Symbolic Dynamical odel of Average Queue Size of Rom Early Detection Algorithm Charlotte Yuk-Fan Ho Telehone: 44 () 788 5555 ext. 4333 Fax: 44 () 788 7997 Email: c.ho@mul.ac.uk School of athematical Sciences Queen ary University of London ile End Road London E 4NS United Kingdom. *Bingo Wing-Kuen Ling Telehone: 44 () 7848 94 Fax: 44 () 7848 93 Email: HTing-kuen.ling@kcl.ac.ukTH Deartment of Electronic Engineering Division of Engineering King s College London Str London WCR LS United Kingdom. Herbert H. C. Iu Telehone: 68 6488 7989 Fax: 68 6488 7989 Email: HTherbert@ee.ua.edu.auTH School of Electrical Electronic Comuter Engineering The University of Western Australia 35 Stirling Highay Craley Perth Western Australia WA69 Australia. ABSTRACT In this aer a symbolic dynamical model of the average ueue size of the rom early detection (RED) algorithm is roosed. The conditions on both the system arameters the initial conditions that the average ueue size of the RED algorithm ould converge to a fixed oint are derived. These results are useful for netork engineers to design both the system arameters the initial conditions so that internet netorks ould achieve a good erformance. Keyords Transmission control rotocol rom early detection algorithm internet congestion roblem symbolic dynamics Lyaunov stability. I. INTRODUCTION There is no doubt that internet netorks lay an imortant role in our daily life. Hoever as
the traffic of internet netorks gros raidly congestion roblems become very serious. Poor managements of internet netorks ill result to artly or fully inaccessible netorks hence degrade general erformances of netorking alications []-[6]. To address this issue various aroaches have been roosed. The commonest aroach to address the congestion roblems is via active ueue management (AQ) mechanisms in hich the RED algorithm is a idely deloyed algorithm for AQ mechanisms []-[6]. The goal of the RED algorithm is to detect an early sign of the congestion rovide a feedback by either droing or marking segments of messages so that the congestion can be avoided. Although the RED algorithm is concetually simle the interaction beteen the transmission control rotocol (TCP) the RED algorithm at the router s gateay is actually governed by a first order ieceise nonlinear difference euation in hich comlex behaviors such as limit cycle behaviors chaotic behaviors could be exhibited. For the commonest oeration the average ueue size of the RED algorithm is reuired to converge to a fixed oint these comlex behaviors degrade general erformances of netork alications []-[6]. As these comlex behaviors deend on both the system arameters the initial conditions of the nonlinear difference euation netork engineers reuire to design both the system arameters the initial conditions of the nonlinear difference euation so that the average ueue size of the RED algorithm ould converge to a fixed oint. Nevertheless no result has been reorted on characterizing the conditions on both the system arameters the initial conditions of the nonlinear difference euation that the average ueue size of the RED algorithm ould converge to a fixed oint. This aer is to address this issue. The outline of this aer is as follos. The orking rinciles of the TCP the RED algorithm are revieed in Section II hile nonlinear behaviors of the average ueue size of the RED algorithm are revieed in Section III. In Section IV a symbolic dynamical model is roosed as ell as the conditions on both the system arameters the initial conditions of the nonlinear difference euation that the average ueue size of the RED algorithm ould converge to a fixed
oint are derived. Finally a conclusion is dran in Section V. II. REVIEW ON WORKING PRINCIPLES OF TCP AND RED ALGORITH This section describes a brief summary of the orking rinciles of the TCP the RED algorithm. For interested readers lease refer to the details in []-[6]. A. Working rinciles of TCP The transmission rate of a TCP connection is controlled by the size of the congestion indo at the sender end denoted as cnd. The cnd size deteres the number of segments of messages to be sent to the receiver end. The cnd size is adjusted to imize the utilization of the link to avoid the congestion. To adjust the cnd size TCP congestion control algorithms emloy the folloing four hases: the slo start hase the congestion avoidance hase the fast retransmit hase the fast recovery hase. The descrition of the slo start hase is as follos. When a ne connection is first established the cnd size at the sender end is initialized to the size of one segment of messages. Uon a receit of every segment of messages a acket of an acknoledgement (ACK) is sent to the TCP sender by the TCP receiver. Uon a receit of every acket of an ACK at the sender end the TCP sender increases the cnd size by the size of one segment of messages. To segments of messages can no be sent. When both segments of messages are acknoledged the cnd size is increased to the size of four segments of messages. These rocedures are iterated in an exonential manner the TCP sender oens u the indo size exonentially that is 4 8 etc. When the cnd size exceeds a threshold denoted as ssthresh the TCP sender enters the congestion avoidance hase. During the congestion avoidance hase the cnd size is incremented by the size of one segment of messages er a round tri time regardless of the number of the ackets of an ACK has been received. Hence the TCP sender oens u the indo size linearly that is 3 4 etc until it reaches the receiver s advertised indo size denoted as rnd. 3
P iteration International Journal of Bifurcation Chaos The descrition of the fast retransmit hase is as follos. A retransmission timer is set every time hen the TCP sender sends a acket of messages. A acket loss is detected by the timeout mechanism if the timer exires before receiving the acket of an ACK. In this case the TCP sender adjusts the ssthresh sitches back to the slo start hase. In the congestion avoidance hase uon receiving an out of order segment of messages the TCP receiver generates a acket of an ACK is immediately folloed by a dulicate acket of an ACK. When three dulicate ackets of an ACK have been received by the TCP sender it is assumed that a segment of messages has been lost. The TCP sender halves the cnd size retransmits the lost segment of messages ithout aiting the exiration of a retransmission timer. The descrition of the fast recovery hase is as follos. Until the retransmitted segment of messages is received the TCP receiver ill continue to receive the out of order segments of messages generate the dulicate ackets of an ACK to the TCP sender. After the fast retransmit hase sends the missing segment of messages the TCP sender increases the cnd size henever each dulicate acket of an ACK is received. Each dulicate acket of an ACK is an indication that one acket of messages has reached the TCP receiver the number of outsting ackets of messages has decreased by one. Therefore the TCP sender is alloed to increment the cnd size. The TCP sender sitches back to the congestion avoidance hase hen the retransmitted segment of messages is received a nondulicate acket of an ACK is sent to the TCP sender. B. Working rinciles of RED algorithm The RED algorithm is a gateay based algorithm for AQ mechanisms. It estimates the congestion level by monitoring udating the average ueue size. In order to maintain a relatively small average ueue size rather than aiting for buffer overflos it dros a acket of messages ith a certain robability to rovide an early sign of the congestion hen the average ueue size exceeds a threshold. Denote the imum ueue threshold the imum ueue threshold the imum acket dro robability the average ueue size the dro robability at the k P th k as 4 resectively. The dro
P P iteration P iteration International Journal of Bifurcation Chaos robability deends on the average ueue size it is governed by the folloing euation: H ( ) < < k. Denote the exonential average eight of the RED algorithm as. The average ueue size at the k th is governed by an exonential la it deends on both the average ueue size the dro robability at the k P in hich th k as follos: ( k ) ( ) ( k ) G( ) ( ( )) ( ) G k k k N rnd here the caacity of the link beteen to routers the acket size the number of TCP connections a constant beteen N K d resectively. 8 as ell as the round tri roagation delay are denoted as C 3 It is trivial to see that the dynamical model of the average ueue size can be further reresented by a first order ieceise nonlinear difference euation as follos: ( ) ( ) k ( ) ( ) k k k. ( ) ( ) N rnd k Due to the hysical nature of both the arameters the variables of the RED algorithm it is assumed that all the arameters ( C N K d rnd ) are nonnegative real-valued. Also it is assumed that < <. < ( ) In this model the dynamics of the average ueue size of the RED algorithm at the gateay is considered. The first order ieceise nonlinear dynamical model reflects the TCP congestion 5
control mechanism takes into account the slo start hase timeout events. III. NONLINEAR BEHAVIORS OF AVERAGE QUEUE SIZE OF RED ALGORITH It is ell knon that the average ueue size of the RED algorithm could exhibit a bifurcation behavior. Figure shos the bifurcation diagram as varies hen N 3 K 6 C.54. 8 ( ) ( ) d 4 rnd 5 5.. Figure shos the freuency sectrum of the steady state dro robability the steady state hase diagram the steady state dro robability the steady state average ueue size hen. hile Figure 3 Figure 4 Figure 5 sho the corresonding numerical comuter simulation results hen. 3. 5. 75 resectively. It can be seen from Figure to Figure 5 that as increases the steady state dro robability the steady state average ueue size exhibit the limit cycle the rom like chaotic behaviors consecutively. When. 7 the steady state dro robability at some time instants is eual to one. Hence it can be seen from Figure 5 that there are to straight lines one located at another one located at hen <. 7 exhibited on the steady state hase diagram. On the other h it can be seen from Figure 3 that there is only one single straight line located at exhibited on the steady state hase diagram. The imortance of observing the above nonlinear henomena is that netork engineers could design both the system arameters the initial conditions so that the average ueue size of the RED algorithm ould not exhibit these nonlinear behaviors. 6
Figure. Bifurcation diagram as varies...5 P(ω)..5 - -.8 -.6 -.4 -. Freuency ω (π) 6.6 Average ueue size 6.4 6. 6 5.8 5.6 5.4.4.6.8...4.6 Probability.6.4 Probability...8.6.4 3 4 5 Time index k 7
6.6 Average ueue size 6.4 6. 6 5.8 5.6 Figure.. 5.4 3 4 5 Time index k. (a) Freuency sectrum of the steady state dro robability. (b) Steady state hase diagram. (c) Steady state dro robability. (d) Steady state average ueue size..8.7.6 P(ω).5.4.3.. - -.5.5 Freuency ω (π) 7 Average ueue size 6.5 6 5.5.5..5. Probability.5. Probability.5..5 3 4 5 Time index k 8
7.5 Average ueue size 7 6.5 6 5.5 5 3 4 5 Time index k Figure 3.. 3. (a) Freuency sectrum of the steady state dro robability. (b) Steady state hase diagram. (c) Steady state dro robability. (d) Steady state average ueue size. 4 3 P(ω) -.6 -.4 -...4.6 Freuency ω (π) 9 8.5 Average ueue size 8 7.5 7 6.5 6 5.5 5.5..5..5.3.35.4 Probability.4.35.3 Probability.5..5..5 3 4 5 Time index k 9
9 8.5 Average ueue size 8 7.5 7 6.5 6 5.5 5 3 4 5 Time index k Figure 4.. 5. (a) Freuency sectrum of the steady state dro robability. (b) Steady state hase diagram. (c) Steady state dro robability. (d) Steady state average ueue size. 6 5 4 P(ω) 3 - -.5.5 Freuency ω (π) 5 Average ueue size 5 5..4.6.8 Probability.8 Probability.6.4. 3 4 5 Time index k
5 Average ueue size 5 5 3 4 5 Time index k Figure 5.. 75. (a) Freuency sectrum of the steady state dro robability. (b) Steady state hase diagram. (c) Steady state dro robability. (d) Steady state average ueue size. IV. SYBOLIC DYNAICAL ODEL AND CONDITIONS FOR EXHIBITING FIXED A. Symbolic dynamical model POINT BEHAVIORS It is obvious to see that different values of corresonds to different dynamical euations. To analyze the behaviors of the average ueue size of the RED algorithm the set of is artitioned into fifteen different subsets denoted as i reresented by fifteen different symbols denoted as S for i L 5. These subsets are s i for i L 5 k in hich only one symbol is activated in each subset. That is if Si then s i ( k ) j i k. Denote ( ) [ s s ] T k L 5 s j for s k here the suerscrit T denotes the transose oerator. The model of the average ueue size of the RED algorithm can be analyzed via a symbolic dynamical model here symbolic dynamics is a system dynamics in hich some signals in the system are multileveled. The dynamics of the average ueue size of the RED algorithm in each subset is as follos: A. Dynamics of the average ueue size in the first subset N rnd If < ( k ) < N rnd then
( k ) ( ) N rnd ( k ). Denote s j. s j for It is obvious to see that if < then. Hence e have ( k ) ( ) N rnd. As ( k ) N rnd < N rnd e have ( ) N rnd <. This imlies that ( ) have ( k ) k <. Hence e. This comletes the roof. A. Dynamics of the average ueue size in the second subset If < ( k ) N rnd then ( k ) ( ) N rnd ( k ). Denote s j. s j for It is obvious to see that if < then. Hence e have ( k ) ( ) N rnd. As ( k ) N rnd ( ) N rnd. This imlies that ( k ) ( k ) e have. Hence e have. This comletes the roof. A.3 Dynamics of the average ueue size in the third subset N rnd If < > ( k ) ( k ) ( ) Nrnd N rnd then
( ) N rnd. ( k ) Denote s 3 s j for j 3. It is obvious to see that if < then. Hence e have ( k ) ( ) N rnd. As > ( k ) N rnd N rnd e have > ( ) N rnd. This imlies that ( k ) Hence e have ( k ) ( k ) ( ) N rnd >.. This comletes the roof. A.4 Dynamics of the average ueue size in the fourth subset Denote P 4 ( ) ( )( ) k : ( ) 3 ( ) ( ) ( )( ) ( ( )) 3 k ( ) ( ) < Q : P. If < 4 4 Q4 then ( k ) ( ) ( k ) ( k ). Denote 4 ( k ) s j for j 4. s 3
It is obvious to see that if < then. Since this imlies that N N K K. Hence e have. ( k ) ( k ) ( ) ( k ) As Q4 e have P4. In other ords e have ( )( ) ( ( )) 3 k ( ) ( ( )) k < This imlies that as ell as Hence e have ( )( ) ( ( )) 3 k ( ) ( ( )) k ( )( ) 4 ( ) < ( )( ) ( ) ( ) <. ( ) k ( ) ( ) <.
( ) < ( k ) < ( k ). Conseuently e have. This comletes the roof. A.5 Dynamics of the average ueue size in the fifth subset Denote ( ) ( )( ) : ( ( )) 3 P ( ) ( ( )) 5 k k k Q : P. If < 5 5 Q5 then ( k ) ( ) ( k ) ( k ). Denote 5 ( k ) s j for j 5. It is obvious to see that if < then s. Since this imlies that N N K K. Hence e have. ( k ) ( k ) ( ) ( k ) 5
As Q5 e have P5 This imlies that. In other ords e have ( )( ) ( ( )) 3 k ( ) ( ( )) k ( )( ) ( ) ( ) ( ) ( k ). Hence e have ( k ).. This comletes the roof. A.6 Dynamics of the average ueue size in the sixth subset Denote P 6 ( ) ( )( ) k : ( ) 3 ( ) ( ) ( )( ) ( ( )) 3 k ( ) ( ) < Q : P. If < 6 6 ( k ) Q6 then ( k ) ( k ) ( ) ( k ) 6
( ). ( k ) Denote s 6 s j for j 6. It is obvious to see that if < then. Since this imlies that N N K K. Hence e have As Q6 e have P6 This imlies that as ell as. ( k ) ( k ) ( ) ( k ). In other ords e have ( )( ) ( ( )) 3 k ( ) ( ( )) k ( )( ) ( ( )) 3 k ( ) ( ( )) k < ( )( ) ( )( ) ( ) ( ) <. 7
Hence e have ( ) <. ( ) ( ) < < ( ) ( k ) <. Conseuently e have ( k ) comletes the roof. ( ) ( k ) ( k ). This A.7 Dynamics of the average ueue size in the seventh subset If < > < then ( ) k ( k ) ( ) ( k ). Denote s for j 7. It is obvious to see that if < then 7 s j. Since > this imlies that > > 8
N N > <. Hence e have ( k ) ( ) ( k ). As K e have ( ) < ( k ) < ( k ) K A.8 Dynamics of the average ueue size in the eighth subset <. This comletes the roof. If < > then ( ) k ( k ) ( ) ( k ). Denote s for j 8. It is obvious to see that if < then 8 s j. Since > this imlies that > > N N > K K <. Hence e have ( k ) ( ) ( k ). As e have ( ) ( k ) ( k ) A.9 Dynamics of the average ueue size in the ninth subset. This comletes the roof. If < > > ( ) k ( k ) ( ) ( k ) s 9 s j for j 9. ( ) It is obvious to see that if < then then. Denote. Since > this imlies that > > N N > K K <. Hence e have ( k ) ( ) ( k ) 9. As
> e have > ( ) ( k ) ( k ) ( k ) ( ) >. This comletes the roof. A. Dynamics of the average ueue size in the tenth subset k If ( ) < then ( k ) ( k ) ( ) ( k ) ( k ). Denote s j. that It is obvious to see that if then ( k ). Since N N K As < K ( k ) < ( k ) s j for this imlies.. Hence e have ( k ) ( ) ( k ) e have ( ) <. This comletes the roof. A. Dynamics of the average ueue size in the eleventh subset k If ( ) then ( k ) ( k ) ( ) ( k ) ( k ). Denote s j. that As It is obvious to see that if then ( k ). Since N N K K s j for this imlies.. Hence e have ( k ) ( ) ( k ) e have ( ) ( ) k
( k ). This comletes the roof. A. Dynamics of the average ueue size in the telfth subset k > ( ) ( k ). If ( ) ( k ) ( k ) ( ) ( k ) then Denote s that s j for j. It is obvious to see that if then ( k ). Since N N K As > K > ( k ) ( k ) this imlies.. Hence e have ( k ) ( ) ( k ) e have > ( ) ( k ) ( ). This comletes the roof. A.3 Dynamics of the average ueue size in the thirteenth subset k < If ( ) < then ( k ) ( k ) ( ) ( k ). Denote s 3 s j for j 3. that < this imlies It is obvious to see that if then ( k ). Since N N > K K <. Hence e have ( k ) ( ) ( k ). As
< e have ( ) < ( k ) < ( k ) comletes the roof.. This A.4 Dynamics of the average ueue size in the fourteenth subset k < If ( ) then ( k ) ( k ) ( ) ( k ). Denote s 4 s j for j 4. that < this imlies k k. As It is obvious to see that if then ( k ). Since N N > K K <. Hence e have ( ) ( ) ( ) e have ( ) ( k ) ( k ) A.5 Dynamics of the average ueue size in the fifteenth subset k <. This comletes the roof. If ( ) > then ( k ) ( k ) ( ) ( ) ( ) k. Denote s s j for j 5. that 5 < this imlies It is obvious to see that if then ( k ). Since N N > K K <. Hence e have ( k ) ( ) ( k ). As > e have > ( ) ( k ) ( k ) ( k ) ( ) >. This comletes the roof. A could be sitched among these fifteen subsets according to the value of s ( k ) B N rnd [ ] ( ). Denote B [ ]
B [ ] B [ ] [ ] 3 4 B 5 ( ) [ B B ( ) B B ] B 3 4 B5 then the dynamics of the averages ueue size of the RED algorithm can be reresented by ( k ) A u here u B ( ) s. This model can be reresented via a closed loo feedback system having a linear time-invariant lant ith the four state sace constants A a ositive nonlinear feedback system ith its inut-outut relationshi governed by u B ( ) s. This roosed symbolic dynamical model is useful for designing both the system arameters the initial conditions so that the average ueue size of the RED algorithm ould converge to a fixed oint. Also the boundedness of the average ueue size of the RED algorithm could be detered easily via the roosed symbolic dynamical model. As < < A is strictly stable. If both the system arameters the initial conditions is designed so that u is bounded then is guaranteed to be bounded. It is orth noting that not all subsets contain a fixed oint. The conditions on both the system arameters the initial conditions that the average ueue size of the RED algorithm ould converge to a fixed oint are derived in Section B as follos. B. Conditions for exhibiting fixed oint behaviors Since the average ueue size of the RED algorithm is reuired to converge to a fixed oint it is imortant to characterize the conditions on both the system arameters the initial conditions so that the average ueue size of the RED algorithm ould converge to a fixed oint. These conditions are summarized in the folloing lemmas: Lemma The fixed oint of the average ueue size of the RED algorithm ould not be located at S S3 U S4 U S5 U S7 U S8 U S9 U S U S U S3 U S4 U S5 U. 3
As ( k ) for S U S3 U S4 U S5 U S7 U S8 U S U S U S3 U S5 the fixed oint of the average ueue size of the RED algorithm could not be located at S U S3 U S4 U S5 U S7 U S8 U S U S U S3 U S5 RED algorithm for S4 is governed by ( k ) ( ). As the dynamics of the average ueue size of the if the fixed oint is located at S then the fixed oint has to be located at the origin but it contradicts to ( ) 4 k. Hence the fixed oint of the average ueue size of the RED algorithm could not be located at S 4. Similarly the fixed oint of the average ueue size of the RED algorithm could not be located at S 9. This comleted the roof. U S U S Although the dynamics of the average ueue size for ( ) 6 k S is characterized it is not guaranteed that ( k ) S for S ( k ) S6 for S6 ( k ) S for S. Further conditions are reuired to be imosed the details are discussed in the folloing lemmas. Lemma If k such that i) N rnd < ii) N rnd iii) ( k ) then s i) k k s ii) k k 4
iii) N rnd as k. As s ( k ) this imlies that ( ) k < 5. Since N rnd e have ( ) N rnd N rnd ( ) N rnd Nrnd Nrnd. On the other h as N rnd < e have N rnd N rnd < Nrnd ( ) N rnd < N rnd Nrnd N rnd ( k ) Nrnd < N rnd Nrnd Nrnd ( k ) <. As ( k ) < Nrnd. Since ( k ) s e have is a convex combinational of Nrnd Nrnd N rnd e have ( k ) <. This imlies that ( k ) S. Similarly e have S e have k ( k k ) ( ) ( k ) k k. As N rnd N rnd k N rnd as k. This comletes the roof. This lemma characterizes the condition on both the system arameters the initial conditions that the average ueue size of the RED algorithm ould converge to a fixed oint in the first subset. It is orth noting that if the second condition in Lemma is changed to N rnd the same result ould been obtained. Hoever due to the hysical nature of k. Since N rnd as k the hysical nature of is
violated. Hence this case has not been considered in Lemma. Lemma 3 If k such that i) ii) ( k ) then s i) ( k ) k k s ii) ( k ) k k iii) as k. As s ( k ) this imlies that ( k ). Since e have ( ). As s ( k ) e have ( k ) ( k ) is a convex combinational of ( ) 6 k. Since e have ( k ). This imlies that ( k ) S. Similarly e have S k. As ( k k ) ( ) ( k ) k k ( k ) k e have as k. This comletes the roof. This lemma characterizes the condition on both the system arameters the initial conditions that the average ueue size of the RED algorithm ould converge to a fixed oint in the
eleventh subset. Lemma 4 If k such that i) ( k ) s 6 ii) ( ) ( k ) ( k ) > iii) ( ) ( k ) ( k ) ( k ) iv) ( ) ( k ) then ( k ) k k. s 6 As s ( k ) e have ( ) 6 e have ( k ) ( ) ( k ) > k. Since ( k ). As 7
( ) ( k ) ( k ) > ( ) ( k ) ( k ) e have < ( k ). This imlies that < ( k ). This further imlies that < ( k ). Conseuently e have This further imlies that the convex combinational of <. ( k ) ( k ) is larger than or eual to that is ( ) ( ) k. Similarly the convex combinational of ( k ) is smaller than that is ( ) <. ( k ) Since > ( k ) e have ( ) ( k ) < ( k ) ( ) ( k ) ( ) k. In other ords e have 8
( ) ( k ) < ( k ) ( ) ( ) ( ) k < ( k ) k ( ) ( ) ( ) ( k ) ( ) k ( ) ( ) ( ) ( k ) ( ) <. Hence e have ( ) ( )( ( )) 3 ( ) ( ( ) k k ) ( ) ( )( ( )) 3 ( ) ( ( ) k k ) < imlies that ( k ) P6 ( k ) Q6. Similarly e have S6. This further k k. This comletes the roof. Although Lemma 4 characterizes the conditions on S6 k k if ( k ) S6 it does not guarantee the existence of a fixed oint. To guarantee the existence of a fixed oint the folloing lemma is reuired: Lemma 5 Define ( ) such that. ( ) ( ). Then this As ( ) ( ) comletes the roof. Obviously ( ) is a fixed oint based on the dynamics defined in the sixth subset. 9
Hoever even though there exists a fixed oint it does not guarantee that this fixed oint ould be located at S 6 satisfy Lemma 4. Further conditions are reuired to be imosed the details are discussed in folloing lemma. Denote the set { ( ) : Lemma 4 is satisfied. } Lemma 6 If Ω k S 6. i) ii) < then Ω. Since < this imlies that < < <. This further imlies that is the fixed oint in Ω. This comletes the roof. 3
Although Ω in general it does not guarantee that ( k ) Ω ( ) ould converge k to. Further conditions are reuired to be imosed the details are discussed in folloing lemma. Define a ma Δ V : Ω R such that ΔV define Lemma 7 ( ( k )) ( ) ( k ) ( k ) ( k ) Ω ( ( k )) ΔV ΔV. ( ( k ) ) ( k ) Suose that satisfies the conditions in Lemma 6 ( k ) Ω ΔV < then ( ) ( ). Denote ( ) ( ) V k k. Obviously k k. Since V satisfies the conditions in Lemma 6 Ω. As 3
V ( k ) V ( k ) ( k ) ΔV ( ) ( k ) ( ) ( k ) ( ) ( k ) ( ) ( k ) Δ < ( ) ( ( k ) ) ( ( k ) ) ( ( k ) ) ( k ) ( k ) ( k ) ( k ) ( ) ( k ) ( ) ( k ) ( k ) ( ( k )) ( ( k ) ) ( k ) ( k ) ( ) ( ) ( k ) ( k ) V this imlies that V ( k ) V( k ) ( k ) Ω. Since ( ) Ω < of the Lyaunov stability theorem e have ( k ) ( k ) ( ) ( ) k by the rincile. This comletes the roof. Lemma 4 to Lemma 7 characterize the conditions on both the system arameters the initial conditions that the average ueue size of the RED algorithm ould converge to a fixed oint in the sixth subset. As there is no fixed oint in other subsets the state vectors in these other subsets ill not stay in these subsets. V. CONCLUSION This aer rooses a symbolic dynamical model of the average ueue size of the RED algorithm characterizes the conditions on both the system arameters the initial conditions that the average ueue size of the RED algorithm ould converge to a fixed oint. By emloying 3
the symbolic dynamical system aroach both the system arameters the initial conditions can be designed so that the average ueue size of the RED algorithm ould converge to a fixed oint behavior. ACKNOWLWDGEENTS The ork obtained in this aer as suorted by an Australian research grant a research grant from the Queen ary University of London. REFERENCES [] Priya Ranjan Eyad H. Abed Richard J. La Nonlinear instabilities in TCP-RED IEEE Transactions on Netorking vol. no. 6. 79-9 4. [] Priya Ranjan Eyad H. Abed Richard J. La Nonlinear instabilities in TCP-RED The Tenty-First Annual Joint Conference of the IEEE Comuter Communications Societies IEEE INFOCO vol.. 49-58 June 3-7. [3] L. Chen X. F. Wang Z. Z. Han Controlling bifurcation chaos in internet congestion control model International Symosium on Circuits Systems ISCAS vol. 3. III3-III35 ay 5-8 3. [4] J. H. C. Nga H. H. C. Iu B. W. K. Ling H. K. Lam Analysis control of bifurcation chaos in average ueue length in TCP/RED model International Journal of Bifurcation Chaos vol. 8 no. 8. 449-459 8. [5] Feng Liu Zhi-Hong Guan Hua O. Wang Imulsive control bifurcation chaos in internet TCP-RED congestion control system The IEEE International Conference on Control Automation ICCA. 4-7 ay 3-June 7. [6] Kirk Chang Gitae Kim Larry Wong Sunil Samtani Aristides Staikos itesh Patel Jeffrey Bocock Netork layer congestion control to ensure uality of service (QOS) in secure battlefield mobile ad hoc netorks The IEEE ilitary Communications Conference 33
ILCO. -7 October 9-3 7. 34