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SRR:AnO(1)Time-ComplexityPacketSchedulerforFlowsinMultiservicePacketNetwork

Ghuanxiong GuoIEEE/ACMTransactiononNetworking,Vol.12,No.6,December2004

Presenters:Taeju Park,Yibo Pi

Introduction• DifferenttypesofservicesontheInternet:• Delayinsensitive:email• Delaysensitive:videoandaudioconferencing

• Resourceisolationisneededtoprovidequalityofservice(QoS)• Flowsareservedbasedontheirrequirements

• PacketScheduler• Decidewhichpackettobetransmittedwhentheoutputlinkisidle

Schedulerw1

w2

w3

Fairness:thenumberofbitsservedforeachflowisproportionaltotheirweights.

IntrotoFairQueueing

Firstinfirstout(FIFO):noisolationamongdifferentflows

12 2 2Scheduler

Schedulerw1

w2

w3

Bit-by-bitroundrobin

round1round2

round3

Ineachround,Flow1:w1 bitsFlow2:w2 bitsFlow3:w3 bits

GeneralizedProcessorSharing(GPS)

GPS:idealfairness,butnotpracticaltouse

PacketizedQueueingSchemes

• Weightedfairqueueing(WFQ)• Deficitroundrobin(DRR)

120

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

0

0

100 200

50100150 100

PacketizedQueueingSchemes

• Weightedfairqueueing(WFQ)• Deficitroundrobin(DRR)

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

0

100

50100150 100 400

PacketizedQueueingSchemes• Weightedfairqueueing(WFQ)• Deficitroundrobin(DRR)

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

0

100

ProblemsofDRR:1)bursty outputand2)short-termunfairness

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

4packets

HowtoimproveDRR?

120

150

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100

0

0

100

100150

0

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

0

HowtoimproveDRR?

120

150

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100

0

0

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100150

0

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

100

HowtoimproveDRR?

120

150

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100

0

0

100

100150

100

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

100

HowtoimproveDRR?

120

150

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100

0

100

100

100150

100

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

100

HowtoimproveDRR

120

150

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

100

200

100

100150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

HowtoimproveDRR

120

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

100

50

100

100150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

HowtoimproveDRR

120

150

w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

200

50

100

100150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2, flow3,flow3

HowtoimproveDRR

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

50

100

100150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

HowtoimproveDRR

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

150

100

100150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

HowtoimproveDRR

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

50

100

150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3, flow3

HowtoimproveDRR

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

150

100

150

Sequenceofservice:flow1,flow2,flow3,flow3,flow3,flow3

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

HowtoimproveDRR?

120w1 =1

w2 =2

w3 =4

DeficitcounterQuantum=100100

50

0

100

sequenceofservice:flow1,flow2,flow3,flow3,flow2,flow3,flow3

#serviceofflow1:1#serviceofflow2:2#serviceofflow4:4 4consecutivepacketsfromflow3à 2consecutivepacketsfromflow3

2packets 2packets1packet

ThedesigngoalofSmoothedRoundRobin

WeightedFairQueueingPro:Short-termfairnessCon:highcomplexityO(#ofactiveflow)

RoundrobinPro:lowcomplexityO(1)Con:short-termunfairness

CombinetheprosofWFQandroundrobin

SmoothedRoundrobinShort-termfairness+lowcomplexityO(1)

WeightSpreadSequence(WSS)

• WSSisaspeciallydesignedsequencethatdistributestheoutputtrafficofeachflowevenly.• AsetofWSSsisdefinedrecursivelyasfollows:• !" = 1

• !% = &'

%= !

%(", *, !

%("

• Totalnumberoftermsin*+, WSSis-./% = 2%− 1

• WSSExample• !2 = 1,2,1,3,1,2,1

• !4 = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• -./4 = 24− 1 = 31

!9

!9

*

WeightMatrix

• Eachflowisassignedaweightinproportiontoitsreservedrate.• (:" =64kb/s,:9 =256kb/s,:2 =512kb/s,:; =192kb/s)=>(<" =1,<9 =4,<2 =8,<; =3)

• Weightof=-><? isencodedasbinarynumber 4 = 1009 inweightmatrix

Thenumberofcolumns=orderofWSS

AB =

AC"

⋮

ACE

=

&",(%(") ⋯ &",I

⋮ ⋱ ⋮

&E,(%(") ⋯ &E,I

Row:weightvectorofaflowIfweightis10,then[1010]wherek=4

Columnnumber

SmoothedRoundRobinScheduler

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<2 = 8,<; = 3

• CorrespondingWSS,!; = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• BasicIdeaofSmoothedRoundRobin(SRR)Scheduler1. scanWSSsequencetermbyterm2. WhenthevalueofthetermisL,M>-NO/%(' oftheWMischosen.3. Inthecolumn,theschedulerscanthetermsfromtoptobottom.4. Ifthetermis1,theschedulerservethecorrespondingflow.

M2 M9 M" MI

scan

scan

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<2 = 8,<; = 3

• CorrespondingWSS,!; = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• Threeasynchronousaction• Schedule,Del_flow,Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4

w3 =8

PQRS =100

0

0

0

100

50100150 100

w4 =3 50100 1000

Deficitcounter

Scheduler

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<2 = 8,<; = 3

• CorrespondingWSS,!; = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• Threeasynchronousaction• Schedule,Del_flow,Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4

w3 =8

PQRS =100

0

0

100

100

50100150 100

w4 =3 50100 1000

Deficitcounter

Scheduler

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<2 = 8,<; = 3

• CorrespondingWSS,!; = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• Threeasynchronousaction• Schedule,Del_flow,Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4

w3 =8

PQRS =100

0

0

50

100

100150 100

w4 =3 50100 1000

Deficitcounter

Scheduler50

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<2 = 8,<; = 3

• CorrespondingWSS,!; = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• Threeasynchronousaction• Schedule, Del_flow, Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4

w3 =8

PQRS =100

0

0

100

100

100

w4 =3 50100 1000

Deficitcounter

Scheduler

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<2 = 8,<; = 3

• CorrespondingWSS,!; = {1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

• Threeasynchronousaction• Schedule, Del_flow, Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4

w3 =8

PQRS =100

0

0

0

100100

w4=3 50100 1000

Deficitcounter

Scheduler

empty

• Fourflowswithfixedpacketsize(=", =9, =2, =;) withcorrespondingweights(<", <9, <2, <;)• <" = 1,<9 = 4,<; = 3

• CorrespondingWSS,!2 = {1,2,1,3,1,2,1}

• Threeasynchronousaction• Schedule, Del_flow, Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4PQRS =100

0

0 100

w3 =3 50100 1000

Deficitcounter

Scheduler

AB =

AC"

AC9

AC2

=

0 0 1

1 0 0

0 1 1

• Fiveflowswithfixedpacketsize(=", =9, =2, =;, =4) withcorrespondingweights(<", <9, <2, <;, <4)• <" = 1,<9 = 4,<2 = 8,<; = 3,w4 = 17 =>CorrespondingWSS,!4

• Threeasynchronousaction• Schedule, Del_flow,Add_flow

SmoothedRoundRobin

100

150

w1 =1

w2 =4

w3 =8PQRS =100

0

100

50

100

100150 100

w3 =3 50100 1000

Deficitcounter

Scheduler

AB =

AC"

AC9

AC2

AC;

AC4

=

0 0 0

0 0 1

0 1

0 0

0 1 0

0 0 0

1 0 0

0 0

1 1

0 1

added

w5 =17 50100 1000

PropertiesofSRR

• Work-conserving• Ifthereareactiveflows,theSRRalwaysforwardit.

• Theorem1:=-><? isvisited<? timesbySRRinaround• Thenumberofreceivedservicebyschedulerofeachflowisproportionaltoitsweight• ThenumberoftheoccurrencesofelementL in!%(1 ≤ L ≤ *) is2%('

• Thenumberofelement3isin!4 is24(2 = 4

{1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1}

PropertiesofSRR:Fairness

• Lemma2(Long-termfairness):Foranypairofbackloggedflows= andW,attheendofaroundinSRR,then

• Corollary1(Short-termfairness):Foranypairofbackloggedflows=andW inSRR,wehave

C?(0, X)

<?

−

CY(0, X)

<Y

Z = 0

C?(0, [) isthenumberoftimesthat=-><? isvisitedbySRRfromtime0to[

!?(0, [)

<?

−

!Y(0, [)

<Y

<* + 2 PQRS

2min(<?, <Y)

!?(0, [) isservicereceivedbyflows= fromtime0to[

Endofaround

PropertiesofSRR:SchedulingDelayBound

• SchedulingDelayBound b?• Schedulingdelay:timebetweenqueuingpacketandtransmittingthepacket.

• Theorem3:Theschedulingdelayboundof=-><? is

• Inverseproportionaltotheweight,proportionaltototalnumberofactiveflows• Cannotprovideastrictlyrate-proportionaldelaybound.• TheyclaimthatthedelayboundismuchbetterthanthatofDRR.

b? <2PQRS

<?

+ (c − 1)2PQRS

dN:thenumberofactiveflows

PropertiesofSRR:Scalability

• DifferentraterangescanbeaccommodatesbyWSSofthesameorderbyadjustingtherategranularity• 1kb/srategranularity• 1Mb/srategranularity

• SRRcanbeusedforvariablebandwidthcapacity• SRRworkswellregardlessofthenumberofflows.• TimecomplexityisO(1)

PropertiesofSRR:Complexity

• Spacecomplexity• -./% = 2

%− 1 becomesverylargeifkislargenumber.

• TheyclaimthateQRS = 32 isenough• Itcanprovide4Tb/sratewithgranularityof1kb/s

• Since 2* +, WSScanbeconstructedbyusing*+, WSSand * + 1 +, WSS,thespacecomplexityofSRRis2"f + g(c×eQRS)

• Timecomplexity• g(1) timetochooseapacketfortransmission• g(*) timetoaddordeleteaflow,where* istheorderofWSScurrentlyusedbySRR.

TostoreeQRS doublelinks.

Evaluation

2videostreamflows

10flowsofself-similartraffic

2ftpflows

Simulationtool:NS2multipleconstantbitrate(CBR)flows

WeightsofCBRflowsarepowersof2

SRRbetterthanWFQ SRRworsethanWFQ

WeightsofCBRflowsarerandomlychosen

SRRbetterthanWFQ SRRworsethanWFQ

WeightsofCBRflowsareequal

Discussion

• Weakness• Thepaperisnotwellwritten• Badworst-casefairness• IgnoretimeoverheadtoconstructhighorderWSS(32+,A!!) usingloworder(16+,&17+,A!!)

• Extension• Singleschedulertomulti-schedulerfairness?• Singeresource(bandwidth)tomulti-resourcefairness?• Queue-independentfairnesstoqueue-dependentfairness?

!?(0, [)

<?

−

!Y(0, [)

<Y

<* + 2 PQRS

2min(<?, <Y)

g(P./k)