An Implementation Parallel Monte Carlo Method for Traffic Flow Simulation

Transcription

1 An Impementation Parae Monte Caro Method for Traffic Fow Simuation Hsun-Jun Cho and Fan-Yu Lai Department of Transportation Enineerin and Manaement Nationa Chiao Tun University Ta Hsueh Road, Hsinchu Taiwan Abstract: A traffic dispersion mode soves with a parae Monte Caro simuation method is proposed. Based on conservative traffic vector fied theor the spatia and capacity dependant mode is derived. In addition, with this new mode, traffic fow simuations usin a messae passin interface is impemented. As a resut, the run time is reduced sinificanty. Key-Words: Monte Caro method, Parae computin, Traffic dispersion mode, Traffic simuation, Transportation system. Introduction Transportation system provide excusivey for throuh movement of traffic at norma speed under enera situation. However, whether the traffic jam is caused by incidents, roadwork, or traffic peak, may reduce the capacity of roadway and enerate traffic conestion. Soon after an somethin happened, if the speed information for the roadway coud be iven to the upstream drivers immediate which can reduce the emission and fue consumption, in addition to owerin traffic conestion and trave cost []. In order to decrease traffic jams and to make efficient traffic manaement, it is necessary to offer rea time information for driver []. However, traffic mode are difficut to et rea with enera computin. So we use messae passin interface technique in the proposed mode to et resut quicky [3]. In this paper, a traffic dispersion reation mode is considered usin the continuity equation for vehice number conservation in traffic fow system [4]. And reaxin the traffic mode by Baker [5], which assumed that the mode had no source traffic density. In order to study the traffic interaction behaviors for two motion objectives, the ravity function mode is aso discussed in this mode. Furthermore, the main overnin equation, so caed the Poisson equation is estabished usin the potentia theory proposed by Sheppard [6]. Numerica soutions usin Monte Caro method are aso presented for a more compete mode study. In order to anayze compex mode usin a sma amount of computer time, the messae-passin interface are impemented. This paper emphasizes Monte Caro method paraeism for sovin the traffic dispersion mode based on spatia, and capacity. The proposed simuation uses severa computers for executin the parae cacuations to reduce run time. This paper is oranized as foows: in section, we formuate the proposed mode usin the conservative vector fied theory. Monte Caro methods and parae one are described in section 3. Some comparisons of resut that use various numbers of computers to execute Monte Caro methods are discussion in section 4. The concusion and remarks are provided in the ast section. Dispersion Traffic Mode Probem In this stud we adapted vector fied methodooy to anayze the traffic fow

2 propert and estabish a mode for traffic fow motion behavior usin diverence theorem. Accordin to the vehice number conservation aw, we derive the fow continuity equation and the ravity function is appied to study the interaction between individua and roup motions. The proposed mode is aso suested as a way to master the oca interaction reationships. Furthermore, usin the potentia theory studies by Sheppard [6], a nove traffic dispersion mode is presented. Suppose q ( x, is a vector function of traffic fow that has three components q x,q y, q c of q which are functions of a coordinate system x, c, namey: q( x, i q x ( x, + j q y ( x, + k qc ( x,, = () where x, y are spatia dependent variabes, c is a capacity function. In enera, the capacity is a maximum fow rate, we note that the function c is not ony a desin capacit but aso it is an operationa capacity in the mode. Where i, j, and k are unit vectors. Accordin to Sheppard [6] discussed potentia theory that describes the notion of infuence between a set of points iven a mode of interaction between motorists. The mathematics of potentia theory indicatin the spatia radient of the potentia or the infuence function between two points is iven by the interaction function. Definin φ as the potentia function in traffic vector fied, then we et () by methodooy of mathematics [7]. q = φ () The term () combines the traffic continuity equation and diverence of q. Then we can obtain or φ φ φ + + = k x y c (3) φ = φ = k (4) In this paper, the traffic dispersion equation, (4), is so caed the Poisson equation. The proposed mode is an eiptic partia differentia equation. And the mode has a soution and this soution is unique. By the maximum principa [8] to prove that the Dirichet probem is one for which a soution not ony exists, but in addition, the soution is unique and depends continuousy on the data. 3 Parae Monte Caro Method Monte Caro Methods, MCM, are simuated with statistics. It uses continuity random numbers to simuate. And Mont Caro methods are suitabe to sove Poisson equation [9], for exampe, the traffic dispersion equation. First of the Monte Caro methods is to determinate the traffic dispersion equation and set the boundary condition. Then mesh is enerated. Some methods need separation. Next point random wak unti wakin to boundar repeat N times of random wak step, then we can obtain N boundary vaues. Computin these boundary vaues to et resut. We briefy present the theoretica backround for Exodus method [-]. We specificay appy the method to Dirichet probems in rectanuar reions. To appy the Exodus method in findin the soution of Poisson Equation invoves three steps: () First obtain the random wak probabiities from the finite difference equivaent of the partia differentia equation describin the probem. () Use the Exodus method aon with the random wak probabiities in cacuatin the transition probabiities. (3) The potentia at the point of interest is determined usin the transition probabiities, the boundary conditions, and the source function. The major disadvantae of the Exodus method is that it ony permits cacuatin the potentia at one point at a time. In our study is to appy the Exodus method in sovin Poisson equation φ = k( x, in reion Ω (5)

3 subject to boundary conditions φ = φ b on boundary Ω (6) The rectanuar soution reion Ω is divided into a mesh and the finite difference equivaent is derived. In order to compute the soution of the probem defined in (5) and (6), and determine the potentia at a specific point ( x, y ). The transition probabiity p is defined as the probabiity that a random wak startin at the point of interest ( x, y ) in Ω ends at a boundary point ( x, y ) with prescribed potentia φ ( b ), i.e. approach is imited to homoeneous rectanuar soution reions. The Exodus method offers a numerica means of findin p and q. A steps in the procedure for sovin the partia differentia equation with Exodus method are shown in Fi.. In Monte Caro method, a process must record passin node. So every wak can et N step, in other words when we require smaer error, N is bier. Then it needs more time to run. But a points to compute are independent. So we use messae passin interface technique [3] in Monte Caro method. The purpose of parae computin is a work of one computer to divide equay other computers. That can reduce run time to compute resut. p = prob x, y > x, y ). (7) k ( The transient probabiity q is the probabiity that a random wak startin at point ( x, y ) passes throuh point ( x, y ) on the way to the boundar i.e. Determinate the probem s PDE Set the boundary condition φ = φ b on boundary Ω Rectanuar soution reion φ = k in reion Ω Axisymmetric soution Reion q prob( x y = > boundary Ω). (8), If there are m boundary or fixed nodes (excudin the corner points since a random wak never terminates at those points) and M f free nodes in the mesh, the potentia at the startin point x, ) of the random wak is where ( y m, y ) = p b( ) = φ ( x φ + q G, (9) M f = G = w ( x, y ) / 4. () Since φ ( b ) is specified and source term G is known, the probem is reduced to findin the probabiities p and q. The vaue φ ( x, y ) woud be exact from (9) if we know the transition probabiities p and the transient probabiities q exacty. Usin an expansion technique described in [] to obtain the vaues of p anayticay. But this Compute Transition and Transient Probabiities The potentia at start point Transition Probabiity p = prob x, y > x, y ) ( Fi. A procedures of Exodus methods Genera, computin of parae Monte Caro beons to computin partition without data partition that is shown in Fi.. This parae method is the easiest to compute. And it has increased computin efficiency many times. In this stud we use point partition of parae Monte Caro method, it divided into two partitions that is shown in Fi. 3. Transient Probabiity q prob( x y = ---->bounday Ω), 3

4 istart iend ntota Cpu ======= istart iend ntota Cpu ======= istart iend ntota Cpu ======= istart iend Cpu ======= Fi. Parae computin of computin partition without data partition N km k(x,= 3km Fi. 4 Traffic mode without source traffic density This probem can then be formuated as a traffic dispersion mode. In order to find the anaytica soution, we simpify this reion into a unit square (,3) (,3) and consider the probem of findin an unknown function φ=φ(x, which satisfies the foowin mode φ( x, = φ(, =, φ (3, = sin πc 3 φ( x,) = φ( x,3) = c 3 x 3 In this probem, the soution of Monte Caro Method and parae MCM approximate the anaytica soution. The potentia soution is shown in Fi. 5. D partition D partition Fi. 3 Point parae partition It divides into one dimension and two dimension partition. In this stud we use one dimension partition. 4 Numerica Exampe In this section we compute two probems usin Monte Caro method and parae Monte Caro method. First exampe is a traffic mode without source traffic density. Computin the proposed mode convenient we simpify the probem to two-dimensiona mode. Consider a on section of roadway shown in Fi. 4. We consider ony fowin traffic on boundar but with no source traffic density in this reion. For the convenience, the foowin numerica exampes are examined with dimensioness and normaized measurement units. Fi. 5 Numerica soution for the simuated traffic mode without source traffic density In process of computin, since it is imited to four computers. So we execute the proposed mode with one, two, and four computers. And CPU of computer is PIII-45, RAM is 8M. Comparin the resuts in Monte Caro Method and parae MCM, in tabe, we find Monte Caro method needs much more CPU time than paraeism to run the simuations for this probem. Tabe Comparison of simuation Monte Caro Method Run Time One computer :59:44.43 Two computers ::5.5 Four computers :3:43. 4

5 seconds Run time computer(s) Fi. 6 Comparin run time of various numbers of computer The second exampe is a traffic mode with source density. Consider a on section of roadwa shown in Fi. 7. We considered ony traffic fowin on the boundar and with source density in this reion. Fi. 8 Numerica soution for the simuated traffic mode with source traffic density seconds 3 Run time 4 computer(s) Fi. 7 Traffic mode with source traffic density This probem can be then transformed into another traffic dispersion mode. In order to find the anaytica soution, we simpify this reion into a unit square (,) (,) and consider the probem of findin an unknown function φ=φ(x, which satisfies the foowin mode. φ ( x, = 4xc φ(, = sinπc, φ( x,) = φ( x,) = φ (, = c x In this probem, the soution of Monte Caro Method and parae MCM approximate the anaytica soution. The potentia soution is shown in Fi. 8.Moreover, comparin the resuts with various numbers of computers in Monte Caro Method that are shown in tabe. Fi. 9 Comparin run time of various numbers of computer Tabe Comparison of simuation Monte Caro Method Run Time One computer :4:7.6 Two computers ::.54 Four computers ::58.48 From Fi. 6 and Fi. 9, we find parae Monte Caro method is ood method to sove the traffic dispersion mode. Computin Foows numbers of computer to increase, it ets more efficiency. 5 Concusion Conventiona the traffic dispersion mode is function of spatia and density. In this stud the traffic fow mode based on the interreations between spatia distributions of traffic fow and its roadway capacity was proposed. It provided an efficient too to study fow variation with traffic capacities. The traffic dispersion mode is a conservative vector fied, which is proved by the vector fied theory. An effective parae Monte Caro method was proposed to sove the traffic dispersion mode. 5

6 To discuss efficienty and exacty the variation of traffic fow in various traffic phenomena, a nove traffic dispersion mode was proposed and simuated by a parae Monte Caro method. Some computation resuts were aso iven to iustrate the accuracy and efficiency for various numbers of CPU in this simuation. In this paper, we found parae Monte Caro method that is an efficient method to sove the traffic dispersion mode. It is conducive to anaysis rea time traffic information. In our numerica experience, the run time costs of Monte Caro method is very expensive than parae MCM for both exampes, which so a distributed computation technique shoud be deveoped to perform more numbers of computer simuation which is an efficient approach for the modem traffic fow simuation. The Monte Caro method obtained resuts accuratey and efficienty usin variance reduction except paraeism. In this stud there are ony fixed boundary mode probems have been considered. To appy the proposed mode for more reaistic traffic fow phenomena, a more enera boundary vaue probem, such as Dirichet- Neumann Mixed boundary mode probem shoud be estabished. And the numerica experiences in this study ony consider twodimensiona probem, so we can simuate three-dimensiona probem to anayze more reaistic traffic situations. This paper was partiay supported by Nationa Science Counci, R.O.C., under contract number NSC89--E Reference [] Ross, P., Traffic Dynamics, Transportation Research B, Vo., Issue: 6, 988, pp [] Hiies, M. and W. Weidich, A phenomenooica mode for dynamic traffic fow in networks, Transportation Research Part B, Vo. 9, Issue: 6, 995, pp [3] Johnston, C. M. and Chronopouos, A. T., A communication atency hidin paraeization of a traffic fow simuation, Parae and Distributed Processin, 999, pp [4] Ocott, E. S., The Infuence of Vehicuar Speed and Spacin on Tunne Capacity, Transportation Science, Vo. 3, 955, pp [5] Baker, Robert G. V., A Mode of Traffic Dispersion From A Conested Road Transportation Research part B, Vo. 5, No. 5, 98, pp [6] Sheppard, E. S., Georaphic potentias, Annas of the Association of American Georaphers, Vo. 69, 979, pp [7] Grossman S. I., Cacuus, fifth edition, Harcourt Brace Compan 986. [8] Strauss, W. A., Partia Differentia Equations: an introduction, John Wiey & Sons, Inc., New York, 99. [9] R. Schott, A Monte Caro Method For The Dirichet Probem Of Dieectric Wedes, IEEE Tracsactions on Microwave Theory and Techniques, Vo.36, No.4, 988, pp [] Sadiku, M. N. O. and Hunt, D. T., Soution of Dirichet Probems by the Exodus Method, IEEE Transactions on Microwave Theory and Techniques, Vo. 4, No., 99, pp [] Matthew N. O. S., Ajose S. O., and Zhibao F., Appyin the Exodus Method to Sove Poisson s Equation, IEEE Transactions on Microwave Theory and Techniques, Vo. 4, No. 4, 994, pp [] Mccrea W. H. and Whippe F. J. W., Random paths in two and three dimensions, Proc. Roy. Soc. Edinb, Vo. 6, 94, pp [3] Pacheco, P. S., Parae prorammin with MPI, Moran Kaufmann Pubishers,