Intersection Models and Nash Equilibria for Traffic Flow on Networks

Transcription

1 Intersection Models and Nash Equilibria for Traffic Flow on Networks Alberto Bressan Department of Mathematics, Penn State University (Los Angeles, November 2015) Alberto Bressan (Penn State) Traffic flow on Networks 1 / 38

2 A conservation law describing traffic flow (Lighthill - Witham - Richards, ) ρ = density of cars a b x ρ t + [ρ v i (ρ)] x = 0 v i (ρ) = velocity of cars on road i (depends only on the density) f i (ρ) = ρ v i (ρ) = flux on the i-th road of the network f i < 0, f i (0) = f i (ρ jam i ) = 0 Alberto Bressan (Penn State) Traffic flow on Networks 2 / 38

3 Modeling traffic flow at a junction incoming roads: i I outgoing roads: j O j i Boundary conditions account for: θ ij = fraction of drivers from road i that turn into road j. c i = relative priority of drivers from road i (fraction of time drivers from road i get green light, on average) θ ij = 1 j Alberto Bressan (Penn State) Traffic flow on Networks 3 / 38

4 Boundary conditions at junctions incoming roads: i I outgoing roads: j O i j Boundary conditions should relate ρ i (t, 0 ) i I ρ j (t, 0+) j O depending on drivers turning preferences θ ij Conservation equations: f i (ρ i )θ ij = f j (ρ + j ) j O i Alberto Bressan (Penn State) Traffic flow on Networks 4 / 38

5 Boundary conditions for several incoming and outgoing roads H.Holden, N.H.Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal. 26, G.M.Coclite, M.Garavello, B.Piccoli, Traffic flow on a road network, SIAM J. Math. Anal. 36, M.Herty, S.Moutari, M.Rascle, Optimization criteria for modeling intersections of vehicular traffic flow, Netw. Heterog. Media 1, M.Garavello, B.Piccoli, Conservation laws on complex networks, Ann.I.H.Poincaré M.Garavello, B.Piccoli, Traffic Flow on Networks, AIMS, A.B., S.Canic, M.Garavello, M.Herty, and B.Piccoli, Flow on networks: recent results and perspectives, EMS Surv. Math. Sci. 1 (2014), Alberto Bressan (Penn State) Traffic flow on Networks 5 / 38

6 Construction of a Riemann Solver (Coclite, Garavello, Piccoli) { ρ1,..., ρ N = initial densities (constant on each road) θ ij = fraction of drivers from road i that turn into road j f i f j = imum flux on the incoming road i I = imum flux on the outgoing road j O Feasible region Ω R n. Vector of incoming fluxes (f 1,..., f n ) Ω iff f i [0, f i ] i I i f iθ ij f j j O 1 f 2 f f 3 4 f Alberto Bressan (Penn State) Traffic flow on Networks 6 / 38

7 The feasible region Ω f 2 1 f 2 f 3 f 4 f Ω 0 f 1 Riemann solver rule for selecting a point in the feasible region Ω. Natural choice: imize the total flux through the node: i I f i Alberto Bressan (Penn State) Traffic flow on Networks 7 / 38

8 Continuity of the Riemann Solver Selection rule: imize the total flux i I f i If the turning preferences θ ij remain constant, the fluxes f i depend Lipschitz continuously on the Riemann data ρ 1,..., ρ N. f 2 Ω 0 f 1 The Riemann solver is discontinuous w.r.t. changes in the θ ij f 2 f 2 Ω 0 f 1 0 Ω f 1 Alberto Bressan (Penn State) Traffic flow on Networks 8 / 38

9 Why can the θ ij vary in time? Drivers turning preferences θ ij must be determined as part of the solution 2 1 A 3 B 4 5 # of vehicles on road i that wish to turn into road j is conserved: (ρθ ij ) t + (ρv i (ρ)θ ij ) x = 0 Alberto Bressan (Penn State) Traffic flow on Networks 9 / 38

10 Traffic flow on a network of roads j i On the i-th incoming road, car flow is described by ρ t + f i (ρ) x = 0 conservation law θ ij,t + v i (ρ)θ ij,x = 0 linear transport equation θ ij are passive scalars, relevant only at intersection Alberto Bressan (Penn State) Traffic flow on Networks 10 / 38

11 Continuous Riemann Solvers (A.B. - F.Yu, Discr. Cont. Dyn. Syst., 2015) f 2 _ f Ω 0 f 1 The selection rule: imize i I f i yields a Riemann solver which is Hölder continuous w.r.t. all variables (ρ i, θ ij ) i I, j O (f i ) i I One can also construct a Riemann solver which is Lipschitz continuous w.r.t. all variables Unfortunately all this is useless, because if the θ ij are allowed to vary the Cauchy problem is ill posed anyway Alberto Bressan (Penn State) Traffic flow on Networks 11 / 38

12 Ill-posedness of the Cauchy problem at intersections Modeling assumptions If all cars arriving at the intersection can immediately move to outgoing roads, no queue is formed. If outgoing roads are congested, the inflow of cars from road 1 is twice as large as the inflow from road Alberto Bressan (Penn State) Traffic flow on Networks 12 / 38

13 Example 1: θ ij with unbounded variation, two solutions f 2 f 1 x f 3 f 4 2x f k (ρ) = 2ρ ρ 2 imum flux on every road: f k = 1 Initial data: ρ k = 1, k = 1, 2, 3, 4 1 if 2 n < x < 2 n 1, n even ˆθ 13 (x) = ˆθ 24 (x) = 0 if 2 n < x < 2 n 1, n odd Alberto Bressan (Penn State) Traffic flow on Networks 13 / 38

14 f 2 f 1 x f 3 f 4 2x Solution 1. Incoming fluxes: f 1 (t, 0) = 1, f 2 (t, 0) = 1 Solution 2. Incoming fluxes: f 1 (t, 0) = 2 3, f 2(t, 0) = 1 3 Alberto Bressan (Penn State) Traffic flow on Networks 14 / 38

15 Example 2: θ ij constant, Tot.Var.(ρ i ) small, two solutions 6 5 cars from roads 5, 8 go to road 3 cars from roads 6, 7 go to road f ( ρ) = ρ(2 ρ) k 8 ρ (x) = ρ (x) 5 8 x ρ (x) = ρ(x) x 0 At some time T > 0, the same initial data as in Example 1 is created at the junction of roads 1 and 2 Alberto Bressan (Penn State) Traffic flow on Networks 15 / 38

16 Example 3: lack of continuity w.r.t. weak convergence f 1 = 2 f = 1 2 _ f = 1 f 3 = 1 _ f = 1 As n, the weak limit is θ 12 = θ 13 = 1 2 f 1 = 2 _ f = 2 f = 1 2 f 3 = 1 Alberto Bressan (Penn State) Traffic flow on Networks 16 / 38

17 An intersection model with buffers q 4 q the intersection contains a buffer with finite capacity (a traffic circle) t q j (t) = queues in front of outgoing roads j O, within the buffer incoming drivers are admitted to the intersection at a rate depending on the size of these queues drivers already inside the intersection flow out to the road of their choice at the fastest possible rate Alberto Bressan (Penn State) Traffic flow on Networks 17 / 38

18 M. Herty, J. P. Lebacque, and S. Moutari, A novel model for intersections of vehicular traffic flow. Netw. Heterog. Media M. Garavello and P. Goatin, The Cauchy problem at a node with buffer. Discrete Contin. Dyn. Syst M. Garavello and B. Piccoli, A multibuffer model for LWR road networks, in Advances in Dynamic Network Modeling in Complex Transportation Systems, Toward the analysis of global optima and Nash equilibria, we need well posedness for L initial data ρ 0 k, θ0 ij continuity of travel time w.r.t. weak convergence ρ k,t + f k (ρ k ) x = 0 conservation laws θ ij,t + v i (ρ i )θ ij,x = 0 linear transport equations Alberto Bressan (Penn State) Traffic flow on Networks 18 / 38

19 Intersection models with buffers (A.B., K.Nguyen, Netw. Heter. Media, 2015) q j (t) = size of the queue, inside the intersection, of cars waiting to enter road j (SBJ) - Single Buffer Junction M > 0 = imum number of cars that can occupy the intersection c i > 0, i I, priorities given to different incoming roads Incoming fluxes f ( i satisfy fi c i M q j ), i I j O Alberto Bressan (Penn State) Traffic flow on Networks 19 / 38

20 Well-posedness of the Cauchy problem with buffers Theorem A.B.- K.Nguyen, Netw. Heter. Media, Assume that the flux functions satisfy f k < 0, f k (0) = f k (ρ jam k ) = 0 k I O Consider any L initial data ρ(0, x) = ρ k (x) [0, ρ jam k ], q j (0) = q j, θ ij (0, x) = θ ij [0, 1] with j O q j < M, θ ij (x) = 1 j O Then the Cauchy problem has a unique entropy admissible solution, defined for all t 0. Moreover, the travel times depend continuously on the initial data, in the topology of weak convergence. ρ n k(x) ρ k ρ n i θ n ij ρ i θij, q n j q j Alberto Bressan (Penn State) Traffic flow on Networks 20 / 38

21 Variational formulation (A.B., K.Nguyen) ρ 1 ρ 2 q 4 q 5 ρ 3 ρ 4 ρ 5 (q, q ) 4 5 length of queues Lax formula boundary values (V, V, V, V, V ) (q, q ) V (t,x) = k x ρ (t,x) dx k If the queue sizes q j (t) within the buffer are known, then the initial-boundary value problems can be independently solved along each incoming road. These solutions can be computed by solving suitable variational problems. From the value functions V k, the traffic density ρ k = V k,x along each incoming or outgoing road is recovered by a Lax type formula. Alberto Bressan (Penn State) Traffic flow on Networks 21 / 38

22 ρ 1 ρ 2 q 4 q 5 ρ 3 ρ 4 ρ 5 Lax formula (q, q ) 4 5 length of queues boundary values (V, V, V, V, V ) (q, q ) ρ = V k k,x traffic densities Conversely, if these value functions V k are known, then the queue sizes q j can be determined by balancing the boundary fluxes of all incoming and outgoing roads The solution of the Cauchy problem is obtained as the unique fixed point of a contractive transformation The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general L data, and continuity of travel time w.r.t. weak convergence Alberto Bressan (Penn State) Traffic flow on Networks 22 / 38

23 The Legendre transform of the flux function f Legendre transform: g(v). = inf u [0, ρ jam ] {vu f (u)} f vu jam f 0 ( ρ ) f (0) _ f v f(u) g(v) 0 u *(v) jam ρ u Alberto Bressan (Penn State) Traffic flow on Networks 23 / 38

24 A variational problem describing traffic on road i I t f i(u) x = v flux across the characteristic = f i(u) vu a x g i (v) = [flux of cars from left to right, across the characteristic] For boundary conditions (SBJ), define h i (q). = min f i, c i ( M j O ) q j i I Alberto Bressan (Penn State) Traffic flow on Networks 24 / 38

25 Incoming roads with boundary condition (SBJ) initial data: V i (x). = x ρ i (y) dy, queue lengths: q j (t), j O To find V i ( t, x), consider the optimization problem: t imize: V i (x(0)) + L i (x(t), ẋ(t)) dt 0 running payoff: L i (x(t), ẋ(t)) = g i (ẋ(t)) if x(t) < 0 h i (q(t)) if x(t) = 0 terminal condition: x( t) = x Alberto Bressan (Penn State) Traffic flow on Networks 25 / 38

26 Optimization Problem 1 t imize: V i (x(0)) + L i (x(t), ẋ(t)) dt 0 among all absolutely continuous functions x : [0, t] R such that x( t) = x, x(t) 0 for all t [0, t] (t, x) t (t, x) t τ x opt τ x(t) opt x x x _ y Alberto Bressan (Penn State) Traffic flow on Networks 26 / 38

27 The Value Function V i an optimal solution x opt exists, and is the concatenation of at most three affine functions ẋ opt [f i (0), f i (ρjam i )] is the speed of a characteristic the traffic density ρ i (t, x) = V i,x (t, x) is an entropy solution of the conservation law, satisfying initial + boundary conditions Alberto Bressan (Penn State) Traffic flow on Networks 27 / 38

28 # _ x (t) (t,x) (t,x) _ t τ τ x y y 0 V i (t, x). = { [ ( x y V i (y) + t g i y 0 t )], [ ( V i (y) + τ y ) τ ( x ) ] } g i h 0 τ τ t, y 0 τ i (q(s)) ds + (t τ) g i. τ t τ V i (t, x) = total amount of cars which at time t are still inside the half line ], x] V i (0) V i (t, 0) = total amount of cars which have exited from road i during [0, t] Alberto Bressan (Penn State) Traffic flow on Networks 28 / 38

29 The limit Riemann Solver for buffer of vanishing size Letting the size of the buffer M 0 one obtains a Riemann Solver which is Lipschitz continuous w.r.t. all variables ρ i, θ ij f2 f2 f2 _ f _ f γ γ _ f γ f1 f1 f1 s γ(s) = (f 1 (s),..., f m (s)), f i (s). = min{c i s, f i } Then the incoming fluxes are f i { where: s = = f i ( s) s 0 ; f i (s) θ ij i I f j for all j O Alberto Bressan (Penn State) Traffic flow on Networks 29 / 38 }.

30 Optimization Problems for Traffic Flow on a Network Existence of a globally optimal solution Existence of a Nash equilibrium solution Alberto Bressan (Penn State) Traffic flow on Networks 30 / 38

31 Basic setting n groups of drivers with different origins and destinations, and different costs Drivers in the k-th group depart from A d(k) and arrive to A a(k) can use different paths Γ 1, Γ 2,... to reach destination Departure cost: ϕ k (t) arrival cost: ψ k (t) A d(k) Aa(k) Alberto Bressan (Penn State) Traffic flow on Networks 31 / 38

32 Basic assumptions (A1) The flux functions ρ f i (ρ) = ρ v(ρ) are all strictly concave down. f i (0) = f i (ρ jam i ) = 0, f i < 0. (A2) For each group of drivers k = 1,..., N, the cost functions ϕ k, ψ k satisfy ( ) ϕ k < 0, ψ k, ψ k < 0, lim ϕ k (t) + ψ k (t) = + t f f(ρ) ϕ(t) ψ(t) 0 ρ ρ jam ρ t Alberto Bressan (Penn State) Traffic flow on Networks 32 / 38

33 Admissible departure rates G k = total number of drivers in the k-th group, k = 1,..., n Γ p = viable path to reach destination, p = 1,..., N t ū k,p (t) = departure rate of k-drivers traveling along the path Γ p The set of departure rates {ū k,p } is admissible if ū k,p (t) 0, p ū k,p (t) dt = G k k = 1,..., n τ p (t) = arrival time for a driver starting at time t, traveling along Γ p (depends on the overall traffic conditions) If this is a k-driver, his total cost is ϕ k (t) + ψ k (τ p (t)). Alberto Bressan (Penn State) Traffic flow on Networks 33 / 38

34 Optima and Equilibria An admissible family {ū k,p } of departure rates is globally optimal if it minimizes the sum of the total costs of all drivers. J(ū) = ( ϕ k (t) + ψ k (τ p (t))) ū k,p (t) dt k,p An admissible family {ū k,p } of departure rates is a Nash equilibrium if no driver of any group can lower his own total cost by changing departure time or switching to a different path to reach destination. ϕ k (t) + ψ k (τ p (t)) = C k for all t Supp(ū k,p ) ϕ k (t) + ψ k (τ p (t)) C k for all t R Alberto Bressan (Penn State) Traffic flow on Networks 34 / 38

35 Existence results Theorem (A.B. - Khai Nguyen, Netw. Heter. Media, 2015) Under the assumptions (A1)-(A2), on a general network of roads, there exists at least one globally optimal solution. If, in addition, the travel time admits a uniform upper bound, then a Nash equilibrium exists. Proof is achieved by finite dimensional approximations + a topological argument (relying on the continuity of the travel time w.r.t. weak convergence of the departure rates) For a single group of drivers on a single road, solutions are unique. Uniqueness is not expected to hold, on a general network. An earlier existence result was proved in A.B. - Ke Han, Netw. & Heter. Media, 2013, with highly simplified boundary conditions at road intersections. Alberto Bressan (Penn State) Traffic flow on Networks 35 / 38

36 Construction of Nash equilibria By finite dimensional approximations + topological methods u k,p Φ (t) = ϕ (t) + ψ (τ ) k,p k k p (t) t u k,p Φ (t) = ϕ k,p (t) + k ψ (τ p(t) ) k t m t l t Alberto Bressan (Penn State) Traffic flow on Networks 36 / 38

37 Can traffic get completely stuck? A A B C C A B B C Assume: at each node, equal numbers of cars are allowed to enter from the two incoming roads. Then: for every two cars entering, only one exits the triangle of roads ABC at any time t, [# of cars that has reached destination] [# of cars inside the triangle ABC] only finitely many cars can reach destination. All the others are stuck forever. Alberto Bressan (Penn State) Traffic flow on Networks 37 / 38

38 References A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction. Discr. Cont. Dyn. Syst. 35 (2015), A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads. Netw. Heter. Media 10 (2015), A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues. Netw. Heter. Media 10 (2015), Alberto Bressan (Penn State) Traffic flow on Networks 38 / 38