Distance- and Time-headway Distribution for Totally Asymmetric Simple Exclusion Process

Size: px

Start display at page:

Download "Distance- and Time-headway Distribution for Totally Asymmetric Simple Exclusion Process"

Ilene Boyd
5 years ago
Views:

1 Avalabl onln a Procda Socal and Bhavoral Scncs h EWGT Mng & 6h MEC & s RH Dsanc- and Tm-hadway Dsrbuon for Toally Asymmrc Smpl Excluson Procss Pavl Hrabá a,*, Mlan Krbál a a Dparmn of Mahmacs, aculy of uclar Scncs and Physcal Engnrng, Czch Tchncal Unvrsy, Trojanova 3, Pragu 0 00, EU - Czch Rpublc Absrac W consdr h on-dmnsonal oally asymmrc smpl xcluson procss on a lac of ss wh boh, h prodc boundary condon and h opn boundary condon. Bcaus of s smplcy, hs modl s ofn usd for hghway raffc smulaons. Our goal s o nvsga h hadway dsrbuon for hs modl n ordr o compar h mcroscopc srucur of h modl wh h ral hghway raffc. d In rcn arcls, h dsanc-hadway dsrbuon d for hs modl has bn prsnd. Usng hs rsul, h m-hadway dsrbuon s drvd, whch rprsns h probably dnsy for h m nrval of h lngh bwn h jumps of wo succssv parcls from h obsrvd s. rsly, h dsrbuon funcon for h random-sunal mdscr upda s drvd, from whch h dsrbuon for h m-connuous dynamcs can b oband. urhrmor, h mhadway dsrbuon for forward- and bacward-sunal upda s sudd by mans of Mon Carlo smulaons. W fnd ou ha for all varans of h consdrd modl, h spac dscr srucur s rflcd n h dsanc-hadway dsrbuon. Th m-hadway dsrbuon rflcs conrary o h ral raffc bhavor h "parcl-hol symmry" of h modl. 0 Publshd by Elsvr Ld. Opn accss undr CC BY-C-D lcns. Slcon and/or pr-rvw undr rsponsbly of h Organzng Comm. Kywords: oally asymmrc smpl xcluson procss; vhcular raffc; m-hadway dsrbuon. Inroducon Th amp o dscrb and undrsand h ssnc of raffc flow dynamcs s as old as h raffc slf. In rcn dcads, h uc dvlopmn of nllgn chnologs nabls o smula a vary of problms by mans of compurs. Morovr, a nw nd of compuaon by mans of h cllular auomaa has appard, and sms o b a vry appropra ool for smulang numrous physcal sysms ncludng h raffc flow; on of h frs cllular modls of raffc appard n agl, Schrcnbrg 99. Th movaon for smulang h raffc by mans of h cllular auomaa s gvn by h smplcy of hs modls, whch nabls vry uc, mayb vn ral-m, * Corrspondng auhor. Tl.: E-mal addrss: hrabapav@fjf.cvu.cz Publshd by Elsvr Ld. Opn accss undr CC BY-C-D lcns. Slcon and/or pr-rvw undr rsponsbly of h Organzng Comm do:0.06/j.sbspro

2 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs smulaons of h raffc sysm. Ths mans w ar loong for a modl as smpl as possbl, bu complx nough o rflc h "mporan" aspcs of h ral raffc sysm. On of h smpls modls usd n h raffc hory s h modl basd on h oally asymmrc smpl xcluson procss TASEP nvsgad n hs arcl. Ths modl s a paradgmac xampl of non-ulbrum sysms wh nars-nghbor nracon. or vry labora summary of hs modls s Blyh, Evans 007. Of cours, h TASEP modl s oo smpl o dscrb fully h complxy of raffc flow; nvrhlss, appars o b an nal pon for a vary of mor complx bu y analycally solvabl sysms. Svral suds abou h nx-nars-nghbor nracon modls show a vry good agrmn wh h raffc flow, for mor dals s Anal, Schüz, 000 or urlhnr, Lasgous, 009. In Rchnbachl al. 007 w may s a wo-lan gnralzaon of h TASEP modl xplanng h raffc jam nducon on hghways and n Karmpour 998 a mul-spcs modl wh ovrlappng possbly s nvsgad. Our goal n hs arcl s o sudy h mcrosrucur of h parcl nracons rprsnd by h dsanc- and m-hadway dsrbuon brngng nsgh no h mcroscopc characrsc of h modl. W prsn h analycal rsuls and compar h mcrosrucur of TASEP wh an xnsv sudy of h ral raffc mcrosrucur prsnd n Hlbng, Krbál 004, Krbál 007, and Krbál, Šba 009. Th arcl s srucurd as follows: h followng scon focuss on h raffc hory bacground of h problm; n h hrd scon, h modl s dfnd; h fourh scon summarzs h nvsgad characrscs of h modl; h ffh scon dals wh h analycal drvaon of h m-hadway dsrbuon; and h concluson focuss on h comparson of our rsuls wh h raffc flow hory.. Invsgad raffc flow characrscs As mnond abov, our nvsgaons ar focusd prdomnanly on hos proprs of cllular modls havng a ralsc nrpraon n vhcular raffc srams. As wll nown from h prvous rsarch s Drrda al., 993, h macroscopc phnomna of TASEP nsmbls corrspond o h ffcs obsrvd n frway sampls. Spcfcally, boh sysms show smlar rnds n h so-calld fundamnal dagram analyzng h dpndnc of raffc flow on raffc dnsy. Indd, f comparng h fundamnal dagrams n gur and gur 3, on can rcognz h ypcal sauraon ffcs sas of hgh aggrgaon of parcls n boh of hm. I mans ha boh sysms mrg from fr phas no h congsd phas n whch h movmn of parcls/vhcls s nnsvly rsrcd by ohr lmns. Ths ffc s accompand n boh cass by h rducon of raffc flow wh ncrasng dnsy. gur : Macroscopc and mcroscopc raffc flow characrscs. Lf graph rprsns h fundamnal dagram of h ral raffc sampl oghr wh h schmac rprsnaon as h mrror mag of h lr. Rgh graph rprsns h scald m-hadway dsrbuon for 4 dffrn dnss of cars. As undrsandabl, h dcd ransons bwn raffc phass can b dsclosd n h mcroscopc srucur as wll. Th dald sascal analyss of raffc daa oband on Europan frways and assocad analycal calculaons hav dsncly rvald ha probably dnsy for no-m nrvals rfrd o as m-claranc or

3 408 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs m-hadway bwn succdng cars can b sasfacorly approxmad by h on-paramrc famly of funcons Axp B, whr h symbol corrsponds o h Havsd sp-funcon, and wo consans 3xp B, A K B B K x assur h propr normalzaon and h scalng o h man nrval. Th funcon rprsns h modfd Bssl funcon of scond nd and of frs ordr. Whras for h raffc sas of small dnss h corrspondng m-hadway dsrbuon s almos xponnal,.., 0, for saurad sas h paramr s ncrasng wh raffc dnsy. In hs cass h non-zro valu of causs h dscn n probably for occurrnc of wo parcls clos o ach ohr. I mans ha for h corrspondng probably dnsy holds ha lm 0. Th progrss n m-hadway dsrbuon s vsualzd n h rgh graph of 0 gur. Hr w no ha h corrspondng rscald-dsanc-hadwaydsrbuon s, n h zro approxmaon, of h sam shap as h m-hadway dsrbuon s Krbál, 00. Also w add ha all dals concrnng h ralsc raffc masurmns and hr valuaons dscussd n hs arcl ar crcumsanally dpcd n h scon Daa analyss n h arcl Krbál, Hlbng, Dfnon of h TASEP modl Consdr a on-dmnsonal lac conanng uvaln clls,,...,. Each cll may b hr occupd by a sngl parcl or mpy. Parcls ar movng along h lac n on drcon, usually from lf o rgh, jumpng o h nghborng cll. Thos jumps ar drvn by h xcluson procss wh gnral paramr p 0, n h followng way: a parcl occupyng h cll was a cran m dpndng on h dynamcs m-connuous or m-dscr, and hn jumps o h cll wh probably p f h arg cll s mpy. Commonly, wo dffrn boundary condons ar dsngushd: h opn boundary condon and h prodc boundary condon. Th opn boundary mans ha a nw parcl can nr h lac jumpng no h frs cll wh probably 0, f h cll s mpy and a parcl can lav h lac jumpng ou of h las cll wh probably 0,. Th prodc boundary mans ha h lac s closd, crang a crcl. Ths mans ha h rgh nghbor of h las cll s h frs cll,.., w us h formalsm j mod, wh noaon 0=. Boh of h boundary condons ar schmacally dmonsrad n gur. gur : Schmac dmonsraon of h dynamcs of TASEP wh opn boundars lf and prodc boundars rgh.

4 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs Whn consdrng h m-connuous dynamcs, h jumps of ach parcl ar drvn by h Posson procss wh paramr,.., h numbr s of how many ms h parcl s asd o jump durng h m nrval, s has h Posson dsrbuon j s s Pr s j. 3 j! Ths mans ha h m h parcl was bfor rs o jump, s xponnally dsrbud wh paramr. L,,..., T, whr T 0, and us dfn a sochasc procss f hcll s occupd, 4 0 f hcll s mpy. As w can s n,.g., Blyh, Evans 007, h saonary dsrbuon P for probably of fndng h sysm n h sa n h opn boundary cas can b calculad as P,,..., W E DV W DE V, 5 whr D, E ar suar marcs and W, V vcors fulfllng and n h prodc boundary cas pde D E, W E W and D V V, M! M! P,,...,, 6! whr M. o ha M s h oal numbr of parcls n h lac, whch rmans consan; h rlaon 6 mpls ha n h prodc cas all confguraons ar ually probabl. Du o h smpl saonary dsrbuon, w wll wor furhr n hs arcl wh h lac wh prodc boundars, alhough whn consdrng h opn boundary cas, vry smlar rsuls hav bn oband whn nvsgang h sysm far nough from h boundars for mor dals s Drrda al. 993, Rajws al. 998, Krbál, Hrabá, 00. or compur smulaons, s mor appropra o us h m-dscr modfcaon of h procss. Svral m-dscr upda procdurs hav bn sudd laboraly n Rajws al., 998, whr w can fnd a soluon of h saonary dsrbuon for four dffrn m-dscr approachs oghr wh h dnsy-flow rlaons for boh, opn boundary and prodc boundary condon; for a brf summary s also Blyh, Evans, 007. Holdng h rmnology n hos arcls, w wll b nrsd n random-sunal, forward- and bacward-sunal, and fully-paralll updang procdur. Th random-sunal upda s, n fac, a dscr ralzaon of h m-connuous dynamcs. In vry sp,,..., s chosn a random, and hn h xcluson rul s appld o h ranson on of h clls,.., f 0, h parcl jumps from o wh probably p h 0, h. On sp of h upda corrsponds o h of h m un. Ths rscalng s ncssary whn comparng h m-dpndn varabls masurd on wo sampls wh a dffrn lac sz. In hs arcl, w wll rspc h followng noaon: whn alng abou h numbr of dscr sps, w wll us h lr,,... 0,,,..., and n h cas of h m masurd n m uns h lr, 0,... T wll b usd. In h 0, cas of h random-sunal upda, h s T of allowd ms s lac-sz-dpndn and rads T 0,,,.... As w may s n Rajws al. 998, h saonary soluon of hs procss s of xacly h sam form as h on for h m connuous procss gvn by 5 or 6. urhrmor, for h coun dsrbuon of how many ms h parcl has bn asd o jump holds ha j j s s s j s j Pr, whr j j!. 7

5 40 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs Thrfor, s mahmacally corrc o us h random-sunal dynamcs for nvsgaon and smulaon of h m-connuous procss. urhrmor, s asy o show ha n hos wo cass h paramr p only rscals h m and has no ohr nflunc on h dynamcs; hnc, whou loss of gnraly w may s p. Th las hr updas mnond abov rprsn h paralll updang approach. Ths mans ha vry parcl n h lac s asd o jump durng on sp of h procdur and hrfor T 0,,.... In h fully paralll upda procdur h xcluson rul s appld o all clls smulanously durng on sp. I s h mos frunly sudd of paralll updas. Ths dynamcs has bn dscrbd n agl, Schrcnbrg, 99 as h agl-schrcnbrg modl wh maxmum spd v max and brang paramr p. Th prncpl of h sunal updas s followng: h xcluson rul s appld o h clls sunally n h ordr, 3,..., n h forward-sunal cas and n h ordr,,..., n h bacward-sunal cas. or h paralll updas h valu of jumpng ra p s crucal for h dynamcs of h procss and sgnfcanly nfluncs h modl bhavor. 4. Traffc flow characrscs of h modl L us now summarz h flow characrscs of h TASEP. Th spac-dscr naur of h modl allows us o dfn h uans n h followng way: h dnsy s undrsood as h avrag occupaon h of h cll,.., Th parcl flow P,,...,. 8 Q hrough cll wll b calculad as Q p P,,...,, 9 whr / p s h man valu of h m a parcl n h cll was bfor jumps o h mpy cll. urhrmor, l d b h probably ha h dsanc bwn h parcl n h cll and h closs parcl n forward drcon s d clls. Th dsanc d s masurd from cnr of on parcl o cnr of h scond parcl,.., hr ar d mpy clls bwn h cll and h closs occupd on. Th valu d s calculad from d A... P,,...,, 0 d d whr A s h normalzaon consan assurng ha. o ha n h larg lm A. d Consdr now h m-connuous or random sunal dynamcs on h crcl of h lngh conanng M parcls. L p. rom h sady sa probably dsrbuon 6 w asly oban M, Q M M M M M M,. Th flow- whr h lm s man n h sns M, fulfllng M dnsy rlaon s hn,.g., M Q.

6 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs Invsgang h hadway dsrbuon d, for d fxd w oban MM M M M 3 d d d d A... M M. 3 Whn nvsgang h sysm far from h boundars,..,, d, h rlaons and 3 hold ru vn for h opn boundary cas, whr h lr rprsns h so calld bul dnsy of h lac gvn by, /,, /, / /, / or h drvaon by mans of h marx-produc-ansaz 5 of h flow-dnsy rlaon s Drrda al. 993 or Blyh, Evans 007, for h drvaon of h dsanc hadway dsrbuon s Krbál, Hrabá 00. Th paralll m-dscr updang procdurs hav bn laboraly xamnd n Rajws al. 998 and Chowdhury al. 998, for a brf summary s also Blyh, Evans 007. or llusraon purposs, w prsn h flow-dnsy and hadway dsrbuon-dnsy rlaons. or h bacward-sunal upda has bn shown ha,. 4 Qp, p, p d d, 5 for h forward-sunal holds ha Qp, p, p d d, 6 And for h fully-paralll upda w hav Qp, py d y y d, d and y d, d, 7 whr y 4 p /p. or llusraon, h fundamnal dagrams for all h updas ar plod n h lf graph of gur 3. gur 3: Macroscopc and mcroscopc characrscs of TASEP. Lf graph rprsns undamnal dagrams of TASEP for random-sunal blac, forward-sunal grn, bacward-sunal rd, and fully-paralll blu upda wh p=0.75. Rgh graph rprsns h normalzd dsanc-hadway dsrbuon of TASEP. 5. Drvaon of h m-hadway dsrbuon Th goal of hs scon s o drv h xac formula for probably dnsy funcon of h m-hadway dsrbuon. W wll procd as follows. rs, w drv h dscr probably funcon f of h sp-

7 4 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs hadway dsrbuon for h random-sunal dynamcs. Thn, h rspcv m-hadway dsrbuon funcon wll b oband from h rlaon f, whr fulfls. 8 Ths mans ha w rscal h m n h sns mnond n h prvous scon sps of h upda corrspond o =/ m uns, so w can compar h dsrbuon funcons n a rasonabl way. Th sunc can b chosn,.g., as blow, h lm dsrbuon funcon can b calculad va. Ths nabls us o calcula h pon lm lm. As shown d d s absoluly connuous, and, hrfor, h rspcv dnsy funcon f. As h procss wh random-sunal upda convrgs, n h sns mnond n Scon 3, o h procss wh m-connuous dynamcs, w can conclud ha h dsrd m hadway dsrbuon has h lm dnsy f. To compar h rsuls wh h normalzd dsrbuon, w wll rscal h m, so f, /. 9 r L us now calcula h valus f for,,.... W us h sam srucur as n Chowdhury al W ar nrsd n h sp-hadway bwn h ladng parcl LP and h followng parcl P, LP P, whr P, P LP, P s h m n sps n whch h parcl P passs jumps ou of h rfrnc cll RC. Th probably ha h P lavs h rfrnc cll RC xacly sps afr h LP jumpd from RC o RC+ wll b calculad as r f p, 0 whr p dnos h probably ha h P nrs h cll RC xacly sps afr h LP jumpd ou of RC and sands for h condonal probably ha h P was xacly sps n h cll RC bfor jumpng ou of f h P nrd h cll RC xacly sps afr h LP jumpd ou of. As w can s, holds ha whr p d d w d, d d d d d, b b a a w a, b a dnos h probably of h parcl P n hs cas bng chosn a -ms durng b sps and bng chosn n h b -h sp. Bcaus h nghborng cll of h P s always mpy, mans ha h P jumps o h nghborng cll any m s b chosn. Th condonal probably can b calculad as v w, v u w,, 3 whr u sands for h probably of h LP lavng h cll RC+ xacly sps afr had jumpd from RC o RC+ and v u dnos h probably of h LP lavng h cll RC+ a las sps afr had jumpd ou of RC,.., v s h probably ha durng h -h sp h cll RC+ wll b mpy RC LP / / 0. ow, whn obsrvng h movmn of hols hrough parcls, h probably

8 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs u corrsponds o h probably ha wll las sps afr h jump of h LP from RC o RC+ bfor a hol approachs h cll RC+, ravlng bacwards. Hnc, from h parcl-hol symmry follows ha u, 4 whr for convnnc w us h noaon. Thrfor, u v. 5 Subsung 4 and 5 no 3 w oban. 3,,, 6 Th uaon 0 hn rads. 3,,, p p p f 7 ow w can calcula h valus of h m-hadway dsrbuon funcon accordng o uaon 8. W no ha h dsrbuon funcon wll b calculad as h sum 3,,,, whr,, 8,, 9. 3, 30 Tan oghr, for 0 w oban lm, 3 whch lads o f '. 3

9 44 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs I s asy o vrfy ha f d. Afr rscalng h m ax accordng o 9 w oban 0 r/ r/ r/ r. 33 r Svral xampls of h dsrbuon dnss f and ar plod n gur 4. r gur 4: Tm-hadway dsrbuon f and rscald m-hadway dsrbuon r for h procss wh m connuous dynamcs. L us now nvsga h m-hadway dsrbuon for m-dscr paralll updang procdurs. In hs cas, h sp-hadway dsrbuon s dncal o h m-hadway dsrbuon. Th fully-paralll cas has bn analycally sudd n Chowdhury al. 998 as h agl-schrcnbrg modl wh maxmum spd v and h brang paramr p. Usng h s-ornd man fld hory h auhors hav drvd max h dsrbuon f,,,... n h form py py py py py py y y y y f p p p, 34 gur 5: Tm hadway dsrbuon f for fully-paralll upda lf, and forward-sunal upda rgh wh p 0.5. Th dsrbuon for h forward-sunal upda s plod for dnss c grn, c / rd, and c /5 blu, whr c dnos h dnsy of maxmum flow. whr y 4 p /p. As w can s, h dsrbuon shows h xpcd symmry n dnsy, f f ; hs symmry s causd, analogcally as n 3, by h parcl-hol symmry of h dynamcs. Ths symmry s bron by h sunal updang procdurs. vrhlss, s asy o obsrv anohr symmry n h moon of h parcls. Th forward-sunal parcl dynamcs wh h dnsy corrsponds o h bacward-sunal hols dynamcs wh h dnsy. Ths symmry s rflcd n h m-hadway dsrbuon as wll. Th numrcal sudy of h uany s prsnd n gur 5. W can s ha h dnsy c

10 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs corrspondng o h mos narrow dsrbuon f corrsponds o h maxmum flow dnsy,.., c p/ p n h bacward-sunal cas, and 6. Concluson c c p / p n h forward-sunal cas. W hav sudd h mcroscopc characrscs of h oally asymmrc smpl xcluson procss - h d hadway dsrbuon. Consdrng h dsanc-hadway dsrbuon, h powr-law dpndnc d has bn drvd n Krbál, Hrabá 00, rflcng h spac-dscr naur of h modl. As a consunc, h comparson of h normalzd dsrbuon s gur 3 wh h ral raffc dsrbuon s gur s no conclusv. On h ohr hand, h m-hadway dsrbuon dnsy r for h m-connuous dynamcs s uaon 33 and gur 4 gvs a dp nsgh n h mcrosrucur of h dynamcs. Comparng h dsrbuon dnss and 33, w conclud ha for h dnsy 0 h dsrbuon s clos o h xponnal dsrbuon; for h dnss fulfllng 0 /, h dsrbuon s Posson-l and h shap shows h sam rnd as h shap of h dsrbuon. On h ohr hand, du o h parcl-hol symmry of h modl, h dsrbuon for h dnsy s h sam as h on for dnsy. Tha mans, h dsrbuon for dnss grar han / dos no corrspond o h rnd obsrvd n h ral raffc suds s Scon. Insprd by h macroscopc rsuls for h bacward- or forward-sunal updang procdurs, for whch h parcl-hol symmry s bron ladng o h asymmry of h fundamnal dagram, w sudd h mhadway dsrbuon of hs modfcaons by mans of h compur smulaons. Alhough h parcl-hol symmry dos no hold, h dsrbuon shows analogcal bhavor as h on for h m-connuous dynamcs. I has h sam rnd-changng a h valu c. Tha mans, conrary o h fundamnal dagram, h mdscr updang procdur dos no lad o a br agrmn of h modl bhavor wh h ral raffc flow. W may obsrv, ha h capacy of a hghway sram 80 vh./m corrsponds o h dnsy / of h occupaon of h lac. Thrfor, would b bnfcal o modfy h modl n h way ha wll manan h shap of h dsrbuon and s dpndnc on h dnsy, bu sgnfcanly changs h flow-dnsy dpndnc. On of h proposd soluons s an addon of a srongr nx-nars-nghbor nracon Anal, Schüz 000, mananng h analycal solvably of h problm. Acnowldgmns Ths wor was suppord by h Mnsry of Educaon, Youh and Spors of h Czch Rpublc whn h projcs LC0600 and MSM Anohr suppor was provdd by h projc SGS0/09/OHK4/T/4. Rfrncs Anal T. & Schüz G. M Asymmrc xcluson procss wh nx-nars-nghbor nracon: Som commns on raffc flow and nonulbrum rnranc ranson. Physcal Rvw E, 6, Bham O., Mddlon A. A., & Lvn D. 99. Slf-organzaon and a dynamcal ranson n raffc-flow modls. Physcal Rvw A, Vol 46, 0, R64-R67, Blyh R. A. & Evans M. R onulbrum Sady Sas of Marx Produc orm: A Solvr s Gud. J. Phys. A Mah. Thor., 40, R333-R44 Chowdhury D., Pasupahy A.& Snha S Dsrbuon of m- and dsanc-hadways n h agl-schrcnbrg modl of vhcular raffc: Effcs of hndrancs. Eur. Phys. J, B5, 78 Daganzo C Th cll ransmsson modl, par II: wor raffc. Transpn. Rs.-B., 9B,, Drrda B., Domany E. & Muaml D. 99. An xac soluon of on-dmnsonal asymmrc xcluson modl wh opn boundars. J. Sa. Phys., 69, Drrda B., Evans M. R., Ham V.& Pasur V Exac soluon of a d asymmrc xcluson modl usng a marx formulaon. J. Phys. A, 6, urlhnr C. & Lasgous J. M A Quung Thory Approach for a Mul-Spd Excluson Procss. Traffc and Granular low 07, Hlbng D. & Krbál M Drmnaon of nracon ponals n frway raffc from sady-sa sascs. Physca A, 333,

11 46 Pavl Hrabá and Mlan Krbál / Procda Socal and Bhavoral Scncs Karmpour V A Mul-Spcs Asymmrc Smpl Excluson Procss and s Rlaon o Traffc low. Phys. Rv. E, 59, 05. Krbál M Eulbrum dsrbuons n a hrmodynamcal raffc gas. J. Phys. A: Mah. Thor., 40, Krbál M. 00. Analycal drvaon of m spcral rgdy for hrmodynamc raffc gas, Kybrna, 46, 6, 08- Krbál M. & Šba P Spcral rgdy of vhcular srams random marx hory approach. J. Phys. A: Mah. Thor. 4, Krbál M. &, Hrabá P. 00. J. Phys. A: Mah. Thor., 44, 7503 Krbs K. & Sandow S Marx produc gnsas for on-dmnsonal sochasc modls and uanum spn chans. J. Phys. A: Mah. Gn., 30, Lghhll, M.J. & Whham, J.B On nmac wavs. I. low movmn n long rvrs. II. A Thory of raffc flow on long crowdd roads. Proc. Royal Soc. A9, agl K. & Schrcnbrg M. 99. A cllular auomaon modl for frway raffc. J. Phys. I ranc,-9. Rajws., Sann L., Schadschndr A. & Schrcnbrg A Th asymmrc xcluson procss: comparson of upda procdurs. Journal of sascal physcs, 9, Rchnbachl T., ry E. & ranosch T Traffc jams nducd by rar swchng vns n wo-lan ranspor. w Journal of Physcs, 9,

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm