows may be tested for TCP conformance. However, a sgnfcant percent of the Internet traffc may be composed of short lved flows. Stochastc Far Blue (SFB) s proposed n [2], and t offers per flow test for responsveness by mappng dfferent flows to parallel bns. The approach s based on the fact that the bns contanng a nonconformng flow are lkely to be overloaded. However, f there are many nonconformng flows n a traffc aggregate, t s lkely that all bns are overloaded, and the algorthm wll not be able to dstngush between conformng and nonconformng flows. There are many other works dealng wth the TCP dynamcs and ts throughput analyss. In [] the authors have offered the throughput model for a TCP traffc under assumpton of statonary random losses. In [6] the authors offer a flow based analyss of TCP dynamcs n the Actve Queue Management (AQM) routers by usng stochastc dfferental equatons. 2

3 III. AGGREGATE PERTURBATION METHOD In ths secton, we ntroduce the Aggregate Perturbaton Method (APM) for quantfyng the responsveness of TCP aggregates. APM works based on nstantaneously droppng a number of packets from an aggregate at some pont and observng the resultng transent decrease n the rate of the aggregate. For the purpose of ths paper, we assume the TCP aggregates are composed of TCP flows that conform to TCP-Reno congeston control mechansm. TCP-Reno has two dfferent phases known as slow start and congeston avodance. Slow start begns after makng a connecton, and upon successful transmsson of every packet and recevng acknowledgement from the recever the wndow sze s ncreased by one. Congeston avodance starts after the wndow sze exceeds a threshold value, and n ths phase the wndow sze s ncreased one per round trp tme, and upon experencng a drop t s decreased to half ts current value. Assume at some router we have an aggregate of TCP flows wth arrval rate of λ(t). In order to test the aggregate for responsveness, at tme t =, we drop D packets from t. It s expected that the aggregate responds to the packet drops by decreasng ts rate for a whle after tme t =. We defne the followng responsveness measure for the aggregate as a response to packet drops: η(d) = tr (λ( ) λ(t)) dt () n whch λ( ) s the nstantaneous rate at the moment before droppng the frst packet, and t r s a nonnegatve fnte tme, and t can be chosen to be the mnmum tme for the recovery of all flows that receved drops (n the order of a few tmes the longest round trp n the aggregate). To acheve better results, λ( ) may be replaced by a short-term average of the rate of the aggregate n a tme nterval earler than t =. η(d) s smply a measure of how many more packets could have been sent by the aggregate f we had not dropped D packets. Ths measure s llustrated n fgure. In [8] we have shown that the expected value of η(d) s a lnear functon of D. Furthermore, ths quantty does not depend on the number of flows contrbutng to the aggregate and the absolute value of the aggregate rate λ(t) f the number of dropped packets D s small compared to the number of actve flows at tme t =. Our approach for quantfyng the responsveness of a TCP aggregate s based on the degradaton measure η(d) as a response to packet drops; under the same value of D for dfferent aggregates, those wth a hgher η(d) are more responsve. In other words, η(d)/d can gve a quanttatve value of the responsveness of an aggregate. IV. CDMA BASED AGGREGATE PERTURBATION METHOD (CAPM) One of the problems of dstrbuted mplementaton for APM s the potental of smultaneous perturbatons; the measurements of a perturbng router on an aggregate can be falsfed Fg.. The shaded area shows η(d), the responsveness measure defned by equaton (). D packet are dropped from the aggregate at t =. by the smultaneous perturbatons beng done on the same aggregate n a downstream or upstream router. Ths phenomenon s llustrated n the fgure 2. As t can be seen n ths fgure, the response of the APM test of a router at t = t s overlapped by the response of the aggregate to another router s test at tme t = t 2, whch causes nterference. Ths nterference happens when t and t 2 are close enough to each other (more precsely t 2 t < t r ). In ths case the measure gven by equaton () does not gve accurate nformaton about responsveness of the aggregate, and nterference causes the results of both tests to be falsfed. In ths secton we ntroduce CAPM to overcome the above problem. In CAPM every perturbng router uses a unque perturbng pattern. We wll show that under proper assgnment of the perturbng patterns and proper defnton of aggregate degradaton measure for each perturbng router, the test and measurement of each router wll be robust to the nterference caused by the other smultaneous perturbng routers. CAPM s dfferent from APM n two ways. The frst dfference s that we spread the packet drops over tme. In other words, nstead of droppng D packets from the aggregate nstantaneously at tme t =, we spread the packet drops over a tme nterval [, T ]. In ths scheme perturbaton s done accordng to the packet drop rate functon r (t) : [, T ] R for the th router. The responsveness test and measurement s done durng the nterval [, T ], and at tme t T, the th router drops r (t) packets per second from the aggregate. We refer to r (t) functon as the drop sgnature of the th router. The second dfference between CAPM and APM s the way we defne the degradaton measure for the th router as the response to droppng wth rate r (t). In ths case nstead of the smple ntegral gven by equaton (), we use a weghted ntegral to measure responsveness of the aggregate under perturbaton: η h (r ) = h(t) λ(t) dt (2) n whch λ(t) = λ( ) λ(t), and h(t) s a weghtng functon that states at what tme nstants the results are more mportant to us, and at what tme nstants we are less nterested n the rate decrease of aggregate. In the next step we try to use an approach smlar to Drect Sequence Spread Spectrum CDMA n multple access communcaton to solve nterferng problems of multple smultaneous perturbng routers [7]. In ths approach, each router perturbs the traffc accordng to ts unque drop sgnature based on a CDMA code assgned to t. The dea s that f we defne the drop sgnature of dfferent routers n a way that they are 3

4 Fg. 2. The nterference effect of smultaneous APM tests done by dfferent routers. Before the aggregate recovers from a router s perturbaton at t = t, another router performs a test at t 2 < t + t r. The results of both tests are falsfed. orthogonal to each other n a certan sense, then by proper defnton of the weght functon h(t) the measure of degradaton n a router defned n equaton (2) wll be ndependent of the perturbatons done by the other routers. Smlar to the CDMA systems, we defne the drop sgnature of the th perturbng router n the followng way: r (t) = A N j= c j p Tc (t (j )T c ) = A s (t) (3) n whch A s a known perturbaton ampltude of the th router, N s a postve nteger called the spreadng factor, T c = T/N, (c, c 2,..., c N ) s a bnary sequence assgned to the partcular router known as the code of the router. In (3), s (t) denotes the normalzed drop sgnature, and p Tc (t) s a realvalued functon known as the chp waveform and t satsfes the followng property: p Tc (t)p Tc (t nt c ) dt =, n =, 2,.... (4) The measurement of the th router about the responsveness of the aggregate s made based on the Matched Flter output. The matched flter output s the value of η h (r ) evaluated at h(t) = s (t): y = s (t) λ(t) dt. (5) Snce n our problem r (t) s a drop rate, t should be nonnegatve, and hence p Tc (t) should be nonnegatve. For ths purpose we suggest the popular smple rectangular chp waveform: { f < t < Tc p Tc (t) = (6) otherwse. Usually, n the CDMA systems assgnment of the codes s very mportant. Users wth a potental of hgh nterference (e.g., neghbor routers n our problem) are assgned to codes that cause ther drop sgnatures to be orthogonal (or close to orthogonal) s (t)s j (t) dt =, for j. (7) Unfortunately, the statement of (7) cannot be satsfed wth the current defnton of drop sgnatures defned n (3). That s because both s (t) and s j (t) are nonnegatve rate functons, and hence the ntegral defned n (7) can never be zero. We can solve ths problem by makng a mnor change of the orthogonalty requrement and the structure of the matched flter. Frst, we replace the orthogonalty condton by a smlar condton n whch the normalzed drop sgnatures are orthogonal after removng ther DC components: s a (t)s a j (t) dt =, for j (8) n whch x a (t) s x(t) after elmnatng ts DC component over [, T ]: x a (t) = x(t) T x(t) dt (9) Furthermore, we change the matched flter output for the th router n the followng way: y = η s a (r) = s a (t) λ(t) dt () y s the value of η h n (2) evaluated for h(t) = s a (t). One mportant fact about notaton η h (r) n () s that n ths equaton r s the total perturbng functon, snce the rate decrease λ( ) λ(t) s affected by ths total drop rate (.e., r(t) = k r k(t), where k s an ndex that covers the set of all router perturbatons that the aggregate experences). It can be shown that f the total drop rate r(t) s small enough compared to the rate of aggregate, then the system wth nput r(t) and output the expected value of rate degradaton E[ λ(t)] can be approxmated by a lnear system. In other words, the system can be lnearzed around ts operatng pont. Now we can state the followng lemma; for the purpose of ths lemma we assume r k (t) are pecewse constant functons as t was defned n (3). Lemma : Assume that the overall drop rate r(t) = k r k(t) s small enough such that the system wth nput r(t) and output E[ λ(t)] can be approxmated by a lnear system. Furthermore assume the holdng tme of the pecewse constant functons r k (t) on each constant nterval s large enough compared to the response tme of the aggregate. Then under the orthogonalty assumpton of (8) we have: E[y ] = E[η s a (r)] = E[η s a (r )] () The proof of ths lemma s gven n the appendx. Note that the mddle term of equaton () s the measure of degradaton wth the weght functon h(t) = s a (t) when all routers perturb the aggregate, however, the rght term s the measure of degradaton wth the same weght functon when only the th router perturbs the aggregate. The sgnfcance of Lemma s that t states under orthogonalty condton of equaton (8) that the expected degradaton measure at router, E[η s a (r)], s ndependent of perturbatons beng done at the other routers. In Lemma we have assumed that the holdng tme of r k (t) on the ntervals on whch t s constant s large enough compared to the aggregate response tme. Generally, the response tme of an aggregate s characterzed by the round trp tme of the flows contrbutng to t. Therefore for the pecewse constant functon r k (t), the length of each constant nterval should be sgnfcantly larger than the typcal round trp tme of the flows n the aggregate. Ths condton can be satsfed 4

5 by makng T c long enough (e.g., to 2 tmes the typcal round trp tme). One useful observaton about () s: s a (t)λ( ) dt = (2) And so we have the followng smple equaton for the output of the matched flter for the th router: y = s a (t)λ(t) dt (3) From (2) and (), we have the followng expresson for the average output of the matched flter of the th perturbng router: E[y ] = E[η s a (r )] Ths equaton gves the bass for quantfyng the responsveness of TCP aggregates. Denote: K = E[y ]/A (4) Notce that K s a coeffcent that descrbes how much the aggregate s responsve to packet drops. We call ths quantty the response coeffcent of the aggregate. Note that y s fully observable, and t can easly be measured by usng (3). The ampltude of perturbng functon A s known to the router that does the perturbaton. Fndng K s the only problem of the estmator. Ths coeffcent can be estmated durng the tmes that there s no congeston n the network. Or t can be estmated by a long term average of y /A based on multple tests. Based on the result of Lemma, the estmaton value of K s not affected by the perturbatons done by the other routers, under the orthogonalty assumpton. There are some key ssues about how to choose the value of T c. As stated before, T c should be long enough such that the rate decrease of the aggregate as a result of packet drops n one chp duraton can show up, and the aggregate rate settles down. On the other hand, too large T c does not mprove the performance n estmatng the response coeffcents, and t only causes longer test and more packet drops, whch causes the test to be more expensve. V. FAIR CONGESTION CONTROL BY USING CAPM In ths secton we suggest a method to use CAPM to do congeston control n a far way. Random Early Drop [2] s one of the popular approaches to proactvely prevent congeston n a router. By utlzng CAPM a router collects nformaton about how responsve dfferent aggregates are -.e., K coeffcents defned n the prevous secton. Knowng these coeffcents helps a router to determne how much t should drop from each aggregate to reduce ts bandwdth to a certan value. Assume a traffc composed of many aggregates s ntended to be forwarded through an outgong lnk that has bandwdth shortage. So t s desred to keep the traffc bandwdth wthn the outgong lnk capacty. If the router apples equal drop probablty governed by a congeston controller such as a RED controller for all aggregates, the aggregates wth hgher response coeffcents wll back off more aggressvely compared to the aggregates wth smaller response coeffcents. A certan degree of farness among aggregates can be acheved by takng nto account ther response coeffcents. Assume the traffc s a combnaton of M aggregates, and let λ (t), and K denote the estmated nstantaneous arrval rate and the response coeffcent of the th aggregate respectvely. Assume that we want to rate lmt the total traffc, and let the output of congeston controller at tme t be p(t). Wth the nformaton of response coeffcents of aggregates the router can estmate how ths total drop probablty should be assgned among the aggregates to get a specfc amount of rate decrease for each one. To llustrate the above approach assume t s desred to have the same amount of rate decrease for all aggregates. Then we can assgn the packet drops among dfferent aggregates n a way that the product of the response coeffcent and the drop rate s equal for all of them. In other words: K θ (t) = K j θ j (t), j M (5) n whch θ (t) and θ j (t) denote the average drop rate of the th and j th aggregate respectvely. In the above equaton subscrpts are ndex of aggregates n the same routers. Heurstcally equaton (5) means that the rate decrease of the aggregates should be equal. It s mportant to note that equaton (4) suggests usng (5) as a heurstc to equalze the rate decreases of the aggregates; however, (5) s not a mathematcal consequence of (4). If p (t) s the drop probablty of the th aggregate, we have θ (t) = λ (t)p (t). Therefore, equaton (5) can be wrtten n the followng way: K λ (t)p (t) = K j λ j (t)p j (t), j M (6) whch gves M lnear equatons. To fnd the numercal values of the drop probabltes we need one other equaton. We use the fact that the total drop probablty of the traffc should be p(t). In other words: M = p (t)λ (t) λ(t) = p(t) (7) n whch λ(t) = λ (t) + λ 2 (t) =... + λ M (t) s the total rate of the traffc. In the above approach we have tred to get the same rate decrease for dfferent aggregates, however, one can apply the response coeffcents n dfferent ways to acheve an arbtrary value of rate decrease for each aggregate. For example, t may be desred to have the same percentage of rate decrease for dfferent aggregates; n ths case t s very easy to wrte equatons smlar to equaton (6) to fnd drop probabltes. VI. SIMULATIONS AND RESULTS We have used the popular network smulator ns2 to perform our experments []. As explaned prevously, our focus s on TCP aggregates. For smulaton we have used a network wth fxed topology as n fgure 3. The nodes S, S 2,..., S n are n sources of TCP traffc. The propagaton delay of the lnk between each source and router R n fgure 3 s dfferent from 5

6 6 Pkt/S 4 2 (a) Fg. 3. The network topology used for smulaton a source to another source, and t has been chosen such that the round trp tme of packets s unformly dstrbuted between 5 and mllseconds under low congeston condtons. The flows at the sources are generated accordng to a brth-death process. Each source starts a TCP flow, and that flow ends after a random tme unformly dstrbuted between and.5 seconds. That source starts a new flow after watng another random tme unformly dstrbuted between and.3 seconds. The packet sze s constant equal to Kbyte for all flows. In ths topology, the lnk between R and R 2, and also the lnk between R 2 and destnaton are bottleneck lnks. The capacty of these bottleneck lnks s Mbps, that translates to 25 packets per second. In the frst experment we show how an aggregate responds to the sgnature based perturbatons. For ths experment we run the smulaton for two cases. In the frst case the aggregate does not experence any perturbaton; n the second case only R perturbs the aggregate by usng drop rate r (t) = A s (t), n whch s (t) s the normalzed drop sgnature generated by pluggng code (,,,,,,,,,,,,,,, ) n equaton (3), and A = 6 packet drops/sec. In the smulaton, the number of sources s 5, T = 32 seconds, N = 6, T c = 2 seconds. Fgure 4-(a) shows the rate of the aggregate when no perturbaton s performed. In fgure 4-(b) the rate of aggregate s shown when R perturbs the aggregate by usng r (t), and fgure 4-(c) shows two perods of the normalzed drop sgnature s (t). By nspectng fgure 4-(b) we can see that the shape of drop sgnature of R has appeared n the rate of the aggregate wth 8 degrees of phase shft. In other words, when s (t) = (e.g., around t = 4), the rate of aggregate decreases, and when s (t) = s (e.g., around t=), the rate ncreases. In the second experment we explore the typcal response of aggregate when two routers perturb t smultaneously. In ths experment R and R 2 perturb the aggregate by usng dfferent CDMA drop sgnatures. In the smulaton, the number of sources s 5, T = 32 seconds, N = 6, T c = 2 seconds. The code of R s (,,,,,,,,,,,,,,, ), and that for R 2 s (,,,,,,,,,,,,,,, ). Under ths assgnment s a (t) and s a 2(t) are orthogonal. Two perods of the resultng normalzed drop sgnatures for R and R 2 are shown n fgure 5-(b) and 5-(c) respectvely. The ampltude of drop sgnatures for the two routers, A and A 2, are the same and equal to 2 drops per second. Fgure 5-(a) shows the rate of aggregate when the two routers R and R 2 perturb the aggregate smultaneously. It can be seen that the addtve shape of the two drop sgnatures appears on top of the aggregate rate wth 8 degree phase shft agan. In other words, the two drop sgnatures modulate the aggregate rate addtvely. For 2 (b) (c) Tme(Sec) Fg. 4. a: The aggregate rate wthout any perturbaton, b: The aggregate rate wth perturbaton of R, and c: the normalzed drop sgnature of R. example, at around tme t = 4, the ampltude of both drop sgnatures s zero, and ths shows up as an ncrease n the rate of aggregate as t can be seen n 5-(a) at t = 4. On the other hand, at tme t = 5 or t = 3, the ampltude of both drop sgnatures s nonzero, and ths shows up as a decrease n the rate at these two tmes. The purpose of next experment s to verfy that under orthogonalty defnton of (8), the matched flter output of a router defned by (3) s not affected by perturbatons done by the other routers. We proved ths fact n Lemma. In ths case we use the same CDMA drop sgnatures as n the prevous experment, but we change A and A 2, the ampltude of the drop sgnatures of the two routers. Fgure 6-(a) shows y, the output of matched flter for R as t s defned by equaton (3), when A changes from to 6 drops per second. In ths fgure, each + represents a test n whch R perturbs the aggregate wth drop sgnature r (t) = A s (t) and at the same tme R 2 s also perturbng traffc wth drop sgnature r 2 = A 2 s 2 (t), and A = A 2. For each value of A several tests have been done, and the average over multple tests has been plotted by the sold lne. It can be seen that the devaton of y for each ndvdual test from the average value shown by sold lne s relatvely small; ths means that the matched flter output shows a small varance. The other observaton about 6-(a) s lnearty n ampltude of drop sgnature A. In the other part of ths experment we turn off the perturbatons done by R 2 by settng A 2 =, and do the same multple test and measurement of y for each value of A. The dashed lne n fgure 6-(a) shows the average of multple tests for each value of A for ths case. It can be seen that the dashed lne s very close to the sold lne showng that perturbatons of R 2 do not affect the output of matched flter of R. Fgure 6-(b) s the same as fgure 6-(a) for the second router. In the next experment we wll show how the response coeffcents can be used to do congeston control n a far way. So we defne two aggregates that pass through R. In ths experment the sources n the smulaton network are dvded nto two groups. The on tme of a flow generated by a source 6

7 Pkt/S (a) (b) Tme (Sec) Fg. 5. a: The aggregate rate under two smultaneous perturbatons, b: the normalzed drop sgnature of R, and c: the normalzed drop sgnature of R2. n group s unformly dstrbuted n [.5,.3] seconds, and after endng a flow, the source starts another flow after beng dle for a random tme unformly dstrbuted n [,.3]. There are 5 sources n group. Sources n group 2 generate larger flows. The on tme of a flow n group 2 s unformly dstrbuted n [.45,.9] seconds, and the dle tme between flows s unformly dstrbuted n [,.5] seconds. There are 2 sources n group 2. We defne the traffc generated by group as the aggregate and traffc generated by group 2 by the aggregate 2. Frst we fnd the response coeffcent of each of the two aggregates by usng equaton (4). The experment shows that K = 77.3, and K 2 = The value of response coeffcents have been found by several tests and averagng the results. The hgher value of the response coeffcent of aggregate 2 s easy to explan by consderng the fact that the flows belongng to ths aggregate are larger, so they show a hgher rate decrease when they experence packet drops. The experment shows that λ, the average rate of aggregate, s about 5234 pkt/sec, and λ 2 = 4547 pkt/sec. The total rate of the traffc s 978 pkt/sec. In the next step of ths experment, we doubled the number of sources n each group, so aggregate needs about = 468 pkt/sec; aggregate 2 needs 994 pkt/sec, and the total demand s 9562 pkt/sec. Ths total demand s more than the lnk capacty whch s 25 packet/sec. The smulaton results show that under drop tal condton n the forwardng queue of R, about 4% of ncomng packets are dropped, and as a result of t the arrval rate of the traffc s reduced to about 257 packets/sec. Under ths drop polcy the rate of aggregate reduces to 82 pkt/sec and the rate of aggregate 2 reduces to 4558 pkt/sec. The above data means that the rate reducton of aggregate s 2456 pkt/sec or 25% of ts demand, whle the rate decrease of aggregate 2 s 4536 pkt/sec that s 52% of ts demand. As t can be seen, aggregate 2 shows much hgher rate decrease as a result of havng a hgher response coeffcent. To keep the farness n rate reducton between the two aggregates, we use the farness scheme explaned n Secton (c) Pkt Pkt 2.5 x (a) x (b) Pkt/S Fg. 6. Matched flter output versus the ampltude of drop sgnature, a: for the frst router, b: for the second router. V to assgn drop probabltes. By usng equatons (6) and (7) we fnd the drop probablty p =.67 for aggregate, and p 2 =. for aggregate 2, so the total drop rate of the traffc s stll 4%. Wth these drop probabltes, the rate of aggregate reduces to 6522 pkt/s, and rate of aggregate 2 reduces to 5523 pkt/sec. In ths case the rate reducton of aggregate s 3946 pkt/sec or 39% of ts demand, and the rate reducton of aggregate 2 s 357 pkt/sec or 42% of ts demand. It can be seen that the rate reductons are much closer to each other than the prevous experment, and we were able to do a far congeston control. In the last experment we wll study how congeston can affect the measurement of response coeffcents. For ths purpose we used an aggregate lke aggregate wth the same condtons that were stated n the prevous experment. The response coeffcent of ths aggregate was measured n ndependent smulaton runs wth dfferent lnk utlzatons of the bottleneck lnks. To ncrease the lnk utlzaton we ncreased the number of sendng sources n group. The results have been shown n fgure 7. In ths fgure each + shows the response coeffcent versus the lnk utlzaton. The response coeffcent shows an almost flat behavor wth reasonable varance up to the pont where the lnk utlzaton s about 9%. After that the measurement of the response coeffcent s not accurate, however, the measurements are stll good approxmatons up to the pont where the lnk utlzaton s about 95%. In ths fgure the sold lne shows the average of the response coeffcent over the experments for whch the lnk utlzaton s less than 9%; ths average s about 75. The degradaton of performance of APM and CAPM n the presence of severe congeston s easy to explan; heavy congeston causes the aggregates to experence hgh rate of drops and as a result of that the aggregates shrnk ther rates. Ths causes them to become less responsve to the packet drops made by APM or CAPM. Although long term congeston s one factor that may degrade the performance of APM or CAPM, the APM and CAPM show a good performance n a wde range of lnk utlzaton before very heavy congeston happens. Ths can be one of the strong ponts about APM 7

8 response coeffcent Lnk Utlzaton(%) Fg. 7. The value of response coeffcent of aggregate measured at dfferent lnk utlzatons and CAPM snce these methods can be appled proactvely to prevent congeston. VII. CONCLUSIONS In ths paper, we ntroduced the Aggregate Perturbaton Method (APM), and CDMA-based APM (CAPM), two technques for quantfyng the responsveness of a TCP aggregate. Both algorthms perform a test on an aggregate by droppng some packets from t and observng the result. APM s the a smpler test but t s not robust to smultaneous tests at dfferent routers. So we ntroduced CAPM that uses some unque drop sgnature for each router to do the test, and the approach s smlar to the Drect Sequence Spread Spectrum CDMA n communcaton theory. We also defned a value called response coeffcent to measure responsveness of TCP aggregates to packet drops. We used these values for the purpose of far congeston control among the aggregates wth dfferent response coeffcents. One mportant advantage of APM and CAPM s that they can be mplemented n a dstrbuted manner wthout needng data exchange between routers, and furthermore, these methods do not need any change n the current protocols. Ths also permts ncremental deployment. One of the strong ponts about APM and CAPM s that these algorthms can perform proactvely, and prevent congeston n an early stage. REFERENCES [] Floyd, S. and Fall, K., Promotng the Use of End-to-End Congeston Control n the Internet, IEEE/ACM Transactons on Networkng, Vol. 7, No. 4, Aug 999. [2] Floyd, S. and Jacobson, V., Random Early Detecton Gateways for Congeston Avodance, IEEE/ACM Transactons on Networkng, Vol., No. 4, pp , Aug 993. [3] Jacobson, V., Congeston Avodance and Control, Proceedng of SIG- COMM 88, Proceedng of SIGCOMM 88, Aug [4] Schulzrnne, H., Long Term Internet Traffc Statstcs, avalable onlne at: hgs/nternet/traffc.html/. [5] Savage, S.; Wetherall, D.; Karln, A. and Anderson, T., Network Support for IP Trace back, IEEE/ACM Trans. on Networkng, June 2,Vol. 9, No 3. [6] Msra, V.; Gong, W. and Towsley, D., A flud based analyss of a network of AQM routers supportng TCP flows wth an applcaton to RED, Proc. of SIGCOMM 2, August 2. [7] Verdu, S. Multuser Detecton, Cambrdge Unversty Press, 998. [8] Kalantar, M; Gallccho, K. and Shayman, M., Usng Transent Behavor of TCP n Mtgaton of Dstrbuted Denal of Servce Attacks, Proc. of IEEE Conference on Decson and Control, Las Vegas, Nevada, December 22. [9] Mahajan, R.; Bellovn, S.; Floyd, S.; Ioannds, J.; Paxson, V. and Shenker S., Controllng Hgh Bandwdth Aggregates n the Network, avalable onlne at: [] Network Smulator ns2, onlne documentaton avalable at: [] Altman, E.; Avrachenkov, K. and Barakat, C., A Stochastc Model of TCP/IP wth Statonary Random Losses, Proc. of SIGCOMM 2, August 2. [2] Wu-chang, Feng; Shn K.G.; Kandlur, D.D. and Saha, D., The BLUE actve queue management algorthms, IEEE/ACM Transactons on Networkng, Vol., No. 4, pp , Aug 22. Appendx: proof of Lemma : Denote λ x (t) to be the rate change of aggregate when t s perturbed wth drop rate x(t). By defnton we wll have: E[η s a (r)] = s a (t)e[ λ r (t)] dt. (8) From the lnearty assumpton E[ λ r (t)] n r(t) we can conclude: E[ λ r (t)] = E[ λ r (t)] + j E[ λ r j (t)]. (9) Substtutng (9) n (8) yelds: E[η s a (r)] = E[η s a (r )] + j E[η s a (r j )]. (2) To complete the proof, t suffces to prove E[η s a (r j )] = for j. We have r j (t) = A j s j (t). Now we use the assumpton that r j (t) changes slower than the aggregate response tme. Hence r j (t) can be approxmated by usng a pecewse constant functon. For an nterval on whch r j (t) s constant, the traffc aggregate responds and settles down to a value. In the next nterval r j (t) jumps to a new value, and so E[ λ rj (t)] responds accordngly, and after experencng a small transent tme settles down to a new steady state value. Accordng to the lnearty assumpton E[ λ r j (t)] on each nterval s proportonal to the constant value of r j (t) on that nterval. Ths means that E[ λ rj (t)] tracks the pecewse constant shape of r j (t). So by gnorng the short transents of E[ λ r j (t)] at the begnnng of each nterval we wll have: E[ λ rj (t)] C j r j (t) = C j A j (s d j + s a j (t)) (2) n whch s d j s the DC component of s j(t) over nterval [, T ]. Recall E[η s a (r j )] = s a (t)e[ λ r j (t)] dt. (22) Substtutng (2) n (22) and usng orthogonalty assumpton of (8) yelds: E[η s a (r j )] =. QED 8