Consistent proportional delay differentiation: A fuzzy control approach

Computer Networks 51 (2007) 2015–2032

www.elsevier.com/locate/comnet

Consistent proportional delay differentiation: A fuzzycontrol approach

Jianbin Wei a,*, Cheng-Zhong Xu b

a Mathematics and Computer Science Department, South Dakota School of Mines and Technology,

Rapid City, SD 57701, United Statesb Department of Electrical and Computer Engineering, Wayne State University, 650 Merrick Street,

Detroit, MI 48202, United States

Received 20 November 2004; received in revised form 15 June 2006; accepted 16 October 2006Available online 9 November 2006

Responsible Editor: M. Buddhikot

Abstract

Proportional delay differentiation (PDD) is an important service model for providing relative differentiated services onthe Internet. It aims to maintain pre-specified packet queueing-delay ratios between different classes of traffic at each hop.Existing rate-allocation approaches for PDD services assume the average queueing delay of a class is inversely propor-tional to its service rate. This assumption is not necessarily valid when the system is not heavily loaded. To provideconsistent PDD services under various load conditions, in this paper, we propose a novel rate-allocation approach thatapplies fuzzy control theory to capture the nonlinear relationship between the queueing delay and the service rate. Inthe approach, a class’s service rate is adjusted according to a set of fuzzy control rules defined over its error (the differencebetween the target delay ratio and the achieved one), the change of error, and the change of service rate. We prove that thefuzzy control system is stable and the service rate of a class converges to its equilibrium point at steady state. Simulationresults demonstrate that, in comparison with other rate-allocation approaches, the fuzzy control approach is able toprovide consistent PDD services under wide range load conditions. It is also shown robust under various system condi-tions, including with multiple classes, changing target delay ratios, changing load conditions, and different traffic patterns.� 2006 Elsevier B.V. All rights reserved.

Keywords: Quality of service; Proportional delay differentiation; Fuzzy control; Rate allocation

1. Introduction

The past decade has seen an increasing demandfor provisioning of different levels of quality of ser-

1389-1286/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.comnet.2006.10.005

* Corresponding author. Tel.: +1 605 394 2470.E-mail addresses: [email protected] (J. Wei), czxu@

wayne.edu (C.-Z. Xu).

vice (QoS) to various network applications and cus-tomers. Differentiated services (DiffServ) [2] is amajor service architecture in order to meet thisrequirement. In the architecture, two service modelshave been proposed: absolute DiffServ, which is toguarantee the end-to-end QoS [12,14]; and relativeDiffServ, which is to quantify the QoS differ-ences between different classes. Although absolute

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2016 J. Wei, C.-Z. Xu / Computer Networks 51 (2007) 2015–2032

DiffServ is desirable for Internet services (e.g., Inter-net telephony) that have hard time constraints, rel-ative DiffServ is sufficient for soft real-timeapplications, such as e-commerce transactions andmultimedia applications, which may last from sev-eral seconds to tens minutes. In relative DiffServ ser-vice model, the network traffic is divided into anumber of classes. A higher ranked class shouldreceive better or at least no worse services than alower ranked class in terms of local service metrics,such as the queueing delay and the loss rate [7]. Inparticular, proportional delay differentiation(PDD) service model is to maintain pre-specifiedqueueing delay ratios between different classes [8].

Existing approaches for PDD services canbe classified into two categories: dynamic-priorityapproaches and rate-allocation approaches. Indynamic-priority approaches, a class’s priority isadjusted according to its current measured states,such as its head-of-line delay [8,16], the averagedelay of its departed packets [9], or the delay ofall arrived packets [19,24]. Although most of theapproaches are capable of providing consistentPDD services under various load conditions, theyintroduce significant overhead because of theirneeds for calculating the priorities of all backloggedclasses to determine the one with the highest prior-ity upon every packet departure. Moreover, thedynamic-priority approaches are incompatible withthose current-in-use routers because they have dif-ferent queueing disciplines, such as class basedweighted fair queueing and modified deficit roundrobin (MDRR) implemented by Cisco [5]. Thisincompatibility makes it difficult to deploy theseapproaches in practice, and necessitate the designof simpler approaches.

In rate-allocation approaches, a class’s servicerate is adjusted periodically based on current systemstates [4,8,17]. For example, in backlog-propor-tional rate (BPR), a class’s service rate is adjustedaccording to its backlogged queue length [8]; in jointbuffer management and scheduling (JoBS), the ser-vice rate is set based on delay predictions of itsbacklogged traffic [17]. In [4], the authors proposeda linear feedback control approach (referred to asLFB in the rest of this paper), in which a class’s ser-vice rate is adjusted according to the differencebetween its normalized head-of-line delay and theaverage of all backlogged classes.

Rate-allocation approaches incur smaller over-head than dynamic-priority approaches becausethey adjust service rates only when needed. More-

over, they are implemented with the same queueingdisciplines as those used in current-in-use routersand thus can be deployed easily.

Existing rate-allocation approaches, however, areable to deliver PDD services only when the system isheavily loaded. For example, BPR showed goodPDD services only when the system load becomeshigher than 90% [8]; performance studies of JoBSand LFB assumed the system load to be as high as99% [4,17]. We tested the approaches in non-heavilyloaded systems and found that none of them wasable to achieve the goal of PDD services consis-tently under light and medium load conditions. Itis because they assume that the average queueingdelay of a class is inversely proportional to its ser-vice rate. This assumption is valid during busy peri-ods of the system. It is not always the case if thereexist idle periods in the queueing system.

We note that consistent PDD services under var-ious load conditions are desirable. It is known thatthe load of a system (e.g., the link utilization) inpractice is often light or moderate [3]. Without guar-antee of the consistency, it would become less incen-tive for high priority customers to pay more for thesame level of QoS most of the time. Dynamic-prior-ity approaches achieved the goal of consistency at acost of frequent priority changes.

For consistent PDD services, a rate-allocationapproach should be able to capture the nonlinearrelationship between the queueing delay and the ser-vice rate. Although lots of work exist for networktraffic characterization (see [21,27] for examples),there is still lack of accurate and solvable mathemat-ical models for the relationship. Classical controlmethods approximate nonlinear systems by linearones with a compromise of accuracy, as we pushthe performance envelope of the controller.

In this paper, we propose a novel rate-allocationapproach based on fuzzy control. Fuzzy control the-ory provides a formal methodology for represent-ing, manipulating, and implementing a human’sheuristic knowledge about how to control a system.In the fuzzy control approach, we quantify heuristiccontrol knowledge developed from system behavioranalysis into a set of control rules. The rule-basestates the relationship between the error (the differ-ence between the target delay ratio and the achievedone), the change of error, and the change of servicerate. A class’s service rate is adjusted according tothe rules.

We prove that the fuzzy control system is stableand the service rate of a class converges to its equilib-

J. Wei, C.-Z. Xu / Computer Networks 51 (2007) 2015–2032 2017

rium point at steady state. Simulation results demon-strate the superiority of the approach, in comparisonwith its main competitors: BPR, JoBS, and LFB. Inparticular, it is capable of providing PDD servicesconsistently under wide range load conditions. Inaddition, the fuzzy control approach is shown robustunder various system conditions, including with mul-tiple classes, changing target delay ratios, changingload conditions, and different traffic patterns.

The remainder of the paper is organized as fol-lows. Section 2 reviews related work in PDD serviceprovisioning. Section 3 discusses the fuzzy controlapproach and design issues. Section 4 proves thestability of the fuzzy control system. Section 5 com-pares the fuzzy control approach with other rate-allocation approaches and shows its robustness.Section 6 concludes this paper.

2. Related work

There have been many efforts in providing PDDservices to network applications and clients. Exist-ing approaches can be classified into two categories:dynamic-priority approaches and rate-allocationapproaches. From the viewpoint of control theory,they can also be classified into open-loop controland closed-loop control (feedback control).

Representatives of dynamic-priority with open-loop control approaches include waiting-time prior-ity [8] and adaptive waiting-time priority [16]. Inwaiting-time priority approach, a backlogged class’spriority is set to be proportional to its head-of-linedelay and inversely proportional to its differentiationparameter [8]. Albeit simple, the performance ofwaiting-time priority approach is poor in non-heavyload conditions. To deliver consistent PDD servicesin light and moderate load conditions, in [16], theauthors proposed an adaptive waiting-time priorityapproach in which a class’s priority is set accordingto its head-of-line delay as well as the distribution ofclass loads. The authors demonstrated the accuracyand adaptivity of the approach under moderate loadconditions in an M/G/1 queueing system.

Dynamic-priority with feedback control app-roaches set a backlogged class’s priority accordingto the current achieved delay ratio or its averagedelay. Proportional average delay [9], hybrid pro-portional delay [9], mean-delay proportional [19],and Little’s average delay (LAD) [24] are four rep-resentatives in this category. Proportional averagedelay approach sets a backlogged class’s priorityaccording to its average delay of departed packets.

It, however, exhibits a pathological behavior causedby its ignorance of backlogged packets. To addressthis issue, hybrid proportional delay was proposedto take into account the average delay of departedpackets and the head-of-line delay simultaneously.This approach is actually a combination of propor-tional average delay approach and waiting-timepriority approach. This combination causes worseperformance under moderate load conditions thanproportional average delay. In mean-delay propor-tional approach, a class’s priority is set accordingto the estimated average delay of all of its arrivedpackets. It results in similar performance as propor-tional average delay approach. One disadvantage ofthese three approaches is that they yield large vari-ance in short time-scales. To overcome this disad-vantage, we proposed LAD based on a proof ofLittle’s law [24]. The scheduler monitors the arrivalrate of the packets in each traffic class and thecumulative delays of the packets, and schedulesthe packets according to their transient queueingproperties. The simulation results demonstrate thatit is able to provide the same level of service qualityin long time-scales and more accurate and robustcontrol over the delay ratio in short time-scales thanother dynamic-priority approaches.

In contrast to dynamic-priority approaches, rate-allocation approaches provide PDD services byadjusting a class’s service rate. Rate-allocation withopen-loop control approaches, as exemplified byBPR and JoBS, adjust a class’s service rate withoutconsidering current achieved delay ratios. In BPR, aclass’s service rate is adjusted upon every packetdeparture event so that the rate is always propor-tional to the load of its backlog [8]. In JoBS, the ser-vice rate is adjusted according to the delayprediction of its backlogged traffic [17].

LFB [4] is a representative of rate-allocation withfeedback control. In light of the fact that the back-log of a class does not change significantly duringbusy periods, LFB adjusts a class’s service rateaccording to the difference between its normalizedhead-of-line delay and the average of all backloggedclasses. In contrast, the fuzzy control approachtakes the achieved delay ratio into account inadjusting a class’s service rate.

In literature, there are other studies on the appli-cation of control theory for providing differentiatedservices. In [1,18,20], the authors used system iden-tification to model Internet servers based on theirpast behaviors, and closed-loop integral controllerswere proposed. In these approaches, the nonlinear


relationship between the allocated resource of aclass and its received service quality is linearized ata fixed operating point. It is well known that lin-ear approximation of a nonlinear system is accurateonly within the neighborhood of the point where itis linearized. In networks with continuously chang-ing load conditions, the operating point changesdynamically and their simple linearization thus isinappropriate.

Note that PDD service provisioning focuses on anon-overloaded system. In the case that the system isoverloaded, admission control should be adopted. Itdrops packets from lower ranked classes so as toguarantee service quality perceived by higher rankedclasses [15]. The work presented in this paper is com-plementary to those on admission control for over-load protection and service differentiation.

Many work have been done on server sides toprovide PDD services using control theory [1],adaptive fuzzy control theory [25], and queueingtheory [26]. This work is complementary to thoseserver-side approaches.

3. Robust resource allocation with fuzzy control

The PDD service model is to maintain pre-speci-fied queueing delay ratios between different classesin a system. Let Wi denote the average delay of classi computed during a service period, and di its pre-specified differentiation parameter. For class i andclass j in the system, it is required that

W i

W j¼ di

dj:

The service period is determined according to theservice level agreements between the service provid-ers and the end-users. For example, the agreementcan be defined as following: the delay ratio of pre-mium customer to basic customer must be 2 forevery 10,000 transmitted packets. The service peri-od, 10,000 packets in this example, is independentof the scheduling approaches. As in [9,22], this fuzzycontrol approach measures the unit of a period interms of elapsed events (i.e., transmitted packets)and assumes the service period to be 10,000 trans-mitted packets in the experiments. This normallylasts a few hundred milliseconds. For example, fornetwork interface with up to 100 Mbps, a serviceperiod with 10,000 transmitted packets has the sameeffect as that with a length of 500 ms when the pack-et rate is 20,000 packets per second. For an averagepacket size of 441 bytes (40% were 40 bytes, 50%

were 550 bytes, and 10% were 1500 bytes), thispacket rate corresponds to 67.3 Mbps of aggregatedflows. This is small enough for soft real-time appli-cations, such as e-commerce transactions and multi-media applications, which normally last for a fewseconds to tens minutes.

In a feedback control system, we need to set itssampling period appropriately. It determines howthe past behavior of the control system is taken intoaccount and how often the controller is invoked.For instance, assuming the sampling period is setto 1000 events. In the fuzzy control approach, theservice rate of a class then is adjusted for every1000 packet departures based on the sampled out-puts (e.g., the delay ratio) for past 100 transmittedpackets.

In the following, we first present the structure ofthe fuzzy controller, followed by the determinationof its inputs and outputs. We then discuss how toquantify heuristic control knowledge into a rule-base. Finally, we discuss converting the fuzzy con-troller output to the service rate adjustment.

3.1. The structure of the fuzzy controller

In the fuzzy control approach, the service rate ofa class in sampling period k + 1, denoted byu(k + 1), is adjusted according to its error e(k)(i.e., difference between the target delay ratio andthe achieved one) and change of error De(k) in sam-pling period k. Such adjustment is controlled by aset of control rules about heuristic knowledgedescribed by fuzzy logic methods. Fig. 1 shows thestructure of the fuzzy control system.

In addition to the error e(k) and change of errorDe(k), the controller takes target delay ratio r(k) as areference input and produces an output of servicerate change Du(k). Du(k) is fed forward into arate-allocation process in the last stage. The processhas an output of achieved delay ratio y(k), which isfed back to the controller.

The fuzzy controller is composed of four com-ponents:

1. A rule-base (a set of If–Then rules), which con-tains quantified control knowledge about howto adjust a class’s service rate according to thee(k) and De(k).

2. A fuzzification interface, which converts control-ler inputs into certainties in numeric values ofthe membership functions that are defined forthe inputs.

Fig. 1. The structure of the fuzzy controller.


3. An inference mechanism, which activates andapplies rules according to fuzzified inputs, andgenerates fuzzy conclusions for defuzzificationinterface.

4. A defuzzification interface, which converts fuzzyconclusions into the change of service rate of aclass in numeric value.

The fuzzy controller shown in Fig. 1 also con-tains three scaling factors, which can be used to tunethe performance of the fuzzy control approach. Ke

and KDe are the scaling factors of the inputs.According to the structure of the fuzzy control sys-tem, the actual inputs of the controller are Ke Æ e(k)and KDe Æ De(k). KDu is the scaling factor of the fuzzycontroller output. It is used to control the actual ser-vice rate of a class. Therefore, a class’s service ratefor sampling period k + 1 is

uðk þ 1Þ ¼ uðkÞ þ KDu � DuðkÞ ¼Z

KDu � DuðkÞdk:

ð1Þ

Note that the scaling factors are always positive soas to ensure the stability of the fuzzy control system.

The fuzzy controller is a proportional integral(PI) like controller with nonlinear operating func-tions. Proportional, integral, and derivative controlsare the most popular control design techniques [11].Derivative control takes into account the change oferrors and has good responsiveness. Its drawback isthat it is sensitive to measurement noises, which arecommon in provisioning of delay differentiationservices. The advantage of proportional control lieson its simplicity and good agility. It, however, can-not eliminate the steady-state error introduced dur-ing the control, which causes the target delay ratiosunachievable. To eliminate such steady-state errors,we need to incorporate an integral controller into it.Therefore, we design the fuzzy controller as a PI likecontroller. For a traditional PI controller, its equa-tion in a discrete form is

uðkÞ ¼ KDe � eðkÞ þ Ke

XeðkÞ:

By taking derivative, it becomes

DuðkÞ ¼ KDe � DeðkÞ þ Ke � eðkÞ;while the inputs of the fuzzy controller are Ke Æ e(k)and KDe Æ De(k) and its output is Du(k).

3.2. The inputs and output of the fuzzy controller

To translate the PDD service model into thefuzzy control framework presented in Fig. 1, thecontrol loops must be determined first, and in everyloop the following needs to be determined: (1) thereference input, (2) the error, (3) the change of error,and (4) the output of the fuzzy controller.

The fuzzy control approach introduces a simplemethod to translate the PDD service model. In thismethod, one class (e.g., class 1) is selected as thebase class. A control loop is associated with everyother class.

In the control loop of class i, the reference inputfor kth sampling period is

riðkÞ ¼diðkÞd1ðkÞ

: ð2Þ

The output of the process is the achieved delay ratioof class i to the base class

yiðkÞ ¼W iðkÞW 1ðkÞ

: ð3Þ

The error and the change of error are therefore de-fined as

eiðkÞ ¼diðkÞd1ðkÞ

� W iðkÞW 1ðkÞ

; ð4Þ

DeiðkÞ ¼ eiðkÞ � eiðk � 1Þ: ð5Þ

When the target delay ratios are the same betweentwo consecutive sampling periods (i.e., ri(k � 1) =ri(k)), the change of error is

DeiðkÞ ¼ yiðk � 1Þ � yiðkÞ: ð6Þ

Table 1The description of linguistic values

Linguistic value Description

‘‘�3’’ Negative large in size‘‘�2’’ Negative medium in size‘‘�1’’ Negative small in size‘‘0’’ Zero in size‘‘1’’ Positive small in size‘‘2’’ Positive medium in size‘‘3’’ Positive large in size

Fig. 2. Fuzzy control rules. (a) Illustration of fuzzy rules. (b) Therule-base for PDD service provisioning.


The output of the fuzzy controller is Dui(k), which isused to calculate the service rate of class i.

Notice that the selection of base class does notaffect the performance of the fuzzy control approachsince we assume no any special requirement on thebase class. In case that there is no sufficient trafficarrivals from the base class, as we shall see fromSection 5.1, the target delay ratio may become infea-sible. Such infeasibility does not depend on the selec-tion of base class, but the system conditions of allclasses.

This fuzzy control method has three advantagesover the one used in [4]. First, there is no tight cou-pling among the control loops because the referenceinput of the control loop associated with class i iscalculated from the pre-specified differentiationparameters di and d1. In [4], a control loop is asso-ciated with every class, and its output is the class’saverage delay. All control loops share the same ref-erence input, which is the normalized average delaycalculated using outputs of all control loops. Sec-ond, this fuzzy control method is more time-efficientbecause the reference input of a control loop is thesame for sampling periods where the differentiationparameters keep unchanged. In [4], however, thereference input needs to be calculated for every sam-pling period to reflect current states of the system.Finally, this method is more flexible. The scalingfactors of a control loop in this method can betuned independently. Essentially, in order to pro-vide consistent PDD services, what the fuzzy con-troller does is adjusting the rate-sharing amongdifferent classes. This meets the work-conservingrequirement in a very simple way. In [4], in orderto ensure the system to be work-conserving, the con-trol parameters of all control loops must be thesame.

3.3. Design of the rule-base

A rule of thumb in packet scheduling is that, themore service rate a class receives, the smaller queue-ing delay it experiences. In the fuzzy controller, thiscontrol knowledge is translated into a set of rulesbased on linguistic variables and values.

The linguistic variables are used to describe eachof the fuzzy controller inputs and output. In thesake of brevity, in the fuzzy controller,

• linguistic variable ‘‘ei(k)’’ describes ei(k),• linguistic variable ‘‘Dei(k)’’ describes Dei(k), and• linguistic variable ‘‘Dui(k)’’ describes Dui(k).

The linguistic variables assume linguistic values,which are represented using integers in the fuzzycontroller. Their meanings are shown in Table 1.For example, the statement ‘‘ei(k) is �3’’ means thatthe achieved delay ratio of class i to class 1 at thekth sampling period is much larger than the targetone. Note that ‘‘�3’’ is not associated with anyparticular value. It simply quantifies the sign ofthe error and indicates the size in relation to theother linguistic values.

Fig. 2(a) illustrates the effect of the fuzzy control-ler on the achieved delay ratio in PDD service pro-visioning. In this figure, five zones of differentcharacteristics can be identified. Zone 1 and 3 arecharacterized with opposite signs of e(k) andDe(k). That is,

Zone 1: e(k) > 0 and De(k) < 0;Zone 3: e(k) < 0 and De(k) > 0.

Fig. 3. Membership functions for the PDD service model. (a)Membership functions for error. (b) Membership functions forchange of error. (c) Membership functions for change of servicerate.


In these two zones, the error is self-correcting, andthe achieved delay ratio is moving towards to thetarget one. Then Du(k) is set to either speed up orslow down the current trend.

Zone 2 and 4 are characterized with same signs ofe(k) and De(k). That is,

Zone 2: e(k) < 0 and De(k) < 0;Zone 4: e(k) > 0 and De(k) > 0.

In these two zones, the error is not self-correcting,and the achieved delay ratio is moving away fromthe target one. Then Du(k) is set to reverse the cur-rent trend.

Zone 5 is related to situations characterized withrather small magnitudes of e(k) and De(k). That is,both e(k) and De(k) are close to zero, and the systemis at a steady state. Thus Du(k) is set to maintaincurrent state and correct small deviations from thetarget delay ratio.

Let Ui(k) denote the equilibrium service rate ofclass i at which the target delay ratio is achieved.Let ~uiðkÞ denote the difference between the equilib-rium service rate and current service rate. It followsthat

~uiðkÞ ¼ U iðkÞ � uiðkÞ: ð7Þ

Based on the identification of the five zones inFig. 2(a), in linguistics, we have

“~uiðkÞ” ¼ �½“eiðkÞ”þ “DeiðkÞ”�; ð8Þ

and ‘‘Dui(k)’’ is set as

“DuiðkÞ” ¼ “~uiðkÞ” ð9Þ

so that the service rate of a class converges to its equi-librium point. Notice that we are using linguistic val-ues of the controller’s inputs and output, and ‘‘�3’’and ‘‘3’’ are their lower bound and upper bound,respectively. Therefore, when ‘‘e(k)’’ = ‘‘De(k)’’ =‘‘�3’’, ‘‘Du(k)’’ is set to ‘‘3’’.

The resulted control rules are summarized inFig. 2(b). A general linguistic form of these rulesshould read as

If premise Then consequent:

Let rule(m,n), where ‘‘�3’’ 6 m,n 6 ‘‘3’’, denote therule of the (m,n) position in the body of the table.As an example, rule(1,1) = ‘‘�2’’ reads that:

If the error is positive small and the change of erroris positive small Then the change of service rate isnegative medium.

The fuzzy control rule-base shown in Fig. 2(b) isconsistent and complete. The rule-base is consistentin the sense that there are no rules that have thesame premise and different consequents. It is com-plete because the controller has two inputs andevery input has seven possible values for the rule-base, which leads to a total of 49 combinations.Any combination of the inputs results in an appro-priate output, according to the 49 rules in the ruletable.

3.4. Fuzzy quantification of the rule-base

In the fuzzy control approach, the meaning of thelinguistic values is quantified using the most popular‘‘triangle’’ membership functions, as shown inFig. 3. The mth membership function quantifiesthe certainty (between 0 and 1) that an input canbe classified linguistically as linguistic value m,‘‘�3’’ 6 m 6 ‘‘3’’. For example, according to the2nd membership function in Fig. 3(a), if e(k) is


1/2, the certainty that it can be classified as ‘‘2’’ is0.5; if e(k) is larger than 1 or smaller than 1/3, thecertainty then is 0.

The fuzzy control approach utilizes a normalizedfuzzy system, in which the left-most membershipfunction peaks at �1 and the right-most one at +1for both the input and output universes of dis-course. For example, in Fig. 3(a), the �3th and3th membership functions peak at �1 and +1,respectively. For different system conditions (e.g.,different target delay ratios), the input and outputscaling factors can be tuned to obtain appropriatelyscaled universe of discourse. For example, in thecase that e(k) = 0.3, if the target delay ratio is 2,then it is certain that the error can be classified as‘‘1’’ in the viewpoint of service requirements andKe may be set to 1; if the target ratio is 8, then itis very certain that the error is ‘‘0’’ because the erroris very small in relation to the target ratio and Ke

may be set to 0.1. Therefore, the scaling factorscan be tuned to improve the fuzzy controller’sperformance.

The fuzzification component translates the inputsinto corresponding certainty in numeric values ofthe membership functions. Let lm(e(k)) denote thecertainty of e(k) of the mth membership function,and ln(De(k)) the certainty of De(k) of the nth mem-bership function. When e(k) = 1/12 and De(k) =1/3, according to the membership functions of e(k)and De(k) as shown in Fig. 3(a) and (b), all member-ship functions yield 0 except that

l0ðeðkÞÞ ¼ 0:75; l1ðeðkÞÞ ¼ 0:25; and

l1ðDeðkÞÞ ¼ 1:

3.5. Inference mechanism and defuzzification

The inference mechanism is used to determinewhich rules should be activated and what conclu-sions can be reached. Let l(m,n) denote the premisecertainty of rule(m,n). The and operation in the pre-mise is calculated via minimum. Suppose e(k) = 1/12and De(k) = 1/3, then only rule(0, 1) and rule(1, 1)should be activated because the premise certaintiesof all other rules are 0 according to the exampleshown in Section 3.4. Consequently,

lð0; 1Þ ¼ minf0:75; 1g ¼ 0:75;

lð1; 1Þ ¼ minf0:25; 1g ¼ 0:25:

Based on the outputs of the inference mechanism,the defuzzification component calculates the fuzzy

controller output using ‘‘center average’’ method.Let b(m,n) denote the center of membership func-tion of the consequent of rule(m,n). In this case, itis where the membership function reaches its peak.The fuzzy control output is

DuðkÞ ¼P

m;nbðm; nÞ � lðm; nÞPm;nlðm; nÞ

: ð10Þ

In the example, according to Fig. 2(b), the conse-quents of rule(0,1) and rule(0, 2) are the �1th and�2th membership functions of Du(k), respectively.According to Fig. 3(c), b(0,1) and b(0, 2), the centersof the membership functions, are �1/3 and �2/3,respectively. Therefore, the fuzzy controller output

DuðkÞ ¼ �1=3 � lð0; 1Þ þ ð�2=3Þ � lð1; 1Þlð0; 1Þ þ lð1; 1Þ ¼ � 5

12:

The fuzzy control output is then substituted into 1to calculate the service rate of the class. After that,the service rate of the class is normalized to the sumof service rates of all classes. At last, the actual net-work bandwidth of the class is calculated accordingto the normalized service rate.

Finally, we remark that the fuzzy controlapproach has small overhead, which mainly comesfrom the service rate adjustment performed at theend of a sampling period. For any input, only twomembership functions lead to nonzero values, andat most four rules on at any time, which requireseight multiply and one divide operations.

4. Stability analysis

Stability is one of the most important concerns ofa control system, because it is often the case that ifthe system is unstable there is no chance that anyother performance specifications will hold. In thissection we prove the stability of the fuzzy controlsystem in a way similar to [6].

Theorem 1. In the fuzzy control system, the service

rate of a class will converge asymptotically to a small

� neighborhood of its equilibrium point.

Proof. By (1), we have

~uiðk þ 1Þ ¼ ~uiðkÞ � KDuDuiðkÞ: ð11ÞWe use the Lyapunov direct method to prove thatthe system converges to the equilibrium point.Choose

V ð~uiðkÞÞ ¼ ~u2i ðkÞ ð12Þ


as the ‘‘Lyapunov function’’. It characterizes thedistance of the service rate of class i from its equilib-rium rate. To prove that the service rate of class i

will converge to the equilibrium point, the difference

V ð~uiðk þ 1ÞÞ � V ð~uiðkÞÞ ¼ ~u2i ðk þ 1Þ � ~u2

i ðkÞ

¼ ½~uiðkÞ � KDuDuiðkÞ�2 � ~u2i ðkÞ

¼ KDuDuiðkÞ½KDuDuiðkÞ � 2~uiðkÞ� ð13Þ

should always be negative.By (9) we know ~uiðkÞ and Dui(k) are always the

same in sign. Recall that KDu is positive. Therefore,by introducing a constraint

jDuiðkÞj <2

KDuj~uiðkÞj ð14Þ

we can ensure (13) is always negative. Therefore, theservice rate ui(k) for class i converges to its equilib-rium point. Note that it is difficult to obtain theequilibrium point Ui(k), so as ~uiðkÞ. Instead, let �be a small positive constant, we use a fix constraintfor Dui(k); that is

jDuiðkÞj <2�

KDu: ð15Þ

Therefore, for j~uiðkÞj > �, jDui(k)j < 2�/KDu leads tothe validity of (14). It implies that ui(k) will convergeasymptotically to an � neighborhood of its equilib-rium point. This concludes the proof. h

5. Performance evaluation

We conducted comprehensive experiments to eval-uate the performance of the fuzzy control approach.In this section, we first compare this approach withJoBS, BPR, and LFB. We then demonstrate itsrobustness under various system conditions, includ-ing with multiple classes, changing target delayratios, changing system loads, and different trafficpatterns.

We assumed distributions of packet inter-arrivalsand sizes similar to those in [8]. The packet inter-arrivals followed a Pareto distribution with theshape parameter to be 1.5. In these packets, 40%were 40 bytes, 50% were 550 bytes, and 10% were1500 bytes.

In these experiments, service rate allocation wasenforced by the use of the deficit round robinmethod [23]. In the method, a class’s service rateshare is controlled by a deficit counter. The largerthe deficit counter, the more service rate it has. Class

1 was assumed to be the base class. Its deficit coun-ter was set to 1, a service rate of one average-sizepacket per round. At the beginning, all classes hadthe same service rate. In order to prevent a trafficclass from starving, the lower bound of a deficitcounter was set to 0.01. In these experiments, theservice period and the sampling period wereassumed to be 10,000 and 1000 transmitted packets,respectively.

It is known that sudden spikes in measured delayratios may introduce a significant random noise intothe feedback loop. In order to filter such noise, inthe experiments, the sampled delay ratio wassmoothed using an exponentially weighted movingaverage with parameter a (0 < a 6 1). Let wi(k)/w1(k) be the delay ratio of class i to class 1 at thekth sampling period. The computed delay ratio thenwas

W iðkÞW 1ðkÞ

¼ a � wiðkÞw1ðkÞ

þ ð1� aÞ � W iðk � 1ÞW 1ðk � 1Þ : ð16Þ

Values of a closer to 0 will increase the horizon overwhich Wi(k � 1)/W1(k � 1) is averaged and viceversa. We carried out experiments with different set-tings of a. In this paper, we only present the resultswhere a was 0.5 for brevity because no qualitativedifferences were observed in the results due to othersettings.

5.1. Comparison with other rate-allocation

approaches

In this part, we compare the fuzzy controlapproach with LAD, JoBS, BPR, and LFB. Inour implementation of the latter four approaches,the service rates of all classes are adjusted uponthe arrival or departure of a packet in order toachieve their best performance [10]. The experi-ments assumed two classes with the same load inthe system. The target delay ratio d2/d1 was set to4. The system load changed from 60% to 95% forevery 200 sampling periods. In these experiments,the scaling factors of the fuzzy controller were setas: Ke = KDe = 0.1 and KDu = 1. The tuning of thesefactors is presented in Section 5.4.2. Fig. 4 presentsthe experiment results.

First of all, we can see from the figure that thefuzzy control approach can provide consistentPDD services if the system load is no less than70%. Meanwhile, from Figs. 4(a) and 5 we can seethat the abrupt increasing and dropping of systemload have little effect on the performance of the

0 200 400 600 800 1000Sampling period

0

2

4

6

8

Del

ay r

atio

Fuzzy approach

load=60% 70% 80% 90% 95%

0 200 400 600 800 1000Sampling period

0

2

4

6

8

Del

ay r

atio JoBS

0 500 1000Sampling period

0

2

4

6

8

Del

ay r

atio

BPRStatic priority

0 200 400 600 800 1000Sampling period

0

2

4

6

8

Del

ay r

atio Linear feedback

0 200 400 600 800 1000Sampling period

0

2

4

6

8

Del

ay r

atio Little’s Average Delay

Fig. 4. Comparison with other rate-allocation approaches. (a) Results of fuzzy approach. (b) Results of JoBS. (c) Results of BPR.(d) Results of LFB. (e) Results of Little’s average delay.


0 200 400 600 800 1000Sampling period

0

2

4

6

8

Del

ay r

atio Fuzzy approach

80% 60%

Fig. 5. Results of fuzzy approach with system load decreases from 80% to 60%.


fuzzy control approach. In comparison, none of thecurrent rate-allocation approaches, including JoBS,BPR, and LFB, was able to achieve the goal. JoBSand BPR can guarantee PDD services only when thesystem load becomes as high as 90%. This is becausethey adjusted a class’s service rate according to itsqueueing delay. In systems with moderate load,the idle periods leads to a deviation of the achieveddelay ratio from the target.

LFB fails to deliver required PDD services in alltest cases. It is because the approach was mainlydesigned for differentiated service provisioning overbusy periods only. In a time interval with mixedbusy and idle periods, a small idle time percentageleads to a great performance deterioration. LFB isa rate-allocation approach with feedback control.It, however, adjusts a class’s service rate accordingto the difference between its normalized head-of-linedelay and the average of all backlogged classes. Incontrast, this fuzzy control approach adjusts the ser-vice rate according to the error of achieved delayratio. It is the fuzzy controller that captures thenonlinear relationship between the queueing delayand the service rate.

From the results we can see that LAD providesmore accurate delay ratio than the fuzzy controller.Such results are expected because LAD determinesthe departure order for every packet based on cur-rent system conditions while while the fuzzy con-troller reallocates network bandwidth for every10,000 packets. Consequently, LAD has much lar-ger overhead than the fuzzy controller.

From the figure, we can also see that when thesystem load becomes lower than 70%, the achieveddelay ratio is always smaller than the target ratio,no matter how processing rate is allocated betweenthe classes. This infeasibility is possible for PDDservices in a work-conserving system due to the con-straints of conservation law. It is known that theaverage queueing delay of a class has a minimum

value due to its inherent load. The minimum valuecan be achieved by the use of a non-preemptive sta-tic priority approach in which packets from a higherranked class will always be transmitted before thosefrom a lower ranked class. Let W sp

i denote the aver-age queueing delay of class i due to the static prior-ity approach. The upper bound of the feasible delayratio in a G/G/1 system is W sp

i =W sp1 [9].

We conducted an experiment using the static pri-ority approach to obtain the upper bound of thefeasible delay ratios for a system of 60% load. Sincethe bound is independent of rate allocation algo-rithms, we plot the bounds in different samplingperiods in Fig. 4(c). From the figure, it can beobserved that the fuzzy control approach can clo-sely approximate the upper bounds. The achieveddelay ratios due to other approaches are much smal-ler than the bounds. This further demonstrates thesuperiority of the fuzzy control approach.

We conducted experiments to evaluate the per-formance of these approaches under abrupt loaddropping. In the experiments, the system load chan-ged from 80% to 60%. Since the results are similar tothose with increasing load, we only plot the resultsfor the fuzzy approach in Fig. 5. From this figurewe observe that the dropping of load has little effecton the performance of the fuzzy control approach.This further demonstrates that the fuzzy controlapproach is robust to the changing of system loadconditions.

5.2. Impact of the sampling period

Fig. 4 also shows that the fuzzy control approachyields larger variance in heavy load conditions thanthe other rate-allocation approaches. This is mainlydue to the effect of sampling frequency. We carriedout simulations to further investigate the impact ofthis factor on the achieved delay ratios. It wasassumed there were two same-load classes and their

0 5 10 50 100 500 1000 5000 10000Size of sampling period

0

0.2

0.4

0.6

0.8

1

Stan

dard

dev

iatio

n

Load = 80%Load = 90%Load = 95%

Fig. 6. The effect of sampling period on the standard deviations of the achieved delay ratios under different load conditions.


target delay ratio was 4. Fig. 6 presents the standarddeviations of the achieved delay ratios with differentsettings of the sampling period. The target delayratio is 4.

This figure indicates that the standard deviationdecreases with the increase of the sampling period.The necessity of large number of experiments isbecause we are controlling a probability. Its vari-ance decreases and the mean converges to the realvalue with the increase of the number of samples.Similarly, in a queueing system, in order to obtainthe effect of the service rate on the average queueingdelay, the sampling period should be large.

When the sampling period continues to increase,the variance rises. It is because that, with a largesampling period the service rate of a class is adjustedless frequently than a small one. Hence, the fuzzycontrol approach becomes less agile, and the vari-ance is increased.

In practice, because the variance is small in rela-tion to the target delay ratio, a large sampling period(e.g., 1000 events) is preferred when the system loadis high, and it can be even larger (e.g., 10,000 event)when the system load is moderate, such as 80%.

One possible way to further reduce the variance isto use more than seven membership functions fore(k), De(k), and Du(k). In this case, the adjustmentof service rate maybe fine-grained. One drawbackis that the fuzzy controller may become more com-plicated and the scaling factors may be difficult totune.

5.3. Robustness of the fuzzy controller

To demonstrate the robustness of the fuzzy con-trol approach, we conducted experiments under var-ious system conditions, including with multipleclasses, changing target delay ratios, changing loadconditions, and different traffic patterns. The scalingfactors were set to be the same as previous experi-

ments. The sampling period was set to 10,000events, the same size as the service period.

5.3.1. Multiple classesIn this experiment, we assumed that there were

three same-load classes in the system. The systemload was assumed to be 90%. The differentiationparameters were set as d1:d2:d3 = 1:2:4. Fig. 7presents the achieved delay ratios of class 2 to class1 and class 3 to class 2. Recall that the service rate ofclass 3 was adjusted based on the achieved delayratio of class 3 to class 1. As shown in Fig. 7, thetarget delay ratio of class 3 to class 2 is achievedas well. It indicates that the fuzzy control approachis able to provide PDD services with different num-ber of classes.

5.3.2. Changing delay ratios

We also evaluated the performance of the fuzzycontrol approach with changing service require-ments. In this experiment, the target delay ratiod2/d1 was set to 2 at the beginning. It was thenincreased to 4 after the 300th sampling period andset back to 2 after the 600th sampling period. Thesystem load was 80%, and both classes had the sameload. As shown in Fig. 8, the fuzzy controllerresponses quickly to the change of the servicerequirements. For example, when the target delayratio changes from 2 to 4 at the 300th sampling per-iod, it implies that

e2ð299Þ ¼ 2� y2ð299Þ � 0;

e2ð300Þ ¼ 4� y2ð300Þ � 2;

De2ð300Þ ¼ 2þ y2ð299Þ � y2ð300Þ � 2:

The service rate of class 2 is then adjusted accord-ingly, which can also be observed from Fig. 8.

5.3.3. Changing load conditions

It is known that the Internet traffic changes fre-quently. We conducted an experiment to further

500 600 700 800 900 1000Sampling period

0

1

2

3

4

Del

ay r

atio class 2/class 1

500 600 700 800 900 1000Sampling period

0

1

2

3

4

Del

ay r


Fig. 7. The fuzzy controller performance with three classes.

0 100 200 300 400 500 600 700 800 900Sampling period

0123456

Del

ay r


0 100 200 300 400 500 600 700 800 900Sampling period

0

0.5

1

1.5

Serv

ice

rate

Κe= 0.1, ΚΔe

= 0.1, ΚΔu= 1

Fig. 8. The fuzzy controller performance (the achieved delay ratio and the change of service rate) for changing target delay ratios.


evaluate the performance of the fuzzy controlapproach under changing load conditions. In theexperiment, the target delay ratio d2/d1 was set to4. Initially, we assumed both classes had the sameload. After the 600th sampling period, the load ratioof class 2 to class 1 was set to 1/2. The load ratiowas then changed back after the 900th samplingperiod. Fig. 9 summarizes the achieved delay ratiosand the change of the service rate of class 2.

It is clear that there is no abrupt change of theachieved delay ratios with the change of class loaddistributions. This implies that the fuzzy controlleris able to capture the small change of achieved delayratios and adjusts the service rate allocation accord-ingly. This adaptivity can be verified from Fig. 9 atthe 600th and 900th sampling period. In addition,recall that the sampled delay ratio is smoothed usingan exponentially weighted moving average. It may

400 500 600 700 800 900 1000 1100 1200Sampling period

0

2

4

6

8

Del

ay r

atio

class 2/class 1

400 500 600 700 800 900 1000 1100 1200 1300Sampling period

0

0.5

1

1.5

Serv

ice

rate

Κe= 0.1, ΚΔe

= 0.1, ΚΔu= 1

Fig. 9. The fuzzy controller performance (the achieved delay ratio and the change of service rate) for changing load conditions.


lessen the effect of the change of class load condi-tions on the achieved delay ratio.

5.3.4. Different traffic patterns

In previous experiments, we assumed that thepacket inter-arrivals followed a Pareto distribution.To evaluate the effectiveness of the fuzzy controlapproach, we further conducted experiments withinput traffic of exponentially distributed inter-arriv-als. The packet size distribution remained the same.The system load was assumed to be 80%. Both clas-ses had the same load. The target delay ratio was setto 4. The scaling factors were set to the same as pre-vious experiments. Fig. 10 shows the experimentresults.

Evidently, the performance of the fuzzy controlapproach is independent of the traffic patterns. Thisis because the fuzzy control approach relies on ageneral nonlinear relationship between the queueingdelay and the service rate: the more service rate a

0 100Sampling

0

2

4

6

8

Del

ay r

atio

Fig. 10. The fuzzy controller performance under exp

class receives, the smaller queueing delay it experi-ences, which is valid in a G/G/1 queueing system.

In summary, from these experiments we can con-clude that the fuzzy control approach is robustunder various system conditions.

5.4. Remarks on the scaling factors

Recall that the fuzzy controller in Fig. 1 involv-ing three scaling factors: Ke, KDe, and KDu. Theyaffect the performance of the controller signifi-cantly. In this following, we discuss the effect of tun-ing the scaling factors.

5.4.1. Effect of tuning the scaling factors

Essentially, tuning the scaling factors is to shapethe control surface (nonlinearity) implemented bythe fuzzy controller. This fuzzy controller is a PI-like controller with nonlinear operating functions.

200 300 period

Exponential inter-arrival

onential distribution for packet inter-arrivals.


For a conventional PI controller, its equation in adiscrete form is

uðkÞ ¼ KDe � eðkÞ þ Ke

XeðkÞ:

By taking derivative, it becomes

DuðkÞ ¼ KDe � DeðkÞ þ Ke � eðkÞ: ð17ÞThe control surface of a PI controller can have thesame shape as the fuzzy controller near the origin.In this case the fuzzy controller may behave simi-larly to the PI controller, provided its inputs aresmall. However, there is no way for a linear PI con-troller to achieve a nonlinear surface of the fuzzycontroller as shown in Fig. 11. It is known thatthe relationship between queueing delay and servicerate is nonlinear. In order to provide PDD servicesconsistently under various system conditions, it isnecessary to utilize a nonlinear control method(e.g., the fuzzy control approach) to adjust the ser-vice rate allocation.

5.4.2. Determination of the scaling factors

It is important to tune the scaling factors in orderto provide PDD services. From Fig. 1 we canobserve that the scaling factors are used to controlthe inputs of the fuzzy controller and the actualchange of service rate. The tuning procedure forthe scaling factors are similar. Therefore, we onlypresent the tuning of KDu in this section.

Fig. 12 shows the simulation results for differentsettings of KDu. In these simulations, both Ke andKDe are set to 0.1. From Fig. 12 we can see that,when KDu is too small, it takes too long before thedesired delay ratio is achieved. For example, whenKDu is 0.1, it takes around 50 sampling period to

Fig. 11. Control surface of the fuzzy contr

approach the desired delay ratio. On the other hand,when KDu is too large, large variance is introducedalthough it can decrease the settling time. This canalso be observed from Fig. 12 where KDu is 5. By set-ting KDu to 1, we have a good balance between sta-bility and responsiveness. To further verify that, wealso present the change of service rate of class 2when KDu is 1 in Fig. 13. From this figure, we cansee that the service rate of class 2 changes accordingto the current system conditions while not abruptly.In summary, we set KDu to 1.

We note that the experimental results in previoussections were due to a parameter setting: Ke =KDe = 0.1, KDu = 1. This setting is not necessarythe only good choice. For example, as pointed outin [28], increasing (or decreasing) the scaling factorsKe and KDe may cause an increase (or decrease) ofthe PI-like controller’s integral and proportionalfactors. Meanwhile, KDu affects both factors simul-taneously. Because Ke and KDe have the same valuein the recommended setting, it is possible to obtainthe same integral and proportional factors of the PI-like controller by setting KDu alone. Fig. 14 showsthe results where Ke = KDe = 1 and KDu = 0.1. Incomparison with the experiment results shown inSection 5.1, it is clear that the fuzzy controllerhas very similar behavior resulted from these twosettings.

We also note that the scaling factors are insensi-

tive to environments. As shown in Section 5.3, thefuzzy controller works very well under very differentenvironments, such as changing load conditions anddifferent traffic patterns, with the same scaling fac-tors. Hence, it is not necessary to tune them fre-quently. This saves a lot of control efforts.

oller for Ke = KDe = 0.1 and KDu = 1.

0 100 200 300 400 500Sampling period

0

2

4

6

8

Del

ay r

atio ΚΔu

= 0.1

0 100 200 300 400 500Sampling period

0

2

4

6

8

Del

ay r

atio ΚΔu

= 1

0 100 200 300 400 500Sampling period

0

2

4

6

8

Del

ay r

atio ΚΔu

= 5

Fig. 12. Tuning of KDu. Ke = KDe = 0.1.

0 20 40 60 80 100Sampling period

0

0.5

1

1.5

2

Serv

ice

rate ΚΔe

= 0.1

Fig. 13. Service rate change of class 2 when Ke = KDe = 0.1 and KDu = 1.

0 100 200 300 400 500Sampling period

0

2

4

6

8

Del

ay r

atio Κ

e= 1, ΚΔe

= 1, ΚΔu= 0.1

Fig. 14. The fuzzy controller behavior for different scaling factors. The system load was assumed to be 80% and both classes have the sameload.



6. Conclusions

We have proposed a novel rate-allocationapproach to provide PDD services. The approachtakes advantage of the fuzzy control theory to cap-ture the nonlinear relationship between a class’squeueing delay and its service rate. The service rateof a class is controlled according to the rule-basedefined over the error, the change of error, andthe change of service rate. We have proved the sta-bility of the fuzzy control system and shown that theservice rate of a class converges to its equilibriumpoint at steady state. Simulation results have dem-onstrated that this fuzzy control approach is ableto provide consistent PDD services under widerange load conditions. We have also shown therobustness of the fuzzy control approach by investi-gating its performance under various system condi-tions, including with multiple classes, changingtarget delay ratios, changing load conditions, anddifferent traffic patterns.

With the increase of network bandwidth, propor-tional loss differentiation services become importantand some methods, such as BoundedRandomDrop[13], have been proposed. The focus of the futurework will be on provisioning of the proportional lossdifferentiation services. Because the dropping rate ofa class has direct relationship with its loss rate, adropping method based on statistical analysis maybeuseful in addition to the fuzzy control theory.

Acknowledgements

We would like to thank the anonymous reviewersfor their numerous helpful comments. This workwas supported in part by US National ScienceFoundation grants ACI-0203592, CCF-0611750,and MCS-0624849, and NASA grant 03-OBPR-01-0049.

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Jianbin Wei received the BS degree incomputer science from Huazhong Uni-versity of Science and Technology,China, in 1997, and the MS and PhDdegrees in computer engineering fromWayne State University in 2003 and2006, respectively. His research interestsare in distributed and Internet comput-ing systems.

Cheng-Zhong Xu is an Associate Profes-sor in the Department of Electrical andComputer Engineering of Wayne StateUniversity. His research interests are indistributed and parallel systems, partic-ularly in resource management for highperformance cluster and grid computingand scalable and secure Internet services.He has published more than 100 peer-reviewed articles in journals and confer-ence proceedings in these areas. He is the

author of the book ‘‘Scalable and Secure Internet Services andArchitecture’’ (CRC Press, 2005) and the leading co-author of the

works 51 (2007) 2015–2032

book ‘‘Load Balancing in Parallel Computers: Theory andPractice’’ (Kluwer Academic, 1997). He serves on editorialboards of three international journals, including the Journal ofParallel and Distributed Computing, and as a guest editor forseveral other journals and transactions. He has also served as theprogram chair or a member of program committees of numerousconferences. His research was supported in part by US NSF,NASA, and Cray Research. He is a recipient of the FacultyResearch Award of Wayne State University in 2000, President’sAward for Excellence in Teaching in 2002, and Career Devel-opment Chair Award in 2003. He obtained his BS and MSdegrees in Computer Science from Nanjing University in 1986and 1989, respectively, and Ph.D. in Computer Science from theUniversity of Hong Kong in 1993. He is a senior member ofIEEE.

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Consistent proportional delay differentiation: A fuzzy control approach