c 2002 Society for Industrial and Applied Mathematics

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1 SIAM J. APPL. MATH. Vol. 63, No. 1, pp c 2002 Society for Industrial Applied Mathematics A RIGOROUS TREATMENT OF A FOLLOW-THE-LEADER TRAFFIC MODEL WITH TRAFFIC LIGHTS PRESENT BRENNA ARGALL, EUGENE CHELESHKIN, J. M. GREENBERG, COLIN HINDE, AND PEI-JEN LIN Abstract. Traffic flow on a unidirectional roadway in the presence of traffic lights is modeled. Individual car responses to green, yellow, red lights are postulated these result in rules governing the acceleration deceleration of individual cars. The essence of the model is that only specific cars are directly affected by the lights. The other cars behave according to simple followthe-leader rules which limit their speed by the spacing between them the car directly ahead. The model has a number of desirable properties; namely, cars do not run red lights, cars do not smash into one another, cars exhibit no velocity reversals. In a situation with multiple lights operating in-phase, we get, after an initial start-up period, a constant number of cars through each light during any green-yellow period. Moreover, this flux is less by one or two cars per period than the flux obtained in discretized versions of the idealized Lighthill Whitham Richards model which allows for infinite accelerations. Key words. traffic flow, follow-the-leader, relaxation models, conservation laws AMS subject classification. 35 PII. S Introduction, model description, statement of results. In this note we examine the behavior of traffic on a unidirectional highway when multiple traffic lights are present. For simplicity we assume the lights operate in-phase. The model postulates the dynamics of individual cars but may also be thought of as a coarse discretization of a continuum model introduced recently by Greenberg [1], Aw Rascle [2], Aw, Klar, Materne, Rascle [3], Zhang [9] details of this correspondence may be found in section 4, 4.6) 4.8)). We assume we are presented with an empirically determined function s Vs) on L s which satisfies 1.1) VL + )=0, 1.2) 1.3) dv ds s) > 0 d2 V s) < 0, ds2 L lim Vs), s s<, dv ds s), d 2 ) V ds 2 s) =V > 0, 0, 0). The independent variable s is interpreted as the spacing between cars, L is the minimum car spacing a lower bound for L is the length of typical car), V > 0 Received by the editors June 22, 2001; accepted for publication in revised form) March 1, 2002; published electronically August 28, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA greenber@rew.cmu.edu). The research of the third author was partially supported by the Applied Mathematical Sciences Program, U.S. Department of Energy by the U.S. National Science Foundation. 149

2 150 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN is the maximum allowable speed of a car. A typical function, one we shall use in simulations, is Vs) =V 1 L ) 1.4), L s<. s In this classic Lighthill Whitham Richards model [4, 5, 6, 7] the function V ) gives the velocity of individual cars; in ours it provides an upper bound for the velocity of an individual car. An extensive discussion of suitable functions V ) may be found in [8, Chapter 4] the references contained therein. Suffice it to say that the functions V ) in our model are consistent with those used in practice. In this model x t), 1 N, denotes the position of the th car at time t, 0 u t) is the velocity of the th car. Throughout, dx 1.5) = u, 1 N, the cars are ordered so that x +1 x )t) L, 1 N 1. During time intervals where the lights are green we assume that 1.6) u = Vx +1 x )t)) + α, 1 N, 1 where α t) 0 satisfies dα 1.7) = α, 1 N. The parameter >0 may be thought of as a relaxation time. Equations 1.6) 1.7) imply that during the green light periods the velocities, u, satisfy 1.7a) du 1.7b) = V x +1 x )u +1 u )+ Vx +1 x ) u ), 1 N 1, du N = V u N ). The interesting feature of our model is how yellow or red lights effect the dynamics of an individual car. Our traffic lights cycle from green to yellow to red, the numbers 0 <TG,0<TY, 0 <TRdenote the duration of the green, yellow, red lights. At time t = 0 we assume we have a sequence of N cars located at 1.8) x 0)= 0 )L 1, 1 N, where L 1 L again L is the minimum allowable auto spacing), we assume these cars are all at rest; i.e., 1.9) u 0)=0, 1 N. Finally, we assume they are at traffic lights located at x = l I, 1 I M, where 1.10) N 0 )L 1 <l 1 <l 2 < <l M. 1 When = N, u N = V + α N.

3 A TRAFFIC MODEL WITH LIGHTS 151 We further assume that each intersection is of wih w>0 let 1.11) t m =m 1)TG+ TY + TR), m =1, 2,..., denote the start of the mth light cycle. During the time interval t m t t m + TG all cars satisfy 1.5) 1.7). At time t y = t m + TG, the green lights turn yellow, this will have an effect on the traffic flow. We start by describing what happens to the lead car, the one indexed by N, when it encounters a light at x = l. We assume that 1.12) x N t y ) <l. If 1.13) x N t y )+u N t y )TY l + w + L, then the lead car will be able to completely clear the intersection if it travels at its current speed u N t y ). We allow it to clear the intersection by following its stard dynamics; that is, over the time interval t y t t m+1 the Nth car satisfies 1.14) where dx N = u N, 1.15) u N = V + α N α N 0 satisfies dα N 1.16) = α N. Following these dynamics the lead car accelerates through the intersection. On the other h, if 1.17) x N t y )+u N t y )TY < l+ w + L, then it will be impossible for the Nth car to clear the intersection during the yellow phase if it continues to travel at its current speed. If 1.18) x N t y )+u N t y )TY + TR) l, then over the time interval t y t t m+1 we require it satisfies the modified dynamics 1.19) dx N = u N du N i.e., we insist that it travels at its current speed. This strategy avoids the Nth car accelerating then possibly having to decelerate as it nears the light. If 1.17) holds 1.18) is violated, the lead car will have to slow down possibly stop. When it satisfies the additional inequality =0; 1.20) x N t y )+u N t y )TY + TR)/2 >l,

4 152 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN the lead car is mated to satisfy 1.21) dx N = u N du N = u 2 N t y) 2l x N t y )),t y t t y + 2l x Nt y )) u N t y ) 0,t y + 2l x Nt y )) u N t y ) t t m+1. 2 This constant deceleration strategy brings the Nth car to rest at x = l at t = t y + 2l x N t y)) u N t y) t m+1, it then sits at the light until t = t m+1. Finally, when 1.22) x N t y )+u N t y )TY + TR) >l x N t y )+u N t y )TY + TR)/2 l, the lead car is mated to satisfy dx N = u N t) du N = 2x Nt y )+u N t y )TY + TG) l) TY + TG) 2 over the whole interval t y t t m+1. This strategy brings the car to the light at x = l at t m+1 with velocity 1.23) u N t m+1 )= 2l x Nt y )) TY + TR) u Nt y ) > 0. We note that if the lead car satisfies 1.17), then the cars with indices N 1 follow their stard dynamics 1.5) 1.7) over [t y,t m+1 ] unless they happen to be influenced by some other light at x = l <l. Having described what happens when the lead car encounters a yellow light at x = l, we turn our attention to what happens when other cars encounter the same light. We let l N 1be the largest integer so that 1.24) x l t y ) <l, we let p l l be the largest integer so that 1.25) x pl t y ) + min u j t y )TY < l+ w + L. p l j l The p l th car will be the first one that does not get through the light at x = l. We first consider the situation when p l < l. We assume the existence of a number λ 1such that cars travelling with the maximum speed V can safely brae at a constant deceleration rate a = V2 2λL over a road segment of length λl. We first focus our attention on the situation in which 1.26) x pl t y ) <l λl. 2 The dynamics described by 1.21) are equivalent to dx N = u N t y)l x N t)) 1/2 2l x N t y)) 1/2, t y t t y + 2l x N t y)) u N t y) dx N =0,t y + 2l x N t y)) u N t y) t t m+1.

5 A TRAFFIC MODEL WITH LIGHTS 153 Our basic strategy is to let cars with indices p l +1follow their stard dynamics 1.5) 1.7) over t y t t m+1. The cars with indices p l +1 l will clear the intersection by t m + TG + TY = t r ; i.e., they will satisfy x t r ) l + w + L. This follows from the observation that local spatial minima in the velocity are nondecreasing in t for details see 2.79) 2.81)). Rules for the p l th car. So long as t y t t r x pl t) <l λl we let the p l th car follow its stard dynamics 1.5) 1.7). If there is a first t pl <t r so that x pl t pl )=l λl, then the driver must decide what to do. In the unliely event that 1.27) u pl t pl )t m+1 t pl ) λl, then over the interval [t pl,t m+1 ] the p l th car is required to satisfy dx pl = min u pl t y ),U pl t)) = u pl t) 1.28) du pl = V x pl +1 x pl )u pl +1 U pl ) + V x p l +1 x pl ) U pl ) U pl t y )=u pl t y ). On the other h, if 1.29) u pl t pl )t m+1 t pl ) > λl, then the p l th car will have to slow down possibly stop. When the p l th car satisfies the additional inequality 1.30) u pl t pl )t m+1 t pl )/2 > λl, the p l th car is required to satisfy dx pl upl t pl )l x pl ) 1/2 1.31) = min 2λL) 1/2 where 1.32) du pl,u pl ) = u pl, = V x pl +1 x pl )u pl +1 U pl )+ Vx p l +1 x pl ) U pl ) 1.33) x pl t pl )=l λl U pl t pl )=u pl t pl ). When 1.31) reduces to 1.34) dx pl = u p l t pl )l x pl ) 1/2 2λL) 1/2 = v pl,

6 154 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN we see that dv pl 1.35) V2 2λL, thus we apply this constant braing strategy over t pl t t pl + 2λL = u2 p l t pl ) 2λL strategy x pl t) =l over t pl + 2λL u pl t pl ) t t m+1. If instead of 1.30) the p l th car satisfies u pl t pl ) the 1.36) u pl t pl )t m+1 t pl )/2 λl, the p l th car is required to satisfy 1.37) dx pl = min u pl t pl )+ 2λL up l tp l )tm+1 tp l ) t m+1 t pl ) 2 t t pl ),U pl ) = u pl,t pl t t m ), again U pl satisfies 1.32) 1.33) 2. The dynamics for U pl postulated in 1.28) 1.32) might seem a bit strange. What we are insisting is that the p l th car must travel no faster than the minimum of its braing speed the speed that it would travel at if it disregarded the light allowed its velocity to be determined by the car ahead. The latter speed U pl is computed from the stard dynamics equation see 1.6), 1.7), 1.7a), 1.7b)). If there is no such time t pl <t r so that x pl t pl )=l λl, then we now that x pl t r ) l λl. In this situation we replace t pl in 1.27) 1.37) by t r the terms λl in all inequalities identities by l x pl t r ). Finally, if 1.26) does not hold, i.e., if 1.38) l λl x pl t y ) <l, we set t pl to t y in 1.27) 1.37) replace λl in these formulas by l x pl t y ). The rules when p l = l are similarly amended. The cars with indices p l 1 p l 1are required to satisfy their stard dynamics over [t y,t m+1 ]. Our first result deals with the model s consistency; we shall show that for all t 0 all indices, L x +1 x )t) 0 u t) < Vx +1 x )t)). We also have the theorem that no cars run any red lights. With two in-phase lights, the number of cars through an intersection during the green yellow phases is, after a start-up period, a constant. This constant is less than the constant obtained with models which allow for infinite accelerations, i.e., discrete Lagrangian versions of the Lighthill Whitham Richards model [4, 5, 6, 7]. One surprising observation about the model just described is that the largest decelerations are not necessarily associated with the cars indexed by p l but rather cars with indices p l 1which are forced to slow down because the p l th car has stopped. Equation 1.7a) implies that the latter cars decelerations are determined by the negative velocity gradients u +1 u. Finally, we note that though we have been quite specific in postulating our stopping rules for the p l th car, it would have sufficed to have chosen any rule of the form dx pl = min v pl,u pl ) = u pl, t pl t t m+1,

7 A TRAFFIC MODEL WITH LIGHTS 155 where U pl satisfies du pl = V x pl+1 x pl )u pl+1 U pl )+Vx pl+1 x pl ) U pl ) U pl t y )=u pl t y )ifp l N 1, du N = V U N ) if p l = N, where v pl 0 is chosen so that if U N t y )=u pl t y ) dx pl = v pl, t y t t m+1 x pl t y ) <l, then x pl t) l, t y t t m Model consistency. In this section we turn our attention to the issue of model consistency. The central issue before us is to show that for 1 N 1 0 t 2.1) L x +1 x )t) 0 u t) < Vx +1 x )t)) that for = N 0 t 2.2) 0 u N t) V. We are also interested in nowing that the distinguished cars indexed by p l do not run the red lights over the intervals t r = m 1)TG+ TY + TR)+TG+ TY t mtg+ TY + TR) = t m+1 that the p l + 1)st car clears the intersection by t r, i.e., satisfies 2.3) x pl +1t r ) l + w + L. Once again x = l is supposed to be the leading edge of the intersection, w the wih of the intersection, L the length of a car. There are two natural approaches that one can tae to establish the above claims. The first is to show that the desired conclusions follow directly from the governing differential equations initial constraining conditions while the second is to show that approximate solutions, generated by numerical discretization, satisfy the desired consistency results. Noting then that these consistency results are sufficient to guarantee that the approximate solutions converge as t 0) to solutions of the original model, we are guaranteed that these limiting solutions satisfy the same consistency results. We adopt the latter procedure here since in the next section we shall perform computations with the discrete approximating system. Throughout, t will denote our time step the quantities x n,un,αn ) will denote the values of the approximate solutions at t n = n t. To eep matters simple we shall assume that the numbers TG/ t, T Y/ t, T R/ t, / t are all integers we shall assume that t min, V L) = max L s V s)) 1 ). Our first result deals with the traffic flow over the time intervals 2.4) t m = m 1)TG+ TY + TR) t n = n t t y = t m + TG

8 156 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN when all lights are green. Over such intervals we replace 1.5) by 2.5) this yields 2.6) where x n+1 = x n + u n t, 1 N, s n+1 = s n +u n +1 u n ) t, 1 N 1, 2.7) s n =x n +1 x n ) s n+1 =x n+1 +1 xn+1 ). The u s s s are related by 2.8) 2.9) u n = Vs n )+α n u n+1 = Vs n+1 )+ 1 t ) α n. These updates hold for indices n satisfying 2.10) m 1)TG+ TY + TR)/ t = n m n n m + TG/ t 1. Theorem 1. Suppose that 2.11) L s nm 0 u nm Vs nm ), 1 N 1, 2.12) 0 u nm N Then, the same inequalities hold for V = lim s Vs). 2.13) n m n n m + TG/ t = n y. Proof. The identity 2.6) implies that if s n L un +1 un 0, then sn+1 s n L. In the situation in which un +1 un < 0, 2.6) implies that 2.14) s n+1 = s n +u n +1 α n Vs n )) t the natural induction hypotheses α n 0, 0 un Vsn ), sn L imply that u n +1 αn 0. In the situation in which 0 un +1 αn < V we are guaranteed a unique s n +1 [L, ) satisfying 2.15) u n +1 α n = V s n +1), here 2.14) reduces to 2.16) or 2.17) s n+1 s n+1 = s n + V s n +1) Vs n ) ) t =1 V s ) t)s n + V s ) t s n

9 A TRAFFIC MODEL WITH LIGHTS 157 for some s min s n, sn +1 ), max sn, sn +1 )). The latter identity, together with 2.18) tv L) 1 min s n, s n +1) L, yields s n+1 L. When u n +1 un < 0 un +1 αn V, the identity 2.14) implies that 2.19) s n+1 s n +V Vs n )) t. The inequality 2.18) 1 guarantees that s s +V Vs)) t is strictly increasing on [L, ) thus 2.19) implies that s n+1 L + V t L as desired. The induction hypothesis α n 0 together with t/ 1 2.9) guarantees that u n+1 Vs n+1 ). What remains to be shown is that u n+1 0. To establish this assertion we combine 2.8) 2.9) to obtain u n+1 = Vs n +u n +1 u n ) t)+ 1 t ) u n Vs n )). Noting that Vs n +u n +1 u n ) t) =Vs n )+V s # )u n +1 u n ) t for some s # L, we find that u n+1 = V s # ) tu n +1 + t Vsn ) u n )+1 V s # ) t)u n. The last identity, when combined with tv s # ) 1, t/ 1, u n 0, u n +1 0, Vs n ) u n 0, yields u n+1 min u n,un +1 ) 0 as desired. We now turn our attention to what happens over the yellow red phases, i.e., when 2.20) t y = m 1)TG+ TY + TR)+TG t n = n t <t m+1 = mtg+ TY + TR). The results of Theorem 1imply that when n = n y = m 1)TG+TY+TR)+TG/ t the following inequalities are valid: 2.21) L s ny 0 <u ny Vsny ), 1 N 1, 2.22) 0 u ny N V = lim Vs). s Our next goal is to show that 2.21) 2.22) hold for indices 2.23) n y n n m+1 = mtg+ TY + TR). For initeness we assume the lights are located at l 1 <l 2 < <l M where M<<N that L<<l I+1 l I,1 I M 1. For 1 I M, I will be the largest integer less than or equal to N, so that 2.24) x ny I <l I

10 158 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN p I will be the largest integer less than or equal to I so that ) 2.25) x ny p I + min u ny j TY < l I + w + L. p I j I It can does happen that for some I<M 2.26) p I = p I+1 = = p M = N. Our first tas is to establish the desired inequalities for indices +1) p I = N for n y n n m+1. This is the situation that is obtained when the lead car, indexed by N, has passed the I 1)st light but not the Ith light. The rules laid out in 1.17) 1.23) imply that x N ) satisfies dx N 2.27) where U N satisfies = min v N,U N ) = u N, t y t t m+1, 2.28) du N = V U N ) v N ) 0 is chosen so that if x N ) satisfies U N t y )=u N t y ), dx N 2.29) = v N x N t y ) <l I, then x N t m+1 ) l I. We replace this system with its discrete analogue, 2.30) 2.31) x n+1 N = xn N + u n N t, n y n n m+1 1, U n+1 N = V + 1 t ) UN n V ), n y n n m+1 1, these are solved subject to the initial conditions 2.32) x ny N <l I 0 u ny N U ny N V. The discrete velocity u n N 2.33) is given by u n N = min v n N,U n N ), v n N 0 is a discretization of v N with the property that if 2.34) x n+1 N = xn N + v n N t x ny N <l I for n y n n m+1 1, then 2.35) x nm+1 N l I. The identities 2.31), 2.32) 2, 2.33) guarantee that 2.36) 0 u n N V, n y n n m+1.

11 A TRAFFIC MODEL WITH LIGHTS 159 If we assume that +1) N 1, then the N 1)st car will follow the stard dynamics 1.5) 1.7) on t y t t m+1, thus for n y n n m+1 1we have the approximating discrete system: 2.37) where 2.38) x n+1 N 1 = xn N 1 + un N 1 t, un N 1 = Vsn N 1 )+αn N 1, u n+1 N 1 = Vsn+1 N 1 )+ ) 1 t α n N 1, s n N 1 = x n N x n N 1 s n+1 N 1 = xn+1 N xn+1 N 1 = sn N 1 +u n N u n N 1) t. The inequalities 2.21) 2.22) imply that α ny The identities 2.37) 2.38) imply that 2.39) s n+1 N 1 = sn N 1 + u n N 1 t 2.37) ), together with 2.40) L s ny N 1, N 1 0, αny N ) n ) α ny N 1 Vsn N 1) t, 0, sny N 1 L. αny N 1 0, un N 0, tv L) 1, t the arguments used to establish Theorem 1, imply that 2.41) L s n N 1, n y n n m+1. The arguments used to establish Theorem 1along with 2.40) 2.41) also yield 0 u n N 1 Vsn N 1 ), n y n n m+1. An induction on for indices +1) then yields 2.42) L s n =x n +1 x n ) 0 u n Vs n ), n y n n m+1. This situation when = N 1is hled similarly, provided that one adopts the proper first order integration scheme for U N 1. The governing equation for U N 1 is 2.43) du N 1 = V x N x N 1 )u N U N 1 )+ Vx N x N 1 ) U N 1 ), where dx N x N 1 ) 2.44) v N 1 0 is chosen so that if = u N u N 1, 2.45) then 2.46) Additionally dx N = v N 1 x N 1 t y ) <l I, x N 1 t m+1 ) l I. 2.47) u N 1 = min v N 1,U N 1 ).

12 160 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN The integration scheme we use is 2.48) where U n+1 N 1 = Vsn N 1 +u n N U n N 1) t)+ 1 t ) UN 1 n Vs n N 1)), ) s n N 1 = x n N x n N 1. To complete the proof one does an induction on the index I, first replacing I by I 1. One nows that the car with index +1) has a velocity u n +1) satisfying 2.50) 0 u n +1) Vsn +1), n y n n m+1. We first focus on the st car note that 2.51) dx p = min v p,u p ) = u p, 2.52) ds p =u p +1) u p ). The rules laid out in 1.7) 1.23) imply that 2.53) du p = V s p ))u p +1) U p )+ Vs +1)) U p ) that the velocity field 0 v p is chosen so that if x p evolves as 2.54) then 2.55) dx p = v p x p t y ) <l I, x p t m+1 ) l. 2.56) The discretization we apply to the st car is x n+1 = x n + u n t s n+1 = s n + u n p +1) un p ) t for n y n n m+1 1. Moreover, for some n y n 0 n y + TY/ t 1 u n+1 = Vs n+1 )+ 1 t ) 2.57) u n p Vs n )) 2.58) U n+1 = u n+1, 3 This scheme is essentially a first order Euler scheme applied to 2.43). The scheme implies that U n+1 N 1 = U N 1 n + tv s n N 1) u n N UN 1 n ) t ) ) + V s n N 1 U n N t) 2.

13 whereas for n 0 n n m+1 1 A TRAFFIC MODEL WITH LIGHTS ) 2.60) u n = min v n,u n ), Up n+1 = Vs n +u n +1) U p n ) t)+ 1 t ) Up n Vs n )), ) Finally v n is chosen so that if Up n0 = u n0 x n0 <l I. 2.62) x n+1 = x n + v n t, n 0 n n m+1 1, then 2.63) x nm+1 l I. The arguments employed to establish Theorem 1guarantee that for n y n n ) L s n 0 u n Vs n ) that for n = n ) 0 u n0 Up n0 Vs n0 ). Lemma 1. For n 0 n n m ) L s n 0 u n U n Vs n ). Proof. The identities 2.56) 2.60) imply that 2.67) Vs n+1 ) Up n+1 = Vs n +u n +1) un ) t) Vs n +u n +1) U p n ) t) + 1 t ) Vs n p ) Up n ) ) = tv s # ) U n p u n p + 1 t ) Vs n p ) Up n ) for some s # min s n +u n +1) un ) t, s n +u n +1) U n ) t). If we now mae the induction hypotheses that 2.68) L s n 0 U n Vs n ), then 2.59) implies that 2.69) 0 u n U n Vs n ) 4 See Footnote 3.

14 162 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN 2.69) 2.42) with = + 1implies that 2.70) min s n +u n +1) un ) t, s n +u n +1) U p n ) t) s n Vs n ) t = Fs n ). This constraint tv s) 1, L s guarantees F ) in nondecreasing on L s, this fact, together with FL) =L, guarantees that s n+1 s # are both greater than or equal to L. Moreover, 2.67) also yields Up n+1 Vs n ). The ining relation 2.60) 2.70) u n +1) 0 also implies that 2.71) Up n+1 = tv s )u n +1) +1 tv s ))Up n + t Vsn ) Up n ) for some s L 2.71) guarantees that Up n+1 0. The last inequality 2.59), with n + 1, guarantees that 2.72) 0 u n+1 U n+1 Vs n+1 ), this completes the proof of Lemma 1. Once again an induction on for indices p I 2 +1) yields 2.73) L s n =x n +1 x n ) 0 u n Vs n ) additionally yields the following theorem. Theorem 2. For n y n n m+1 = mtg+ TY + TR) 2.74) L s n 0 u n Vs n ), 1 N 1, 2.75) Moreover, for 1 I M 2.76) 0 u n N V = lim s Vs). x nm+1 p I l I. Theorems 1 2 go a long way towards establishing the consistency of our model. What remains to be shown is that cars with index p I + 1clear the light, i.e., that they satisfy 2.77) ny+ TY t xp I +1) l I + w + L. The reader should recall that the cars with these indices satisfy ) 2.78) x ny p I+1) <l I x ny p I +1) + min u ny j p I +1) j I TY l I + w + L that cars with indices p I +1) I evolve by the stard discrete dynamics for n y n n y + TY/ t 1; i.e., x n+1 = x n + u n t u n = Vs n )+ 1 t ) n ny u ny Vsny )),

15 A TRAFFIC MODEL WITH LIGHTS 163 where 0 u ny Vsny ) L sn. It is a straightforward calculation to show that cars with these indices also satisfy u n+1 = Vs n +un +1 un ) t)+ 1 t ) u n Vsn )) = tv s # )u n tv s # ))u n + t Vsn ) un ) from some s # L, that this identity, along with implies tv L) 1, t, 0 Vs n ) u n 2.79) u n+1 min u n,u n +1). We now note that at t = t y equivalently n = n y ) the cars with indices p I typically satisfy 2.80) min u ny j = u ny p I j 0, where p I +1) 0 I, I 2.81) u ny +1 uny 0, 0 #, where # is greater than I. Moreover, if the spacing of the lights is sufficiently large, then the spatial monotonicity of the velocities is preserved for n y n n y + TY/ t 0 #. When this is the case, the inequalities 2.78) 2.81) guarantee 2.77). 3. Simulations. In this section we present some simulations of the system outlined in section 1. We chose V =50f/s, L =20f, L 1 =25f, λ =5, =5s N = 600. Our maximal velocity was given by Vs) =V 1 L s ), L s. We restrict our attention to a roadway with two in-phase lights located at l 1 = 1mile = 5280f l 2 = 2 miles = 10, 560f, we assume that the wih of each intersection is w =20f. Finally, the durations of the green, yellow, red lights were chosen to be TG =25s, T Y =5s, TR =30s.

16 164 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN Fig. 1. Our initial data are taen to be x 0) = ) u 0)=0, Snapshots of the solution are shown at times 30, 147, 151, 179, 191 seconds in Figures 1 5, respectively, a film may be seen at users/plin/21380/traffic.html. In the first frame of each snapshot we plot the auto velocity u in miles/hour) versus current auto position x in miles), in the second frame we plot the empirical density ρ = 1 x +1 x in cars/mile) versus current auto position x in miles). After an initial startup period we are able to get 18 cars through each light during each green-yellow-red cycle. This number should be contrasted with what one obtains in the singular limit, where =0 +, TY =0s, TG =30s, w =0f, λ = 5. In this limit L u V 1 x +1 x if, perchance, we have a car satisfying ), x t m + TG) )=l I, I = 1or 2, u t m + TG) ) > 0,

17 A TRAFFIC MODEL WITH LIGHTS 165 Fig. 2. Fig. 3.

18 166 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN Fig. 4. Fig. 5.

19 then for times t m + TG<t t m + TG+ TR, A TRAFFIC MODEL WITH LIGHTS 167 x t) =l u t) 0. For this singular model we declare a car through the light at l if x t y ) >l. The singular model has the potential for infinite accelerations. In steady state the singular model allows us to get 20 cars through an intersection during each green-red cycle. We note that our choice of which car must stop is made at times t y = t m + TG when a green light turns yellow) is conservative when the car chosen to stop satisfies x pl t y ) <l λl. A more aggressive strategy would have been to allow the p l th car to follow its stard dynamics until time t pl <t y + TY, where x pl t pl )=l λl, then reevaluate whether the p l th car can get through the light in the remaining time t y + TY t pl, i.e., chec whether x pl t pl ) + min u t pl )t y + TY t pl ) l + w + L. p l l t pl ) If the latter inequality holds, the aggressive strategy would allow the p l th car through stop the p l 1)st car. We avoided this strategy because it did not seem to be worth the effort to get one more car through the intersection during the green-yellowred cycle. The attentive reader will by now realize that once we have determined which car will slow down or stop at a given light the particular braing strategy adopted is immaterial; all that is required is that the velocity associated with the braing strategy, v pl, be such that if x pl satisfies dx pl = v pl x pl t y ) <l, then x pl t m+1 ) l. We adopted constant braing strategies here because they were simple realistic. 4. Concluding remars. There are some obvious connections between the discrete model studied in this paper the continuum or macroscopic models of Aw, Klar, Materne, Rascle [3]. If one assumes that the maximal velocity V ) introduced in 1.1) 1.3) is actually a function of γ = s L ined on γ = s L 1, i.e., 4.1) Vs) =W s L), then 1.1) 1.7) tae the form dx = u du ) = W u+1 u 4.2) γ ) L where again + W γ ) u ), 4.3) γ = x +1 x ) L 4.4) dγ = u +1 u. L

20 168 ARGALL, CHELESHKIN, GREENBERG, HINDE, AND LIN The connection between the follow-the-leader system 4.1) 4.4) is now clear. One introduces reference coordinates 4.5) X = L, lets 4.6) X X,t)=x t) ux,t)=u t), interprets γ u +1 u L as the downwind finite difference approximations to X u X X at the reference point X ; i.e., X X X,t)=γ = x +1 x U L X X,t)= u +1 u 4.7). L With these identifications one obtains, at least formally, the Lagrangian traffic equations 4.8) where 4.9) X X, t) =ux, t) X t γ t = u X X u t = W γ) u X = γx, t), W γ) u) +. This correspondence is faithful if one restricts one s attention to initial value problems exclusively. We have not seen how to incorporate the traffic light problem into a continuum format. REFERENCES [1] J.M. Greenberg, Extensions amplifications of a traffic model of Aw Rascle, SIAM J. Appl. Math., ), pp [2] A. Aw M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., ), pp [3] A. Aw, A. Klar, T. Materne, M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., to appear. [4] M.J. Lighthill G.B. Whitham, On inematic waves. I: Flood movement in long rivers, Proc. Roy. Soc. London Ser. A, ), pp [5] M.J. Lighthill G.B. Whitham, On inematic waves. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, ), pp [6] P.I. Richards, Shoc waves on the highway, Oper. Res., ), pp [7] G.B. Whitham, Linear Nonlinear Waves, Pure Applied Mathematics, Wiley- Interscience, New Yor, [8] R. Rotherby, Car following models, in Traffic Flow Theory - A State of the Art Report, Special Report 165, Transportation Research Board, Washington, DC, 1975, Chapter 4; also available online from [9] H.M. Zhang, A non-equilibrium traffic model devoid of gas-lie behavior, Transportation Res. B, ), pp