Download
# Stability of Rate Control System with eraged Feedback and Netw ork Delay Richard J PDF document - DocSlides

liane-varnes | 2014-12-12 | General

### Presentations text content in Stability of Rate Control System with eraged Feedback and Netw ork Delay Richard J

Show

Page 1

Stability of Rate Control System with eraged Feedback and Netw ork Delay Richard J. La and Priya Ranjan Abstract study the stability of ariant of elly rate contr ol scheme in simple setting with single ﬂo and single esour ce. The feedback signal fr om the esour ce is function of an erage rate of the ﬂo obtained using lo pass ﬁlter deri sufﬁcient condition or asymptotic stability in the pr esence of an arbitrary ﬁxed communication delay fr om the esour ce to the sender sho that sufﬁcient condition deri ed earlier or system without eraging sufﬁces. alidate our esult using simulation with family of popular utility and price functions. Index erms Rate contr ol, asymptotic stability delay ith emer ging netw ork ed control systems, control of distrib uted system in the presence of communication delays is emer ging as an interesting and important issue. Examples of such systems include sensor netw ork consisting of arrays of sensors that collect and pro vide feedback information for control system, and communication netw ork with man end users that indi vidually adjust their transmission rates based on the feedback information pro vided by acti queue management (A QM) schemes [1], [4]. In [12 we study the stability of rate control system proposed by elly et al. [10] (called primal algorithm), and deri asymptotic stability criteria in the presence of arbitrary ﬁx ed netw ork delays. These stability conditions are obtained using in ariance-based stability results for nonlinear delay dif ferential equations [7], [8 ]. In this article we study ariant of the rate control system proposed in [10 in simple setting where single ﬂo tra- erses single resource. This as ﬁrst studied in [17 ]. elly model adopts ﬂuid model, and the feedback signal generated by resource is function of users instantaneous rates. ﬂuid model is sho wn to be good approximation when the number of ﬂo ws is lar ge and resource capacities are high [2], [13 ], [14 ], [15 ]. In practice, ho we er an instantaneous rate through resource is dif ﬁcult to measure accurately; in order to obtain good estimate of the aggre ate rate, resource may need to rely on an erage rate er period. model this scenario by assuming that the feedback signal from the resource is based on an eraged rate obtained using lo pass ﬁlter [17 ]. This is similar in spirit to Random Early Detection (RED) [4] or Proportional controller [6 type schemes with one or more stages of lo pass ﬁltering. also assume that the feedback signal from the resource to the user is delayed due to netw ork delay The authors are with the Department of Electrical and Computer Engineer ing and the Institute for Systems Research, Uni ersity of Maryland, Colle ge ark, MD 20742 USA. Email: yongla,priya @isr .umd.edu. ﬁrst sho the xistence of unique solution of the system. Then, we demonstrate that an approach similar to that emplo yed in [12 can be xtended to the current problem. deri asymptotic stability criteria on utility and resource price functions with an arbitrary ﬁx ed netw ork delay Stability results based on linear analysis are deri ed in [17 ], and depend on both the netw ork delay and user ain parameter The rest of this article is or anized as follo ws: present our main results on asymptotic stability in Section II. Simu- lation results are presented in III. A. eedbac signal based on low pass ﬁlter ed load In elly optimization frame ork for rate control [9] user recei es utility when it gets rate of This utility function could represent either the user true utility or some function assigned to the user through the selected end user algorithm. tak the latter vie and assume that the utility functions of the users are used to select the desired rate allocation among the users. The utility function is an increasing, strictly conca and continuously dif ferentiable function of er the range 0. denote by the user rate at time Since user rate is bounded in practice, we assume that the user rate belongs to compact set min max := (0 Here the lo wer (resp. upper) bound min (resp. max can be arbitrarily close to zero (resp. arbitrarily lar ge). elly primal algorithm [10 based on ﬂuid model, assumes that the price char ged by resource is function of users instantaneous rate. Although ﬂuid model is sho wn to be good approximation for lar ge scale netw ork [2], [13 ], [14 ], [15], it does not capture the pack et le el dynamics and stochastic nature of real netw ork. In particular the notion of instantaneous rate used in ﬂuid model is an idealized concept that cannot be measured in real netw ork. The rate of ﬂo er round trip can be computed by di viding the number of successfully deli ered bytes by the round-trip time. Ho we er there is no such pre-determined timescale at which resource can measure its rate; the rate seen by resource er period depends on man actors, e.g., round-trip times of the ﬂo ws, details of pack et transmis- sions by end users, and queueing delays at the bottlenecks. Instead, resource can di vide time into short, contiguous timeslots and measure the noisy aggre ate rate of the ﬂo ws during each timeslot. Then, an erage rate can be computed This results in self-clocking beha vior of an end user congestion control mechanism, e.g., TCP

Page 2

using, for xample, an xponentially weighted mo ving erage to obtain better (or smoother) estimate. Let denote the erage rate at the resource at time The price char ged by the resource at time is gi en by := )) assume that the price function is non-ne ati e, increasing and continuously dif ferentiable. assume that the forw ard delay from the sender to the resource is ne gligible and that the feedback signal generated by the resource is returned to the user after ﬁx ed round trip time As mentioned in Section we model the eraging at the resource by introducing lo pass ﬁlter In the presence of lo pass ﬁltering and feedback delay the interaction between the user and the resource is gi en by the follo wing delay dif ferential equation [10], [12 ]. dt )) )) ))) (1) dt where and := )) denotes the user willingness to pay at time It is clear from (1) that the user al ays attempts to reach an equilibrium where its willingness to pay equals its total price. In this article we are interested in ﬁnding conditions on user utility and resource price functions for asymptotic stability of the system in (1). First, for notational simplicity let us deﬁne and to be user willingness to pay and total price, respecti ely as function of its rate as follo ws: := and := (2) mak the follo wing assumptions on the functions and Assumption 1: (i) The function is strictly decreasing (i.e., for all (ii) the function is strictly increasing (i.e., for all and (iii) both and are Lipschitz continuous on min max ], and the in erse function is Lipschitz continuous on max min ]. Note that the in erse function xists from Assump- tion 1(i). As stated in [12], Assumption 1(i) implies that the user demand is inelastic [16 ]. An xample of amily of utility and resource price functions that satisﬁes Assumption is used in Section III for simulation. re write (1) in form more amenable to analysis: Deﬁne := )) (3) This allo ws us the follo wing change of coordinate. )) ))) (4) Deﬁne )) := ))) Clearly )) under Assumption 1. re write (1) in terms of dt )) )) )) dt )) (5) study the system in (5) and sho that there is close correspondence between the in ariance and asymptotic stability properties of discrete-time map +1 )) =: (6) and those of (5) for ﬁx ed orbits as done in [12] with no eraging. Assumption 2: Suppose that := a; ([ min max ]) is compact interv al in ariant under the map i.e., Let := ([ 0] be the Banach space of con- tinuous functions mapping the interv al 0] to or all we deﬁne := for all 0] B. Existence of unique solution asymptotic stability The functions in (5) are Lipschitz continuous by Assump- tion 1. Therefore, unique solution of (1) xists for all with an initial function i.e., and for all 0] Moreo er our ﬁrst result (Theorem 1) sho ws that the solution stays positi and within the interv al This is consequence of the in ariance property of the interv al under the map (Assumption 2). Before stating this in ariance result, we ﬁrst introduce fe lemmas used in the proof. Lemma 1: Suppose that := is compact interv al. Assume [0 is continuous function and [0 is continuous, strictly positi e, and bounded function. If is solution of follo wing equation dt (7) with an initial condition (0) then for all Pr oof: pro this lemma by contradiction. Suppose that the lemma is not true. Then, let be the ﬁrst time at which the solution lea es First, suppose and ery interv al contains point such that and Ho we er if from (7) we must ha because This is contradiction. The case is handled similarly This completes the proof. Lemma is essentially based on [5]. Lemma 2: Suppose that := ([ min max ]) is closed interv al in ariant under the map If and for all then for all Pr oof: ﬁrst re write the ﬁrst equation in (5): )) dt )) )) (8) Then, from Lemma 1, in order to pro Lemma it suf ﬁces to sho that the right-hand side of (8) lies in for all This follo ws directly from the assumed in ariance property of under the map and monotonicity properties of the functions and First, note that the map )) Recall that the price

Page 3

function is non-decreasing and the function is monotonically decreasing. Thus, for all )) )) )) )) because and The in ariance property of no tells us )) ))] and the claim follo ws. Theor em 1: Suppose that Assumptions and hold and that the initial function Then, the corresponding solution ); satisﬁes and for all Pr oof: By Lemma 2, if and then for all [0 Applying this and Lemma to the second equation in (5) tells us that belongs to the in erse image for all [0 No the claim follo ws from an induction ar gument on time (called the method of steps [3]). no consider the case where the map has an attracting ﬁx ed point with domain of attraction In other ords, for an Assumption 3: There is sequence of compact interv als satisfying (i) int +1 +1 int for all and (ii) =0 Here int denotes the interior of It is clear that the ﬁx ed point is the user willingness to pay at the equilibrium. In other ords, if is the equilibrium rate that satisﬁes then Lemma 3: Fix Let a; be an open interv al that satisﬁes Suppose that and for all 0] Then, there xists ﬁnite time such that and for all Pr oof: ﬁrst sho that there xists ﬁnite time such that for all Let be an open interv al containing and whose closure is contained in i.e., pro that there xists ﬁnite time such that Suppose this is not true, i.e., 62 for all First, assume that sup for all This implies that there xists such that because )) )) sup sup This means 1 and contradicts the assumption that sup for all Hence, there xists ﬁnite such that The case where inf for all can be handled similarly No follo wing the same ar gument in the proof of Lemma one can sho that for all Since clearly for all Thus, there xists ﬁnite such that for all no sho that there xists ﬁnite such that for all Let be an open interv al such that and := From abo we kno that for all ollo wing the same ar gument abo we can sho that there xists ﬁnite such that Then, by Lemma 1, it follo ws that for all Since this implies for all Thus, there xists ﬁnite such that for all No letting the lemma follo ws. Theor em 2: Suppose that Assumptions hold. If the initial function then lim !1 ); )) Pr oof: By repeatedly applying Theorem and Lemma we can construct an increasing sequence such that, for all Then, the theorem follo ws directly from the assumption =1 quick look at Theorem abo and Theorem in [12] re eals that the same stability criterion is suf ﬁcient with or without lo pass ﬁlter in the control loop. In other ords, the stability of the map is suf ﬁcient in both cases, pro vided that the initial function lies in an appropriate space determined by Also, note that the stability condition does not depend on the eraging parameter in (5). ha sho wn in [12] that without eraging, when the map is unstable, the delay dif ferential system loses its stability for suf ﬁciently lar ge delays. In Section III we sho using simulation that en with ﬁltering, when the map is not stable, the system in (5) is unstable when the delay is suf ﬁciently lar ge. Also, it is clear that the system is globally stable if can be made arbitrarily lar ge in An xample of such case is pro vided in Section III. C. Multiple sta ﬁltering In this subsection we xtend our results in the pre vious subsection to the case where more than one lo pass ﬁlter are added to the system. sho that similar stability condition suf ﬁces en with multiple stages of ﬁltering. Suppose that there are stages of lo pass ﬁltering. This scenario is modeled by the follo wing set of dif ferential equations: dt )) ))) dt (9) dt where is the output of the -th stage lo pass ﬁlter and are positi constants. Cor ollary 1: Suppose that Assumptions hold. As- sume the initial function satisﬁes and for all Then, lim !1 ); )) Corollary states that the stability of the discrete time map is suf ﬁcient for the stability of (9) as long as the initial function belongs to an appropriate space determined by the domain of attraction of the map In this section we present simulation results using amily of popular utility and price functions. simulate ﬂuid

Page 4

model as opposed to using pack et-le el ent dri en sim- ulator This can be justiﬁed as follo ws: Consider the model described in Section II-A in which resource measures its rate during each (short) timeslot. In this model the number of bytes that arri during timeslot will be random. This system can be modeled as discrete time stochastic system where unit time corresponds to the duration of timeslot (e.g., [14 ]). As mentioned in Section I, when the number of ﬂo ws is lar ge, the gr gate beha vior of the ﬂo ws (normalized by the number of ﬂo ws) in this system can be well approximated using deterministic model that resembles ﬂuid model [13 ], [14 ], [15 ]. Hence, the single ﬂo studied in this article can be seen to represent the erage beha vior of lar ge number of ﬂo ws. This pro vides justiﬁcation for using ﬂuid model for simulation. The ﬂo has utility function with 2.0, and the resource price function is gi en by x=C with 0.7. The link capacity 5. It is easy to sho that the price elasticity of demand of the ﬂo decreases with increasing [12], [16 ]. This amily of utility functions is also used to achie wide range of dif ferent airness in rate control [11 ]. sho in [12 that with the assumed utility and price functions, the map is stable (i.e., has an attracting ﬁx ed point) if and only if with as the re gion of attraction. Since the stability condition holds in this xample. Therefore, Theorem tells us that the user rate will con er ge to the equilibrium rate, starting with an positi e, continuous initial function. The equilibrium user rate is b= (1+ 1.3559. The re erse delay from the resource to the sender is set to 100 in the simulation. The ain parameter 1. The lo pass ﬁlter parameter is set to 0.05. The initial function is set to )] [3 3] for all [-100, 0]. Fig. 1(a) sho ws the olution of the instantaneous rate and the erage rate Clearly the system xhibits stable beha vior; both the instantaneous and erage rates approach the equilibrium alue 1.3559 as goes to In Fig. 1(b) we sho the rate with the utility function parameter which violates our stability condition ). In this case, the system is unstable and the user rate sho ws oscillatory beha vior [1] S. Athuraliya, H. Li, S. H. Lo and Q. in. REM: acti queue management. IEEE Network 15(3):48–53, May/June 2001. [2] Baccelli, D. R. MacDonald, and J. Re ynier mean-ﬁeld model for multiple TCP connections through uf fer implementing RED. erformance Evaluation 49:77–97, Sep. 2002. [3] R. D. Dri er Or dinary and Delay Dif fer ential Equations Springer erlag, Berlin, 1977. [4] S. Flo yd and Jacobson. Random early detection ate ays for congestion oidance. IEEE ans. on Networking 1(7):397–413, 1993. [5] Jack K. Hale and A. Iv ano On high order dif ferential delay equation. ournal of Mathematical Analysis and Applications 173:505 514, 1993. [6] C. Hollot, Misra, D. wsle and Gong. On designing impro ed controllers for aqm routers supporting TCP ﬂo ws. In Pr oc. of IEEE Infocom Anchorage AK, 2001. [7] A. Iv ano E. Liz, and S. I. roﬁmchuk. Global stability of class of scalar nonlinear delay dif ferential equations. Accepted for Publication, Pr eprint fr om uthor 2003. 500 1000 1500 2000 2500 3000 0.5 1.5 2.5 3.5 4.5 Plot of x(t) and (t) (a = 2.0, b = 0.7) Rate Time (t) x(t) (t) (a) 500 1000 1500 2000 2500 3000 Time (t) Rate Plot of x(t) and (t) (a = 1.4, b = 0.7) x(t) (t) (b) Fig. 1. Plot of rate and ﬁltered load (a) 2.0 and 0.7, (b) 1.4 and 0.7 [8] A. Iv ano M. A. Pinto, and S. I. roﬁmchuk. Global beha vior in nonlinear systems with delayed feedback. In Pr oc. of IEEE CDC Sydne Australia, 2000. [9] elly Char ging and rate control for elastic traf ﬁc. Eur opean ansactions on elecommunications 8(1):33–7, Jan. 1997. [10] elly A. Maulloo, and D. an. Rate control for communication netw orks: shado prices, proportional airness and stability ournal of the Oper ational Resear Society 49(3):237–252, Mar 1998. [11] J. Mo and J. alrand. air end-to-end windo w-based congestion control. IEEE/A CM ans. on Networking 8(5):556–567, Oct. 2000. [12] Priya Ranjan, Richard J. La, and Eyad H. Abed. Global stability condi- tions for rate control with arbitrary communication delays. IEEE/A CM ans. on Networking 14(1):94–107, Feb 2006. [13] innak ornsrisuphap and R. J. La. Characterization of queue ﬂuc- tuations in probabilistic QM mechanisms. erformance Evaluation Re vie 32(1):283–294, Special Issue, Jun. 2004. [14] innak ornsrisuphap and R. J. La. Asymptotic beha vior of heteroge- neous TCP ﬂo ws and RED ate ay IEEE/A CM ans. on Networking 14(1):108–120, Feb 2006. [15] innak ornsrisuphap and A. M. Mak wski. Limit beha vior of ECN/RED ate ays under lar ge number of TCP ﬂo ws. In Pr oc. of IEEE Infocom San Francisco CA, Mar 2003. [16] Hal R. arian. Intermediate Micr oeconomics Norton, Ne ork, 1996. [17] Glenn innicombe. On the stability of netw orks operating TCP-lik congestion control. Pr oc. of IF 2002.

La and Priya Ranjan Abstract study the stability of ariant of elly rate contr ol scheme in simple setting with single 64258o and single esour ce The feedback signal fr om the esour ce is function of an erage rate of the 64258o obtained using lo pass ID: 22410

- Views :
**134**

**Direct Link:**- Link:https://www.docslides.com/liane-varnes/stability-of-rate-control-system
**Embed code:**

Download this pdf

DownloadNote - The PPT/PDF document "Stability of Rate Control System with er..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

Stability of Rate Control System with eraged Feedback and Netw ork Delay Richard J. La and Priya Ranjan Abstract study the stability of ariant of elly rate contr ol scheme in simple setting with single ﬂo and single esour ce. The feedback signal fr om the esour ce is function of an erage rate of the ﬂo obtained using lo pass ﬁlter deri sufﬁcient condition or asymptotic stability in the pr esence of an arbitrary ﬁxed communication delay fr om the esour ce to the sender sho that sufﬁcient condition deri ed earlier or system without eraging sufﬁces. alidate our esult using simulation with family of popular utility and price functions. Index erms Rate contr ol, asymptotic stability delay ith emer ging netw ork ed control systems, control of distrib uted system in the presence of communication delays is emer ging as an interesting and important issue. Examples of such systems include sensor netw ork consisting of arrays of sensors that collect and pro vide feedback information for control system, and communication netw ork with man end users that indi vidually adjust their transmission rates based on the feedback information pro vided by acti queue management (A QM) schemes [1], [4]. In [12 we study the stability of rate control system proposed by elly et al. [10] (called primal algorithm), and deri asymptotic stability criteria in the presence of arbitrary ﬁx ed netw ork delays. These stability conditions are obtained using in ariance-based stability results for nonlinear delay dif ferential equations [7], [8 ]. In this article we study ariant of the rate control system proposed in [10 in simple setting where single ﬂo tra- erses single resource. This as ﬁrst studied in [17 ]. elly model adopts ﬂuid model, and the feedback signal generated by resource is function of users instantaneous rates. ﬂuid model is sho wn to be good approximation when the number of ﬂo ws is lar ge and resource capacities are high [2], [13 ], [14 ], [15 ]. In practice, ho we er an instantaneous rate through resource is dif ﬁcult to measure accurately; in order to obtain good estimate of the aggre ate rate, resource may need to rely on an erage rate er period. model this scenario by assuming that the feedback signal from the resource is based on an eraged rate obtained using lo pass ﬁlter [17 ]. This is similar in spirit to Random Early Detection (RED) [4] or Proportional controller [6 type schemes with one or more stages of lo pass ﬁltering. also assume that the feedback signal from the resource to the user is delayed due to netw ork delay The authors are with the Department of Electrical and Computer Engineer ing and the Institute for Systems Research, Uni ersity of Maryland, Colle ge ark, MD 20742 USA. Email: yongla,priya @isr .umd.edu. ﬁrst sho the xistence of unique solution of the system. Then, we demonstrate that an approach similar to that emplo yed in [12 can be xtended to the current problem. deri asymptotic stability criteria on utility and resource price functions with an arbitrary ﬁx ed netw ork delay Stability results based on linear analysis are deri ed in [17 ], and depend on both the netw ork delay and user ain parameter The rest of this article is or anized as follo ws: present our main results on asymptotic stability in Section II. Simu- lation results are presented in III. A. eedbac signal based on low pass ﬁlter ed load In elly optimization frame ork for rate control [9] user recei es utility when it gets rate of This utility function could represent either the user true utility or some function assigned to the user through the selected end user algorithm. tak the latter vie and assume that the utility functions of the users are used to select the desired rate allocation among the users. The utility function is an increasing, strictly conca and continuously dif ferentiable function of er the range 0. denote by the user rate at time Since user rate is bounded in practice, we assume that the user rate belongs to compact set min max := (0 Here the lo wer (resp. upper) bound min (resp. max can be arbitrarily close to zero (resp. arbitrarily lar ge). elly primal algorithm [10 based on ﬂuid model, assumes that the price char ged by resource is function of users instantaneous rate. Although ﬂuid model is sho wn to be good approximation for lar ge scale netw ork [2], [13 ], [14 ], [15], it does not capture the pack et le el dynamics and stochastic nature of real netw ork. In particular the notion of instantaneous rate used in ﬂuid model is an idealized concept that cannot be measured in real netw ork. The rate of ﬂo er round trip can be computed by di viding the number of successfully deli ered bytes by the round-trip time. Ho we er there is no such pre-determined timescale at which resource can measure its rate; the rate seen by resource er period depends on man actors, e.g., round-trip times of the ﬂo ws, details of pack et transmis- sions by end users, and queueing delays at the bottlenecks. Instead, resource can di vide time into short, contiguous timeslots and measure the noisy aggre ate rate of the ﬂo ws during each timeslot. Then, an erage rate can be computed This results in self-clocking beha vior of an end user congestion control mechanism, e.g., TCP

Page 2

using, for xample, an xponentially weighted mo ving erage to obtain better (or smoother) estimate. Let denote the erage rate at the resource at time The price char ged by the resource at time is gi en by := )) assume that the price function is non-ne ati e, increasing and continuously dif ferentiable. assume that the forw ard delay from the sender to the resource is ne gligible and that the feedback signal generated by the resource is returned to the user after ﬁx ed round trip time As mentioned in Section we model the eraging at the resource by introducing lo pass ﬁlter In the presence of lo pass ﬁltering and feedback delay the interaction between the user and the resource is gi en by the follo wing delay dif ferential equation [10], [12 ]. dt )) )) ))) (1) dt where and := )) denotes the user willingness to pay at time It is clear from (1) that the user al ays attempts to reach an equilibrium where its willingness to pay equals its total price. In this article we are interested in ﬁnding conditions on user utility and resource price functions for asymptotic stability of the system in (1). First, for notational simplicity let us deﬁne and to be user willingness to pay and total price, respecti ely as function of its rate as follo ws: := and := (2) mak the follo wing assumptions on the functions and Assumption 1: (i) The function is strictly decreasing (i.e., for all (ii) the function is strictly increasing (i.e., for all and (iii) both and are Lipschitz continuous on min max ], and the in erse function is Lipschitz continuous on max min ]. Note that the in erse function xists from Assump- tion 1(i). As stated in [12], Assumption 1(i) implies that the user demand is inelastic [16 ]. An xample of amily of utility and resource price functions that satisﬁes Assumption is used in Section III for simulation. re write (1) in form more amenable to analysis: Deﬁne := )) (3) This allo ws us the follo wing change of coordinate. )) ))) (4) Deﬁne )) := ))) Clearly )) under Assumption 1. re write (1) in terms of dt )) )) )) dt )) (5) study the system in (5) and sho that there is close correspondence between the in ariance and asymptotic stability properties of discrete-time map +1 )) =: (6) and those of (5) for ﬁx ed orbits as done in [12] with no eraging. Assumption 2: Suppose that := a; ([ min max ]) is compact interv al in ariant under the map i.e., Let := ([ 0] be the Banach space of con- tinuous functions mapping the interv al 0] to or all we deﬁne := for all 0] B. Existence of unique solution asymptotic stability The functions in (5) are Lipschitz continuous by Assump- tion 1. Therefore, unique solution of (1) xists for all with an initial function i.e., and for all 0] Moreo er our ﬁrst result (Theorem 1) sho ws that the solution stays positi and within the interv al This is consequence of the in ariance property of the interv al under the map (Assumption 2). Before stating this in ariance result, we ﬁrst introduce fe lemmas used in the proof. Lemma 1: Suppose that := is compact interv al. Assume [0 is continuous function and [0 is continuous, strictly positi e, and bounded function. If is solution of follo wing equation dt (7) with an initial condition (0) then for all Pr oof: pro this lemma by contradiction. Suppose that the lemma is not true. Then, let be the ﬁrst time at which the solution lea es First, suppose and ery interv al contains point such that and Ho we er if from (7) we must ha because This is contradiction. The case is handled similarly This completes the proof. Lemma is essentially based on [5]. Lemma 2: Suppose that := ([ min max ]) is closed interv al in ariant under the map If and for all then for all Pr oof: ﬁrst re write the ﬁrst equation in (5): )) dt )) )) (8) Then, from Lemma 1, in order to pro Lemma it suf ﬁces to sho that the right-hand side of (8) lies in for all This follo ws directly from the assumed in ariance property of under the map and monotonicity properties of the functions and First, note that the map )) Recall that the price

Page 3

function is non-decreasing and the function is monotonically decreasing. Thus, for all )) )) )) )) because and The in ariance property of no tells us )) ))] and the claim follo ws. Theor em 1: Suppose that Assumptions and hold and that the initial function Then, the corresponding solution ); satisﬁes and for all Pr oof: By Lemma 2, if and then for all [0 Applying this and Lemma to the second equation in (5) tells us that belongs to the in erse image for all [0 No the claim follo ws from an induction ar gument on time (called the method of steps [3]). no consider the case where the map has an attracting ﬁx ed point with domain of attraction In other ords, for an Assumption 3: There is sequence of compact interv als satisfying (i) int +1 +1 int for all and (ii) =0 Here int denotes the interior of It is clear that the ﬁx ed point is the user willingness to pay at the equilibrium. In other ords, if is the equilibrium rate that satisﬁes then Lemma 3: Fix Let a; be an open interv al that satisﬁes Suppose that and for all 0] Then, there xists ﬁnite time such that and for all Pr oof: ﬁrst sho that there xists ﬁnite time such that for all Let be an open interv al containing and whose closure is contained in i.e., pro that there xists ﬁnite time such that Suppose this is not true, i.e., 62 for all First, assume that sup for all This implies that there xists such that because )) )) sup sup This means 1 and contradicts the assumption that sup for all Hence, there xists ﬁnite such that The case where inf for all can be handled similarly No follo wing the same ar gument in the proof of Lemma one can sho that for all Since clearly for all Thus, there xists ﬁnite such that for all no sho that there xists ﬁnite such that for all Let be an open interv al such that and := From abo we kno that for all ollo wing the same ar gument abo we can sho that there xists ﬁnite such that Then, by Lemma 1, it follo ws that for all Since this implies for all Thus, there xists ﬁnite such that for all No letting the lemma follo ws. Theor em 2: Suppose that Assumptions hold. If the initial function then lim !1 ); )) Pr oof: By repeatedly applying Theorem and Lemma we can construct an increasing sequence such that, for all Then, the theorem follo ws directly from the assumption =1 quick look at Theorem abo and Theorem in [12] re eals that the same stability criterion is suf ﬁcient with or without lo pass ﬁlter in the control loop. In other ords, the stability of the map is suf ﬁcient in both cases, pro vided that the initial function lies in an appropriate space determined by Also, note that the stability condition does not depend on the eraging parameter in (5). ha sho wn in [12] that without eraging, when the map is unstable, the delay dif ferential system loses its stability for suf ﬁciently lar ge delays. In Section III we sho using simulation that en with ﬁltering, when the map is not stable, the system in (5) is unstable when the delay is suf ﬁciently lar ge. Also, it is clear that the system is globally stable if can be made arbitrarily lar ge in An xample of such case is pro vided in Section III. C. Multiple sta ﬁltering In this subsection we xtend our results in the pre vious subsection to the case where more than one lo pass ﬁlter are added to the system. sho that similar stability condition suf ﬁces en with multiple stages of ﬁltering. Suppose that there are stages of lo pass ﬁltering. This scenario is modeled by the follo wing set of dif ferential equations: dt )) ))) dt (9) dt where is the output of the -th stage lo pass ﬁlter and are positi constants. Cor ollary 1: Suppose that Assumptions hold. As- sume the initial function satisﬁes and for all Then, lim !1 ); )) Corollary states that the stability of the discrete time map is suf ﬁcient for the stability of (9) as long as the initial function belongs to an appropriate space determined by the domain of attraction of the map In this section we present simulation results using amily of popular utility and price functions. simulate ﬂuid

Page 4

model as opposed to using pack et-le el ent dri en sim- ulator This can be justiﬁed as follo ws: Consider the model described in Section II-A in which resource measures its rate during each (short) timeslot. In this model the number of bytes that arri during timeslot will be random. This system can be modeled as discrete time stochastic system where unit time corresponds to the duration of timeslot (e.g., [14 ]). As mentioned in Section I, when the number of ﬂo ws is lar ge, the gr gate beha vior of the ﬂo ws (normalized by the number of ﬂo ws) in this system can be well approximated using deterministic model that resembles ﬂuid model [13 ], [14 ], [15 ]. Hence, the single ﬂo studied in this article can be seen to represent the erage beha vior of lar ge number of ﬂo ws. This pro vides justiﬁcation for using ﬂuid model for simulation. The ﬂo has utility function with 2.0, and the resource price function is gi en by x=C with 0.7. The link capacity 5. It is easy to sho that the price elasticity of demand of the ﬂo decreases with increasing [12], [16 ]. This amily of utility functions is also used to achie wide range of dif ferent airness in rate control [11 ]. sho in [12 that with the assumed utility and price functions, the map is stable (i.e., has an attracting ﬁx ed point) if and only if with as the re gion of attraction. Since the stability condition holds in this xample. Therefore, Theorem tells us that the user rate will con er ge to the equilibrium rate, starting with an positi e, continuous initial function. The equilibrium user rate is b= (1+ 1.3559. The re erse delay from the resource to the sender is set to 100 in the simulation. The ain parameter 1. The lo pass ﬁlter parameter is set to 0.05. The initial function is set to )] [3 3] for all [-100, 0]. Fig. 1(a) sho ws the olution of the instantaneous rate and the erage rate Clearly the system xhibits stable beha vior; both the instantaneous and erage rates approach the equilibrium alue 1.3559 as goes to In Fig. 1(b) we sho the rate with the utility function parameter which violates our stability condition ). In this case, the system is unstable and the user rate sho ws oscillatory beha vior [1] S. Athuraliya, H. Li, S. H. Lo and Q. in. REM: acti queue management. IEEE Network 15(3):48–53, May/June 2001. [2] Baccelli, D. R. MacDonald, and J. Re ynier mean-ﬁeld model for multiple TCP connections through uf fer implementing RED. erformance Evaluation 49:77–97, Sep. 2002. [3] R. D. Dri er Or dinary and Delay Dif fer ential Equations Springer erlag, Berlin, 1977. [4] S. Flo yd and Jacobson. Random early detection ate ays for congestion oidance. IEEE ans. on Networking 1(7):397–413, 1993. [5] Jack K. Hale and A. Iv ano On high order dif ferential delay equation. ournal of Mathematical Analysis and Applications 173:505 514, 1993. [6] C. Hollot, Misra, D. wsle and Gong. On designing impro ed controllers for aqm routers supporting TCP ﬂo ws. In Pr oc. of IEEE Infocom Anchorage AK, 2001. [7] A. Iv ano E. Liz, and S. I. roﬁmchuk. Global stability of class of scalar nonlinear delay dif ferential equations. Accepted for Publication, Pr eprint fr om uthor 2003. 500 1000 1500 2000 2500 3000 0.5 1.5 2.5 3.5 4.5 Plot of x(t) and (t) (a = 2.0, b = 0.7) Rate Time (t) x(t) (t) (a) 500 1000 1500 2000 2500 3000 Time (t) Rate Plot of x(t) and (t) (a = 1.4, b = 0.7) x(t) (t) (b) Fig. 1. Plot of rate and ﬁltered load (a) 2.0 and 0.7, (b) 1.4 and 0.7 [8] A. Iv ano M. A. Pinto, and S. I. roﬁmchuk. Global beha vior in nonlinear systems with delayed feedback. In Pr oc. of IEEE CDC Sydne Australia, 2000. [9] elly Char ging and rate control for elastic traf ﬁc. Eur opean ansactions on elecommunications 8(1):33–7, Jan. 1997. [10] elly A. Maulloo, and D. an. Rate control for communication netw orks: shado prices, proportional airness and stability ournal of the Oper ational Resear Society 49(3):237–252, Mar 1998. [11] J. Mo and J. alrand. air end-to-end windo w-based congestion control. IEEE/A CM ans. on Networking 8(5):556–567, Oct. 2000. [12] Priya Ranjan, Richard J. La, and Eyad H. Abed. Global stability condi- tions for rate control with arbitrary communication delays. IEEE/A CM ans. on Networking 14(1):94–107, Feb 2006. [13] innak ornsrisuphap and R. J. La. Characterization of queue ﬂuc- tuations in probabilistic QM mechanisms. erformance Evaluation Re vie 32(1):283–294, Special Issue, Jun. 2004. [14] innak ornsrisuphap and R. J. La. Asymptotic beha vior of heteroge- neous TCP ﬂo ws and RED ate ay IEEE/A CM ans. on Networking 14(1):108–120, Feb 2006. [15] innak ornsrisuphap and A. M. Mak wski. Limit beha vior of ECN/RED ate ays under lar ge number of TCP ﬂo ws. In Pr oc. of IEEE Infocom San Francisco CA, Mar 2003. [16] Hal R. arian. Intermediate Micr oeconomics Norton, Ne ork, 1996. [17] Glenn innicombe. On the stability of netw orks operating TCP-lik congestion control. Pr oc. of IF 2002.

Today's Top Docs

Related Slides