Studying Traffic Flow Using Simulation Cellular Automata Model

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Iter atioal Joural of Pure ad Applied Mathematics Volume 113 No. 6 2017, 413 421 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu ijpam.

1 Iter atioal Joural of Pure ad Applied Mathematics Volume 113 No , ISSN: (prited versio); ISSN: (o-lie versio) url: ijpam.eu Studyig Traffic Flow Usig Simulatio Cellular Automata Model S.Rajeswara 1 ad R.Karthik 2 1 Departmet of Mathematics, Velammal Egieerig College, Cheai hod.maths@velammal.edu.i 2 Departmet of Mathematics, Velammal Egieerig College, Cheai karthikr@velammal.edu.i February 13, 2017 Abstract As traffic demad o road etworks steadily icreases every year, the eed for alleviatio of global cogestio arises. I this paper, we explore the usefuless of cellular automata to traffic flow modelig. NaSch model with modified cell size ad variable acceleratio rate is exteded to two-lae cellular automato model for traffic flow. A set of state rules is applied to provide lae-chagig maeuvers. S- t-s rule give i the BJH model which describes the behavior of jammed vehicle is implemeted i the preset model ad effect of variability i traffic flow o lae-chagig behavior is studied. Flow rate betwee the sigle-lae road ad twolae road is compared uder the ifluece of s-t-s rule ad brakig rule. AMS Subject Classificatio: 68Q80, 37B15 Key Words ad Phrases: NaSch Model, Modified NaSch Model, Brakig probability, Slow-to-start rule, ad Lae chagig behavior, Two- lae CA. ijpam.eu

2 1 Itroductio This Efficiet vehicular trasport of people ad goods is of vital importace to ay moder society. I desely populated areas the capacity of the road etwork is ofte at its limit ad frequet traffic jams ad cogestios cause a sigificat loss of lives ad materials. Modelig traffic trasport problem is very iterestig ad importat for its dyamics with serious cosequeces i real life. Mathematical modelig ad computer simulatios play vital roles i studyig the impacts of various policies o vehicular traffic. Traffic flow problems have attracted cosiderable attetio of researchers because of maifold icrease i traffic desity i cities. Amog those, Cellular Automata (CA) models have emerged i the last few years as a very promisig alterative to model the traffic flow ad uderstad its behavior. 2 Traffic Flow Modelig I Literature A Cellular Automata (CA) is a extremely simplified program for the simulatio of complex trasportatio systems. The first applicatio of the CA for simulatio model of traffic flows o street ad highways was itroduced by Nagel ad Schreckeberg popularly kow as NaSch Model [4].This model is based o the homogeeous traffic flow. Chowdhury et al. [1] made a attempt to model two kids of vehicles usig CA. This model is a two-lae traffic flow model with two differet types of vehicles, characterized by two differet values of the maximum speed( v k max )for kth type of vehicle. The two-lae cellular automata model based upo the sigle-lae CA itroduced by Rickert et al. was examied [6]. The set of laechagig rules suggested by Chowdhury et al. [2] was revised to take the Hok effect ito accout. I the preset study the cell size is reduced ad variable acceleratio rate (rather tha 1) is take ito accout [3]. ijpam.eu

3 3 Mathematical defiitio of Cellular Automata CA models represet a discrete dyamic system, cosistig of four igrediets: CA =(L,, N, δ), where the physical eviromet is represeted by the discrete lattice Lad the set of possible states deoted by. Each ith cell of the lattice has at time step t a stateσ i (t). Furthermore, the associated eighborhood with this cell is represeted by N i (t). Fially, the local trasitio rule δ is represeted as: δ : N : j N i (t) σ j (t) σ i (t + 1) (1) Equatio (1) shows that the state of the ith cell at the ext time step t +1 is computed by δ based o the states of all the cells i its eighborhood at the curret time step t. 3.1 Stochastic Nasch CA Model Lagrage form is suitable whe o vehicle overtakes the vehicle i frot i the same lae. The Lagrage represetatio of Rule-184 is give as x i x i + mi { 1, d i } (2) where d i = x p x i 1 deotes the distace (umber of empty cells) i frot of the vehicle i positio x i. From equatio (2), we ca see that a vehicle ca move ahead oe site, whe its frot site is empty. If the maximum velocity of vehicle is v max, the a vehicle ca move forward to at most v max sites per uit time step. For the maximum velocity v max, equatio (2) ca be writte as x i x i + mi { v max, d i } (3) The equatio (3) is the case whe a vehicle follows oly the vehicle i the frot. After icludig the aticipatio parameter A, the equatio (3) ca be modified as x i x i + mi { v max, d t A } (4) ijpam.eu

4 where d t A = x i + A x i A at time t. The equatio (4) is called the Lagrage form of traffic flow models. Here aticipatio parameter A represets the iteractio horizo of drivers. 3.2 Stochastic NFS Model (S-NFS) A ew determiistic model which icluded both s-t-s rule ad aticipatio rule with stochastic effects usig Lagrage form was preseted by Nishiari et al.[5] This model is kow as S-NFS model. 4 CA Model with Modified Cell Size I NaSch model ad other previous models a defiite cell size of 7.5 meter was take for all type of vehicles ad acceleratio rate was also same at differet speed of vehicles i.e. 1 cell/sec2. A fier cell size is modeled to icorporate Idia traffic flow coditios more realistically. Whe modelig realistic traffic stream that cosists of vehicles, havig variable speed ad acceleratio, fier discretisatio is useful. Table 4.1 presets the differet parameters used i developig CA model with modified cell size. 4.1 Table 4.1 Vehicle Characteristics Legth 12 Cells Width 1 Cell Maximum Speed 60 Cells/Sec Acceleratio (Speed 5.5m/Sec) 4 Cells/Sec 2 Acceleratio (5.5m/Sec < Speed 11m/Sec) 3 Cells/Sec 2 Acceleratio (Speed > 11m/Sec) 2 Cells/Sec 2 To represet these coditios a slightly modified S-NFS model alog with both slow to start rule ad aticipatio rule ad modified cell size is used. Updatig the rules the followig oce emerge. Rule 0 : If V (t+1/4) with probability q = V (t+1/2) = V (t+3/4) Rule 1: V ( t + 1/4 ) mi {V max, V t + a } = V (t+1) 0 ijpam.eu

5 Rule2 (a): where DA t = X +A X A V (t+1/2) mi{v (t+1/4), DA t } with probability r Rule2 (b): V (t+1/2) mi{v (t+1/4), DA1 t } with probability (1 - r), where DA1 t = x + A x 1 Rule 3: V (t+3/4) max{0, V (t+1/2) 1} with probability p Rule 4: V ( t + 1) ( t+ 3/4 ) mi { V, D + V ( t + 3/4 ) Rule 5: X t + 1 Xt + V ( t + 1). + 1 } Here q deotes the probability of s-t-s effect is o ad r deotes the probability of aticipatio. I S-NFS model slow-to-start rule is applied to all the vehicles where as i the preset model slow-to-start rule affects oly statioary vehicles i.e. the vehicle havig velocity 0. Here the value of aticipatio parameter is take as A = 2. 5 Simulatio I the preset model the size of the cell is chaged to 0.5 m ad a vehicle occupies 12 cells. The time iterval t take is 1 secod, the typical driver s reactio time. Further, the basic relatio coectig the desity ρ, flow q ad the average velocityv, for the curret model is q = V ρ. I the simulatio process, the umber of cells is 10,000 which is correspods to 5 Km ad the driver s reactio time is take as 1 secod which is same as the time iteral. Values of q are recorded for the differet values ofρ, p, q ad r after a trasiet period of 10,000 time steps. 6 Fudametal Diagrams Ad Results The effect of 3 parameters p, q ad r through the fudametal diagrams are aalyzed. As p icreases from 0, the stochastic fluctuatios ehace ad flow drops. I Figures 1 ad 2 it ca be easily see how the slow to start effect is o; there appear some meta-stable braches, whose size icreases with the icreasig value ijpam.eu

of q. Therefore, oce the flow drops at critical desity, it agai rises eve if the desity surpasses the critical desity because of the uexpected behavior of stopped vehicles. Fig.

6 of q. Therefore, oce the flow drops at critical desity, it agai rises eve if the desity surpasses the critical desity because of the uexpected behavior of stopped vehicles. Fig.1 Fudametal diagram betwee Flow ad Desity with p = 0 ad Fig. 2 for p = MODIFIED NaSch CA MODEL I the preset work the step cocerig deceleratio of the vehicle due to NaSch model metioed i sectio 3.1 has bee modified with the followig step: DECLARATION: If roud (di +(1 β)νp ) < νi, the velocity of vehicle i is reduced to roud (di + ( 1 β ) νp ). νi mi (νi, roud (di + ( 1 β ) νp ), β [0, 1], where the fuctio roud trucates its argumet to the closest iteger. Here β deotes the safe distace parameter ad v p is the positio of the vehicle ahead at a give time. 8 LANE CHANGING RULE A vehicle will oly be allowed to chage to the lae if the followig coditios are satisfied. Icetive Criteria: d i < mi { v i + a, v max } Improvemet Criteria: d i, other > d i Safety Criteria: d i, back > v max whered i, other, d i, back deote the umber of empty cells betwee the ith ad its two eighbor vehicles i the other lae at ijpam.eu

7 time t, respectively. Here a deotes the acceleratio assig to the ith vehicle. 9 SIMULATION RESULTS I order to simulate the curret model, L = cells are cosidered i a sigle-lae loop. The total umber N of vehicles cosidered ad are distributed alog the loop with a startig speed rages i (0, V max ). Note that the average desity ρ is give byρ = N/L, is a costat i time. Simulatio is carried out with differet values of N, p, ad β ad 60,000 time steps. Keepig p fixed at 0.4, N ragig from 0 to L ad varyig the values of β discretely, the fudametal diagram is costructed as show i Figure 3, to fid the relatio betwee the flow ad desity. Also Figure 3 suggests that the maximum flow icreases whe the value of β decreases. Notice that the shape of the curves correspodig to β = 0.25 ad β = 0.5 differ sigificatly. Whe β 0.25, ρ m, the desity for maximum flow reduces for lower values of β. Fig. 3 Relatioship betwee flow ad desity whe β takes differet values. Fig. 4. Fudametal diagram betwee velocity ad desity whe β takes differet values Figure 4 reveals the relatioship betwee mea velocity ad desity for p = 0.4 ad for the differet values of β ad that the maximum velocity of 5 is ever reached. Whe β decreases, the desity iterval for free-flow icreases. ijpam.eu

8 10 CONCLUSION I this study a ew oe dimesioal CA model with modified cell size ad variable acceleratio rate is applied to simulate traffic flow. As a result of reductio i cell size, jam coditio prevails whe critical desity is greater tha 0.5 ad hece traffic flow icreases. Also, slow to start rule ad aticipatio rule are used to preset model. This model shows that whe q icreases the metastable braches aroud the critical desity go higher. Simulatio results show that this model captures the essetial features of traffic flow. Refereces [1] Chowdhury, D., Wolf, D.E, ad Schreckeberg, M., 1997, Particle hoppig models for two-lae traffic with two kids of vehicles: Effects of lae chagig rules, Joural of Physica A, 235, pp [2] Chowdhury, D., Sata, L., ad Schadscheider,A., 2000, Statistical physics of vehicular traffic ad some related systems, Physics Report, vol.329, o.4-6, pp [3] Mallikarjua,C., ad Rao, K., 2007, Idetificatio of a suitable cellular automata model for mixed traffic, Joural of the Easter Asia Society for Trasportatio Studies, vol. 7, pp [4] Nagel,K., ad Schrekeberg, M.,1992, A Cellular Automata for freeway traffic, Joural of Physics I Frace, 2, pp [5] Nishiari, K., Fukui, M., ad Schadscheider, A., A stochastic cellular automato model for traffic flow with multiple metastable states, Joural of Physics A: Mathematical ad Geeral, vol. 37 (9), pp [6] Rickert, M., Nagal, K., Schreckeberg, M., Latour, A.,1996, Two lae traffic simulatios usig cellular automata, Physica A, vol.231, o.4, pp ijpam.eu

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Microscopic traffic flow modeling

Chapter 34 Microscopic traffic flow modelig 34.1 Overview Macroscopic modelig looks at traffic flow from a global perspective, whereas microscopic modelig, as the term suggests, gives attetio to the details