A Proportional Differentiation Model Based on Service Level

Transcription

1 ppl. ah. If. c. 6 o. pp ppld ahmacs & Ifomao ccs Iaoal aual ccs ublshg Co. opooal Dffao odl d o vc Lvl K-o Cho Dpam of Idusal & aagm gg Hau Uvsy of og uds Yog Koa Cospodg auho: mal: cho96@hufs.ac. cvd Ju ; vsd ach ; ccpd Ju ublshd ol: Jauay bsac: I hs pap w cosd a popooal dffao modl bd o h pfomac mc of svc lvl whl mos xsg sachs o h popooal dffao us oh mcs such avag wag m ad pac loss pobably. us a mc calld ufll a o mu h svc lvl of h affc clss. s a mplmao mhod of h popooal dffao modl w suggs a m-dpd poy schdulg whch xpsss h poy of h affc cls a la fuco. fd ou how o adjus h poy fuco paams o achv h dd popooal dffao hough h ympoc aalyss of h quu wag m dsbuo a wo-cls quug sysm wh a osso aval pocss. umcal xpms show ha h schdulg mhod s ffcv fo mplmg h popooal dffao modl spcally wh h qud svc lvls a hgh. Kywods: opooal Dffao Dffad vc Tm-Dpd oy Ioduco opooal dffao s a yp of lav dffad svcs appoach fo I applcaos. Th lav dffad svcs appoach goups h wo affc o sval clss ad ods hm bd o h poy. o a slcd pfomac mu h wo opao s o ma su ha a hgh-poy cls xpcs b svc ha a lowpoy o ms of h mu. h popooal dffao pfomac mus should b popooal o h dffao paams. uppos m s a pfomac mu fo cls. Th popooal dffao s ha m. m j wh δ s h dffao paam fo cls. Th a sval lau o popooal dffao modls. Dovols ad amaaha [] compa vaous appoachs fo dffad svcs ad dscb schdulg mhods fo j sval popooal dffao modls. Dovols al. [] cosd dlay dffao whch uss h avag quug dlay h pfomac mu ad how h popooal dlay dffao ca b appoxmad by sval schduls. Lug al. [3] sudy a wocls c wh boh aval pocsss a osso. Thy fd sval pops o schdulg paams fo popooal dlay dffao bw h wo clss. os suds o popooal dffao ad h aalycal suls cosd a dffao bd o avag wag m whch s calld popooal dlay dffao. lhough avag wag m s h mos commo mc boh hoy ad pacc h a oh sg pfomac mus ca ccumsacs. o xampl call cs o of h mpoa pfomac mus s svc lvl whch s dfd a popoo of cusom calls wd a spcfc m fd o accpabl wag m T [4]. Ths d of pfomac mu

2 454 K-o Cho: opooal Dffao odl s had o aalyz bcaus qus fomao o jus o h avag of wag m bu o s dsbuo. I hs pap w cosd a pfomac mu calld ufll a u whch s dfd o b h pobably ha h oal wag m of cls s log ha T ; u := >.. c h ufll a s dfd h pobably ha h wag m s log ha T o ca g h svc lvl by ag ou h ufll a fom %. o ha h ufll a dcs T cs. I oh wods wh h cusoms a wllg o wa fo a log m lag h ufll a svc lvl bcoms low hgh. Usg h pfomac mu of h ufll a w cosd a popooal dffao modl bd o svc lvl ad suggs a schdulg dscpl whch ffcvly mplms. I h x sco w dscb h schdulg dscpl usd hs pap o mplm h popooal dffao. co 3 shows aalycal suls o h pfomac mu ad suggss how o us h schdulg dscpl o achv popooal dffao bw clss. f gvg umcal xampls llusag how ffcvly h schdulg dscpl wos co 4 w d hs pap wh cocludg m co 5. Tm-Dpd oy To mplm popooal dffao may sachs cosd h m-dpd poy TD schdul whch s a opmpv pac schdulg dscpl cg h poy of a pac wh s wag m. Th a may vsos of TD schdulg ad w f h ad o ssaf ad olch [5] fo dals o hm. I hs pap w us a TD schdul whos poy fuco q s dfd follows; f a aggd cls pac avs a m τ h s poy a m τ s q = + τ.. aam dms h poy bw clss ad a hgh-poy cls s sgd wh a lag. lhough s fom a hgh-poy cls a pac dos o hav hgh poy ov vy pac fom a low-poy cls. c h poy cs wh h lapsd m h sysm low-poy cls pacs hav hgh poy ha hgh-poy cls pacs whch hav o sp much m wag h sysm. gu. shows a plo of h poy fuco.. ssum ha cls h a hgh poy ha j > j. If a cls pac avs a τ always h a hgh poy ov a cls j pac whch h o avd bfo τ j τ. I addo s poy q s hgh ha a cls j pac whch h avd bfo f q j < q.. τ j > τ j. poy j τ j j τ q j q gu.: oy of a cls pac s s o wh avs a τ h cs laly wh m. 3 opooal Dffao I hs sco w cosd a wo-cls quug sysm wh a..d. aval m pocss ad a..d. svc m pocss. Cls s sumd o hav hgh poy ha cls ad h popooal dffao s sad by h ao of h ufll : u 3. u To mplm h ufll a dffao w cosd h TD poy fuco 3. wh >. h h poy fuco a cls pac h hgh poy ha cls pacs whos avg m τ s al ha h cls pac aval m τ bu wh a val of Δ :=.. τ Δ < τ < τ. I od o aalyz h wag m dsbuo fs w compa h svc ods ud h TD schdulg wh h pac aval ods. gu 3. llus ha cls pacs hav hgh poy ov a umb of cls pacs whch hav avd bfo hm. Ud h TD schdulg wh poy fuco 3. h cls pac whch h avd a τ calld τ -cls pac could b svd bfo h cls pac whch h avd bw τ Δ ad τ calld [τ Δ τ ]-cls pac. c h TD schdulg s o-pmpv h cls pac cao b svd bfo h cls pac whos svc h alady sad. I oh wods h τ -

3 vc d ud TD K-o Cho: opooal Dffao odl cls pac cao cach up h [τ Δ τ ]-cls pac whch s o b svd bfo τ. s fo hos cls pacs havg avd bfo τ Δ h τ -cls pac h a low poy ad cao cach hm up a all. Compad wh h wll-ow s-i-s- u I polcy wh h svc od s h sam h aval od h TD schdulg ca duc h wag m of a cls pac by much h aval m whv cachs up a cls pac. L do h umb of h [τ Δ τ ]-cls pacs. Hc s h maxmum umb of h cls pacs ha h τ - cls pac ca cach up. I h xampl of gu 3. s qual o h umb of wh ccls bw τ Δ ad τ. Ud I polcy h τ -cls would b svd a h 5-h poso amog h 5 pacs gu 3. bcaus h pacs a svd accodg h aval ods. m cls cls τ Δ Δ val d τ poy gu 3.: Ud TD polcy h svc od ca b dff fom h aval od. Th cls pac avg a τ would b svd bfo h cls pac whos aval m s [τ Δ τ ] ulss s o b svd bfo τ. Isad ud h TD schdulg would cach up wo cls- pacs ad b svd a h 3- d poso f o of hos cls- pacs do o sa o b svd ul τ. Th dffc of wag m of h 3-d ad 5-h posos ca b lag wc h aval m. Thus w ca s ha h TD schdulg ca duc h wag m of h τ -cls pac by a mos. If w ma fuh sumpos gadg h pu pocss ha h pac aval follows a osso pocss ad h pcag of h cls affc s < θ < h h a osso dsbuo wh ma θ Δ. d w hav h followg sul o h ufll a of cls. Th poof ca b foud ppdx. Thom If h xss γ > sfyg wh 3. dos h cumula gag fuco: : log[ ] h fo a cosa C whch dos o dpd o Δ o θ lm u C. 3.3 Thom suggss ha h ufll a fo hghpoy cusoms ca b appoxmad follows; [ ] u C 3.4 ca us h smla agums fo Thom o oba a sul o h low-poy cls wll. L do h umb of cls pacs whch av dug m slo [τ τ + Δ ]. Ths umb pss how may cls- pacs ca cach up h cls- pac whch h avd a τ calld τ -cls pac. s o cls- pac cachs up h τ -cls pac h wag m of h τ -cls pac cs by much h cls- pac svc m. Thus ud h TD schdulg h wag m of h τ -cls pac ca b cd by a mos. Ud h sumpos ha h pac aval follows a osso pocss ad h pcag of h cls affc s θ h a osso dsbuo wh ma θδ. d w hav h followg sul o h ufll a of cls. Is poof s gv ppdx. Thom If h xss γ > sfyg 3. ad h wh h sam cosa C 3.3 lm u C. 3.5 Thom suggss ha h ufll a fo lowpoy cusoms ca b appoxmad follows; u C. 3.6 Usg h abov ympoc suls o h ufll w xam how h wo svc povd ca cool h svc lvl dffao bw h wo affc clss. om Thom ad w ca ly g h followg sul gadg h popoo of h ufll. Coollay If h xss γ > sfyg 3. h u lm. 3.7 u Th gh-had sd of 3.7 calld h ympoc ao gvs a h o how o popooally dffa bw h hgh- ad low-poy clss. To achv h popooal dffao gv by 3. h poy fuco paam

4 456 K-o Cho: opooal Dffao odl ad ca b slcd such ha h dffc Δ = sfs h followg quao: 3.8 oh wods l. 3.9 c h ympoc ao 3.7 s o lag ha h ufll a of hgh-poy cls u s o ga ha ha of low-poy cls u. I bcoms oly f Δ = whch m q = q ad vually o poy dffc bw h clss. o ha h ympoc ao dos o clud θ h pcag of h hgh-poy cls affc. Thus Coollay mpls ha h pcag of h hgh- ad low-poy cls affcs h ll fluc o h popoo of h ufll wh h T s lag ough ad oh wods a hgh svc lvl s qud. 4 umcal xpms I hs sco w coduc umcal xpms o vfy h suls o h popooal dffao dscussd h pvous sco. Th oal affc aval a λ s sumd o b.9. y sval svc m dsbuos wh ma. Th poy paam gap Δ s s o 5 ad w cosd wo dff combaos of h affc clss θ =.3.8. gu 4. ad 4. plo h ufll obsvd by smulao fo boh clss ad h ao wll h ympoc ao dfd 3.7 fo θ =.3 ad.8 spcvly. Thy show ha h ao of h ufll u /u covgs o h ympoc ao T cs. I vy c h gap bw h obsvd ad ympoc aos bcoms small wh T s log ough o ma h ufll ud %. Th umcal suls also show ha h appoxmao of u /u by h ympoc ao wos b fo a svc m wh a lag squad coffc of vaao c. h h svc m h small vaao.g. c < b of gu 4. ad 4. s aly obsvd ha h wag m s log ha T whch m ffc o vfy a ympoc sul hough smulao. Th ufll of boh hgh- ad low-poy clss c θ cs fom.3 gu 4. o.8 gu 4.. u h ao sms o ma a h sam lvl clos o h ympoc ao wh T s lag. s Coollay mpls h a xpoal vc Tm b amma vc Tm c =.5 c amma vc Tm c = gu 4.: ao of ufll θ =.3 pcag of hgh- ad low-poy clss h ll ffc o h ao of h svc lvls wh h svc lvls a hgh ough. Coollay also suggss ha h wo svc povd ca cool h lav svc lvls bw affc clss hough paam Δ. ow coduc aoh umcal xpm ad dmosa how wll h ympoc ao appoxm h acual ao of h ufll fo dff valus of Δ. Th svc m s sumd o hav a xpoal dsbuo ad oh paams a s o h sam gu 4. ad

5 K-o Cho: opooal Dffao odl Tabl 4.: cual ao of h ufll obsvd by smulao alog wh h ympoc ao Δ u = u u /u u = 3 u u /u a xpoal vc Tm b amma vc Tm c =.5 c amma vc Tm c = gu 4.: ao of ufll θ = Tabl 4. lss h xpm suls such u u ad h ao fo wo dff Ts = 3. choos Ts such ha h ufll fo boh clss a lss ha %. Tabl 4. shows ha mos cs h acual ao u /u obsvd by smulao s vy clos o h ympoc ao suggsd Coollay. h a sho T = h gap bw h acual ad ympoc aos gs bgg Δ cs γ =. hs xampl. h oh had cs = 3 h ympoc ao appoxm mo closly h popooal dffao of h ufll. I od o ma h ufll a of h hgh-poy cls low say 6% of ha of h low-poy cls h svc povd ca fd a pop valu of Δ usg 3.9. I hs xampl lu /u /γ = l.6/. 5. ad h umcal suls Tabl 4. show ha wh Δ = 5 acually h obsvd ao s clos o 6% u /u = fo = 5 3 spcvly. 6 Cocludg m I hs pap w suggs a popooal dffao modl bd o a pfomac mu of ufll a. s h mplmg mhod of h popooal dffao w suggs a TD schdulg dscpl wh poy paam. c s dffcul o mu h xac pfomac ud h TD schdulg dscpl w dv ympoc suls o h wag m dsbuo ad appoxma h ao of h ufll. dmosa ha h appoxmad ao ca b usd ffcvly o cool h dd popooal dffao bw h hgh- ad low-poy affc clss wh h pac aval follows a osso pocss. Th appoach usd hs pap ca b clss. I qus addoal aalyss o h svc od ad wag m of h pacs fom ach cls whch s lf fo fuu sudy. lso mgh b vfd by mo umcal xpms whh h TD schdulg dscpl suggsd h pap m ffcv wh h a may affc clss. xamg oh yps of TD poy fucos s aoh possbl aa fo fuh sachs. cowldgms Ths wo w suppod by Hau Uvsy of og uds sach ud. ppdx

6 K-o Cho: opooal Dffao odl 458 us h followg oaos fo h poof of Thom hs sco ad Thom ppdx. H: s of aval dcs of hgh-poy cls pacs L: s of aval dcs of low-poy cls pacs : oal wag m of h -h avg pac : wag m h quu of h -h avg pac : svc od of h -h avg cusom ud h TD schdulg dscpl : svc m of h -h dpag pac : wag m h quu of h -h avg pac ud I polcy Ud h TD schdulg dscpl h -h avg pac s svd h od of whch would b dff fom s aval od. If h - h avg pac s cls h dffc bw ad s qual o h aval m bw h -h ad -h avg pacs; fo H.. df aoh oao lad o h hghpoy cls svc lvl; : : L fo H.. s mod co 3 a hgh-poy pac ca cach up wh oly h low-poy pacs whch hav avd a mos Δ m us al. Th l m. dos h umb of hos low-poy pacs. Thus m h als svc od ha h -h avg pac could a wh s h hgh-poy cls. Hc fo H..3 If > m h -h svc h sad bfo h -h pac avs; fo H..4 d bfo gg svd h -h pac ds o wa a mos fo h pacs svc od of hough o fsh h svc; fo H..5 ow w show ha fo H.6 wh s..d. ad h h sam dsbuo...7 om.4 ad.5 w hav..8 om h aov s qualy xp xp.9 fo a abay. om h dfo of. w hav ha h h sam dsbuo m wh s dfd follows; ; L. ad s a..d. squc whch h h sam dsbuo. Thus fo wh f ] [xp.. c h gh-had sd of h qualy. dos o dpd o covgs o ufomly. h.6 hs mpls ha.. I od o compl h poof w d o show ha fo som cosa C xp lm C

7 K-o Cho: opooal Dffao odl.. L..3 ad df = ad..4 If [ ] h covgs waly o a adom vaabl whch h h sam dsbuo max [6]. Usg h sul h poof of Thom of lsma ad ag [7] w coclud covgs dsbuo o ad s dpd of ad. om h abov agum wh w oaos w hav ha lm.5 wh s a adom vaabl havg h sam dsbuo ad dpd of ad. Usg.4 w df f : T wh T = +. Th max..6 us xpoal wsg [6]. pcally w us γ-wsg of ad ad γ-wsg of ad do h us of a wsd mu compug xpcaos by. o ha τ s a soppg m fo ad s a soppg m fo. om Thom II.4. of smuss [6] w hav ha xp xp xp ;.7 wh h smcolo sd h xpcao dc ha h xpcao s valuad ov h v af h smcolo. Usg h dfo of γ w duc h abov quao fuh; xp ; xp T ;.8 wh. c a cumula gag fuco s covx [8] ad.9 ad hus h v of τ < h pobably o. Th adom vaabl T s dpd of ad T. om Coollay 8.33 of gmud [9] w hav ha T Z 459 C lm. wh Z s a ladd vaabl. c h dsbuo of Z s dpd of w hav lm C C. wh C : C. f γ-wsg h a xpoal dsbuo wh ma /λ + γ. I m ha ud h wsd mu h a osso dsbuo wh ma λ + γ θ Δ. Thus xp xp xp ad w hav.3. /!. ppdx If h -h avg pac s cls h dffc bw ad s qual o h oal svc m of h -h hough -h dpag pacs; followg fo L.. o h aalyss of h low-poy cls svc lvl w oduc a w vaabl whch s smla o. fo h hgh-poy cls. o h -h avg pac whch h low poy.. L : : j j H fo L.. Th mag of s h l svc od ha h -h avg pac could a wh s h low-poy cls. If < h xss a l o hgh-poy pac whch avs wh Δ m us af h -h pac avd bu cao cach up wh. Ths m h low-poy pac sad o g svd bfo h hgh-poy pac avs. Is wag m h quu mus hav b lss ha Δ < Δ. om. h followg qualy holds; fo L..3 ow w show ha fo L /

8 K-o Cho: opooal Dffao odl wh s..d. ad h h sam dsbuo...5 om.3 w hav. / /.6 om h aov s qualy xp / /.7 fo a abay. om h dfo of. w hav ha h h sam dsbuo whch s a osso adom vaabl dfd follows; ; H.8 ad s a..d. squc whch h h sam dsbuo. Thus fo wh f ] [xp / / /..9 c h gh-had sd of h qualy.9 dos o dpd o covgs o ufomly. h.4 hs mpls ha.. I od o compl h poof w d o show ha fo som cosa C xp lm C.. Usg a smla mhod h poof of Thom w ca show ha covgs dsbuo o ad s dpd of. Thus lm. wh s..d. wh h sam dsbuo ad dpd of ad. Usg.4 w df : f T wh T =. Th max..3 h γ-wsg of ad w hav ha ; xp xp..4 Usg h dfo of γ w duc h abov quao fuh;. ; xp ; xp T.5 h h wsd mu ] [ ad hus h v of τ < h pobably o. Th adom vaabl T s dpd of ad T. om Coollay 8.33 of gmud [9] w hav ha Z T C xp lm..6 c h dsbuo of Z s dpd of w hav ha C C lm

9 K-o Cho: opooal Dffao odl wh C : C. o ha h cosa s h sam Thom. c s dpd of boh ad xp xp ad w hav.. xp /!.8 fcs [] C. Dovols ad. amaaha c fo lav dffad svcs ad h popooal dffao modl I wo Vol [] C. Dovols D. lads ad. amaaha opooal dffad svcs: dlay dffao ad pac schdulg I/C T. wog Vol. o. -6. [3]. K. H. Lug J. C.. Lu ad D. K. Y. Yau Chaaczao ad fomac valuao fo opooal Dlay Dffad vcs ocdg of h I Iaoal Cofc o wo oocols ov saa Japa. [4] L. ssaf ad. olch Tm dpd pos call cs Ia. J. of mulao Vol.6 o [5] L. ssaf ad. olch fomac valuao of oy d chduls h I ocdgs of h 7h uopa mulao ulcofc Ju 3 ogham UK. [6]. smuss ppld obably ad Quus ly w Yo 987. [7]. lsma ad Y. ag Ladm-voy adoffs smbl o od sysms paos sach Vol.46 o [8]. Kdall dvacd Thoy of cs Vol. 5h d. xfod w Yo 987. [9] D. gmud qual alyss : Tss ad Cofdc Ivals pg w Yo 985. K-o Cho cvd.. ad.. dgs dusal gg spcvly fom oul aoal Uvsy Koa 99 ad fom KIT Koa 993 ad h.d. dg dusal ad sysms gg fom oga Isu of Tchology U 3. H s culy a ssoca ofsso a Dpam of Idusal ad aagm gg Hau Uvsy of og uds Koa. vously h wod a lcocs ad Tlcommucao sach Isu ad amsug Daa ysms. Hs sach ss a lcommucao wos ad qualy of svcs.