Towards a Flexible Air Traffic Management: Dealing With Conflicts

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1 Towrds Flexle Ar Trff Mngement: Delng Wth Conflts Stefn Resmert C. Doppler Lortory for Emedded Softwre Systems Unversty of Slzurg A-5020 Slzurg, Austr E-ml: Mhel Heymnn Deprtment of Computer Sene Tehnon-Isrel Insttute of Tehnology Hf 32000, Isrel E-ml: George Meyer NASA Ames Reserh Center Moffett Feld, CA 94035, USA E-ml: Astrt We propose mult-gent frmewor for sfe nvgton n dsrete envronment, whh llows employment of dstruted onflt vodne proedures whle preventng the ourrene of domno effets (propgton of onflts n the system s result of onflt resoluton). The nvgton envronment s modeled y grph. Eh gent hs set of lternte trjetores from ts soure vertex to ts destnton vertex, where trjetory represents sequene of resoures (grph vertes nd oupny tmes) to e suessvely ouped y the gent. A sfety requrement, tht spefes legl oupny of resoure y the gents n the system, must e stsfed. Resoures where the sfety ondton s volted re lled ontested resoures nd represent onflts etween the nvolved gent trjetores. Contested resoures re prortzed over the ompetng gents, nd n gent n hve dfferent prortes for dfferent ontested resoures. We present methodology for seleton y eh gent, of suset of ts set of vlle trjetores, suh tht the seleted trjetores of dstnt gents re onflt-free. The proposed methodology yelds mxml solutons; tht s, solutons n whh n gent nnot mprove ts seleted suset wthout retng onflts wth seleted trjetores of other gents. Three types of sfety requrements re onsdered: (1) Mutul exluson, where no two gents my oupy resoure t the sme tme, (2) Resoure pty, where eh resoure hs fxed pty nd the numer of gents tht shre the resoure must not exeed the resoure's pty, nd (3) Agent pty, where eh gent hs fxed pty nd the sfety ondton requres tht the numer of gents tht shre resoure s t most the pty of eh nvolved gent. The methodology, whh s motvted y the Free-Flght prdgm n Ar Trff Mngement, s lso pplle to vrous ground trnsport settngs. 1

2 1 Introduton Motvted y the Free-Flght prdgm n Ar Trff Mngement (ATM), we desre n ths pper multgent frmewor for sfe nvgton of rrft equpped wth dstruted ollson vodne pltes. The urrent ATM system s sed on entrlzed ontrol, wth lmted flexlty for ndvdul rrft to hoose trjetores. The system reles on humn opertors n lol r trff ontrol (ATC) enters (tht re further prttoned nto setors), to tr ll rrft long ther nomnl pthwys nd ontrol ther trjetores so s to nsure dequte rrft seprton. Consequently, n order to eep the system mngele, there re strt lmts on the numer of rrft tht re llowed nto setor, retng pty nd flexlty onstrnts, whh re now eng hllenged y the nresng numers of rrft n the sy. An ntense reserh effort s urrently under wy to overome some of the ove mentoned lmttons nd etter utlze exstng tehnologl pltes. A sgnfnt prt of ths reserh dels wth md-r ollson vodne, where rrft seprton s ensured y requrng rrft to follow sfe mneuvers [4][9][10][14][17]. A roder pproh tht ms lso t nresng ndvdul rrft flexlty s the Free-Flght/Free-Routng prdgm [11], n whh plots would e llowed to hoose optml trjetores, nd possly e responsle for mntnng flght sfety. In ddton to the potentl svngs tht ould e heved y rlnes (n terms of fuel onsumpton nd trvel tme), deentrlzton of ontrol ould redue the worlod t ATC's (whh would then ply more supervsory role). As onsequene, t mght e possle to nrese the numer of rrft supervsed y ATC, therey hevng etter utlzton of the rspe. One possle pproh to Free Flght mplementton s the Dstruted Ar-Ground Trff Mngement (DAG- TM) opertonl frmewor [1][2], whh nludes ondtons under whh seprton responslty my e delegted from ground-sed ontrol enters to ndvdul rrft whh re dequtely equpped wth on-ord onflt deteton nd resoluton pltes. The mn onerns re hevng sfety (rrft seprton), effeny (mnml trjetory devton) nd stlty under dstruted onflt resoluton rules. Here, stlty refers to the so-lled domno effet, where resoluton of lol onflt nvolvng smll numer of rrft my trgger system-wde propgton of onflts: n rrft resolves onflt y sutle devton from ts nomnl pth, whh my rete new onflts wth neghorng rrft flyng long onflt-free routes. Ths stuton my reursvely our for the neghorng rrft, ultmtely ledng to trjetory devtons of lrge numers of rrft. Chllenged y the Free-Flght de, we proposed n prevous wor [16] frmewor for dstruted ATM, whh, n ddton to sfety nd ndvdul optmlty, ddressed expltly the ssue of effent utlzton of the r spe. In tht frmewor, onflts were resolved off-lne, suh tht the on-lne operton of the system ws onflt-free (so the 2

3 system ws nherently stle). The frmewor ws sed on generl methodology for onflt resoluton n multgent systems ntrodued n [15]. The methodology ws ppled to ATM y onsderng rrft s utonomous gents nd y vewng the rspe s prttoned nto ells, so tht the requred rrft seprton onstrnt s stsfed y gurnteeng tht t most one gent oupes ell t ny tme. Thus, the rspe s modeled y n undreted grph, where vertex orresponds to ell nd n edge represents n djeny relton etween two ells. An gent must trvel from n ntl vertex to destnton vertex (oth of whh re spef to eh gent). A resoure s pr (vertex, tme step) nd n gent trvel s represented y trjetory: fnte sequene of resoures to e suessvely ouped y the gent durng the trvel. The numer of gents n the system s not spefed nd my hnge wth tme. An gent n enter the system t ny rtrry (nteger) nstnt of tme nd exts the system upon ts ompleton. An gent s requred to nnoune the set of ll ts optml trjetores (whh re of equl qulty to the gent). We ll ths the gent's model. Optml trjetores re determned y spefed set of performne rter, omputton not further dsussed n the present pper. We ssume tht n gent n do tht. Two trjetores of dstnt gents re n onflt f they ontn ommon resoure. To stsfy sfety, n gent's movement s restrted to legl suset of ts model, lled legl pln, suh tht legl trjetores of dfferent gents re onflt-free. An gent follows n rtrry trjetory n ts legl pln. An nomng gent enters the resoure system (strts up ts trvel) upon determnng nonempty legl pln, t whh tme t eomes tve. An tve gent nnot e stopped or suspended whle exeutng ts trvel. Thus, lveness spefton, tht nsures tht n tve gent lwys hs nonempty legl pln, s stsfed. At the hert of the proposed frmewor s the mehnsm y whh gents selet ther legl pths, so s to nsure sfety, lveness, nd effent resoure utlzton. The proposed methodology onssts of two lgorthm phses, preeded y n ntlzton phse. Frst, n nomng gent determnes the suset of ts optml trjetores tht re onflt-free wth the legl trjetores of the tve gents (lredy n the system). Then, n the onflt resoluton phse, the gent resolves potentl onflts wth trjetores of ll other nomng gents. If no legl trjetory s otned, then the ommodton phse s exeuted, where the gent n request nd otn resoures owned y tve gents (who wll try to ommodte n nomng gent). In ths pper, we desre sgnfnt extensons of the ove frmewor tht llow tve gents to shre resoures, sujet to pty onstrnts. Regrdng ATM, n the new frmewor vertex represents gger ell of rspe, wthn whh the seprton onstrnt etween multple rrft n e stsfed y employng lol onflt vodne proedures. A vrety of methods n e used n ths respet (see for exmple [4][9][10][14][17]). To heve stlty, we requre stsfton of pty onstrnts tht represent upper ounds on the numer of gents tht my oupy resoures. 3

4 The fous of ths pper s the onflt resoluton phse, whh now must gurntee sfe operton under pty onstrnts. Thus, we onsder here two relxtons of the orgnl sfety requrement: At most q gents my oupy resoure q. The numer q s lled the resoure pty ssoted wth q. In the se of ATM, resoure ptes n e used to ensure stlty of the system under dstruted onflt vodne methods. Moreover, ths orresponds to the present stuton where opertors n ATC enters re delng wth multple rrft n setor nd there s n upper ound on the numer of rrft tht re n the sme setor t ny tme. Eh gent hs ertn onflt resoluton ltes expressed y the gent's pty: The pty of gent represents the mxmum numer of gents wth whh gent s le to resolve onflts. It s requred tht the numer of gents oupyng resoure does not exeed the pty of eh nvolved gent. In ATM, the gent pty s mesure of the rrft s on-ord pltes for resolvng onflts wth neghorng rrft. It n e seen tht the orgnl frmewor s spel se of oth extensons, where ll ptes re equl to 1. Ths pper presents the non-trvl generlzton of the prevous onflt resoluton phse to oth ses. In relton to ATM, the mn propertes of the orgnl pproh (sfety, flexlty of pth hoes, effent utlzton of the r spe) re preserved. In ddton, the new frmewor ensures stlty under vrety of dstruted onflt vodne proedures, wth ll the tthed enefts: more rrft n the system, roustness, menlty to grdul mplementton n the urrent ATM system. A lterture revew on onflt resoluton n ATM n e found n [7]. A stlty evluton of two dstruted onflt resoluton methods s presented n [1]. Deentrlzed onflt vodne rules tht gurntee stlty of ntersetng rrft flows re presented n [13]. Mult-gent sed pprohes to Free Flght n e found for exmple n [8][18]. A toen-sed frmewor for deentrlzed ATM s desred n [3]. 2 Agent Models Let V e the set of grph vertes. A resoure s pr (,) vt, where v V nd t N. An gent model s fnte set P of trjetores, where n gent trjetory s sequene of the form: p :( v,0) ( v,1) ( v, m 1) ( v, m), 0 1 m 1 m where vj V, =0, j m. The numer {0,, m} represents the tme step of ths trjetory's exeuton t whh v 4

5 s ouped y the gent. The sme resoure my e ontned n more thn one trjetory n the gent model. The vertex v 0 represents the ntl vertex, whh s the sme n ll trjetores of the sme gent. Also, v m s the fnl vertex, whh s the sme n ll trjetores of the sme gent. We use n to denote the numer of gents nd P to represent the model of gent ( = 1,..., n). A resoure s lled shred resoure etween two trjetores of dstnt gents f oth trjetores ontn the resoure. Gven sfety requrement, shred resoure s lled ontested resoure, or onflt, f the sfety requrement s volted t the resoure. A olleton of sets of trjetores ( L 1,, L n ) wth L P s onflt-free f t ontns no ontested resoures. Gven set of n gent models nd sfety requrement, for the purpose of onflt resoluton t s suffent to onsder only gent trjetores tht ontn ontested resoures. Moreover, resoures tht re not ontested re of no nterest (nd onsequene) to the resoluton of onflts. Therefore, nsted of worng wth full gent trjetores, we shll onsder redued trjetores, ontnng only the ontested resoures. For smplty of presentton, ntl nd fnl resoures re lso nluded n the redued trjetores. The redued trjetory ssoted to trjetory p s the sequene r( p):( v0,0) ( x1, t1) ( x2, t2) ( xh, th) ( vm, m), where ( x 1, t 1 ),( x 2, t 2 ),,( xh, th) re ll the ontested resoures ontned n p (exept for the ntl nd fnl resoures), wth t > t 1, for =2,..., h. To llustrte some of the ove termnology, onsder the followng exmple. Fgure 1 represents porton of 2D spe prttoned nto ells nd three gents, denoted y R, S, nd T. Agent R must trvel from ell A1 to ell C21, S from C1 to B21, nd T from B1 to D21. The gent models re represented n Tle 1, where eh olumn orresponds to tme step nd eh row lsts the sequene of ells representng n gent trjetory. Shred resoures re outlned n oldfe. A B C D R T S S R T E Fgure 1. Cell prttonng exmple 5

6 R S T A1 A1 A2 A2 A3 A3 A3 A4 A5 A6 A7 A8 A9 A10 A10 A11 A1 A2 A3 A4 A5 A6 B6 B6 B7 C7 C8 C8 D8 D9 D9 D10 A1 A2 A3 A4 B4 B5 B6 B6 B7 B8 B8 B8 B9 C9 C10 C11 A1 A2 A3 A4 A5 A6 B6 B6 B7 B8 C8 C8 C9 C9 D9 D10 C1 C2 B2 B3 B4 A4 A5 A6 A7 A8 A9 A10 B10 B11 B12 B13 C1 C2 C3 C4 C5 C6 B6 B7 A7 A8 A9 A10 A10 A11 A11 A11 C1 C2 C3 D3 D4 D5 D6 D7 D7 D7 D8 D9 D10 D11 D11 C11 C1 C2 D2 D3 D4 D5 E5 E6 E6 E7 E7 E8 E8 E9 E9 E10 C1 C2 C3 C4 C5 C6 C7 C8 B8 B9 B9 B9 B9 B9 B10 B11 B1 B2 B3 B4 B4 C4 C5 C6 D6 D6 D7 D8 D9 D10 D10 D11 B1 B2 B3 C3 C4 C5 C6 C7 C8 C9 C9 C9 C10 C10 C11 C11 B1 B2 B3 B4 B4 B4 A4 A5 A6 A7 A8 A9 B9 B10 B11 B12 B1 B2 B3 C3 C4 C5 C6 C7 C7 C8 C9 C9 B9 B10 B11 A A12 A13 A13 A14 A14 B14 B15 B16 B17 C17 C18 C18 C18 C18 C19 C20 C21 E10 E11 E12 D12 D13 D14 C14 C15 C16 B16 B17 B18 C18 C18 C19 C20 C21 C11 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 D10 D11 D12 D12 D13 D14 D15 D15 D16 D16 C16 C17 C18 C18 C19 C20 C21 B14 B15 B15 B16 B16 B17 B17 B18 B18 B18 C18 C19 C20 C20 B20 B21 B11 B11 B12 B13 B14 B15 B16 B17 B18 B18 B18 B18 B19 B20 B21 C11 B11 B12 B13 B14 B15 B16 B17 B18 B19 B20 B21 E10 E10 E11 E12 E13 E13 E14 E15 E16 E17 D17 D18 D19 C19 C20 B20 B21 B11 B11 B12 B13 B14 B15 B16 B17 B18 B19 B20 B21 D12 D13 D14 D14 D15 D15 D15 D16 D17 D18 D19 D20 D21 D11 D12 D13 D13 E13 E14 E15 E16 E17 E18 D18 D19 D20 D21 B12 C12 C13 C14 C15 C16 C17 C18 D18 D19 D20 D21 A11 A12 A12 B12 B13 B13 B14 C14 C15 C16 C17 D17 D18 D19 D20 D21 Tle 1. Exmple of full gent models 3 Conflt Resoluton Consder the stuton when severl nomng gents wnt to enter the system t the sme tme. They hve ess to ommon dtse, ontnng the models nd urrent legl plns of ll tve gents n the system. They must lso regster ther optml trjetores n the dtse. Thus, fter regstrton, n nomng gent wll now the models of ll other nomng gents. Intlly, n nomng gent determnes the suset of ts trjetores tht re onflt-free wth the legl plns of the tve gents. However, ths suset my hve resoures tht re ontested wth the orrespondng susets of other nomng gents. Thus, n nomng gent must determne whh of these trjetores re ultmtely legl, so tht legl trjetores of dfferent nomng gents re onflt-free. Ths s the onflt resoluton phse. If the resultnt legl pln of the nomng gent ontns t lest one legl trjetory, the gent eomes tve. Otherwse, t 6

7 exeutes the ommodton phse, whh s not further onsdered n ths pper. A soluton of the onflt resoluton prolem for gven set of nomng gents s olleton of legl plns tht re onflt free. A soluton S s less restrtve thn nother soluton S f, for eh gent, the legl pln n S s nluded n the legl pln gven y S, nd the nluson s strt for t lest one gent. A soluton s lest restrtve, or mxml, f no other less restrtve soluton exsts. Whle mxml solutons need not e unque, mxml soluton mens tht no gent n unlterlly mprove ts legl pln wthout retng onflt wth some other gents' legl plns (nd thus voltng the sfety onstrnt). An lgorthm tht lwys fnds mxml soluton s lled optml. The followng sfety requrements re onsdered n ths pper: Resoure pty: Eh resoure hs spefed pty. The numer of gents tht my oupy resoure must not exeed the pty of the resoure. Agent pty: Eh gent hs spefed pty. The numer of gents tht my oupy resoure must not exeed the pty of eh gent nvolved. A spel se of oth the ove requrements s the mutul exluson requrement: no two gents re llowed to oupy vertex t the sme tme. In ths se, every shred resoure s ontested. Our lgorthm pproh to onflt resoluton s sed on prortzton of gents over ontested resoures. Ths prortzton n e otned y evluton of domn spef ttrutes t every ontested resoure. A possle evluton method s to use n ordered set of rules tht s ppled to eh pr of gents nvolved n onflt, suh tht the prortzton of n gent pr s estlshed y the frst rule tht n prortze the two gents. In other words, rule r l+1 s used to prortze only those gent prs tht ould not e prortzed y rules r 1, r 2,..., r l. One n otn ondtons under whh prwse pplton of rule set to onflt yelds totl orderng of ll gents nvolved n the onflt. These ondtons re not presented here due to spe onstrnts. In ATM, prortzton of rrft for lol onflt resoluton hs een employed n vrous settngs [8] [9] [12]. Prwse prortzton ws employed to resolve onflts etween rules n log progrmmng [5][6]. For exmple, the followng rules n e used to prortze the gents gven n Tle 1 under the mutul exluson requrement. We ll trjetory ontnng resoure q q-trjetory. The gent tht hs prorty for ontested resoure q s the one whh: 1. Hs q-trjetory y whh t spends smller mount of tme n the vertex of q thn ts opponent, regrdless of the q-trjetory tht s used y the ompetng gent. 2. Hs q-trjetory y whh t leves frst the vertex of q, regrdless of the q-trjetory tht s used y the ompetng gent. 7

8 3. Hs more q-trjetores thn the ompetng gent. 4. Hs shorter q-trjetory to ts destnton thn ny of the q-trjetores of the ompetng gent. The prortzton otned y these rules s denoted y π nd s gven n Tle 2. We employ here the onventon tht lower numer mens hgher prorty. For exmple, oth R nd S spend one unt of tme n (B4,4), therefore eh of them hs prorty over T. Agents R nd S re prortzed for (B4,4) y rule 4, whh gves prorty to R euse R hs shorter trjetory from (B4,4) to ts destnton. For (B9,12), oth R nd T hve prorty over S y rule 1, nd T hs prorty over R y rule 3. For (A11,15), rule 1 suffes to prortze ll the three gents. Agents S nd T re prortzed for (C11,15) y rule 2. For smplty n the sequel, ontested resoures re denoted y,,,..., s shown n the tle. Resoure Notton π ( R) π ( S) π ( T ) (B4, 4) (B6, 6) 2 1 (B9, 12) (A11, 15) d (C11, 15) e (C11, 16) f 2 1 (E10, 16) g 1 2 (E13, 20) h 2 1 (D15, 22) 1 2 (C18, 26) j 2 1 (B18, 27) 1 2 Tle 2. Prortzton exmple For the remnder of the pper, unless otherwse stted, y resoure we shll men ontested resoure. Our pproh to onflt resoluton n the mutul exluson se (see [15] [16]) ws sed on the followng optmlty prnple. An gent qures resoure f nd only f the followng ondtons re oth met: O1) The gent hs the hghest prorty for the resoure mong ll gents tht hve ess to the resoure. O2) The gent n me suessful use of the resoure y ompletng legl exeuton. The noton of ess s defned ndutvely: n gent hs ess to ts ntl resoure nd t hs ess to non-ntl resoure q f ondton O1 s stsfed for ll resoures preedng q on n gent s trjetory. For resoure q tht stsfes O1, ondton O2 mens tht ll resoures sueedng q on n gent s trjetory lso stsfy ondton O1. Ths, n prtulr, mples tht n gent s not permtted to reserve resoure (for whh t my hve prorty) f the reservton nnot e ppled towrd suessful ts ompleton. Indeed, t s esly seen tht f ondton O2 hd not een mposed, the otned soluton mght not e optml. Ths s euse f n gent qures resoure sed only on ondton O1, t s possle tht the resoure my tully not e used n legl exeuton, euse eh trjetory 8

9 ontnng the resoure nludes lso susequent resoures qured y other gents. Thus, lthough qured y the gent, the gent would not e le to utlze suh resoure. The system would onsequently e under-utlzed nd hene, suh n lgorthm would not e optml. In ths pper, we show how to pply the ove prnples for hevng optml onflt resoluton n the ses of resoure pty nd gent pty. To ths end, we frst revew the mutul exluson lgorthm. 3.1 Mutul exluson In [15], we presented resoluton lgorthms for the mutul exluson se under dfferent ondtons regrdng the nowledge vlle to n gent out the other gents' models nd prortzton. Here, we refly present only the stuton where eh gent nows the models of the other gents s well s the prortzton. The optmlty prnple s mplemented y rule for qurng full trjetores, s follows. An gent lms resoure q f nd only f t hs the hghest prorty for q mong ll gents tht hve ess to q. An gent hs ess to q f t hs lmed ll resoures preedng q on t lest one of the gent's trjetores. An gent qures full trjetory f the gent n lm ll resoures ontned n the trjetory. An tertve exeuton of the lgorthm onssts of the followng steps: 1. Resoure lmng: Eh gents lms resoures untl no further lms n e mde on the urrent worng sets of gent trjetores. At ths stge, eh resoure s ether lmed y some gent or unlmed (f no gent ould lm the resoure). 2. Trjetory quston: Eh gent qures (desgntes s legl) ll of ts trjetores tht hve ll resoures lmed (y the gent). 3. Trjetory removl: All trjetores tht hve onflts wth the newly qured trjetores re desgnted s llegl. All legl nd llegl trjetores re removed from the worng sets of gent trjetores. Ths opens the wy for new possle lms of resoures ontned n undeded trjetores, whh re nether legl nor llegl t ths step. 4. Stoppng ondton: If no undeded trjetory remns n the system, then stop. Else, go to step 1 (egn new terton). We llustrte the resoluton lgorthm on the gents gven n Tle 1. The redued models for the mutul exluson se re shemtlly depted n Fgure 2, whh employs the resoure notton gven n Tle 2. 9

10 d j R p 1 p 2 p 3 p 4 e f g j S p 5 p 6 p 7 p 8 p 9 d e f g h T p 10 p 11 p 12 p 13 d e h π( )=1 π( )=2 π( )=3 Fgure 2. Redued gent models Agent R hs ess to nd d for whh t lso hs prorty; therefore R lms nd d. Smlrly, S lms, nd T lms nd e. Sne S s loed t (hvng ess to ut no prorty), t hs no ess to j. Thus, R s the only gent wth ess to j, nd onsequently R lms j. R s loed t (y S), hvng no ess to g. Therefore, S s the only gent wth ess to g, so t lms the resoure. T hs ess to h (hvng lmed e) nd hs lso prorty for the resoure, so T lms h. No further lms n e mde t ths step. Resoures f, nd remn unlmed. Sne gent R hs lmed ll ontested resoures on trjetory p 1, R qures p 1. Smlrly, T qures p 11. These re ll trjetores tht n e desgnted s legl n the frst terton. Sne p 5 nd p 6 hve onflts wth p 1 (t j nd d, respetvely), they re now desgnted s llegl. Also, p 13 s llegl due to the onflt wth p 1 t d nd p 3, p 7 nd p 8 re llegl due to the onflts wth p 11 t e nd h, respetvely. All the legl nd llegl trjetores re now removed. The seond terton strts wth the sets of undeded trjetores s shown n Fgure 3. Note tht the only ontested resoures here re nd. In the lmng step of the seond terton, R lms, g, nd, whle T lms nd. The trjetores p 2, p 4 nd p 12 re desgnted s legl, whle p 9 nd p 10 re llegl. No undeded trjetory remns nd the lgorthm stops. The mxml soluton of the onflt resoluton prolem s s follows: LPR ={ p1, p2, p 4}, LPS =, LPT ={ p11, p 12}, whh s shown lso n Fgure 4. Note tht gent S otned no legl trjetory. 10

11 R p 2 g p 4 S p 9 T p 10 π( )=1 p 12 π( )=3 Fgure 3. The seond terton n the mutul exluson exmple d j R p 1 p 2 g p 4 S T p 11 p 12 e h Fgure 4. The soluton for the mutul exluson exmple In [15], we dsussed vrety of possle pprohes, s follows. On one hnd, the most tht n gent n do s to te nto onsderton ll the gents nd onflts n the system, nludng those n whh t s not dretly nvolved (ssumng t nows the prortzton for ll of them). For ths se, the resoluton lgorthm s optml. On the other hnd, the lest tht n gent should do s to te nto onsderton only the gents wth whh t hs onflts, nd to gnore the others. Conflts re resolved wth eh of the ompetng gents, prwse, the fnl result eng the nterseton of the prtl results. Sne the numer of gents onfltng wth gven one s usully muh lower thn the numer of ll gents n the system, the prwse pproh s omputtonlly effent, ut the result s, n generl, not mxml. 3.2 Resoure pty Let us onsder the stuton where eh resoure hs pty, expressed s (nonzero) nturl numer. The sfety onstrnt requres tht the numer of gents tht my oupy resoure must not exeed the resoure's pty. 11

12 In ATM, the resoure pty n e used to represent the mxmum numer of rrft tht re llowed to e n the sme rspe ell t the sme tme. Ths numer depends on the lty of ground ontrollers to sfely route r trff n the orrespondng ontrol setor. Ths upper ound my vry wth tme, dependng on wether, tme of dy, et. Ths tme vrne s ommodted n our frmewor y ssgnng ptes to resoures (whh hve tme omponent) rther thn to grph vertes. The struture of the onflt resoluton lgorthm s mostly the sme s n the mutul exluson se. To heve the sfety requrement, the lmng prnple nd the trjetory removl re modfed s follows. At the egnnng of n terton, every ontested resoure n the urrent worng sets of trjetores hs n ssoted vlle pty, whh represents the dfferene etween the ntl pty nd the numer of gents tht qured the resoure n prevous tertons. It n e redly seen tht the sfety requrement s stsfed f, n every terton, the numer of gents tht qure resoure does not exeed the resoure s vlle pty t tht terton. If the numer of gents tht qure resoure n n terton rehes the vlle pty of the resoure, then the ntl resoure pty s rehed nd the vlle pty of the resoure eomes zero, long further qustons n susequent tertons. Consequently, ll unqured trjetores ontnng the resoure re desgnted s llegl nd elmnted from the worng sets of trjetores. Ths mples, n prtulr, tht the vlle ptes of ll ontested resoures onsdered n n terton re postve. Note tht n gent qures resoure only fter t hs lmed t (long wth ll the other resoures on trjetory). To ensure tht the numer of gents tht qure resoure n n terton does not exeed the resoure s vlle pty, we requre tht the numer of gents tht lm the resoure does not exeed ths pty. Ths n e esly heed f gents tht hve ess to the resoure lm t n order of prortes (hghest prorty frst). Thus, n gent lms the resoure f nd only f the numer of ll hgher-prorty gents tht hve lmed the resoure s smller thn the vlle pty of the resoure. Rememer tht the onflt resoluton lgorthm nvolves only dle gents,.e., gents tht re not urrently exeutng tss. In our onflt resoluton frmewor, lveness of tve gents s gurnteed y ensurng tht no legl trjetory of n tve gent onflts wth trjetores desgnted s legl for dle gents n the ensung onflt resoluton phse. In the mutul exluson se, ths ws done y smply elmntng from the nput of the onflt resoluton phse ll trjetores of dle gents tht shred resoures wth legl trjetores of tve gents. Ths must e hnged, sne now dle gents my shre resoures wth tve gents so long s resoure ptes re not exeeded. To ensure lveness of tve gents, the ntl vlle pty of eh resoure s set to e the dfferene etween the resoure pty nd the numer of tve gents tht hve legl trjetores whh shre the resoure. Then, ll the trjetores of dle gents tht hve t lest one resoure whose ntl vlle pty equls zero, re removed. The onflt resoluton lgorthm s exeuted on the remnng trjetores. Thus, only the resoure slots whh re not ouped y tve 12

13 gents re mde vlle to the dle gents for onflt resoluton. The onflt resoluton lgorthm for the resoure pty se hs the followng terton struture: 1. Resoure lmng: Eh gent lms resoures untl no further lms n e mde on the urrent worng sets of gent trjetores. An gent lms resoure f t hs ess to the resoure nd f the numer of ll gents tht hve ess to the resoure nd hve hgher prorty thn the gent, s smller thn the urrent vlle pty of the resoure. 2. Trjetory quston: Eh gent qures (desgntes s legl) ll of ts trjetores tht hve ll resoures lmed (y the gent). 3. Cpty updte: For eh resoure on trjetory qured n the prevous step of the urrent terton, new vlle pty s omputed s the dfferene etween the prevous vlle pty nd the numer of gents tht hve qured the resoure n the prevous step. 4. Trjetory removl: Eh trjetory tht hs not een qured t step 2 nd tht hs resoure wth vlle pty equl to zero s desgnted s llegl. All legl nd llegl trjetores re removed from the worng sets of gent trjetores. 5. Stoppng ondton: If no undeded trjetory remns n the system, then stop. Else, go to step 1 (egn new terton). For exmple, onsder the gent models represented n Fgure 2, wth the ntl ptes of,, g, h, nd j equl to 1 nd the ntl ptes of, d, e, f nd equl to 2. The prortzton s ordng to Tle 2. For smplty, n ths exmple y pty we men vlle pty. Smlrly to the mutul exluson se, gent R lms nd d, S lms nd T lms, e nd h. Note tht only S nd T hve ess to, whh hs pty of 2. The numer of gents tht hve hgher prorty thn S for equls 1, whh s smller thn the pty of, therefore lso S lms. All three gents hve ess to d, whh s lmed y R nd T. S lms g, eng the only gent wth ess to g. Sne S hs no ess to j, R lms j. No further lms n e mde n the system t ths step. Resoures f, nd remn unlmed. Agent R qures p 1, S qures p 9 nd T qures p 11 nd p 13. All these re desgnted s legl. The vlle ptes of, d, h nd j eome zero. The vlle pty of e hnges to 1. The vlle ptes of ll the other resoures remn unhnged. The followng trjetores re llegl: p 3 (due to ), p 5 (euse of j), p 6 (due to d) nd p 8 (due to h). All legl nd llegl trjetores re removed, nd the seond terton strts wth the undeded trjetores shown n Fgure 5. 13

14 R p 2 g p 4 S e f T π( )=3 Fgure 5. The seond terton for the resoure pty exmple The lmng step of the seond terton proeeds s follows: R lms, g, nd, nd S lms, e, f nd (whh hs pty of 2). No further lms n e mde. Agent R qures p 2 nd p 4, whle gent S qures p 7. The vlle pty of eomes zero, therefore trjetores p 10 nd p 12 re desgnted s llegl. No undeded trjetory remns, so the lgorthm stops wth the followng soluton: LPR ={ p1, p2, p 4}, LPS ={ p7, p 9}, LPT ={ p11, p 13}, whh s shown lso n Fgure 6. Note tht ths mxml soluton nnot e ompred n terms of set nluson wth the one otned for the mutul exluson se, even though the mutul exluson s prtulr stuton where ll resoure ptes re equl to 1. However, s expeted, ths soluton provdes more flexlty due to the legl shrng of resoures etween dfferent gents. j R p 1 p 2 g p 4 S p 7 e f p 9 p 11 e h T p 13 d Fgure 6. The soluton for the resoure pty exmple 14

15 3.3 Agent pty In ths seton we del wth the se where eh gent hs pty, expressed s (nonzero) nturl numer. The sfety onstrnt requres tht the numer of gents tht my oupy resoure does not exeed the pty of eh gent nvolved. In ATM, the pty of n rrft represents, for exmple, the mxmum numer of rrft wth whh the rrft s le to vod md-r ollsons. Dfferent rrft my e equpped wth dfferent genertons of on-ord ollson vodne systems, where newer genertons re le to resolve onflts nvolvng more rrft. Clerly, one hs to ensure tht newer genertons wll e omptle (nd hene e le to oexst) wth older ones. One mght try to pproh the onflt resoluton prolem for gent ptes y stng t nto prolem nvolvng only resoure ptes nd then usng the lgorthm outlned n Seton 3.2: set the pty of eh ontested resoure s the mnmum of ll the ptes of the gents nvolved n the onflt. It n e seen tht mxmlty s not gurnteed n ths se, sne n gent wth mnmum pty my not neessrly qure the resoure. Smlrly to the se of resoure pty, we ssote n vlle pty to eh resoure qured y some gent. At the egnnng of n terton, the vlle pty of resoure s equl to the mnmum of the ptes of ll gents tht hve qured the resoure pror to the urrent terton mnus the numer of these gents. For exmple, suppose tht pror to the urrent terton resoure q hs een qured y three gents: R wth pty 5, S wth pty 6, nd T wth pty 7. Then, for the urrent terton, the vlle pty of q equls 2. The sfety requrement s stsfed y mposng the followng ondton: The gents tht qure resoure n n terton re suh tht the vlle pty of the resoure for the next terton s nonnegtve. In the ove threegent exmple, ths ondton s volted f the resoure s qured n the urrent terton y: - n gent of pty 3, or - two gents, one of whh hs pty 4, or - more thn two gents. The quston ondton s enfored y n dequte lmng of resoures. Smlrly to the resoure pty se, gents lm resoure n order of ther prortes (hghest prorty frst). An gent lms resoure q f nd only f the followng ondtons re ll met: C1) The gent hs ess to q. Tht s, the gent hs prevously lmed ll resoures preedng q on t lest one of the gent s trjetores. C2) The gent s pty s greter thn the numer of gents whh hve qured q pror to the urrent 15

16 terton plus the numer of (hgher prorty) gents tht hve lredy lmed q n the urrent terton. C3) The mnmum of the ptes of ll gents tht hve qured q pror to the urrent terton or hve lredy lmed q n the urrent terton s greter thn the numer of these gents. Smlrly to the prevous lgorthms, llegl trjetores re determned t the end of n terton, fter the trjetory quston step. An unqured trjetory of n gent s desgnted s llegl f t ontns resoure q suh tht one or oth of the followng ondtons re met: I1) The pty of gent s smller thn or equl to the totl numer of gents tht hve qured q (nludng prevous tertons). Ths mens tht gent wll never qure q n susequent tertons. I2) The vlle pty of q eomes zero for the next terton. Ths mens tht the totl numer of gents tht hve qured q s equl to the pty of one of these gents. Thus, no gent wll qure q n susequent tertons. All llegl trjetores re elmnted from the worng sets of trjetores for the next terton. Note tht lveness of tve gents s ensured y the ove rules, sne tve gents shrng resoure re gents tht hve qured the resoure pror to ny terton of the onflt resoluton lgorthm. The struture of the onflt resoluton lgorthm s gven elow. Intlly, ondtons I1 nd I2 re heed to determne nd elmnte ntl llegl trjetores of dle gents whh re n onflt wth legl trjetores of tve gents. Then the followng tertve exeuton s ppled on the remnng sets of gent trjetores of dle gents: 1. Resoure lmng: Resoures re lmed ordng to rules C1, C2, nd C3, untl no further lms n e mde on the urrent worng sets of gent trjetores. 2. Trjetory quston: : Eh gent qures (desgntes s legl) ll of ts trjetores tht hve ll resoures lmed (y the gent). 3. Cpty updte: For eh resoure on trjetory qured t step 2 of the urrent terton, determne the vlle pty of the resoure for the next terton. 4. Trjetory removl: Illegl trjetores re determned ordng to ondtons I1 nd I2. All legl nd llegl trjetores re removed from the worng sets of gent trjetores. 5. Stoppng ondton: If no undeded trjetory remns n the system, then stop. Else, go to step 1 (egn new terton). Let us pply ths lgorthm on the prolem onsstng of the gent models n Fgure 2, the prortzton n Tle 16

17 2, wth ptes of gents R, S, T eng 1, 2, 2, respetvely. Agent R lms resoure, for whh t hs the hghest prorty. Aordng to rule C3, gents S nd T nnot lm (whh hs een lmed y one gent of pty 1). Resoure s frst lmed y gent S. Agent R nnot lm (C2 s not stsfed). Resoure s lmed y gent T v p 13. Unle R, gent S hs ess to nd t lms (C1 C3 re stsfed). Resoure d s lmed y R. Consequently, gents S nd T nnot lm t (C3 s not stsfed). Resoure e s lmed y T (whh s the only gent wth ess to t). Agent S s the only gent wth ess to g, hene t lms g. Smlrly, gent R lms j. Resoure h s lmed y oth S nd T. No gent hs ess to ny of the resoures f,, nd, whh remn unlmed. At ths step, no further lms n e mde n the system. Trjetores p 1, p 8, p 9 nd p 11 re qured y the orrespondng gents. Next, llegl trjetores re determned y heng ondtons I1 nd I2 on the unqured trjetores: - Trjetory p 2 ontns resoure g, whh hs een qured y S. Sne the pty of gent R s equl to 1, R wll never e le to qure g (regrdless of the pty of S). Condton I1 pples, hene p 2 s llegl. - Smlrly, p 3 s llegl due to resoure e hvng een qured y T. - Trjetory p 4 hs no qured resoure t ths step, so t s undeded. - Trjetory p 5 ontns resoure j, whh ws qured y gent R. Sne R hs pty 1, the vlle pty of j for the next terton s zero. Thus, no gent n further qure j (wthout voltng the pty of R). Aordng to I2, p 5 s llegl. - Smlrly, p 6 s desgnted s llegl due to resoure d hvng een qured y R. - Trjetory p 7 ontns one qured resoure: e, whh ws qured y T. It n e esly seen tht I1 nd I2 re not stsfed: sne oth S nd T hve pty 2, S mght stll e le to qure e n further tertons. Thus, p 7 remns undeded. - Trjetory p 10 hs no qured resoure t ths step, so t s undeded. - Trjetory p 12 (of gent T) remns undeded even though t ontns resoure () qured y nother gent (S). As n the se of p 7, ondtons I1 nd I2 re not stsfed. - Both resoures of trjetory p 13 were qured, ut not y T: ws qured y S nd d ws qured y R. The vlle pty of d eomes zero, I2 pples nd p 13 s llegl. The legl nd llegl trjetores re removed nd the seond terton egns wth the sets of trjetores shown n Fgure 7. Note tht two resoures on the undeded trjetores were qured n the frst terton: e nd. At the 17

18 egnnng of the seond terton, eh of them hs n vlle pty equl to 1. In the lmng step, gent R lms nd, S lms, e, f,, nd T lms nd. Consequently, gent R qures trjetory p 4, S qures p 7 nd T qures p 12. Both resoures of p 10 re qured y other gents. The vlle pty of eomes zero nd therefore p 10 s llegl. The lgorthm stops wth the followng mxml soluton: LPR ={ p1, p 4}, LPS ={ p7, p8, p 9}, LPT ={ p11, p 12}, s shown n Fgure 8. R p 4 S p 7 e f π( )=1 p 10 π( )=2 T p 12 π( )=3 Fgure 7. The seond terton for the gent pty exmple p 1 d j R p 4 S p 7 p 8 p 9 e f g h T p 11 p 12 e h Fgure 8. The soluton for the gent pty exmple 4 Conludng Remrs A onflt resoluton methodology whh ddresses oth the resoure pty nd the gent pty requrements n e redly otned y omnng the lgorthms of Setons 3.2 nd 3.3. Ths lgorthm n e exeuted ndependently y eh gent, f the gent hs urte nformton out the other gents models, gent nd 18

19 resoure ptes, nd out the prortzton rules. Sne the lgorthm fnds legl trjetores nrementlly, n gent n termnte ts exeuton of the lgorthm t ny step, efore fndng ll ts legl trjetores. Thus, dstnt gents my perform dfferent exeutons of the lgorthm wthout srfng sfety of the glol soluton (ut possly srfng mxmlty). Regrdless of how the lgorthm s ppled (entrlzed or dstruted), ts outome gurntees system stlty when usng dstruted onflt vodne methods, due to stsfton of pty onstrnts. Ths s espelly relevnt to Ar Trff Mngement, where setor ptes ply sgnfnt role n trff ontrol nd the domno effet s n mportnt onsequene of usng dstruted methods for onflt vodne. Moreover, f sfety s to e delegted to ndvdul rrft, one hs to te nto ount stutons where dfferent rrft re equpped wth dfferent omptle on-ord onflt vodne systems. Tng nto ount oth resoure nd gent pty n the sme frmewor llows for grdul, supervsed shftng from the present, setor pty sed r trff ontrol, to the future, gent pty sed Free Flght. The dstruton of ontrol heved y mens of ptes nd lol onflt resoluton hs lso dvntges regrdng the omputtonl omplexty of the lgorthm. As ompred to the mutul exluson se, the resoure system s modeled t orser grnulrty (the sme volume of rspe orresponds to smller numer of gger ells). Thus, the numer of ontested resoures nd the numer of gent trjetores n e sgnfntly smller thn n the mutul exluson se. In ft, grnulrty of spe prttonng s prmeter whh estlshes the trdeoff etween entrlzed nd dstruted ontrol n our frmewor. If ptes re suffently lrge, most of the onflt resoluton s exeuted n dstruted wy. For the exmple of Fgure 2, f ll resoure nd gent ptes re equl to 3, then there s no ontested resoure nd ll trjetores re mmedtely desgnted s legl the tul onflt vodne s delegted to the dstruted proedures. On the other hnd, f ll resoure ptes re equl to one (the mutul exluson se), ll onflts re resolved off-lne y the lgorthm presented n ths pper nd no onflt ours n the susequent operton of the system. There re mny ssues whh wll e further nvestgted n relton to the frmewor presented n ths pper. An mportnt one s frness of trjetory lloton to gents. Vrous optmlty rter my e dded to selet mxml soluton, e.g., mxmum numer of legl trjetores, mnmum numer of shred resoures, et. These my e ddressed y pproprte prortzton funtons nd/or sutle lloton of resoure ptes. We pln to develop smulton envronment sed on ths frmewor, whh wll llow performne evluton of vrous prortzton poles nd dstruted onflt vodne methods. 19

20 Referenes [1] K. D. Blmor, K.S. Sheth, H.Q. Lee, nd S.R. Gre, Performne evluton of rorne seprton ssurne for Free Flght. AIAA [2] DAG-TM we pge: [3] S. Devs, M. Heymnn nd G. Meyer, Automton proedures for Ar Trff Mngement: toen-sed pproh. Proeedngs of the ACC02. [4] V. N. Duong, K. Zeghl, Conflt resoluton dvsory for utonomous rorne seprton n lowdensty rspe. Proeedngs of the CDC97. [5] B.N. Grosof, Courteous log progrms: Prortzed onflt hndlng for rules. Tehnl report RC28036, IBM T.J. Wtson Reserh Center. [6] B.N. Grosof, Prortzed onflt hndlng for log progrms. In Jn Mluszyns, edtor, Log Progrmmng: Proeedngs of the Interntonl Symposum (ILPS-97), MIT Press, Cmrdge, MA, pp [7] J. K. Kuhr nd L. C. Yng, A revew of onflt deteton nd resoluton modelng methods. IEEE Trns. Intell. Trnsp. Syst. 1(4), pp [8] J. C. Hll, J. K. Arhld, W. C. Strlng, nd R. L. Frost, A Mult-Agent System Arhteture for Dstruted Ar Trff Control. AIAA [9] J. Hu, M. Prndn, A. Nlm, nd S. Sstry, Optml oordnted mneuvers for three dmensonl rrft onflt resoluton, AIAA [10] I. Hwng nd C. Tomln, Protool-sed Conflt Resoluton for Fnte Informton Horzon, Proeedngs of the ACC02. [11] R. Josen, NASA's Free Flght Ar Trff Mngement Reserh, NASA Free Flght/DAG-ATM Worshop. [12] M. R. Jrdn, Ar trff onflt models. AIAA

21 [13] Z.-H. Mo, E. Feron, nd K. Blmor, Stlty of ntersetng rrft flows under deentrlzed onflt vodne rules. AIAA [14] P.K. Menon, G.D. Swerdu, nd B. Srdhr, Optml strteges for Free Flght r trff onflt resoluton, Journl of Gudne, Control nd Dynms 22(2). [15] S. Resmert nd M. Heymnn, Conflt Resoluton n Mult-Agent Systems, Proeedngs of the CDC03. [16] S. Resmert, M. Heymnn, nd G. Meyer, A Frmewor for Conflt Resoluton n Ar Trff Mngement, Proeedngs of the CDC03. [17] C. Tomln, G.J. Ppps, nd S. Sstry, Conflt Resoluton for Ar Trff Mngement: A Study n Mult-Agent Hyrd Systems, IEEE Trnstons on Automt Control, 43(4). [18] J.P. Wngermnn nd R.F. Stengel, Prnpled negotton etween ntellgent gents: model for r trff mngement. Artfl Intellgene n Engneerng (12). 21