Modelling and control of urban bus networks in dioids algebra

Transcription

1 Modelling and control of urban bus networks in dioids algebra Laurent Houssin, Sébastien Lahaye, Jean-Louis Boimond To cite this version: Laurent Houssin, Sébastien Lahaye, Jean-Louis Boimond. Modelling and control of urban bus networks in dioids algebra. WODES 04, Se 2004, France. IFAC roceedings volume, <hal > HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 10 Mar 2009 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 MODELLING AND CONTROL OF URBAN BUS NETWORKS IN DIOIDS ALGEBRA L. Houssin S. Lahaye J.L. Boimond Laboratoire d Ingénierie des Systèmes Automatisés, 62, avenue Notre Dame du Lac, Angers, France Abstract: We aim at alying dioids algebraic tools to the study of urban bus networks. In articular, we show how they can be used for timetables synthesis. Keywords: Discrete Event System, dioid, bus networks, timetables synthesis 1. INTRODUCTION Discrete Event Dynamic Systems (DEDS) subject to synchronization henomena can be modeled by linear equations in a articular algebraic structure called dioid or idemotent semi-ring. For about twenty years, this roerty has motivated the elaboration of a new linear system theory. Indeed, concets such as state reresentation, transfer matri, otimal control, correctors synthesis and identification theory have been transosed from conventional linear system theory to the considered algebraic structure (Baccelli et al., 1992). Alications of this theory have essentially concerned manufacturing systems (Menguy et al., 2000; Lahaye et al., 2003a), communication networks (Le Boudec and Thiran, 2001) and transortation networks (Braker, 1993; de Vries et al., 1998). In the latter, the focus has been on modelling, erformance evaluation and stability analysis of railway networks. For our concern, we are interested in alying dioids algebraic tools to the study of urban bus networks. The iloting of these networks can be seen as a two stes rocess. Firstly, a so-called oerating schedule is defined for mean conditions. In articular, timetables are defined at each sto to secify times at which buses should theoretically run. These timetables are used to inform assengers and, at articular stos, to re-synchronize buses. The second ste consists in regulation oerations: in reaction to current conditions (breakdown of a bus, modifications of traffic flows, etc.), a suervisor 1 may decide of adjustments from the oerating schedule (transfer assengers, sto or reroute buses, etc.). In (Lahaye et al., 2003b), we have roosed a first aroach aiming at modelling some of these regulation oerations. In ractice, such a model may be used as a tool to aid decisions since it allows evaluating relevance of adjustments. In this aer, the focus is rather on the first stage described above. In fact, we roose a model for urban bus networks functioning according to their oerating schedule. We then show how dioid techniques 2 can be used for the synthesis of timetables. As described in (Ceder et al., 2000) this roblem is crucial, and aims notably at maimizing connections between buses from different lines. The outline of the aer is as follows. In 2, we recall elements of dioid theory as well as rinciles of DEDS descrition and control over dioids. In 3, we describe how bus networks oerate, and a model is introduced. In 4, the method for timetables synthesis is roosed. An alication to an elementary bus network is roosed in 5. 1 Visualizing evolutions inside the network and communicating with bus drivers. 2 The roosed aroach is mainly based on residuation theory.

3 2. PRELIMINARIES 2.1 Elements of dioid theory Definition 1. (Dioid). A dioid is a set D with two inner oerations denoted and. The sum is associative, commutative, idemotent ( a D, a a = a) and admits a neutral element denoted ε. The roduct is associative, distributes over the sum and admits a neutral element denoted e. The element ε is absorbing for the roduct. In a dioid D, the equivalence : a b a = a b defines a artial order relation. Definition 2. (Comlete dioid). A dioid is said to be comlete if it is closed for infinite sums and if roduct distributes over infinite sums too. The sum of all its elements is generally denoted. A comlete dioid has a structure of comlete lattice (Baccelli et al., 1992, 4), i.e. two elements in a comlete dioid always have a least uer bound namely a b and a greatest lower bound denoted a b = a, b}. Eamle 1. The set Z = Z +, } endowed with the ma oerator as sum and the classical sum as roduct is a comlete dioid, denoted Z ma and usually called (ma,+) algebra, with ε = and e = 0. Theorem 1. The imlicit equation = a b defined over a comlete dioid admits = a b as least solution with a = + i=0 ai and a 0 = e. The star oerator is usually called Kleene star. 2.2 Residuation theory A maing f defined from a comlete dioid D into a comlete dioid C is said to be isotone if a, b D, a b f(a) f(b). Residuation theory (Blyth and Janowitz, 1972) defines seudo-inverses for some isotone maings defined over ordered sets such as dioids (Cohen, 1998). More recisely, if the greatest element of set D f() b} eists for all b C, then it is denoted f (b) and f is called residual of f. The isotone maing L a : a defined in a comlete dioid is residuated. The greatest solution to a b eists and is equal to L a(b), also denoted b a or a \b. We furthermore have the following formulæ (see (Baccelli et al., 1992, 4.4)). ab = a \ b (1) a (2) a \ a = a (3) Theorem 2. (Baccelli et al., 1992, Th. 4.56) Let f : D C and g : C B. If f and g are residuated then g f is residuated and (g f) = f g. Definition 3. (Maing restriction). Let f : D C a maing and A D. We denote f A : A C the maing defined by equality f A = f Id A where Id A : A D is the canonical injection. A constrained residuation roblem (Cohen, 1998) consists in finding the greatest solution to f() b not in the whole D but only in a subset A of D. More recisely, being given a residuated maing f we aim at solving: f A () = f Id A () b. (4) Proosition 1. If the canonical injection Id A is residuated, then the constrained residuation roblem (4) admits an otimal solution denoted f A (b). Proof : Thanks to theorem 2, if Id A is residuated, then so is f Id A, and the greatest solution to (4) is given by f A (b) = (f Id A) (b) = Id A f (b) (5) 2.3 Reresentation of DEDS using dioids Dioids algebra enables to model DEDS involving (only) synchronization henomena. For instance, by dating each event, i.e. by associating with each event a discrete function (res. u and y for an inut and an outut) called dater 3, it is ossible to get a linear state reresentation: (k) = A(k 1) Bu(k) y(k) = C(k). (6) An analogous transform to Z-transform (used to reresent discrete-time trajectories in classical theory) can be introduced for daters. The γ- transform of a dater (k) is defined as the following formal ower series: = k Z (k)γk. Indeed, variable γ can be interreted as the backward shift oerator in event domain. The set of formal ower series in one variable γ and coefficients in Z ma is a dioid denoted Z ma γ. State reresentation (6) becomes in Z ma γ : 3 (k) is the time of the k + 1-th occurrence of event.

4 = γa Bu y = C. (7) Considering the earliest functioning of the system, we select the least solution of the first equation of (7) which is given according to theorem 1 by = (γa) Bu. This leads to y = Hu = C(γA) Bu, in which H = C(γA) B is called the transfer matri. 2.4 Just In Time control The Just In Time (JIT) control of a DEDS consists in comuting the latest inut dates which cause outut dates to be earlier than or equal to a given target. Let us define a dater secifying the desired oututs and z its γ-transform. From the reresentation in dioid algebra, this control roblem is formulated as an inequality, and residuation theory leads to otimal solution u ot : u ot = u L H(u)=Hu z} u = z H = 3. MODELLING OF URBAN BUS NETWORKS z C(γA) B. (8) Many transortation systems may be studied as DEDS. With this oint of view, their evolution is conditioned by events such as arrivals or deartures of vehicles. In net sections, the focus is on urban bus networks. We first describe how they oerate. Then, we roose a model in dioid Z ma. 3.1 Functioning of urban bus networks As resented in (Hayat and Maouche, 1997), iloting of ublic transortation networks can be decomosed in two stages : the oerating schedule and the regulation. Definition of an oerating schedule The oerating schedule is established with the aim of otimizing the offer of service according to objectives and constraints (bus fleet, line layouts, staff hours of work, etc). It is calculated for mean functioning conditions. In ractical terms, this otimization results in: (1) The definition of the bus routes and the bus stos for each line. (2) The choice of a level-of-service for each line: distribution of resources (buses, drivers), definition of the minimum and maimum headways 4 (i.e. the eected minimum and maimum time searations between two buses). 4 A minimum headway is defined to counteract the natural tendency of transit vehicles to bunch u. Thus, if a bus falls (3) The synthesis of timetables defining times at which buses should theoretically run at each sto. Timetables are used to inform assengers and, at some stos, to re-synchronize buses. Regulation This stage corresonds to adjustments or adatations from the oerating schedule in reaction to current functioning conditions. Common conditions leading to such adjustment oerations are disturbances: breakdown of buses, modifications of traffic flows (for instance due to accidents), etc. A suervisor may then decide to transfer assengers, sto or reroute buses... In the following, we are only interested in modelling the oerating schedule. In 4, we furthermore show how dioid techniques can be used for the synthesis of timetables. 3.2 Modelling of a bus network In this section, we roose a model for urban bus networks functioning according to their oerating schedule. We assume that each line i includes n i bus stos. The network is comosed by M lines and N = n n M stos. In the following, let i (k) denote the dearture time of the (k + 1)-st bus at sto i. Without loss of generality, we assume that at the beginning of oeration a bus dearts from each sto 5. Suose that the bus coming from sto j reaches sto i. Then we have the following inequation: i (k) a ij + j (k 1),k > 1, in which a ij denotes the travelling time from sto j to i. Let (k) = ( 1 (k), 2 (k),..., N (k)) t, for the whole network this inequation can be written in maalgebraic matri notation (k) A (k 1), (9) in which A ij = a ij if sto j recedes sto i, otherwise A ij = ε. In ractice, a ublic transortation network oerates under a timetable which schedules the dearture times of every bus. Buses resect timetables slightly behind schedule for any reasons, it will have more than the average number of assengers to ick u at the net station, which causes further delays. Thus, it kees failing further behind schedule. Conversely, the bus behind it encounters fewer assengers than usual, allowing it to catch u with the receding bus. Such bunching tends to aear if no minimum headway is defined. Symmetrically, a maimum headway secifies the desired level-of-service since it defines the minimum frequency of buses on the line. 5 If no or several bus(es) initially deart from stos, then this results only in indees modification. These cases can be dealt eactly as cases of laces initially containing no or several token(s) for the modelling of timed event grah (Baccelli et al., 1992, 2.5.2).

5 only at secific stos 6 such as terminus or dearture of lines or main stations. At the other stos, timetables are only used to inform assengers. We denote u l (k) the scheduled dearture time for the (k + 1)-st bus at sto l and S u the set of secific stos, this leads to (k) Bu(k), (10) in which B is a N N matri where B ii = e if sto i S u, B ij = ε otherwise. Considering inequations (9) and (10) and assuming that buses deart as soon as ossible, a scheduled transortation network is modelled in dioid Z ma by a state equation as in (6). 4. TIMETABLES SYNTHESIS OF URBAN BUS NETWORKS 4.1 Problem descrition In a bus network, it may be rofitable to minimize waiting times at a given sto for assengers arriving from another given sto. Such stos, generally close from a geograhic oint of view 7, belong to an itinerary which should be romoted 8 and are called connection stos. With the aim of doing so, we consider a roblem formulated in (Ceder et al., 2000) 9. This roblem consists in finding timetables which (i) maimize synchronizations at connection stos, (ii) ensure an eected level-of-service at articular stos of the network, (iii) satisfy lanning constraints (i.e. minimum and maimum headways, see 3.1). In order to comute timetables at each sto so as to synchronize bus deartures at connection stos (objective mentioned at item (i)), we have to find inut u(k)} k Z such that state (k)} k Z satisfies: (k) = A (k 1) B u(k) (11) where evolution matri A takes into account connections between lines and B = B. More recisely, A ij = A ij if sto j recedes sto i on the same line; A ij = w ij denotes the walking time 6 Where buses can ark without blocking or disturbing traffic. 7 Walking time between these stos is small (they are sometimes located at a same node of the network). 8 Due to the imortant flow of users and/or for commercial reasons. 9 Tools used in (Ceder et al., 2000) are essentially based on linear rogramming methods from sto j to sto i if i is in connection with j; A ij = ε otherwise. The desired level-of-service (mentioned at item (ii)) is secified for some articular stos (we denote S o the set of these stos). More recisely, for such a sto i S o, a dater z i (k)} k Z can be defined: z i (k) secifies the latest date at which the k + 1-st bus should deart from sto i. From equation (11), dearture times at stos belonging to S o are given by: y(k) = C (k), (12) where C is a q N matri (q = Card(S o )) in which C ij = e if i,j are indees in y and of a same sto in S o, C ij = ε otherwise. We then aim at finding u(k)} k Z such that y(k) = C (k) z(k). (13) According to item (iii), timetables given by dater u(k)} k Z must furthermore satisfy the lanning constraints, i.e. u(k + 1) min u(k) (14) ma u(k) u(k + 1), (15) where min and ma are diagonal matrices traducing minimum and maimum headways for each line. For all stos s of line i, the element min ss (res. ma ss maimum) headway min i ) is equal to the minimum (res. (res. ma i ) of line i. 4.2 Formalization as a constrained residuation roblem From γ-transforms of equations (11) and (12), we obtain the eected inut/outut behavior (cf. 2.3): y = Hu = L H (u), with H = C (γa ) B. Equation (13) can then be written y = L H (u) z. (16) More recisely, we are interested in finding the greatest solution to this inequation. This solution allows buses to stay at stos belonging to S u as late as ossible while satisfying desired deartures dates at stos in S o (given by z). With this formulation, the roblem given by equation (16) then aears to be a JIT control roblem (cf. 2.4). Moreover, the searched solution u must satisfy inequations (14) and (15). Denoting A γ the subset of Z ma γ comosed of series satisfying γ-transforms of (14) and (15), we can rewrite the

6 considered roblem as a constrained residuation roblem (cf. 2.2): LH (u) z u A γ (admissible domain). (17) Canonical injection of the subset A γ in Z ma γ is denoted Id A and L H A is the maing L H restricted to the domain A γ. Regarding definition 3, the constrained roblem is equivalent to solve: L H A (u) = L H Id A (u) z. (18) 4.3 Solving the roblem From equation (18), we can aly roosition 1 if Id A is residuated: L H A (z) = Id A L H (z). (19) The considered roblem will then be solved in two stages: (i) comute the solution to the relaed roblem u ot = L H (z) (i.e without lanning constraints), (ii) find the best aroimation of (i) in admissible domain u ota = Id A (L H (z)). To aly this reasoning, it remains to show that canonical injection Id A is residuated. With that goal, we first characterize the subset A γ. Subset A γ is comosed of formal ower series which satify lanning constraints (14) and (15). Since ma is diagonal, it is invertible and we obtain the two following inequations: min u(k 1) u(k) ( ma ) 1 u(k + 1) u(k), The γ-transforms of these inequations lead to min γu u ( ma ) 1 γ 1 u u, and ( min γ ( ma ) 1 γ 1 ) u u. Since roduct is residuated, we equivalently have u u = min γ ( ma ) 1 u. γ 1 The greatest solution is given by (see (Baccelli et al., 1992, Th )) u = u (γ min γ 1 ( ma ) 1 ). (20) With the aim of abbreviating notations, we below denote = γ min γ 1 ( ma ) 1. Equality (20) allows the following definition of subset A γ : A γ = Z ma γ = }. In order to show the residuability of canonical injection Id A, we study the quotient of Z ma γ by a articular equivalence relation (this idea was insired by (Cohen, 1993, Th. 30)). Proosition 2. Let Π be the maing from Z ma γ into itself defined by Π :. We define the following equivalence relation R y} = y }. (1) Each equivalence class of Z ma γ /R contains one and only one element belonging to Π(Z ma γ ) and this element is elicitly given by for any in the class. (2) Element is the least element in [] R, and it is the greatest element among those of Π(Z ma γ ) which are less than. Proof : (1) For a given element Z ma γ, the element clearly belongs to Π(Z ma γ ) and since \ = (using (3)), it also belongs to [] /R. Moreover, suose that there is another element 1 [] R which also belongs to Π(Z ma γ ): Π( 1) = 1 1 = 1 so 1 = \ 1 = = 1 hence 1 =, however 1 R, which roves uniqueness of the element belonging to [] R and Π(Z ma γ ). (2) Since e, for all y [] R, we have = y y (see (2)). Consequently is the least element of [] R. For all z Π(Z ma γ ) such that z, since Π is isotone, we obtain z, hence is the greatest element of Π(Z ma γ ) which is less than. Note that for any element Π(Z ma γ ) there eists y such that =, and we have \y = y y = = (using (3)), which shows that Π(Z ma γ ) A γ. The converse is obvious and subsets Π(Z ma γ ) and A γ are then identical. Last assertion of roosition 2 consequently roves that canonical injection Id A is residuated and with Z ma γ, its residual is Id A ( ) = A γ } =.

7 %! # & & ",, 5. EXAMPLE We now resent an eamle of bus network comosed of two lines including two connection stos (see the grah below). We assume that 5 and 7 are resectively in connection with 1 and 3 (null walking times for these connections). We then obtain the following matri A. E A N " N # N N N! N % N $ E A A = ε ε 4 ε ε ε ε 3 ε ε ε ε ε ε ε 5 ε ε ε ε ε ε ε ε ε ε ε 1 e ε ε 7 ε ε ε ε ε ε ε 8 ε ε ε ε e ε ε 8 ε The desired outut is secified for only one sto 7 (S o = 7 }), this leads to C = ( ε ε ε ε ε ε e ). Moreover, the following tabular reresents the desired level-of-service of 7. k z(k) Planning constraints are identical for both lines: min = 7 and ma = 10. We solve the roblem in two stages (cf. item (i) and (ii) of 4.3). The first ste consists in comuting the solution of the relaed roblem u ot. We show the comuted dater u ot6 of the generated timetables. Note that lanning constraints are not satisfied. k y(k) u ot6 (k) Finally we find the best aroimation of u ot in A γ. k # "!! " # $ y(k) u ota6 (k) # "!! " # $ K F J $ K F J $ K F J ) $ Fig. 1. Comutation of dater u ota6 (k) in two stages Remark 1. For the comutation, we have used software tools dedicated to calculus in dioids (SW2004, 2004). 6. CONCLUSION E = N the functioning of such systems. A (ma,+) modelling is established to simulate such articular systems which are conditioned by timetables. We suggest a solution using residuation theory to the roblem of timetables generation. We are currently trying to etend this work by considering more general synchronizations at connection stos (e.g. not occurring at each dearture). REFERENCES F. Baccelli, G. Cohen, G. J. Olsder, and J. P. Quadrat. Synchronization and Linearity. Wiley, T. S. Blyth and M. F. Janowitz. Residuation Theory. Pergamon ress, H. Braker. Algorithms and alications in timed discrete event systems. PhD thesis, Delft University of Technology, Dec A. Ceder, B. Golany, and O. Tal. Creating bus timetables with maimal synchronization. Transortation Research Part A: Policy and Practice, 35: , G. Cohen. Two-dimensional domain reresentation of timed event grahs. In Summer School on DES, Sa, Belgium, June G. Cohen. Residuation and Alications. In Algèbres Ma-Plus et alications en informatique et automatique : Ecole d informatique théorique, Noirmoutier, INRIA. R. de Vries, B. de Schutter, and B. de Moor. On ma-algebraic models for transortation networks. In Proceedings of WODES 98, Cagliary, Italy, S. Hayat and S. Maouche. Régulation du trafic des autobus : amélioration de la qualité des corresondances. Technical Reort LI-TU0192, INRETS, S. Lahaye, L. Hardouin, and J. L. Boimond. Models combination in (ma,+) algebra for the imlementation of a simulation and analysis software. Kybernetika, 39(2): , 2003a. S. Lahaye, L. Houssin, and J. L. Boimond. Modelling of urban bus networks in dioids algebra. In Proceedings of POSTA 2003, ages 23 30, Roma, Italy, aug. 2003b. Sringer Verlag. J. Y. Le Boudec and P. Thiran. Network Calculus. Sringer Verlag LNCS 2050, E. Menguy, J.-L. Boimond, L. Hardouin, and J.- L. Ferrier. Just in time control of timed event grahs: Udate of reference inut, resence of uncontrollable inut. IEEE TAC, 45(11): , SW www-rocq.inria.fr/malusorg/soft.html., We show that dioids algebra can be suitable for bus networks modelling. First, we have described