Covariances of Linear Stochastic Differential Equations for Analyzing Computer Networks *

Transcription

1 SINGHUA SCIENCE AND ECHNOLOGY ISSNll7-4ll6/6llpp64-7? Volume 6, Number 3, Jue Covaraces of Lear Sochasc Dffereal Equaos for Aalyzg Compuer Neworks * FAN Hua ( 樊华 ),,**, SHAN Xumg ( 山秀明 ), YUAN Ja ( 袁坚 ), REN Yog ( 任勇 ). Deparme of Elecroc Egeerg, sghua Uversy, Bejg 84, Cha;. Deparme of Cemaography, Bejg Flm Academy, Bejg 88, Cha Absrac: Aalyses of dyamc sysems wh radom oscllaos eed o calculae he sysem covarace marx, bu hs s o easy eve he lear case f he radom erm s o a Gaussa whe ose. A uversal mehod s developed here o hadle boh Gaussa ad compoud Posso whe ose. he quadrac varaos are aalyzed o rasform he problem o a Lyapuov marx dffereal equao. Explc formulas are he derved by vecorzao. hese formulas are appled o a smple model of flows ad queug a compuer ework. A sably aalyss of he mea value llusraes he effecs of oscllaos a real sysem. he relaoshps bewee he oscllaos ad he parameers are clearly preseed o mprove desgs of real sysems. Key words: covarace marx; sochasc dffereal equao (SDE); compoud Posso whe ose; rasmsso corol proocol (CP) flow Iroduco Sochasc dffereal equaos (SDE) ca be used o descrbe sysems affeced by radom processes; however, he aalyses of he SDE are very dffcul, especally wh ozero ose a he equlbrum po of he deermsc soluo. Such ose causes a o-eglgble oscllao, so he esmae of hs oscllao becomes very mpora. hs paper preses formulas for he covarace marx of a lear Iô SDE wh addve compoud Posso ose or Gaussa whe ose. he quadrac varao [,] s drecly aalyzed o derve he Lyapuov marx dffereal equao whch he covarace marx obeys. he aalyss Receved: --5; revsed: -4- ** Suppored by he Naoal Naural Scece Foudao of Cha (Nos , 67753, 6674, ad 6935) ad he Naoal Key Basc Research ad Developme (973) Program of Cha (Nos. 7CB37 ad 7CB375) ** o whom correspodece should be addressed. E-mal: fahua@gmal.com; el: he gves explc formulas for he covarace marx. he resuls are used o aalyze a compuer ework. he addve crease ad mulplcave decrease (AIMD) algorhm used by he rasmsso corol proocol (CP) evably causes sysem oscllaos. Alhough he SDE for he wdow sze for a sgle CP coeco s kow [3,4], researchers sll do o kow how o esmae he oscllaos of he ere sysem. he ma dffculy les wrg he SDE for he aggregae flow ad dervg he equaos for he covarace marx. he aalyss hs paper uses a approxmao up o he secod order mome o derve he SDE wh Posso ose for he aggregae flow. he he resul s learzed o ge a lear SDE. A sably aalyss gves s covarace marx whch shows he effec of queue legh varaos o he sysem sably rage. Alhough he covarace marx of he lear Sraoovch SDE wh addve Gaussa whe ose s kow [5], ha of he Iô SDE wh addve Posso ose s sll ukow. Zygadlo [6] proved he Iô SDE covarace marx cao be derved explcly from

2 FAN Hua ( 樊华 ) e al.:covaraces of Lear Sochasc Dffereal Equaos 65 he Sraoovch resul. Grgoru [7] proved ha a real sysem wh jumpg ose coforms o he Iô SDE, sead of he Sraoovch SDE, o model he real physcal process. he marx equaos preseed here ad he mehod used o hadle he aggregae flow boh provde ew sghs o he desgs of real sysems. Covarace Marx Cosder he followg lear Iô SDE wh addve compoud Posso ose: d X( ) AX( )d+ Fd C ( ) () X() X () where X ( ) s he sae vecor of a -dmesoal sysem, A s a cosa marx, F s a m cosa marx, C() ( C(),, Cm()) are m-dmesoal depede compoud Posso processes defed he followg, he dervave coforms o Iô s defo, ad X s he al value whch s a radom vecor depede of he compoud Posso processes. ( [ ] deoes he raspose of [ ].) Each eleme of C ( ) s a compoud Posso process defed as, N ( ) ; N () () ( ) C C Y, λ,,, m (3) Yk, N () > ; k where C () depeds o he depede decally dsrbued γ -valued radom varables { Y k} + whch k have he same dsrbuo as he radom varable Y ad he homogeeous Posso coug process N () has esy λ. { C ( )} m are muually depede. { Y } m are also muually depede. Use he followg symbols ˆ () [ ()] [ [ ] [ ]] C E C λe Y,, λ E Y G, where G [ λe[ Y],, λme [ Ym]] s a m cosa marx. Here, C ˆ () s he predcable quadrac varao (or predcable compesaor) of C ( ), so C () C() C ˆ () s a margale [,]. heorem he mea of sysem {(), ()} s E[ X( )] e A ( E [ X ] + A FG) A FG (4) Iff A s sable (.e., all egevalues of A have egave real pars), he mea coverges ad he lm s lm E [ X( )] A FG (5) + Proof akg he expecao of boh sdes of Eq. (), m m d E[ X( )] ( AE [ X( )] + FG)d (6) he soluo o hs equao gves he heorem. heorem he covarace of sysem {(), ()} s Cov[ X()] E[( X() E[ X()])( X() E [ X()]) ] A A A( s) A ( s) e Cov[ X ]e + e FΓF e ds (7) where Γ dag( λe[ Y ],, λme [ Ym]) deoes a dagoal marx whose dagoal eres are λ E [ Y ],, λme [ Ym]. Iff A s sable, he covarace marx coverges ad he lm of he vecorzao of he covarace marx s lm vec(cov[ X ( )]) + ( ) ( )vec( ) A I + I A F F Γ (8) where s he Kroecker produc, I s a dey marx of sze, ad vec( ) s he colum vecorzao fuco. Proof Usg symbols Ĉ ad C, Eq. () ca be rewre as d X( ) ( AX( ) + FG)d+ Fd C ( ). Subracg Eq. (6) from boh sdes gves: d( X E[ X]) A( X E [ X])d + FdC (9) From ow o, he symbol s omed whe he meag s clear o make he expresso more compac. he erm FdC s he Iô calculus over a margale, so he egral s also a margale. Iegrag by pars usg Iô calculus gves d(( X E[ X])( X E[ X]) ) d( X E[ X]) ( X E[ X]) + ( X E[ X]) d( X E[ X]) + d[ X E[ X],( X E[ X]) ]. akg he expecao of boh sdes wh Eq. (9) ad he margale propery of Iô calculus over he margale C gves dcov[ X] ( ACov[ X] + Cov[ X] A ) d + F d E[[ CC, ]] F () he quadrac varao of C s [,] [ C, C ]() [ C, C]() ( Δ C()) s < k N ( ) < s k, λ, Y C( Y ), [ C, C j], f j. hus, he quadrac varao [ C, C ]( ) self s also a compoud Posso process based o he same Posso coug process N () as C (), bu wh a radom jump scale of Y sead of Y. hus,

3 66 E[[ C, C ]] λe [ Y ]. Usg he defo of Γ, Eq. () ca be wre as dcov[ X] ( ACov[ X] + Cov[ X] A + FΓF )d () hs s a Lyapuov marx dffereal equao. he soluo gves Eq. (7). Whe +, Eq. () becomes ACov[ X( + )] + Cov[ X( + )] A + FΓF () Vecorzao he gves Eq. (8). Corollary If he ose sgals are all pure Posso ose, le Y Y m a.s. (so ha E [ Y ] E [ Y ]), he heorem sll works. Proof Because ff Y a.s., he compoud Posso process C s a pure Posso process. Corollary If he sysem Eq. () s chaged o d X( ) AX( )d+ Fd C ( ) (3) where C () C() E [ C()] C() G as defed before. he equaos for he covarace marx are sll Eqs. (7) ad (8). Proof Noe ha AX ()d + Fd C () ( AX () FG)d+ Fd C ( ). Sce FG s a cosa vecor, remas he equao for d E [ X ] bu cacels he equao for d( X E [ X]). herefore, FG does o affec he equaos for he covarace process, bu does chage he mea value process. Now cosder he followg lear Iô SDE wh addve Gaussa ose: d X( ) AX( )d+ Dd B ( ) (4) X() X (5) where X ( ) s he sae vecor of a -dmesoal sysem, A s a cosa marx, D s a m cosa marx, B ( ) s a m-dmesoal depede Browa moo where he dervave sasfes Iô s defo, ad X s he al value whch s a radom vecor depede of he Browa moo. heorem 3 he mea value of sysem {(4), (5)} s E[ X( )] e A E [ X] (6) he covarace of sysem {(4), (5)} s A A A( s) A ( s) Cov[ X( )] e Cov[ X ]e e DD e ds + (7) Iff A s sable, he mea coverges o ad he covarace marx coverges o lm vec(cov[ X ( )]) + sghua Scece ad echology, Jue, 6(3): 64-7 ( A I + I A) ( D D)vec( I m) (8) Proof he proof s smlar wh hose of heorems ad. he oly dfferece les he quadrac varao of he Browa moo: d[( X( ) E[ X( )]),( X( ) E[ X( )]) ] d[ DB(), B () D ] Dd[ B(), B ()] D DD d. Replacg FΓF Eq. () by DD he gves Eqs. (7) ad (8). SDE Model for he Aggregae Flow Usg CP Oe of he mpora applcaos of covarace esmaes egeerg sysems s he aalyss of CP flows compuer eworks. Wh he AIMD algorhm, boh he CP flows ad he rouer queue legh oscllae evably. he effcecy of he acve queue maageme (AM) algorhm he depeds o esmaes of hese parameers, especally he rouer queue legh. Accurae esmaes of o oly he mea queue legh bu also s varace wll provde more precse operag parameer rages o more effecvely avod cogeso he rouers. he ypcal sgle boleeck ework show Fg. s used as a smple example o llusrae he mehod. Fg. A sgle boleeck ework here are a oal of rasmers S,, S sedg edless fp flows o he correspodg recevers D,, D uder CP Reo. RA ad RB are wo rouers coeced by a commucao lk wh badwdh C ad rasmsso delay R /. We gore he me delays bewee S ad RA ad bewee D ad RB. Furher assume ha he processes o RB ad all he recevers very fas ad here s o cogeso of he reur ACK flows. I s he a ypcal sysem wh a sgle boleeck a rouer RA. Msra e al. [3,4] gave he SDE wh Posso ose o descrbe he evoluo of he wdow sze for each sgle CP coeco: () () W () dw d dn,,, (9) R

4 FAN Hua ( 樊华 ) e al.:covaraces of Lear Sochasc Dffereal Equaos 67 where meas he lef lm of (because here are jumps he Posso process), R s he roud- () rp me for sace, { N } are depede decally-dsrbued Posso processes deog he arrval processes of cogeso messages se back o each rasmer, ad { F } s he flrao (or referece famles) geeraed by hese Posso processes. hese Posso processes have he same eses () λ λ/ because he rasmers are homogeeous. I addo, alhough λ s me-vara, chages much more slowly ha he sysem saes due o he expoeally weghed movg average (EWMA) roduced he followg. hus, whe λ s calculaed over a shor perod of me or ear he fxed mea sysem value, λ ca be reaed as a me-vara parameer. o furher smplfy he problem, Eq. (9) uses he delay-free assumpo, gores he slow sar erm, ad oly calculaes he rasmsso delay ad he queug delay for he boleeck ode RA. Despe hese smplfcaos, Msra e al. [3,4] have show ha her SDE model ca quaavely descrbe a real CP coeco. hs SDE for a sgle CP coeco s ow exeded o a SDE for he aggregae flow o reduce he umber of SDEs for he ere sysem whle modelg oscllaos as precsely as possble. Summg up all he wdows gves W W. () Dffereag boh sdes ad usg Eq. (9) gves () () () dw dw d ( W d N ) R + C () Sce he Posso processes have depede cremes, E () () λ () λ ( W d N ) W d W d F () () () λ () Var ( W d ) N F ( W ) d () By he defo of he faress dex [8] ( W ) () ( W ) a (3) where a s he faress dex. May researchers have calculaed hs dex usg smulaos. For geeral CP flows, he faress dex s bewee.5 ad, whle for CP/RED, s ofe larger ha.75 [9]. herefore, a s bewee ad.3. he Eq. () ca be rewre as () () aw Var ( W d N ) λ F d where Var N N λ ds ad s aw d N F (4) N N. hus, N s a Posso process wh esy λ ad wh N as s correspodg Posso margale. Combg Eqs. () ad (4) gves a approxmao of he radom par of he oal creme o he d order mome, () () λ aw ( d ) W N Wd dn (5) hus, Eq. () ca be rewre as λ aw dw W d dn (6) R + C hs descrbes he umber of packes arrvg a RA a me, whch s he wdow sze for he aggregae flow. I pracce, rouers use some algorhm o adjus λ accordg o he queue legh o avod cogeso, such as radom early deeco (RED) []. CP/RED ca be modeled as a feedback corol sysem []. he equao for he sadard RED algorhm wh EWMA s W W λ f ( A) f( A) (7) R R + C (), A < qm; A ( ) qm max, m max; qmax qm, qmax < A f A p q A q (8) da l( h) ( A ) (9) d R + C where s he rouer queue legh, A s he expoeally weghed movg average of, h s a wegh whch geerally equals. ad q, q, ad m max p max are preassged parameers [3]. he RED queue legh performace ca be mproved by usg explc cogeso ofcao (ECN) [], where he orgal packe s rasmed forward as usual wh oly a ofcao se back o

5 68 sghua Scece ad echology, Jue, 6(3): 64-7 s rasmer. ECN s used here o smplfy he equao so ha ca be solved explcly. W max C,, ; R + C d W C, < < buff; (3) d R + C W m C,, buff R + C he prmary equaos Eqs. (6), (3), ad (9) plus he assocaed equaos Eqs. (7) ad (8) gve a egraed descrpo of he aggregae flow he ework Fg. for CP/RED corol. 3 Sably Aalyss of he Mea Value he sably aalyss of he sysem mea seeks o fd he rage of he umber of coecos for whch he sysem has a sable fxed mea. he expecaos of boh sdes of Eq. (6) are dw λ W (3) d R + C he fxed operag po ( WA ˆ, ˆ, ˆ) of he sysem {(3), (3), (9)} he hree-dmesoal space [, + ) [, buff] [, buff] s Aˆ ˆ (3) Wˆ ˆ + CR (33) Oly < Aˆ ˆ qmax wll gve a reasoable fxed operag po. For hs codo, ˆ obeys he equao ˆ ˆ ( qmax qm ) ( + CR) ( qm ) (34) pmax ˆ ca he be deermed eher by usg Cardao s formula or umercally. From codo < ˆ qmax ad Eq. (34) gves Proposo. Proposo o esure a reasoable fxed operag po for he sysem mea, he umber of coecos has he upper boud pmax ( qmax + CR) (35) For he sably aalyss, learze he SDE {(6), (3), (9)} ear ( WA ˆ, ˆ, ˆ). Noe ha he radom erm Eq. (6) s o zero a ( WA ˆ, ˆ, ˆ), so he soluo cludes he lear erm for he deermsc erm of he SDE ad he cosa erm for he radom erm of he SDE. hus, X ( W Wˆ, ˆ, A Aˆ), K ( A qm ) W W l( h) ( W ) ( ) A C A,, R+ R+ R+ C C C ( W ) ( ˆ ˆ ˆ,, A W,, A) Φ,, ˆ ˆ ˆ K ˆ K K( A q ˆ m ) W ( A qm) W W ˆ ˆ ˆ R + C R R C + + C C Wˆ φ φ φ ˆ 3 φ3, ˆ R + C CR+ φ4 φ 4 C Aˆ R + l( h) l( h) C ˆ ˆ R R + + C C where K pmax /( qmax qm ), φ ˆ CK( qm), φ aw ˆ a ( + CR) CK( ˆ + CR)/( ), φ ˆ 3 C/( + CR), φ4 C l( h) / ( ˆ + CR). hs soluo has subsued Eqs. (7) ad Ψ. (8) o Eq. (6), ad he zero he frs row of he ( ˆ ˆ ˆ WA,, ) secod colum s due o Eqs. (3), (33), ad (34).

6 FAN Hua ( 樊华 ) e al.:covaraces of Lear Sochasc Dffereal Equaos 69 ˆ ˆ W λ K ( A ˆ q ˆ m ) CK ( q m ). ˆ R + C he, he learzed equao of he sochasc sysem {(6), (3), (9)} s dx Φ X d+ Ψ dn (36) Some approxmaos gve [3] Proposo. Proposo o esure all he egevalues of Φ have egave real pars, should sasfy he followg codo: 3 Kq ( m + CR) > (37) ( qm + CR+ h ) If here are o radom oscllaos he sysem,.e., Eqs. {(3), (3), (9)} are he exac sysem equaos, Eqs. (35) ad (37) ca be used as he upper ad lower bouds for he admsso corol pracce. 4 Covarace of he CP/RED Sysem Real CP/RED sysems always have oscllaos whch affec he sably of he fxed operag po. he covarace marx gves he amplude of he radom oscllaos o udersad he operaos of real sysems. For he sysem Eq. (36), le V lm Cov[ X ]. + From Corollary, V obeys Eq. (8). o furher clarfy, Eq. () he proof of heorem ca be used drecly o ge ΦV + VΦ + Ψ Ψ ˆλ. Subsug Ψ ad ˆλ o he equao ad usg Eq. (34) gves ac ΦV + VΦ + (38) hus, he covarace marx V s approxmaely proporoal o he badwdh C (eglecg he olear effec Φ ). hus, whe he badwdh creases, he sysem loses s sably as show by Low e al. [4] who dd o clearly expla he cause because her model oly cluded he deermsc par. he resul Eq. (38) s compared wh a real sysem for a smulaed ework usg s-. he ypcal evromeal parameers are: packesze KBye 8 Kb, C 4 Mbps 3 Kpackes/s, R. s, q m 8 Kb packes, q max. 4 Mb 3 packes, p max., h,. buff 4. 8 Mb 6 packes, ad a.. From Eqs. (35) ad (37), he upper boud o he umber of coecos s 34 ad he lower boud s. he, calculae he covarace marces V Eq. (38) for s 5, 75, ad 3. V ad he modfedcorrelao marx ϒ for each, whch s he same as he correlao marx excep ha he elemes of he dagoal are he relave sadard devaos (all hese dagoal elemes would be for he correlao marx.), are: V , ϒ ; V , ϒ ; V , ϒ he relave sadard devao of a radom varable Z s defed as: RSD[ Z] Var[ Z] /E[ Z]. Boh he varaces ad he relave sadard devaos of W ad decrease dramacally as creases. For ear he lower boud, he varace of ca o be egleced. Boh he varace ad he RSD of he EWMA varable A are much smaller ha ha of, whch meas ha he oscllaos of A are much smaller ha ha of, whch s he objecve of EWMA. he relaoshp bewee he rouer queue legh ad he umber of coecos predced by he prese model are compared wh hose of a smulaed sysem based o he mea ( ˆ ) ad he varace (he secod row he secod colum of V ) as he umber of coecos creases. I he smulaos, was

7 7 creased from 5 o 5 seps of 5. Each sep was smulaed for s wh he las 5 s of he rajecory used o calculae he mea ad he varace (he smulao me sep was.5 s). he resuls are show Fg. where he abscssa s he umber of coecos,, ad he ordae s he queue legh,. he wo horzoal doed les deoe q m ad q max. he wo vercal doed les deoe he lower boud low from Eq. (37) ad he upper boud up from Eq. (35). he upper par of curves s he mea calculaed by he smulao (dashdoed curve) ad by Eq. (34) (sold curve). he lower par of curves are he sadard devaos (he square roo of he varace) of calculaed by he smulao (doed curve) ad Eq. (38) (dashed curve). sghua Scece ad echology, Jue, 6(3): 64-7 slow he crease of he rasmsso wdow sze afer recevg a ECN, whch s egleced he aalyses. Also whe he varace s large, he learzao precso decreases. Despe hese smplfcaos, hs model ca sll provde some useful sghs o real sysems. Aoher mpora queso s how he queue legh sadard devao s affeced by smulaeous sysem parameer chages. hs effec s bes show a mul-dmesoal graph whch s usually draw from smulaos whch are very me cosumg. he reds Fg. 3 calculaed usg Eq. (38) llusrae he relaoshp bewee he queue legh relave sadard devaos, RSD, he umber of coecos,, ad he badwdh, C. he resul shows ha RSD creases o a sgfca value as decreases or C creases. Fg. Comparso of predced ad smulaed queue legh meas ad sadard devaos he resuls hs graph ad he modfed- correlao marces show ha he varace ca o be egleced relave o he mea, especally whe s ear he lower boud. hs meas he radom oscllao ca desroy he sysem sably whe s slghly larger ha he sable boud. hs s he reaso why prevous sudes have used some suffce bu o ecessary assumpos for he parameer rages for CP/RED sably aalyses whe usg he flud flow model [5,6]. hs sudy quaavely shows he effecs of he oscllaos. he resuls also show he curre equaos accuraely model he real sysems. he predced sadard devao s always above he smulaed curve, bu s usually que close. he predced meas are very close o he smulaed values. he dffereces are due o may smplfcaos he curre dervaos. For example, he real CP uses he slow sar algorhm o Fg. 3 Varao of he queue legh relave sadard devao for varous ad C 5 Coclusos hs paper preses equaos for he covarace marx of he lear SDE wh addve Gaussa or compoud Posso ose. he equaos ca be used for may sochasc sysems. A sgle boleeck compuer ework usg CP/RED s used o llusrae he aalyss of he varace. he sysem SDE s aalyzed o deerme he sably rage of he mea queue legh. he covarace marx s he calculaed for sable meas. he resuls show how he sable sysem rage esmaed by he mea feld mehod eed o be adjused, whch s very mpora he desg of real egeerg sysems. he curre resuls descrbe he sascal characerscs of he sysem operag sae ha are eeded for beer sysem desgs.

8 FAN Hua ( 樊华 ) e al.:covaraces of Lear Sochasc Dffereal Equaos 7 Refereces [] He Shegwu, Wag Jagag, Ya Jaa. Semmargale ad Sochasc Aalyss. Bejg: Scece Press, 995: 4-5. ( Chese) [] Ikeda N, Waaabe S. Sochasc Dffereal Equaos ad Dffuso Processes. d ed. Norh-Hollad, 989: [3] Msra V, Gog Webo, owsley D. Sochasc dffereal equao modelg ad aalyss for CP-wdow sze behavor. ECE, Uv. of Massachuses, MA, ech. Rep. ECE-R-CCS-99--, 999. [4] Msra V, Gog Webo, owsley D. Flud-based aalyss of a ework of AM rouers supporg CP flows wh a applcao o RED. I: Proc. ACM SIGCOMM. New York, NY, USA, : 5-6. [5] Shafee M, Razzagh M. O he soluo of he covarace marx dffereal equao for sgular sysems. Ier. J. Compuer Mah., 998, 68: [6] Zygadlo R. Margale egrals over Possoa processes ad he Io-ype equaos wh whe sho ose. Phys. Rev. E, 3, 68: 467. [7] Grgoru M. he Io ad Sraoovch egrals for sochasc dffereal equaos wh Posso whe ose. Prob. Egg. Mech., 998, 3(3): [8] Chu D M, Ja R. Aalyss of he crease ad decrease algorhms for cogeso avodace compuer eworks. Compuer Neworks ad ISDN Sysems, 989, 7(): -4. [9] Reddy B, Ahammed A. Performace comparso of acve queue maageme echques. J. Compuer Scece, 8, 4(): -3. [] Floyd S, Jacobso V. Radom early deeco gaeways for cogeso avodace. IEEE/ACM ras. New., 993, : [] Frou V, Borde M. A sudy of acve queue maageme for cogeso corol. I: Proc. IEEE INFOCOM. el Avv, Israel, : [] Chug J, Claypool M. Aalyss of acve queue maageme. I: Proc. IEEE NCA. Cambrdge, Massachusee, USA, 3: [3] Fa Hua, Sha Xumg. Usg SDE o acheve he sable ad sascal aalyses for CP/RED flows. I: Proc. WCICA. Ja, Cha, : [4] Low S H, Paga F, Wag J, e al. Dyamcs of CP/RED ad a scalable corol. I: Proc. IEEE INFO- COM. : [5] Hollo C, Msra V, owsley D, e al. A corol heorec aalyss of RED. I: Proc. IEEE INFOCOM. Achorage, AK, USA, : [6] Hollo C, Msra V, owsley D, e al. Aalyss ad desg of corollers for AM rouers supporg CP flows. IEEE ras. Auom. Corol,, 47: