Switching resilient PI controllers for active queue management of TCP flows

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Proceedings

of the 2007 IEEE International

Conference

on

TuesM02

Networking,

Sensing and Control, London, UK, 15-17 April 2007

Switching

Resilient PI

Controllers for

Active Queue

Management of

TCP

Flows

Deniz

Ustebay

Hitay Ozbay

Department of Electrical and Department of Electrical and

ElectronicsEngineering Electronics Engineering

BilkentUniversity BilkentUniversity

Ankara 06800,Turkey Ankara 06800, Turkey

Email: deniz@(d'ee.bilkent.edu.tr Email: hitay@bilkent.edu.tr

Abstract-Active Queue Management (AQM) is used in comr- complicated to implement in real networks. Therefore we

puter networks to increase link utilization with less queueing consider simpler PI controllers and switch among them. delays. The fluid flowmodel of TCP based on delay differential In this

paper

weapply the

PI

controller proposed in

1118]

to a

equations supplies the mathematical background for modelling w.

the AQM as a feedback system. Recently various PI and PID network with time varying round trip times(RTT). We propose controllers are designed for this feedback system, [7], [18]. In that when RTT (i.e. time delay) varies with time, dividingthe thispaper, we consider the case for which the Round TripTime operatingrangeto smallerregionsanddesigningPI controllers (RTT) is time varying and we propose switching resilient PI for each of theseregions give better results than using a single

controllers usingthe design method introduced in [18].

PI

controller for the entire operating range. This proposition

I.

INTRODUCTION is supported by ns-2 [12] simulation results.

Remaining of the

paper

is organized as follows. The AQM

One of the most persistent problems confronted in the model and the details of PI controller design for AQM are Internet is the congestion. When congestion is present in the given in Section II. The results and analysis of the ns-computernetwork, buffersattherouters arefilled withpackets. simulations can be found in Section III. Concluding remarks Any packet that reaches to these routers are lost and to be aremade in Section IV.

resent. If lots of packets are lost and tried to be resent, a

considerableamountofdelaywillbe observedbythe endusers II. PlCONTROL FOR AQM

of the

Internet,

[4]. AQM is a congestion control mechanism A. Mathematical Modelof AQM Supporting TCP Flows

that is used for preventing buffer overflows and such large The dynamical model of TCP was developed using fluid delays. On the other

hand,

buffers that have an amount of flow approximation in [11] and used by

[2], [7], [10], [14],

packets

less than a desired level are also unwanted because

[15],

[20]. In this paper we use a

simplified

version of this

they signal under utilization oflink capacities. In

fact,

AQM model introduced in [7]. The model consists of the following triesto maintain acertain desired level of queue length atthe nonlinear

differential,

time-delayed equations:

routerbuffersbyavoiding both buffer overflow andemptiness. I

W(t)

W R

For this

objective

AQM schemes mark the

packets passing

147(t 1 WC) RI) throughthelink according to acertain probabilistic rule. This R(t) 2 R(t - R(t))

packet marking probability can be a static function of queue q(t) N(t) W(t)

length as in RED [3] and REM [1] or a dynamic function R(t)

of queue length as in [7], [18]. In [11]7 a fluid flow model where W is the TCP window size, q is the queue length, N of AQM was developed. In

[6],

[7], the fluid flow model is is the number of TCP

flows,

c is the link

capacity,

and p is linearized and a design method for PI control of AQM is the probability ofpacket mark.

Here,

RTT (total delay in the proposed. In

[18],

new PI and PID controllers for AQM are feedback path) is expressed by

developed using techniques introduced in

[5],[13]

and it is (t)

shown that the proposed PI controller performs better than

R(t)

=

To(t)

+

q(t)

(3)

the PI controller

designed

in [7] in terms of

tracking

and

robustness. where

7O(t)

is the propagation delay. Note that we consider

Different methods ofAQM have beenproposed in

2],

[9],

time

varying

propagation

delay.

The variation of lo t) will be

[14],

[16],

[19],

[20].

With the

exception

of

[20]

where

Ho,

taken to be slow

compared

to the variations of

q(t

_c(t) but the based AQM techniques are

used,

the papers mentioned above magnitude of the variations of thepropagation delay islarger do not consider time

varying propagation

delays,

which may than the variations of

queueing delay.

occur due to changes in the communication channels. The Equation(1)specifies the TCPwindowdynamic int-corporat-proposed switched HOO

conltrol

method of [20] is relatively ing the additive irncrease arnd multiplicative decrease (AIMD)

(2)

behavior ofTCP. Likewise, equation (2) describes the queue C.

Switching

Control

length dynamic.Itispossibleto usetheseequationstodescribe The

PI

controller design given above assumes that RTT TCP as a feedback control system, where p is the control is time invariant.

However,

in

computer networks

RTT is

inputgenerated by feedback from q. This nonlinear feedback

probably

time varying

oruncertain.

system can then be linearized around an equilibrium point Forthe case shown in Fig.

1,

the nominal value of RTT is

using small signal analysis. Let qo,W0,co,N0o,

tRo

be the 0 and we assume that RTT takes values between

ft-Af

nonrlLinal

~ ~ ~ ~ ~ ~ ~ ~

~

Rvalue we assum

thatbrl

RTTn

take value bewe

Rq

dA(t

nominal values atthe

equilibrium point.

For

q(t)

q +° q

(t),

and Ro+ AR. If we are to

design

a

single

PI controller for WV(t) = WO+

6w(t),

c(t) c0 +

0c(t), N(t)

=

No

+

N(t),

this plant we canassume

p(t) =po+

Ap(t)

and Rt(t)

tRo

+

6R(t),

with

Ro

=o+ co (i) the plant is nominal let R = Ro andimplement

Kpio0

atransfer functiont-~~pqs

Gp()from

input

6p

to outPUt

6q

can be (ii fo fti:-0- Af i- tII <t0 lt ft(i)frR R<RY<R,ltR=R-t:i. :;< - 2 andl

o-ARan

obtained,

see e.g.

[2],

[6], [7],

[15],

implement

K21

G

(Y)-e

Ros

Roco

K 1

Roco0

(iii) for

Ro

< RTT <

Ro

+

tR,

let = t

R2

=

Pfo

+

AR

pq

ft-0Ros

+

f-

t. - 1

2NO

and

implement

Kpi2.

Since it is shown that considered PIcontroller is robust to

Now,

with

plant dynamics expressed by

this transfer func- thechanges

in

RTT,

[18],

these three controllers are expected

tion, we can

design

a PI controller for the

plant.

to have

good performance

in the

neighborhood

of RTT values

B. Resilient PI Controller Design they are designed for. In this paper, we illustrate that it is

possible to improve the performance in the case of time

Recently a study on allowableP1and PD control parameters

varyinlrg

RTT by

applyinlg

switching control. Two

differenlt

for time delayed systems with at most two unstable poles is

done in

[5],

[13]. The

objective

of this

study

was to find tbe

configurations

are

investigated:

donest

allowable

in].Ters

objectiveroftain

sty

wan

D

cntorollher

a)Usingtwo of thePIcontrollers

above,

we performmid-point

largest aloal itraS fo cerai P1 an PD cnrller swthn. Wbe RT is in [ft0 - Aft...is: :,ft,] intra K-parameters. Controllers obtained with this method is expected active and when RTT is in

[Ro,

ft

Af

tR]

interval

Kpi2

is

to work for a wide range of system parameters. Hence they active.

will be resilient in the sense of [17]. In

[18],

this controller b)Instead of dividing

[fO

-ARfRo

+

AfR]

interval into

two,

parameter

design

method is

applied

to the AQM

problem by

we divide it into N » 1 intervals. Therefore we

design

N

selecting

the center of the

largest

allowable intervals as the

different

PI controllers for each of these

intervals

andas RTT

optimal gains of the controllers. We now summarize the PI varies among these intervals the controller parameters switch

control design of [18].

For K > 1 the transfer function of the plant can be

rewritten as III. SIMULATIONS

c)K eRos The

performance

of the

designed

PI controllers are tested

Cpq() w-f) here s

(Rs

1) with ns-2 simulations. The network topology (Fig. 2) of the

simulations consists ofasinglebottlenecklinkandtwo routers

~

f(s)

-1 at the ends of this link which support N TCP flows as in

0

=0 11 s

ll'[19].

The buffers of both routers can hold 300

packets

where

the packets are of size 1000 Bytes. All the links

inl

tbe According to

[5], [13],

the optimal proportional gain that network has the same capacity C0 = C1t

=h10

Mbps. The

maximizes the allowable integral gain interval is ₂₀I and the _propagation_{delays of the}_links _vary _with _time._{As RTT}_{of the}

maximum value of the maximal interval of the

integral

gain network

vanis

in the interval [0.16s 0.48s] th

propagation

is 1 .nTo make the controller

robustly

stable with respect to delay of the bottleneck

link, To,

takes values

in the interval

largest perturbations in the controller parameters, we choose [4ms 36ms] the

propagation delay

of the links between

the

proportional

gain

as -₂₀ andthe

integral

gain

as ₈₀-

which

_{routers and TCP}

sources/destiantions,

_'l_l,_{takes values between}

is the midpoint of the maximal

interval,

[18]. Then the PI controller for the plant

(5)

is obtained as

TCP Sources TCP Destinations

Kpi(s)

₂₀

_c0K

(1+ _80s

)(6)

In order to implement these controllers in ns-2 we use a Q

digital implementation

method

suggested

in

[8].

[6]. COJO

Ro-AR

Ro Ro+AR

Fig. 1. RTT

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TABLE I TABLE II

THEANALYSISOFSIMULATION RESULTS FORRTTi THEANALYSISOFSIMULATION RESULTSFORRTT2

Controller Mean Std RM[S error s Controller Mean Std RMS error £

Kpio 153.34 39.64 0.27 0.33 Kpio 152.28 46.03 0.31 0.42

Kpil 142.60 39.14 0.27 0.27 Kpil 134.68 43.60 0.31 0.42

Kpi2 159.91 45.96 0.31 0.49 Kpi2 159.47 51.80 0.35 0.52

Kpij-Kpi2 Switching 152.52 35.07 0.23 0.22 Kpij-Kpi2 Switching 150.67 41.69 0.28 0.32

16 SwitchingControllers 148.77 34,63 0.23 0.23 16SwitchingCon_oilers 150,57 40.31 0.27 0,32

[8ms,72ms]. The nominal values of the system parametersare: better than the benchmarkPI design [7] which is available in

No = 30 TCP flows,

c,

= 1250 packets/s,

q,

= 150 packets, the current version ofns-2).The performances of the designed

Ro = 0.32 s. controllerswere tested via ns-2 simulations. Simulations show

We investigate the performance of five different configura- that switching between two controllers gives better results

tions: compared to a single controller case.

(a) single controller,

Kpio,

Note that theoreticalproof of performance improvement by

(b) single controller,

Kpil,

using switched controllers in such a complicated nonlinear

(c) single controller,

Kpi2,

system (packet level simulation setting) is not easy to obtain.

(d) two switching

controllers,

Kpil

and

Kpi2,

In fact, even for simplified flow model it can be shown

(e) N=16 switching controllers. that arbitrary switching between two controllers may even

This

experiment

is done

twice,

first for an RTT function destabilize the feedback

system.

Therefore our simulation

RTT1,

as in Fig.

3(a)

and then for a more

quickly changing

results illustratethe value of

mid-point

switched

PI

controllers

RTT function, RTT2 as in Fig. 4(a). The plots of queue for this AQM problem.

lengths are given in Fig. 3 and Fig. 4. For the purpose of ACKNOWLEDGMENT

evaluating the simulation results,we compare someproperties

of thequeuelengths for allcases. We do the comparison with This work is supported in part by the European

Commis-mean, standard deviation (abbreviated std) RMS error and E sion (contract no. MIRG-CT-2004-006666) andbyTUBITAK

as in [18]. RMS error can be formulated as (grant no.

EEEAG-105E]56).

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Fig. 3. ns-2 simulation:: (a) RTT1 (h) single controiler, Kpo (c) single con toiler, Kpi1 (d) single controller, Kpi2 (e) two :witching controIlers,

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300 0250 0 so 1 00 1 50 2(00 300 2300 2500 100 1|>l 0 so 1 00 1 50 2(00 300 2005 015 0 E- ~ ~ ~ ~ ~ ~ ~ ~ ~ Tm (:sec)

Fi.4 s2smltos:()RT b igecntolr p;()snl

cotolrc p d)snl otolr,Ki e w wtcigcnrles