Queue Dynamics of RED Gateways Under Large Number of TCP Flows

Transcription

1 Queue Dynamcs of RED Gateways Under Large umber of TC Flows eerapol Tnnakornsrsuphap, Armand M. Makowsk Department of ECE and and Insttute for Systems Research Unversty of Maryland, College ark, MD Abstract We consder a stochastc model of a RED gateway under competng TC-lke sources sharng the capacty. As the number of competng flows becomes large, the queue behavor of RED can be descrbed by a two-dmensonal recurson. We confrm the result by smulatons and dscuss ther mplcatons for the network dmensonng problem. I. ITRODUCTIO One of the key mechansms for the operaton of the besteffort servce Internet s the congeston-control mechansm n TC. Whle there are several varatons on the basc TC congeston-control mechansm, they all have n common the addtve ncrease/multplcatve decrease (AIMD) algorthm. The AIMD algorthm enables TC congestoncontrol to be robust under dverse condtons. However, t s well known that wth tal-drop gateways ths congestoncontrol also leads to undesrable behavor,.e., global synchronzaton. When several TC flows compete for bandwdth n a tal-drop gateway, t has been observed expermentally that packets from many flows are usually dscarded smultaneously [], resultng n a poor utlzaton of the network. Actve queue management algorthms such as Random Early Detecton (RED) [2] were ntroduced to help allevate ths problem by randomly droppng packets dependng on the queue sze, thereby avodng heavy congeston and prevent global synchronzaton. Whle there are many efforts to model TC throughput under a tal-drop assumpton [3] [4] [5] [6], only a few studes have focused on modelng the nteracton of RED gateways wth TC congeston-control. In [7], an analytcal framework for multple TC flows sharng a RED gateway s developed under several potentally unrealstc assumptons. In [8], a smple analyss has been done wth TC connectons operatng as osson processes under slow and fast rates. Fxed pont solutons to average TC wndow szes and queue occupancy are dscussed n [9]. However, a model that can provde a good analytc understandng of TC and RED s yet to be found. The dffcultes arse from the complex behavor of TC congeston-control, and are further compounded by the random drop mechansm and queue averagng. Detaled modelng of these characterstcs results n a number of states whch explodes when the number of TC flows ncrease, makng the analyss untractable. In ths paper, we present a stochastc model that captures the essental features of TC,.e., the gradual adaptve ncrease and the sudden decrease of transmsson rate, combned wth a random drop algorthm smlar to RED. We analyze ths ersatz model as the number of competng TC flows becomes large, and show that the stochastc model smplfes n the lmt to a two-dmensonal recurson. Ths result suggests that wth a large number of flows, t s easy for network operators to estmate the aggregate behavor of TC flows and to dmenson network resources accordngly. The remander of the paper s organzed as follows. Secton II descrbes the stochastc model. Secton III present the man asymptotc results for the large number of TC flows whereby the stochastc model smplfes nto a smplfed lmtng recurson. Smulaton results supportng ths behavor are shown n Secton IV. The conclusons of the paper are gven n Secton V. II. THE MODEL TC congeston-control utlzes the AIMD algorthm to provde TC flows a far bandwdth share [0] by usng feedback obtaned through acknowledgement packets (ACKs). If an ACK packet s receved (.e., a packet s successfully transmtted and acknowledged), TC ncreases ts transmsson rate by a small, conservatve amount. Otherwse, TC nterprets a lack of acknowledgement as a sgn of congeston and reduces ts transmsson rate by half. We present an algorthm smlar n sprt to the AIMD congeston-control n TC and apply t to the model de-

2 scrbed earler. A. Defntons and notaton Tme s assumed dscrete and slotted n contguous tmeslots of equal duraton normalzed to a packet transmsson tme. We consder traffc sources, all transmttng through a bottleneck RED gateway. The capacty of ths bottleneck scales wth the number of flows,.e., t has capacty C packets/slot for some postve constant C. We model the RED buffer as an nfnte queue, so that packet losses are due only to the random drop algorthm B. Dynamcs Fx =; 2;:::and t =0; ;:::. For any quantty X, we wrte X ( ) to ndcate the explct dependence of X on the number of connectons. Let Q ( ) denote the number of packets n the buffer at the begnnng of the tmeslot [t; t +). Suppose that each source (or equvalently, TC connecton) generates at most one packet n each tmeslot. So let B ( ) be a f0; g-valued rv that encodes the number of packets generated by source at the begnnng of the tmeslot [t; t+ ). The packet from source, upon arrval at the RED gateway, may be rejected by the random drop algorthm (to be specfed shortly). We represent ths possblty by the f0; g-valued rv wth the nterpretaton that = (resp. =0) f the packet s rejected (resp. accepted nto the RED buffer). Gven that sources are actve, the total number of packets whch are accepted nto the RED buffer at the begnnng of tmeslot [t; t +)s gven by A ( ) := X = (, : If Q ( ) denotes the number of packets n the buffer at the begnnng of the tmeslot [t; t + ), then Q ( ) +A ( ) packets are avalable for transmsson. Snce the outgong lnk operates at the rate of C h + packets/tmeslot, Q ( ) +A ( ), C packets wll not be transmtted durng tmeslot [t; t +],and reman n buffer, ther transmsson beng deferred to subsequent tmeslots. The number Q ( ) of packets n the buffer at the begnnng of the tmeslot [t +;t+2)s therefore gven by 2 h + Q ( ) = Q ( ), C + A ( ) : () We can account for the fnteness of the buffer by modfyng the queue dynamcs as s done n the footonote to recurson (). 2 The fnteness of the buffer n () can be replaced by Q () = C. Statstcal assumptons In order to fully specfy the model, we need to specfy the statstcs of the rvs fb ( ) ; ; = ;:::;; t =0; ;:::g for each =; 2;:::. To do so we ntroduce the collecton of..d. [0; ]- unform rvs fv (t+);u (t+); =;:::; t =0; ;:::g. For each =;:::;,wetake B ( ) =[U ( ) ] (2) where ( ) s an [0; ]-valued rv whch denotes the (condtonal) transmsson rate (to be specfed shortly) of traffc source at the begnnng of the tmeslot [t; t +). We also set =[V ( ) f ( ) (Q ( ) )] (3) where f ( ) : IR +! [0; ] denotes the drop probablty functon of the RED gateway. To select the transmsson rates we argue as follows: Suppose that source generates no packet durng tmeslot [t; t +)(.e. B ( ) =0), then the transmsson rate of source n the next tmeslot remans unchanged. If on the other hand, a packet s produced by source at the begnnng of tmeslot [t; t +), then ether the packet s successfully transmtted ( (t+) = 0), or t s dropped (t+) = ). In the former case, the transmsson rate ( of source n the next tmeslot s ncreased to ( )," s de- (wth 0 < " < ). In the latter case, ( ) creased by a factor (or ( ) = ( ) ), where 0 <<. These two stuatons attempt to emulate (under the constrant that transmsson rates are bounded to the unt nterval) the addtve ncrease and multplcatve decrease, respectvely, of the TC congeston-control. They can be summarzed nto the sngle equaton ( ) = ( )," (, + ( ) B ( ) + ( ) (, B ( ) (t + )): (4) For each t = 0; ;:::,letf t denote the -feld generated by the rvs fq ( ) (0); ( ) (0);V (s);u (s); = ;:::; s = ;:::;tg. ote the rvs Q ( ) and ( ) ( =;:::;)areallf t -measurable, so that E hb ( ) jf t = ( ) mn([q (),C + = (,R() (t + ))B () (t + )] + ;B) where B denotes the buffer sze per connecton.

3 for all =;:::;,and E h jf t = f ( ) (Q ( ) ): III. MAI RESULTS In ths secton, we analyze the model presented n Secton II as the number of traffc flows ncrease to nfnte and dscuss the mplcatons of the result. Q () / A. The asymptotcs The dscusson s carred out under the followng assumptons: There exst a contnuous functon f : IR +! [0; ] and a constant n (0; ) such that for each = 0; ;:::, (A) f ( ) (x) =f (x= ) (x 0); (A2) Q ( ) (0) = 0 and ( ) (0) = ( =;:::;). We begn wth an easy consequence of these assumptons. Lemma : Assume (A)-(A2) to hold. Then, for each t = 0; ;:::;, the rvs f ( ) ;:::; ( ) g are exchangeable for all =; 2;:::. The next proposton presents the asymptotcs for the normalzed buffer content as the number of TC sources becomes large. Theorem : Assume (A)-(A2) to hold. Then, for each t =0; ;:::, there exst a non-random constant q and a rv such that Q ( ) and for every p>0, Moreover, and! q and ( )! (5) = (( ) ) p! E [ p ] : (6) q=[q, C +(, f (q))e []] + (7) =," [V >f(q)][u ] +[V f (q)][u ] +[U > ] (8) for..d. [0; ]-unform rvs fv ;U; t = 0; ;:::g. Wth a p =E [ p ] (p >0), we readly get a p = a p +(, f (q)) a (,")p +( p f (q), ) a p+ : (9) t Fg.. The normalzed queue length of Smulaton. B. Dscusson Theorem suggests that a bottleneck queue wth random-drop algorthm, under large number of TC-lke sources, can be characterzed by a two-dmensonal recurson gvng the evoluton of the normalzed queue length q and the lmtng transmsson rate. Ths result s not a straghtforward consequence of the Law of Large umber due to the fact that () the transmsson rates of traffc sources are correlated and () they vary wth. However, as the number of sources ncreases, the dependency between any par of sources becomes weaker so that the aggregate behavor eventually becomes determnstc. Thus, as the aggregate queue behavor scales lnearly wth the number of sources, the network provder could effectvely dmenson network resources by tackng the normalzed queue behavor. The sequence f(q; ); t = 0; ;:::g n Theorem defnesanir + [0; ]-valued Markov process, and we expect that t admts a steady-state regme. Ths wll be dscussed elsewhere. IV. SIMULATIO We smulate the system descrbed earler for = 0; 00; 000 wth " = 0:, = 0:5 and C = 0:5; the ntal condtons are Q ( ) (0) = 0 and ( ) =0:5 for all =; ::;. The drop probablty s calculated through the pecewse lnear functon f :IR +! [0; ] gven by f (x) = 8 >< >: 0 x< x, 4 x<5 5 x: (0)

4 (/) Σ α = Average queue sze per user n tmeslot t Fg. 2. The average transmsson rate per user of Smulaton = tmeslot Fg. 3. The average queue sze per user of ns smulaton. The smulaton results are shown n Fgure and 2. It s clear that the fluctuaton of Q ( ) = decreases as the number of sources ncreases, and the same s true for the average transmsson rate. Wth a hundred or more flows, our analytcal result seems to hold reasonably well. Moreover, ths smulaton result also suggests the exstence of the steady-state, whch happens quckly after only around a hundred teratons. We also smulate a smlar system n ns by generatng TC Reno connectons, each of whch havng 00 ms round-trp delay, all competng to transmt through a bottleneck gateway wth lnk capacty Mbps. The TC packet sze s set to be 500 bytes. The buffer management scheme n the gateway s RED wth the followng parameters: thresh =2, maxthresh =5, p drop =when queue sze s greater than 5 and w q =0:. The tmeslot that we observed the average queue length n the bottleneck gateway s second. Fgure 3 shows the smulaton result; a trend smlar to that of Fgure s observed n that as the number of TC connecton ncreases, the fluctuaton n the average queue sze decreases. As tme passes, the range of fluctuaton settles to a certan lmted range. The prelmnary fndngs comfort our belef that our ersatz model captures some of the essental features of TC and RED and then llustrate the mportant behavor of the nteracton between these two mechansms. V. COCLUSIOS We have developed a stochastc model for a RED gateway under competng TC flows. We have shown that, as the number of flows grow large, the aggregate behavor of the queue can be descrbed by a two-dmensonal recurson. We have also dscussed the mplcatons of our results to the dmensonng of the network. Although we have yet to prove the exstence of a steadystate regme for the lmtng recurson, the lmted smulaton results () are compatble wth the exstence of such a steady-state and () suggest that the rate of convergence s fast n ether the number of sources (to acheve lmtng behavor) and the tme (to reach the steady-state). Future work on ths class of models ncludes () a proof of the exstence of a steady state for the lmtng dynamcs and ts evaluaton; () a dervaton of a CLT complement to the basc convergence result; and () the development of more accurate models (e.g., More than one packet generated per tmeslot; asynchronous updatng of the transmsson rates; non-homogeneous populaton of TC flows and contnuous-tme versons.) It s also nterestng to see how the shape of the drop functon affect the rate of convergence. Furthermore, we should be able to nvestgate the farness of the competng TC flows n ths model. And fnally, we beleve that ths model s smple enough to be extended to the scenaro where the sources are stochastc, say on-off sources. REFERECES [] L. Zhang and D. Clark, Oscllatng behavor of network traffc: A case study smulaton, Internetworkng: Research and Experence, vol., no. 2, pp. 0 2, 990. [2] S. Floyd and V. Jacobson, Random early detecton gateways for congeston avodance, IEEE/ACM Transactons on etworkng, vol., no. 4, pp , 993. [3] M. Maths, J. Semke, J. Madhav, and T. Ott, The macroscopc behavor of the tcp congeston avodance algorthm, Computer Communcaton Revew, vol. 27, no. 3, pp , 997.

5 [4] J. adhye, V. Frou, D. Towsley, and J. Kurose, Modelng tcp reno performance: A smple model and ts emprcal valdaton, IEEE/ACM Transactons on etworkng, vol. 8, no. 2, pp , [5] Th. Bonald, Comparson of TC Reno and TC Vegas va flud approxmaton, Tech. Rep. 3563, IRIA, ov [6] T. V. Lakshman and U. Madhow, The performance of TC/I for networks wth hgh bandwdth-delay products and random loss, IEEE/ACM Transactons on etworkng, June 997. [7] A. Abouzed and S. Roy, Analytc understandng of red gateways wth multple competng tcp flows, n roceedngs of Globecom 2000, [8] H. M. Alazem, A. Mokhtar, and M. Azzoglu, Stochastc modelng of random early detecton gateways n tcp networks, n roceedngs of Globecom 2000, [9] A. Msra, Dynamcs of TC Congeston Avodance wth Random Drop and Random Markng Queues, h.d. thess, Unversty of Maryland, [0] V. Jacobson, Congeston avodance and control, n roceedngs of SIGCOMM 88 Symposum, Aug. 988, pp []. Tnnakornsrsuphap and A.M. Makowsk, Queue Dynamcs of RED Gateways Under Large umber of TC Flows: Lmt Behavor and Steady-State Regme, Techncal Report, Insttute for Systems Research, Unversty of Maryland, College ark, MD [2] G. R. Grmmett and D. R. Strzaker, robablty and Random rocesses, Oxford Scence ublcatons, 992. VI. OUTLIE OF AROOF OF THEOREM A complete proof of Theorem s avalable n []. Before we outlne the key elements of ths proof, we ntroduce the followng notaton: For each t = 0; ;:::,the statements [A:t], [B:t] and [C:t] refer to the convergences [A : t] : Q ( ) [B : t] : ( )! ; [C : t] :! q wth q non, random; = [( ) ] p! a p non, random: The equalty a p =E [ p ] n [C:t] readly follows under [B:t] snce the rvs f ( ) ;:::; ( ) g are exchangeable and bounded. Snce the statements [A:t], [B:t] and [C:t] do hold for t = 0, Theorem wll be proved by nducton f the followng nducton step can be establshed. roposton : Assume (A)-(A2) to hold as n Theorem. If for some t = 0; ;:::, [A:t], [B:t] and [C:t] hold, then so do [A:t+], [B:t+] and [C:t+]. Ths proposton can be proved wth the help of a seres of lemmas. The frst two lemmas are elementary, and ther proofs are therefore omtted. Lemma 2: Let U denote a [0,] unform rv whch s ndependent of the [0; ]-valued rvs fx; X n ; n =; 2;:::g. If X n! n X,then [U X n ]! n [U X]. Lemma 3: For each = ; 2;:::, assume the rvs f ( ) ; = ;:::;g to be bounded, say j ( ) j for all =;:::;, and the rvs f ( ) ; = ;:::;g to be exchangeable and bounded, say j ( ) j for all = ;:::;.If ( )! 0,then ) = ( ( )! 0. The next lemma takes the frst step towards provng roposton. Lemma 4: Under (A), f [A:t], [B:t] and [C:t] hold for some t =0; ;:::,then[b:t+] holds wth (t+) related to by (8). roof: The contnuty of f andtheassumedconver- gence [A:t] readly lead [2, p. 326] to f ( ) (Q ( ) ) = f (Q ( ) = )! f (q); () and Lemma 2 thus yelds! [V f (q)]: (2) Also by Lemma 2, we have B ( )! [U ] (3) where the rv U s ndependent of the rv. Fnally, under [B:t], we obtan (8) drectly from (4). The next two lemmas provde the fnal steps n the proof of roposton. Lemma 5: Under (A), f [A:t], [B:t] and [C:t] hold for some t =0; ;:::,then A ( )! (, f (q))a (4) and [A:t+] holds. Lemma 6: Under (A) (A2), f [A:t], [B:t] and [C:t] hold for some t =0; ;:::,then[c:t+] holds. The proofs of Lemmas 5 and 6, whle nvolved, are not very dffcult and follow the same pattern: The rvs (,, B ( ) and, B ( ) are ndcator functons of mutually exclusve events. Hence, [ ( ) (t+)] q equals to the rghthand sde of (9) wth ( ) replaced by [ ( ) ] q.we can expand A ( ) and ) = [( (t + )] q by centerng each term by subtractng and addng back the ap- proprate condtonal mean, say [V f (q)] for and ( ) for B ( ). By repeated applcatons of Lemma 3, each centered term wll converge n probablty to zero, and the desred results follow.