A (Max, +)-Linear Model for the Analysis of Urban Traffic Networks

Transcription

1 A (Max, +)-Linear Model for the Analsis of Urban Traffic Networks Aurélien Corréïa, Abdel-Jalil Abbas-Turki, Rachid Bouekhf and Abdellah El Moudni SeT-EA3317 UTBM, site de Belfort Belfort Cedex, France Abstract Inthispaper,weintroduceanewmodelofurbantraffic networks b using Timed Event Graphs(TEG). Based on dioid algebra, the behavior of the sstem is described b (Max, +)-linearequations. Thiswillallowtoapproach the traffic control problems which rise issues of ccletimesandofsnchronization.asaresult,weshowhow we can solve a non-null initial marking problem inherent in dioid modeling. Besides, the proposed model provides us with interesting performance evaluations, such as realtimecountsofvehiclesandboundsofsojourntimeineach partofthestudiedstreet.anexampleisworkedouttovalidate the model. 1. Introduction Despite several existing models of traffic networks [12,6,10,11],thereisstillplentofscopeforthetraffic modeling improvement. Indeed, the traffic signal control with variable ccles is often treated b means of averaged models, which are not suitable for short sections generall encountered in factor sites or in downtowns. Moreover, on the one hand, macroscopic models remain rough to consider the real-time signal problems. This is due to thefactthatsuchmodelsdonottakeintoaccounttheindividual arrival of vehicles. On the other hand, microscopic modelsbecomecomplicatedwhenwetrtostudthebehaviorofasetofvehicles.itisbasicallthesefactswhich motivate us to introduce a new traffic model. Since through the problem of the traffic control we have to tackle ccle time and snchronization issues, a modelbasedontimedeventgraphs(teg)seemstobea judicious choice to carr out our modeling objective. Indeed,itiswidelknownthattheTEGiswelladaptedto deal with periodic and snchronization phenomena. Besides,theTEGcanbedescribedasasstemoflinear equations b using the dioid algebra. This has motivated several researches dealing with modelization, performance evaluation[3, 1] and the computation of control lawsfor (Max, +)-linearsstems[4,8]. Thegoalofthispaperistodefineamax-plusalgebra modelforurbantrafficsstems. Todothis,aTEGthat takes into account a non-null initial marking is proposed. Toshowthatourmodelisjustasimportantintrafficmodelingasintrafficlightcontrolproblems,itwillbeused to evaluate macroscopic performances(densit and flow rate) and microscopic ones(vehicle sojourn times). Thenextsectionpresentsanoverviewofthedioidalgebra and the TEG. Section 3 describes the traffic sstem anditsmodel.thelatterisanalsedinsection4.section 5 show the validit of our approach through an example. 2. Notions about dioid algebra and TEG Since the traffic modeling presented in this paper is based on dioid algebra, we give here a brief introduction of the necessar background Dioid algebra Thedioid Z max,commonlknownas (Max, +)algebra,istheset Z = Z {, + }endowedwithboth following operations[1, 5]: a b = Max(a, b), a b = a + b, for scalars a, b Z,and [A B] ij = a ij b ij = Max(a ij, b ij ), n [A C] ij = a ik c kj = Max ik + c kj ), k=1,...,n k=1 formatrices A, B Z m n and C Z n p. Thedualdioid Z min,alsocalled (Min, +)algebra, isobtainedbreplacing Maxwith Mininallprevious equations. Each dioid D has b definition the following intrinsic properties,forall a, b D: aneutralelement,denoted ε: a ε = aand a ε = ε a = ε; anidentitelement,denoted e: a e = e a = a; anorderrelation,givenb: a b a b = b.

2 2.2.StateequationsofaTEG Weareabletowritethe (Max, +)-or (Min, +)-linear state equations that describes the firings of the transitions ofategsinceitrespectsthejustintime(jit)operational rule. For each transition x, we can either evaluate thedaterfunction x(k),whichgivesthedateofits k th firing, or the counter function x(t), which gives the number oftimesithasbeenfireduntilthedate t. We illustrate this b means of the following example. u 1 u 2 2 Figure1.AsimpleTEG. ConsidertheTEGmodeloffigure1. Wewanttodescribe the behavior of transition, then: theassociateddaterisgivenb (k) = Max(u 1 (k) + 2, u 2 (k 1)), = 2 u 1 (k) u 2 (k 1) in Z max ; theassociatedcounterisgivenb (t) = Min(u 1 (t 2), u 2 (t) + 1), = u 1 (t 2) 1 u 2 (t) in Z min. Like the z-transform for series in classical algebra, the γ- and δ-transforms allow to translate daters and counters functions, respectivel, to formal series: γisthebackwardshiftoperatorintheeventdomain, γ e.g. x(k 1) γx(γ), δisthebackwardshiftoperatorinthetimedomain, δ e.g. x(t 1) δx(δ). Thebehavioroftransition offigure1isdescribedb thefollowingequationin M ax in γ, δ : (γ, δ) = δ 2 u 1 (γ, δ) γ 1 u 2 (γ, δ). ThebehaviorofawholeTEGcanbedescribedb stateequationsthatarelinearinadioidofformalseries intwocommutativevariables γ, δwithexponentsin Z and with boolean coefficients[3]. This dioid, denoted b γ, δ allowstostudbothtimeandeventdomains atatime[1].usuall,wegrouptheseries,whichdescribe thefiringofthe M ax in nsourcetransitionsintotheinputvector U, pwelltransitionsintotheoutputvector Y, mothertransitionsintothestatevector X. State equations look like the following: { X(γ, δ) = A X(γ, δ) B U(γ, δ), Y (γ, δ) = C X(γ, δ). Itispossibletocomputeatransfermatrix H from thoseequations: H = CA B [1]. Itrepresentsthe input-output relation of the sstem in a JIT operational rule.moreprecisel,entries H ij ofthetransfermatrixare causal(see[1,def.5.35p.254]). An informal definition of the causalit is as follows: entries H ij representneitheranticipationineventnorin time,i.e.themonomialsoftheformalseriesof Hareall with positive exponents. Thecausalprojection Pr + allowstocomputeacausal series b removing the monomials with nonpositive exponants.alessnaivedefinitionisgivenin[4] Notations and additional results The operator defined b a = k Z is called Kleene star. Itiswellknownthatthe -multiplicationofadioid D is rarel invertible. However, let us draw the reader s attention to the fact that the residuation theor provides a pseudo-division [1]. We denote R a : R a () = a = a = {x D x a } theresiduatedofthemapping R a : x x a. Insection4,wewillevaluatetheperformancesofour model. Let us remind the following proposition, which allows to evaluate the bounds of cars soujourn times in eachsegmentofroad,i.e. tocomputetheextremaofthe street filling rate. Proposition 1(Bounds of the sojourn times) The boundsoftokens sojourntimesinthepathsofateg can be computed with residuation theor as follows: let x α and x ω betheformalseriesof M ax in γ, δ describing firing sequences of entering and outgoing transitions ofapath(eithercomposedboneorseveralplaces), respectivel. Theupperboundofsojourntimesinthepath x α x ω isgivenbtheinverseofdater D (xα x ω)(0), i.e. iofthemonomial γ 0 δ i oftheseries x α x ω, Thelowerboundofsojourntimesinthepath x α x ω isgivenbthedater D (xω x α)(0),i.e. iofthe monomial γ 0 δ i oftheseries x ω x α. Theproofcanbefoundin[2]. a k

3 3. Elementar road traffic sstem modeling Figure 2. A regular street. InordertomodelawholeroadlanebaTEG,each direction of the lane is treated separatel and discretized into segments where the characteristic variables of flow dependonlonthetimebutnotonthepositionintothe segment. Each segment is considered as an elementar roadtrafficsstem [c i, c i+1 ].Asitisshowninsection5, oncethetegmodelofasegmentisdefined,itispossible to generalize it b concatenation to deal with an overall sstem. We take into consideration the following parameters of the elementar road traffic sstem: Thelengthofthesegment, theminimaltimetopassthroughthesegment(sojourntimeoftokensinplace P 2 offigure3), the maximal densit, themaximalflowrate, thenumberoftracks, the initial number of vehicles TEG modeling τ 0 0 τ 1 1 P 4 P 5 m0 0 m 1 1 τ 0 1 Table 2 provides more details about how we compute these parameters. Parameter Computation Max capacit length max densit tracks τ i (i+1) minimum time to pass through the segment m i (i+1) initialnumberofvehicles m (i+1) i maxcapacit initialnumberof vehicles m i i numberoftracks 1 τ i i max densit Parameter Meaning τ i (i+1) Minimumtimetopassthroughthesegment m (i+1) i Maximumnumberofvehiclesthatcan beheldbthesegment τ i i Minimuminter-vehicleduration m i i Numberoftracks m i (i+1) Numberofvehiclesinthesegment Table 1. Parameters of the model. Table2.ComputationoftheTEGmodelparameters. Itisimportanttonotethatthesimulationmusthavea startingdate,denoted t 0 = 0.However,before t 0,wecannotaffordtostudthebehaviorofthesstem.Thisleads to assume that tokens of the initial marking are available foraninfiniteamountoftime(aformalexplanationisdiscussed in[1, p.95]). The TEG analsis principle makes all enabled transitions firing before the starting date until the initial tokens ofupstreamplacesdonotallowannewfiringoftransistions. This is due to the following reasons: u m 0 1 P P x 2 1 x 0 1 P3 Once a transition is enabled, it is fired immediatel. ThisisduetotheFIFOrule; m 1 0 P 6 Figure3.TEGmodelofaregularstreet. The sstem behavior is modeled b the TEG presented infigure3. Here,weconsidera P-timedordinarTEG workingunderthefirstinfirstout(fifo)rule(see[9]). Therearethreekindsofplaces:themarkingofplaces P 1, P 2 and P 3 representthenumberofvehiclesintheinput segment, in the considered segment and in the output one respectivel.markingof P 4 and P 5 representthenumber oflanesatthebeginning(c 0 )andattheend(c 1 )ofthe segment respectivel. Since the segment cannot hold an infinitenumberofvehicles,markingofplace P 6 limitsthe numberoftokensin P 2.Themeaningoftheparameters aredetailedintable1. Tokensoftheinitialmarkingareconsideredtobe availableaninfiniteamountoftimebefore t 0 = 0. ThisisduetocausalitneededtostudaTEGbehavior with (M ax, +)-algebra. Hence,themodeloffigure3doesnotallowustoconsidertheinitialstateofthesegmentandthusneedstobe improved. Forinstance,ifwechoose m 0 1 = 5infigure3,thesefivetokensareavailablesince t = and thetransition x 1 isenabledandfiredfivetimesbeforethe date t 0. Consequentl,place P 2 isemptat t 0 = 0and the model does not conserve the initial condition. Nevertheless, from a practical point of view, we ma needanon-nullinitialmarkinginordertomodelanonemptstreetatthesimulation sstartingdate t 0.Toovercome this drawback, a virtual input init connected to each

4 outputofsegmentisaddedasitisshowninfigure4.this virtualinputprevent x 1 fromfiringbeforethedate t 0 and freezetheinitialmarkinguntilthisdate[1].thus,ithas tobeinfinitelfiredat t 0 toavoidinterferenceswiththe behavior of the sstem. theupperboundofthesojourntimeisgivenb D x0 Pr +(γ M init x i)(0), thelowerboundofthesojourntimeis givenb D Pr+(γ M init x i) x 0 (0). init u τ 0 0 m0 0 τ 0 1 m 0 1 m 1 1 x 0 x 1 m 1 0 Figure4.TEGmodelofanon-emptstreet at simulation-starting date. 3.2.Modelinginthedioid M ax in γ, δ Eachtransitionisassociatedtoaseriesofthedioid γ, δ whichrepresentsitsfiringdates. Thus,the following analtical model is obtained directl from the TEG: M ax in «γ m 0 0 δ τ 0 0 γ m 1 0 e ε X= γ m 0 1 δ τ 0 1 γ m 1 1 δ τ X 1 1 ε e Y =`ε e X τ 1 1 ««u init Inordertoobservethebehavioroftheflowofvehicles inanon-emptstreetwithoutanincoming,weuse u = δ = Pr + ( )(see[3],[1, 5.4.3]). 4. Performance measures WecanretrievefromtheproposedTEGmodelofthe elementar traffic sstem some of the main performance measures, which are: 1. Minimum and maximum vehicle sojourn times, 2. Instant count of vehicles driving in each segment of the road, 3. Fundamental diagram Bounds of the sojourn time We begin this subsection b emphasizing the fact that since the virtual input allows a non-null initial marking, thefirstfiringofagiventransitiondoesnotcorrespondto thefirstfiringofthenextone. Thisisbecausethefirst firings of both transitions do not involve the same vehicle. This situation can be avoided b an adjustment, as shown in the following proposition. Proposition2Let M init bethesumofinitialmarkingsof theplaces x i x i+1 with 0 i < n,where nisthe number of state variables. Then: Proof: SinceweconsideraFIFOruleoftheTEGoperation,firstoutputfiringsareduetotokensoftheinitial marking. Hence, the first output transition firing, which corresponds to the first token from the input transition, is enumeratedb M init insteadof 0. Itissuffisentthen to solve our problem, first, offseting all firing enumerationb M init andthenremovingallfiringsoftheinitial markingbthecausalprojection Pr +. Thus,thefiring number 0 of both transitions, i.e. input and output ones, concernthesametoken. Thisallowsustoestimatethe boundsofthesojourntimeasitisproposedpreviousl. Itisimportanttonotethat,theinitialmarkingstillinfluences the sojourn time despite the procedure to obtain the results given in the above proposition. Wedrawthereader sattentiontothefactthattheformulæoftheproposition2giveupperandlowerboundsof the sojourn time of vehicles for each particular situation, i.e. a given {initial marking, input sequence} pair. However, we can extract from the same formulæ the absolute boundsofthesojourntimeforallsituation.indeed,onone handtheabsolutelowerboundofthesojourntimeisobtainedwhenthestreetisemptandwhenthereisatleast onevehiclearrival(u e). Ontheotherhandtheabsoluteupperboundofthesojourntimeisobtainedwitha full-street(initial marking equals maximum capacit) and whenthereisatleastonevehiclearrival(u e). 4.2.Stateofthetraffic SinceinaTEG,eachplacehasexactloneupstream and one downstream transition, we can easil compute the transientmarkingsfromtheseriesof M ax in γ, δ which describe the firing sequence of each transition. Let m init betheinitialmarkingoftheplacewhichhas respectivel T α and T ω asupstreamanddownstreamtransitions.let x α and x ω betheseriesof M ax in γ, δ which describesthefiringsequenceof T α and T ω respectivel. Proposition3Themarkingoftheplace T α T ω atinstant tisgivenb C xα (t) C xω (t) + m init,where C xi (t) isthecounterof x i evaluatedatinstant t. Proof: Themarkingofaplace Patinstant tequalsits initial marking plus the amount of tokens that entered P until t,minustheamountoftokensthatleft Pbefore t. The counters joined to the input and output transitions of P suppl the second and third amounts. Theinstantmarkingoftheplaces T i T i+1 givesexactl the current number of vehicles driving in the segment {i, i + 1}.

5 4.3. Fundamental diagram In order to build the fundamental diagram, we simulate the model of the street several times with different constant input flow rates: Themeandensit ρisgivenbthefollowingequation: ρ i (t i+1 t i ) i ρ = T where ρ i istheinstantaneousdensitinthetimeinterval t i+1 t i and T isthesimulationduration. ρ i is obtained as follows: ρ i = m i l at t = t i where m i isthecurrentmarkingand listhesegment length. Themeanflowrate qisgivenbthecountofthe outputfirings C xi (T )dividedbthesimulationduration: 5. Example q = C x i (T ) T Letusconsiderthestreetoffigure5. Wewanthere to analse the behavior of the two segments delimited b {c 0, c 1 }and {c 1, c 2 }.. init u TheTEGmodelofthisstreetisdepictedinfigure6. τ 0 0 m 0 0 x 0 τ 0 1 m 0 1 m 1 0 τ 1 1 m 1 1 x 1 Figure6.TEGmodelofawholestreet. TheparametersofthisgrapharegiveninTable4. Parameter Value Segment {c 0, c 1 } Segment {c 1, c 2 } Max capacit τ i (i+1) m i (i+1) m (i+1) i m m 0 0 = 2 i i m 1 1 = m 2 2 = 1 τ i i 2 Wecancomputetheboundsofthesojourntimewith respect to the above defined parameter values: τ 1 2 m 1 2 m 2 1 τ 2 2 m 2 2 x 2 c0 c1 c2 Figure 5. Sections of a whole street. The parameters of both segments are grouped in the following table: Value Parameter Segment {c 0, c 1 } Segment {c 1, c 2 } Length(m) Min time to pass through the segment (50km.h (s) ) (40km.h 1 ) Maxdensit 1 (veh.m 1 ) Maxflowrate 1 (veh.s 1 ) Number of tracks 2 1 Initial number of vehicles(veh) Table 3. Parameter values of a whole street. Minimum relative sojourn time: 198 seconds (Pr + (γ 99 x 2 ) x 0 = δ 198 [γδ 2 ] ), Maximum relative sojourn time: 322 seconds (x 0 Pr + (γ 99 x 2 ) = δ 322 [γδ 2 ] ). Table 4. Computation of the model parameters. Wecanalsocomputetheabsoluteboundsofthesojourntime,choosingforexample u = e: Minimum absolute sojourn time(empt street): 93 seconds(x 2 x 0 = δ 93 [γδ 2 ] ), Maximum absolute sojourn time(fulfilled street): 450seconds(x 0 Pr + (γ 225 x 2 ) = δ 450 [γδ 2 ] ). Usingtheexpressionofdensitandflowrate,thefollowing fundamental diagrams are obtained. 1 Weuseheretheusualvaluesfromthelitterature,see[7].

6 References Figure 7. Fundamental diagrams of the sstem. Throughthesimulation 2,onecanimmediatelobserve that the well-known fundamental diagram is obtained. The variation of the flow rate against the densit shows thattheflowraterisesuntilacertainvalueofthedensit. Hence, the diagrams are composed in two parts; fluid and saturated ones. 6. Conclusion Wehaveproposedatooltomodelroadtrafficsstems usingteganddioidtheories.thisnewtoolinthetraffic field allows one to evaluate both kind of performances: macroscopic(densit and flow rate) and microscopic ones (vehicle sojourn times). Hence we have shown through asimulationexamplethatthetegisavalidandpowerful tool when evaluating traffic road parameters. We think that there are several issues that deserve further investigation.oneistheextensionofthemodeltoincludethe traffic light control. Other issues include a stochastic input. 7. Acronms DES Discrete Event Sstems: See[1]. FIFO First In First Out: Operational rule. JITJustInTime:Eachtransitionisfiredassoonasitis enabled. PNPetriNets:See[9]. TEG Timed Event Graphs: Petri Nets(PN) subclass. [1] F. Baccelli, G. Cohen, G.-J. Olsder, and J.-P. Quadrat. Snchronization and linearit, an algebra for discrete event sstems. Wile, [2] G. Cohen, S. Gaubert, R. Nikoukhah, and J.-P. Quadrat. Second order theor of min-linear sstems and its applicationtodiscreteeventsstems. InProceedingsofthe30 th Conference on Decision and Control, IEEE, Brighton, Angleterre, Dec [3] G. Cohen, P. Moller, J.-P. Quadrat, and M. Viot. Algebraic tools for the performance evaluation of discrete event sstems.inproceedingsoftheieee,volume77,pages39 58, [4] B. Cottenceau, L. Hardouin, J.-L. Boimond, and J.-L. Ferrier. Model reference control for timed event graphs in dioids. In Automatica, number 37, pages Pergamon Press, Mar [5] R. Cunninghame-Green. Minimax algebra. In Lecture notes in economics and mathematical sstems 166. Springer-Verlag, [6] A.DiFebbraroandS.Sacone. Hbridpetrinetsfor the performance analsis of transportation sstems. In 37 th IEEEConferenceonDecisionandControl(CDC 98), pages , Tampa, Florida, [7] N. Elloumi, H. Haj-Salem, and M. Papageorgiou. Metacor: Modèle d écoulement du trafic sur un corridor. In Actes de l INRETS, number 52, pages [8] M. Lhommeau. Étude de sstèmes à événements discrets dans l algèbre(max,+): Snthèse de correcteurs robustes dans un dioïde d intervalles. Snthèse de correcteurs en présence de perturbations. PhD thesis, Laboratoire d Ingénierie des Sstèmes Automatisés, Angers, France, [9] T. Murata. Petri nets: properties, analsis and applications. In Proceedings of the IEEE, volume 77, pages , Apr [10] C.Tolba,D.Lefebvre,P.Thomas,andA.ElMoudni.Continuous petri nets models for the analsis of traffic urban networks. In Proceedings of IEEE Sstems, Man and Cbernetics, pages , Tocson, Arizona, [11] C.Tolba,D.Lefebvre,P.Thomas,andA.ElMoudni. Performances evaluation of the traffic control in a single crossroadbpetrinets.in9 th IEEEconferenceonEmerging Technologies and Factor Automation(ETFA 03), volume 2, pages , Lisbon, Portugal, [12] H.Wang,G.F.List,andF.DiCesare. Modelingand evaluation of traffic signal control using timed petri nets. In IEEE conference on Sstems, Man and Cbernetics (SMC 93), Le Touquet, France, Oct This work has been validated b simulations computed with the toolbox MinMaxGD [8] developed for the software Scilab, allowing the manipulation of realizable series in the dioid M ax in γ, δ. The simulation scripts are available at Instant markings and fundamental diagram are computed with a tool available at