Removing Timed Delays in Stochastic Automata*

Transcription

1 Remov Tmed Delay Sochac Auomaa* Fda Kamal Daka, Geo V. Bochma School of Ifomao Techoloy ad Eee SITE Uvey of Oawa Abac: We pee a mehod o emove med delay med eal aco fom a ube of ochac auomaa. Afe emov he eal aco, he eady ae pobably of a ube of he ae of he auomaa peeved. The ochac auomaa codeed h pape have he popey of be Makov eeeave pocee.. Ovevew Ieal o vble aco a poce ofe ae due o commucao ad ychozao bewee he dffee compoe of he poce [7]. Remov hee eal aco mplfe he yem o educe he complcao of pefomace aaly, make deemc, ad eve may ueful applcao uch a ubmodule couco [6]. We code h pape pocee whoe ao ae eed by he occuece of ochacally med eve, hee pocee may be decbed u eealzed em- Makov poce aleba [] o ochac auomaa []. I hee yem, emov he med eal aco a challe ope poblem. The appoach ake o fa o emove eal aco fom hee yem o epaae aco fom he m fomao by pl evey ao o wo ao: oe decb he me delay ad he ohe a mele eal aco [,5], ha way emov eal aco fom hee yem become he val poblem of emov mele eal aco [7]. Howeve, epaa aco fom he m fomao double he umbe of ao fo he pupoe of emov few of hee ao he oe epee eal aco, o h appoach doe o lead o a mplfcao of he poce. Hece he eed fo emov med eal aco, whch a challe ad ope poblem o fa. I h pape, we pee a alohm o emove med eal aco, alo kow a med delay, fom a ube of ochac auomaa. Whle emov he eal aco, he eady ae pobably fo a ube of he ae of he auomaa peeved. The *A exeded abac of h pape appeaed poceed of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

2 ube co of all hoe ae ha do o have a com o ouo eal ao. The pefomace meaue ha ae peeved afe emov he eal move ae he oe obaed fom a ewad model [4] ha a zeo o all he ae ha do o keep he eady ae pobably. I elably aaly, a fal ae aed he ewad whle a up ae aed he ewad. So f he fal ae ad he ecovey ae ae o eached fom aohe ae va a eal ao whch uually he cae he depedably meaue ae peeved. The cla of ochac auomaa codeed h pape have he popey ha a ay me, he e of acve clock wh a eeal dbuo have he ame elaped lfeme, ohe wod hey wee all eabled a he ame me a, whle clock wh a expoeal delay dbuo may be e a abay me. The dea behd he alohm wll be peeed moe deal ubeco., bu we wll f pee ome backoud o Makov eeeave pocee ad ochac auomaa.. Ioduco.. Makov eeeave pocee ad Sochac Auomaa We be ou oduco wh ome backoud o Makov eeeave poce ad ochac auomaa a hee model ae heavly ued h pape. To povde a fomal defo of a Makov eeeave poce MRGP, he oo of a eewal equece mu be oduced. The follow defo ae ake fom [8]: A Makov eewal equece defed a he equece of pa of adom vaable X, T uually X epee he ae of he poce ha wa eeed a me T fo whch he follow popee hold: P, T+ T X, T, X, T,..., X { X +, T} P { X +, T+ T X } P { X, T X } *A exeded abac of h pape appeaed poceed of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

3 whee he f equaly hhlh he Makov popey of he poce, ad he ecod how homoeey. Accod o h defo, he cue ae of he poce aloe deeme pobablcally he ex ae ad he duao of me he cue ae. If a Makov eewal equece X, T aocaed wh a ochac poce Y, whoe behavo bewee a T ad T + of ay kd, bu whoe value T + deemed by X aloe, h wll be called a Makov eeeave poce. Fomally he follow popey hold fo Y : P{ Y T + Y u, u T, X } P{ Y T + X } P { Y X } So hee pocee behave lke a Makov poce elave o a T, whch we efe o a eeeao a. Bu bewee hee a, he poce ca evolve ay way. Fom a uve po of vew, ca be ad ha hee ae a T, T,..., T,... bewee whch he behavo of he poce o affeced by pevou hoy. A he poce homoeou, each of he cycle ca be uded a f he po of eeeao fom whch he poce examed wee T. Two quae capable of decb he evoluo of he MRGP ae defed: The local keel E ad he lobal keel K. Whee E P Y T > X decbe he evoluo of he poce bewee wo eeeao a, ad K P X T X decbe he evoluo of he poce a he eeeao a hemelve. Fo moe deal efe o [8] Sochac auomao SA a ae auomao whoe ao ae eed by he occuece of ochacally med eve. We be f by eumea he compoe of a SA. A SA a uple S,, C, a, k, F whee: S a oempy e of ae wh be he al ae. C a e of clock, A a e of aco, *A exeded abac of h pape appeaed poceed 3 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

4 a S A C C S he e of ede whee C epee f f f he e of fe ube of C, a eleme of a epeeed a whee a, C A C ad E C, f f a, C E ' k : S C he clock e fuco whch epee all he clock ha f ae alzed whe we each a ae, ad F : C S R [,] whee R epee he e of pove eal umbe he clock dbuo fuco uch ha F c fo < ad lm F c. Fo mplcy of oao, we wll deoe F c by c. Noe ha he dbuo of he clock deped o he ae wa alzed. A oo a ae eeed, all clock c k ae alzed accod o he pobably dbuo fuco F c. Oce alzed, clock a cou dow ul hey ehe expe o ae dabled. A clock expe f eache he value. The occuece of a aco coolled by he expao of clock. Thu wheeve hee a ao, C a E ' ad he yem ae, aco a ca happe a oo a all clock e C expe, clock E ae he dabled ad he auomao move o ae '. The acve clock of a ae ae fomed fom he acve clock of he pevou ae ha have o exped o dabled oehe wh he e of clock ha ae alzed : k. So f deoe he acve clock of ae, he acve clock ae ' would he be E C + k '. The clock he e E C ae o eaed ', hey ahe keep whaeve ema of he lfeme, whle eve k ' ae aed a ew lfeme accod o he dbuo. I he e of he pape, whe decb a ao, we wll o clude he dabled clock a hey ca be deduced fom he acve clock he ae. Fo moe fomao o SA efe o []... Ioduco o he Removal of Tmed Delay A meoed befoe we wll code a pecal cae of ochac auomaa ha we efe o a cocue eealzed ochac auomaa o CGSA. CGSA ae SA wh he follow eco: *A exeded abac of h pape appeaed poceed 4 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

5 . A mo Μ, fo ome Μ eeally dbued clock ae eabled mulaeouly, ad o ohe eeally dbued clock ca be eabled ul hey ae all dabled. So ay ae of a CGS f wo o moe eeally dbued clock ae acve, he hee clock have he ame elaped lfeme. I ohe wod hey wee acvaed a he ame me a.. All clock have a couou dbuo. 3. The expao of ay le clock duce a ao, ad duce exacly oe ao. Noe ha he compoo of wo CGSA mh o eul a CGSA Pulafo e al. [8] poved ha wh hee popee -4 he auomaa fac a Makov eeeave poce. To be able o ee h, we code all he ae he CGSA whee. A lea oe eeally dbued clock alzed o,. o eeally dbued clock acve oly expoeally dbued clock ae The h e of ae clealy fom he embedded eewal equece. We call he e of hee ae eeeave ae RS becaue oce you each oe of hem o kowlede of he poce hoy eeded o pedc he fuue. I [8], Pulafo e al. peeed a mehod o deve he ae ad eady ae pobable of uch pocee fom he lobal ad local keel E ad K. I h pape, ve a CGSA M wh eal med ao, we would lke o elmae hee ao fom he auomaa whle keep ome kd of equvalece. We wll poceed by elma he eal ao oe by oe, afe each elmao, he equvalece peeved. So ve a eal ao fom a ae a follow:, c ' whee τ he label ha epee a eal aco, we have oced ee τ Seco 4. ha f ae ad all of dec ucceo ae eeeave, he we ca emove he τ ao ad oba a CGSA ha weakly bmla o he oal oe. Moeove, we deed a alohm o afom ay ae a CGSA o a eeeave oe. Ad h afomao peeve he eady ae pobably. So he τ elmao wll be doe wo ep: *A exeded abac of h pape appeaed poceed 5 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

6 . F we afom ome of he ae of M o eeeave ae, we oba a CGSA M ', wh M ad M ' hav he ame eady ae pobable. I h ep we wll beef fom he fac ha he ae ad eady ae pobable of he CGSA ae ve a meoed befoe, hey ae calculaed u he keel E ad K.. The we wll emove he τ ao fom M ' ; he CGSA M " obaed wll be weakly bmla o M '. The eul equvalece bewee M ad M " wll be deoed by equlbumequvalece. The equvalece defo wll be peeed he ex Seco oehe wh ome pelmay defo. The he τ elmao wll be peeed Seco Defo Defo. Succeo, vble ucceo, ad level ucceo Le be a ae a CGS we call ucceo of, we Succ, he e of all ae he CGSA ha ca be eached fom by oe ao vble ucceo of o ISucc, all he ae he CGSA ha ca be eached fom by oe ao volv a vble aco. Level ucceo of o LSucc, he e Succ Succ. ISucc Defo. Local ace A local ace a CGSA a ace: a, c a, c... whee eeeave ad,..., ae o. Defo 3. Peced eeeave ae of Le be a o eeeave ae a CGSA. We defe R a he maxmal e of eeeave ae fom whch eachable houh a local ace. *A exeded abac of h pape appeaed poceed 6 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

7 Defo 4. Redual dbuo Le be a ae a CGS le c C be a clock uch ha c acve. The edual dbuo of clock c ae : Re M c x he pobably ha clock c wll expe bewee [, x ] me u afe each ae Noe ha he calculao of h quay a complex ad dffcul ak a deped o he ace of ao pefomed ul we each ae. Whe hee o cofuo abou he CGSA we wll mply we Re c x. Noe ha f c k o f c expoeally dbued, he Re c x c x Defo 5. Sucual ace A acual ace a CGSA a ace of he follow fom:... a *, a a *, *, whee he me whe ae wa eeed, a * he aco a peceded by ay umbe of τ ao wh {,..., }. a τ fo all Defo 6. Weak bmulao Le M S,, C, a, k, F ad M ' S', ', C', A', a, k', F' be wo CGSA. A equvalece elao R S S' a weak bmulao f wheeve R' he *, a c mple ha hee ex ' uch ha *, c' ' a ' ad R' *, ' ' a c ' *, c mple ha hee ex uch ha a ad R' Moeove, f wo acual ace T a a a *, *, *,... ad T ' ' ' '... ' wee followed M ad M ' a *, a*, a *, befoe we eached ad ', epecvely, uch ha R' fo all,...,, he *, c he pobably { M }, ha he ao a wll be doe wh me u afe each equal o he pobably { M ' }, ha ao *, c' ' a ' wll be doe wh me u afe each '. Defo 8. Equlbum Equvalece *A exeded abac of h pape appeaed poceed 7 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

8 Le M S,, C, a, k, F ad M ' S', ', C', A', a, k', F' be wo CGSA wh S' S. Le Γ S', he M ad M ' ae ad o be equlbum equvale ove Γ, o M M ', f fo all Γ, he eady ae pobably of M ad M ' he ame. Γ 4. Alohm fo Remov Tmed Delay I h eco, we wll pee he mehod o emove med delay. If, c ' a τ ao a CGS he a dcued befoe, he emoval of he τ τ ao wll be doe wo ep: f we afom LSucc ad o eeeave ae, he we delee he τ ao. I he ex ubeco we wll pee he mehod o afom a o-eeeave ae o a eeeave oe, he Subeco 4., we wll pee he alohm o delee he τ ao wh he aumpo ha LSucc ad ae eeeave. 4.. Fom o-reeeave o Reeeave Le be a o-eeeave ae a CGSA M S,, C, a, k, F, le R {,..., } ad le {,..., m, e,..., el} be he acve clock of ae, whee he have a eeal dbuo ad he e have a expoeal dbuo. Ou am h eco o afom o a eeeave ae. I ohe wod, we eed o fd he expeced dbuo of clock,..., } ae,.e. we eed o deeme { m Re fo {,..., m}. Fo ha, we aume ha a eady ae pobably π ex fo he CGSA ad ha fo all S, π. Theoem. Le be a o-eeeave ae a CGSA M S,, C, a, k, F, le R {,..., } ad le,..., m, e,..., e } be he acve clock of ae, whee he { l have a eeal dbuo ad he e have a expoeal dbuo. *A exeded abac of h pape appeaed poceed 8 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

9 Re λ de d λ E + d whee he local keel E P T >, ad λ he ae of E ee ae equlbum. Poof. We a f wh ome oao: mea ha we ee ae a equlbum mea ha we avel fom o ad o eeeave ae ved NR bewee. mea ha eached he eval [, + d ] ve we eeed NR a me, ad ha o eeeave ae ved bewee. mea ha eached he eval [, + d ] ve we eeed N{,..., } h a me, ad ha o ae amo,..., } ved o he way. { h Be ae a equlbum mple ha he la eeeave ae ved wa, o,., o, ad ha wa eached me u afe ee oe of he ae, whee fe. Re P ha expe wh me u of ee Sce he ae wa eeed a equlbum afe hav bee ae, o,., o, we have *A exeded abac of h pape appeaed poceed 9 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

10 *A exeded abac of h pape appeaed poceed of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5. Re P wll expe wh me u afe ee ad NR Bu h mea ha eached he eval ], [ d + ve we eeed oe of he a me, be fe. So Re { P wll expe wh me u afe ee ad NR } P wll expe wh me u afe ee ad. NR d P NR d > + E d T P λ λ + E d d de λ λ Noe ha λ whch he ae of ee ae ca be calculaed u he lobal keel ad whou eo o calculae he eady ae pobably of he CGSA: f we code he SMP udely he CGSA who defed by he lobal keel, he d π λ whee π he eady ae pobably of ae he SMP, ad d he aveae me we ay ae oce we ee Theoem.

11 Le be a o-eeeave ae a CGSA M S,, C, a, k, F, le { m,..., } be he e of acve clock of. Le M ' S,, C, a, k, F' be he CGSA obaed by afom o a eeeave ae a follow: k {,..., } ad ' F ' Re. The M ad M ' have he ame eady ae pobably π M ad π M ' epecvely. Poof of heoem. Le S ad aume ha a acve clock of. We wll pove f ha ' Re M Re M. Noe ha M ' ha oe addoal eeeave ae, o we M eed o udy effec o he edual me of clock. If RS,..., } he ehe { M ' M ' RS {,...,, } o RS {,..., }. We wll aume whou lo of eealy ha M ' RS {,...,, }. We have ha: I boh M ad M ', f we ee ae equlbum, he: - Ehe we wee oe of he ae,..., } he we eached ae whou pa by ae,..., ad. { - O we ae com fom ae he we eached ae whou pa by ae,..., ad. Ad boh cae wll poduce he ame edual me fo clock boh M ad *A exeded abac of h pape appeaed poceed of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5. M '. ' Hece Re M Re. Bu h mple ha he oou me dbuo M evey ae uchaed, ad ha he pobably of o fom oe ae o aohe equlbum uchaed. Ad ha mple ha he eady ae pobable ae peeved. 4.. Remov delay Le be a ae CGSA M S,, C, a, k, F ad aume ha ad LSucc ae all eeeave ae. Aume ha τ, c '. I h ubeco, we wll pee a mehod o emove h τ ao fom he auomaa ad oba a CGSA M ' S', ', C', A', a, k', F' ha weakly bmla o he oal oe.

12 F, aume ha ' ha ao lead o ae:,..., } ad ha ao { m lead o ae ', l,..., l }, ad ha k { c, c,..., c } efe o Fue. { τ,c ' b,e b m,e m a,c m a,c l l Fue. CGSA M Befoe poceed o emove he τ ao, we oe ha he pobably of o fom o wh me u of ee, p M / M o mply, equal o P M, ' dc de P M ', { c e d} d d d whch mea ha, f clock c expe a me ad clock e expe a me whee [, ] ad [, ] he all acve clock ae.e. c,...,c have o expe afe me, ad all acve clock of ae '.e. e,..., e,..., e, e+ m have o expe afe me. Smlaly, he pobably of o fom o l wh me u of ee : dc, c c d P M l d To emove he τ ao, we eed o ceae m ew ao ou of ae : b e, ', {,..., m } ee Fue. Moeove, we eed o chae he dbuo of *A exeded abac of h pape appeaed poceed of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

13 he clock he oally ex ao b c, b c l o, ' l, {,..., } efe o Fue. b,e' b m,e' m a,c' a,c' l l Fue. CGSA M ' So he ema ak would be o deeme he ew dbuo fo he clock e', {,..., m} ad c', {,... } a way ha peeve weak bmulao bewee he wo auomaa. I ohe wod, we eed o peeve he pobably of o fom ae o ae l epecvely wh me u of ee. Th mea, we have P M ', P M, ad P M ', l P M, l whee: ' de' P M, c' e' d d ad ' dc' P M, l e' c' d d Hece de' d dc' d c' e' d P M e' c' d P M, l, *A exeded abac of h pape appeaed poceed 3 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

14 So we have + m o-lea equao fo + m ukow fo {,..., } ad fo {,..., m}. Fo eu he exece of oluo, he dbuo {,..., } ad, {,... m hould be locally eable. P M, l } P M,, Howeve, ce ae, {,..., } ad l, {,..., m} ae eeeave, o fomao abou clock e', {,..., m} ad c', {,..., } eeded oce we leave ae. I ohe wod, hee clock ae oly acve ad he oly ole o deeme he ex ae oce we each ae. So he dbuo of he clock e', {,..., m} ad c', {,..., } ae oly ued fo deem he pobable of each ae o l, ' ha P M ' l, {,..., } ad P M, {,..., }, epecvely. Howeve, hee, m, pobable wee aleady deemed by equao ad, heefoe we wll o eo hee o olv he yem of equao, ode o fd he clock pobable. Clam. Le be a ae CGSA M S,, C, a, k, F ad aume ha ad, c LSucc ae all eeeave ae. Aume ha τ ' a ao M. Le M ' S,, C', a ', k', F' be he CGSA obaed fom M by emov he τ move u he mehod above. The all ae S { ISucc } keep he eady ae pobably afe he afomao. Idea behd he poof. Noe ha f S { ISucc }, ad f Γ a ace fom o M he hee ex a ace Γ ' fom o M ' uch ha Γ ad Γ ' have he ame vble aco hey alo ed wh he ame ao whehe vble o o ad hey boh have he ame pobably dbuo. Theoem 3. Le M S,, C, a, k, F be a CGS le M ' S',, C', a', k', F' be he CGSA obaed fom M by emov he med eal ao follow he alohm he pevou ubeco. The M ad M ' ae equlbum equvale. *A exeded abac of h pape appeaed poceed 4 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

15 The poof ca be ealy deduced fom Clam ad Theoem. Oe way o udy pefomace auomaa u ewad model. Rewad model ae obaed by a a ewad o evey ae of he auomao. A ae ewad a ee epee he deably of be ha ae. U hee ewad ad he eady ae pobably of he auomao he expeced ewad ae eady ae calculaed. Fo moe fomao efe o [9]. I he CGSA M ' of Theoem 3, he pefomace meaue ha ae peeved ae he oe obaed fom a ewad model ha a zeo o all he ae ha do o keep he eady ae pobably. I elably Aaly, a fal ae aed he ewad whle aed fo he up ae. If he auomaa M ' he fal ae ad he ecovey ae ca o be decly eached houh a eal ao he depedably meaue ae peeved bewee M ad M '. 5. Cocluo The ue of emov eal ao fom ochac pocee ha bee a ope poblem fo que a whle. I h pape, we have peeed a oluo fo h poblem he cae of cocue eealzed ochac auomaa. Whle emov he eal ao, he eady ae pobably of a ube of he ae of he auomaa peeved. The ube co of all ae ha have o com o ouo eal ao. A a fuue wok, we would lke o eealze h mehod o cove a boade ubcla of ochac auomaa. The τ -elmao peeed h pape, could alo be ued a a ba fo ae aeao Makov eeeave pocee. Refeece *A exeded abac of h pape appeaed poceed 5 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.

16 ..M. Bave, M. Beado, R. Goe, Towad Pefomace Evaluao wh Geeal Dbuo Poce Aleba, LNCS, vol. 466, Spe,. P. D'Aeo, J-P Kaoe, E. Bkma, A Compooal Appoach o Geealzed Sem-Makov Pocee, WODES98, P.R. D'Aeo, J.-P. Kaoe, ad E. Bkma. A alebac appoach o he pecfcao of ochac yem Exeded abac. I Poceed PROCOMET'98. Chapma & Hall, Hllo, J., A compooal Appoach o Pefomace Modell, Duhed Deao Compue Scece. I: Cambde Uvey Pe, J-P Kaoe, P.R. D'Aeo, Geeal Dbuo Poce Aleba, LNCS, vol. 9, Spe, 6. P. Mel, G.V. Bochma, O he Couco of Submodule Specfcao ad Commucao Poocol, ACM Taaco o Poamm Lauae ad Syem TOPLAS, R. Mle, Commucao ad Cocuecy, Pece-Hall, A. Pulafo, M. Scapa, ad K. S. Tved, Pe Ne wh k-smulaeouly Eabled Geeally Dbued Tmed Tao, Pefomace Evaluao, K. Tved, ad k. Goeva-Popoaova, Sochac Modell Fomalm fo Depedably, Pefomace ad Pefomably, Lecue Noe Compue Scece 769,. *A exeded abac of h pape appeaed poceed 6 of he 7h Ieaoal Wokhop o Pefomably Model of Compue ad Commucao Syem: PMCCS-7,5.