All-IP 網路上以預算為基礎之品質管理研究－子計畫四:All-IP 網路End-to-End 品質管理之研究(II)

行政院國家科學委員會專題研究計畫 成果報告

子計畫四:All-IP 網路 End-to-End 品質管理之研究(II)

計畫類別： 整合型計畫

計畫編號： NSC92-2219-E-004-004-

執行期間： 92 年 08 月 01 日至 93 年 07 月 31 日

執行單位： 國立政治大學應用數學學系

計畫主持人： 陸行

報告類型： 完整報告

處理方式： 本計畫可公開查詢

中 華 民 國 93 年 8 月 18 日

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MathematicalModels of Pareto Optimal Path Selection on All-IP Networks

: NSC 92-2219-E-004-004
: 920801930731  :
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1 Abstract

We present an approach for the fair resource allo-cation problem and QoSroutingin All-IP networks thatoermultipleservicestousers. Theobjectiveof theoptimizationproblemistodeterminethe amoun-tofrequiredbandwidth foreach linkand eachclass to maximize the sum of the users' utility. In this work, wefocuson approachesthat, while allocating bandwidth,attempt to provide aproportionallyfair treatmentofallthecompetingclasses. First,wewill showthatanachievementfunctioncanmapdierent criteria subject to various utility onto anormalized scale. It may be interpreted as a measure of QoS (Quality of Service) on All-IP networks. Using the bandwidth allocation model, we can nd a Pareto optimalallocationof bandwidthonthenetwork un-der a limited available budget. This allocation can provide the so-called proportional fairness to every class, that is, thisallocationcanprovide thesimilar satisfactiontoeachuser. Next,wepresentarouting schemeunderconsiderationofthedelay. Suchan op-timal path provides the end-to-end QoS guarantees to each user. Finally, a numericalexample is given toillustrate howtosolvethefairresourceallocation problemandhowto modifythenonlinearparts. (Keywords: multiple-objectiveproblems,routing, achievementfunction,proportionalfairness,delay, Paretooptimal,orderedweightedaveragingmethod, fairbandwidthallocation)

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Packet switchednetworks suer threemajor quality problemsin oeringtime-sensitiveservices: long de-laytime,jitter,packetloss. TheUniversalMobile T-elecommunicationsSystem(UMTS) [1]hasspecied

[Š1: UMTSServiceClasses

Traffic Classes

Examples

of Applications

Sensitivity

to Jitter

Sensitivity

to Delay

Sensitivity

to Packet Loss

Conversational

VoIP

high

high

low

Streaming

VoD

high

high

low

Interactive

WWW, Telnet

low

low

high

Background

E-mail, FTP

very low

low

high

fourdierenttraÆcclassesaccordingtotheirquality of service (QoS) requirements for dierent applica-tionsasTable1shows. Dierentpeoplehavedierent expectationstothenetworkQoS.Thereareanumber of characteristics that qualify QoS, including mini-mizing delivery delay, minimizing delay variations, providingconsistentdatathroughputcapacity.

QoSrouting concernsthe selection of apath sat-isfying the QoS requirements of a ow. The path selected most likely is not the traditional shortest path. Dependingonthespecicsandthenumberof QoSmetricsinvolved,computationrequiredforpath selection can become prohibitively expensive asthe network size grows. The path selection process in-volvestheknowledgeofthe ow'sQoSrequirements and characteristics and (frequently changing) infor-mationon the availability of network resources (ex-pressedin termsof standard metrics, e.g., available bandwidthanddelay). Resourceallocationdecisions areconcernedwiththeallocationoflimitedresources soastoachievethebest systemperformances.

In a multi-objective decision-making situation in theabsenceof uncertainty weoftensearchfor Pare-to optimal solutions. One scheme for dealing with multi-objective models that permits more balanced handlingoftheobjectivesissimplytocombinethem in aweightedsum. Multiple objectivefunctions can becombinedintoasinglecompositeoneto be max-imized bysumming objectiveswith positiveweights onmaximizing and negativeweightson minimizing. Ifthecompositeistobeminimized,weightson max-imizing objectivesshould be negative, and those on minimizingshouldbepositive. Signsorientall objec-tivesin thesamedirection,andweightsre ecttheir relativeimportance. Ifasingleweighted-sum objec-tive model derivedfrom amulti-objective optimiza-tionproducesanoptimalsolution,thesolutionisan

Inthiswork,weusethemethodofweightedsumsto solveourproblems.

We dealwith the problem of dimensioning band-widthforelasticdataapplicationsinpacket-switched communication networks, which can be considered as a multiple-objective optimization model. In our work,wewillfocusonthefollowingsubjects: (i)How do we transform the dierent criteria measurement onto anormalizedscale? (ii)Howdo weallocate re-sourceswithproportionalfairnessandndarouting schemeonAll-IPcommunicationnetworks? (iii)How dowemodifythenonlinearmultiple-objective prob-lemsassolvableMixed-Integerprogrammingmodels?

3 Achievement Function

Inordertotransformthedierentmeasurements on-to anormalizedscale, weconstructtheachievement function 

i

for each criteria i which can be viewed as an extension of the fuzzy membership function in terms of a strictly monotonic and concave utili-ty function as shown in Figure 1. We assume that the decision maker species requirements in aspira-tionandreservationlevelsbyintroducingdesiredand requiredvaluesfor several outcomes. Depending on thespeciedaspirationandreservationlevels,a

i and r

i

, respectively, weconstruct ourachievement func-tionofz i asfollows:  i (z i )=log  z i r i ; where= a i r i : (1) Formally,wedene i

(
)overtherange [0;1),with 

i

(0)= 1and 0 i

(0)=1. Dependingonthe spec-ied reference levels, this achievementfunction can be interpreted asa measure of thedecision maker's satisfactionwith the value of thei-th criteria. It is astrictly increasingfunction of z

i ,havingvalue1if z i = a i , and value 0 if z i = r i . The achievement function canmap thedierentcriteriavaluesontoa normalizedscaleofthedecisionmaker'ssatisfaction. Moreover,thelogarithmic achievementfunction will beintimately associatedwiththeconceptof propor-tionalfairness(see[6]and[8]). Wewillformulatethe mathematicalmodelofthefairbandwidthallocation byusingtheachievementfunction.

satisfaction

i

a

i

r

z

i

‹1: TheGraphofanAchievementFunction i

(z i

)

4 Formulation of the Band-width Allocation Model with Proportional Fairness

Given a network topology G =< V;E >, where V andE denotethesetofnodesandthesetoflinksin the network respectively. There is given a set S of m classes, i.e., jSj = m. We denote by S

i

aset of sessions in class i. There is also giventhe maximal possiblenumberK

i

ineachclassi,thatisjS i

j=K i

. Wewillgetthefollowingmathematicalmodel(MP1):

Maximize m X i=1 w i i Subjectto X e2E  e x e =B X i X j  i j (e) i j =x e ; 8e2E X i (K i
c i + i )=B  i j b i ; 8j2S i ; fori=1;:::;m x e U e ; 8e2E i =it i m X k =1 d k i ; 8i=1;:::;m t i d k i f k (x); 8i; k=1;:::;m d k i 0; 8i; k=1;:::;m  i j
X e  e  i j (e)=c i ; 8j2S i ; fori=1;:::;m  i 1 = i 2 =


= i K i ; 8i=1;:::;m x e 0; 8e2E  i j 0; 8j2S i ; fori=1;:::;m  i j (e)=0or1; 8e2E; t i unrestricted,8i=1;:::;m; wherew m = m ,w i = i  i+1 fori=1;:::;m 1,  i

2(0;1)is givenforeachi, and P m i=1  i =1. The individualfunction i

istherstisumoftheordered multipleobjectivefunctions

i

intheallocation pat-ternx=fx

e

je2Egandthebandwidth i

allocated toclassi. Here,weletK

i

in(MP1)beaxednumber forthediscussionunder deterministicassumptionof feasibilityof(MP1). Ingeneral,K

i

mayberandom which isbeyond scopeofthethesis.

5 Modications of Nonlinear Parts

Werewrite(MP1)asthefollowingmodel(MP2).

Maximize m X i=1 iw i t i m X i=1 m X k =1 w i d k i subjectto X e2E  e x e =B X i X j A i j (e)=x e ; 8e2E X i (K i
c i + i )=B x e U e ; 8e2E d k i 0; 8i; k=1;:::;m A i j (e)+b i M
 i j (e); 8e2E; 8j2S i ; fori=1;:::;m A i j (e)M
(1  i j (e)); 8e2E; 8j2S i ; fori=1;:::;m

e  e A i j (e)=c i ; 8j2S i ; fori=1;:::;m t i d k i z i 1 ^ f i (0)+z i 2 ^ f i (b i;1 )+z i 3 ^ f i (1) +z i 4 ^ f i (b i;2 )+z i 5 ^ f i (b i;3 ) z i 6 ^ f i (10) z i 7 ^ f i (b i;4 ) z i 8 ^ f i (M i ); 8i; k=1;:::;m  i =z i 2 b i;1 +z i 3 +z i 4 b i;2 +z i 5 b i;3 +10z i 6 +z i 7 b i;4 +z i 8 M i ; fori=1;:::;m z i 1 y i 1 ; fori=1;:::;m z i k y i k 1 +y i k ; 8k=2;:::;7; i=1;:::;m z i 8 y i 7 ; fori=1;:::;m P 8 k =1 z i k =1; fori=1;:::;m P 7 k =1 y i k =1; fori=1;:::;m y i k =0or1;8k=1;2;:::;7; i=1;:::;m z i k 0;8k=1;2;:::;8; i=1;:::;m x e 0; 8e2E  i j (e)=0or1; 8e2E; 8j2S i ; fori=1;:::;m A i j (e)0; 8e2E; 8j2S i ; fori=1;:::;m t i unrestricted,8i=1;:::;m; wherew m = m ,w i = i  i+1 fori=1;:::;m 1,  i

2(0;1)isgivenforeachi, and P m i=1  i =1. 6 Conclusions

Inthiswork,wepresentanapproachforthefair re-sourceallocationproblemandQoSroutinginAll-IP networksthat oermultiple servicestousers.Users' utilityfunctionsaresummarizedbymeansof achieve-mentfunctions. First,wendthat theachievement functioncanmapdierentcriteriaontoanormalized scale. Theachievementfunctionalsocanworkinthe Ordered WeightedAveragingmethod. Moreover,it may be interpreted as a measure of QoS on All-IP networks.Usingthebandwidthallocationmodel,we canndaParetooptimalallocationx



ofbandwidth onthenetworkunderalimitedavailablebudget,and this allocationcanprovidetheso-called proportion-al fairness to everyclass i. That is, this allocation canprovidethesimilarsatisfactiontoeachuserinall classes. Wealsondthebandwidthallocatedtoeach classi. Moreover,weobtainthemaximalrate,which the link can oer to each class. Next, we present arouting scheme under considering thedelay. This schemeaimsatseekingapathforwhichtheresidual

maximalrate(i.e.,afterestablishingthenew connec-tion) of its bottleneck link is maximal. This opti-mal path provides the End-to-End QoS guarantees toeachuser.



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