Journl of Physics: Conference Series PAPER OPEN ACCESS Bsic model for trffic interweve To cite this rticle: Ding-wei Hung 25 J. Phys.: Conf. Ser. 633 227 Relted content - Bsic sciences gonize in Turkey! Ftm Akdemir, Asli Arz, Ferdi Akmn et l. - Reserch Funding: Bsic is beutiful! - Piston cylinder pressure blnce design with negligible distortion coefficient T J Eswrd, R C Preston nd P N Gélt View the rticle online for updtes nd enhncements. This content ws downloded from IP ddress 37.44.23.43 on 8//28 t 2:36
4th Interntionl Conference on Mthemticl Modeling in Physicl Sciences (IC-MSqure25) IOP Publishing Journl of Physics: Conference Series 633 (25) 227 doi:.88/742-6596/633//227 Bsic model for trffic interweve Ding-wei Hung Deprtment of Physics, Chung Yun Christin University, Chung-li, Tiwn E-mil: dwhung@cycu.edu.tw Abstrct. We propose three-prmeter trffic model. The system consists of loop with two junctions. The three prmeters control the inflow, the outflow (from the junctions,) nd the interweve (in the loop.) The dynmics is deterministic. The boundry conditions re stochstic. We present preliminry results for complete phse digrm nd ll possible phse trnsitions. We observe four distinct trffic phses: free flow, congestion, bottleneck, nd gridlock. The proposed model is ble to present economiclly cler perspective to these four different phses. Free flow nd congestion re cused by the trffic conditions in the junctions. Both bottleneck nd gridlock re cused by the trffic interweve in the loop. Insted of directly relted to conventionl congestion, gridlock cn be tken s n extreme limit of bottleneck. This model cn be useful to clrify the chrcteristics of trffic phses. This model cn lso be extended for prcticl pplictions.. Introduction Prcticl pplictions of trffic dynmics re well pprecited becuse of their importnce to modern society. Due to the complexity of rel trffic, workble model often hs numerous prmeters. A lot of reserch hs focused to explore the fundmentls of complex dynmics []. A bsic model is indispensble in understnding the underlying dynmics. Previously, we proposed cellulr utomton model for trffic dynmics t roundbout, which hs twenty four prmeters [2]. In this work, we im to reduce the number of prmeters nd propose bsic model for the trffic interweve. On simple rodwy without trffic interweve, trffic dynmics is competition between inflow nd outflow which results in two phses: free flow nd conventionl congestion. On the complex rodwy with trffic interweve, there re two more types of congestion: bottleneck nd gridlock. Together, there re three types of congestion. We propose three-prmeter model to present economiclly cler perspective to these distinct phses. The three prmeters cn ech be ssocited to different chrcteristics of trffic phses: free flow, congestion, nd bottleneck. Conventionl wisdom would tke the gridlock s n extreme limit of congestion. We show tht it cn be more pproprite to consider the gridlock s n extreme limit of bottleneck. This model cn be useful to clrify the chrcteristics of trffic phses. This model cn lso be extended for prcticl pplictions. 2. Model The model system consists of rotry with two junctions s shown in Fig.. All the rodwys re single-lne nd unidirectionl. Vehicles move into the system through one of the stright rodwy, go round the loop few times, nd leve the system from the other stright rodwy. Content from this work my be used under the terms of the Cretive Commons Attribution 3. licence. Any further distribution of this work must mintin ttribution to the uthor(s) nd the title of the work, journl cittion nd DOI. Published under licence by IOP Publishing Ltd
4th Interntionl Conference on Mthemticl Modeling in Physicl Sciences (IC-MSqure25) IOP Publishing Journl of Physics: Conference Series 633 (25) 227 doi:.88/742-6596/633//227 Rodwy is divided into discrete cells, on which vehicles hop with cellulr utomton rules. The dynmics is deterministic nd governed by the Asymmetric Simple Exclusion Processes (ASEP) [3]. If n empty cell in front is vilble, vehicle moves forwrd in the next time step. The system configurtion is updted in prllel. In contrst to the deterministic dynmics, the boundry conditions re stochstic. The inflow nd outflow re controlled by prmeters α nd β, respectively. When the first cell of the incoming rodwy is empty, new vehicle is dded stochsticlly with probbility α in the next time step. When vehicle reched the lst cell of the outgoing rodwy, tht vehicle is removed stochsticlly with probbility β in the next time step. Upon entering the system, ech vehicle is ssigned n integer n which specifies the number of turns ( + n) for tht vehicle to move long the loop. The prmeter n is tken from Poisson process, i.e. with probbility (γ n /n!) exp( γ). The verge number of turns in the loop is ( + γ). In summry, the model hs three prmeters: α (, ) controls the inflow, β (, ) controls the outflow, nd γ (, ) controls the interweve. The model configurtion cn lso be tken s T-shped intersection imposed with periodic boundry condition [4], s shown in Fig.. The relted trffic rules cn be pplied directly. On ll cells, except the shded one, vehicle hs only one direction to move. The shded cell indictes the exit junction, on which vehicle hs two choices: to exit or to go nother round. The interweve implies trffic conflict on the cell next to the shded one, i.e. the entry junction. The model prescribes the incoming vehicles to yield to the trffic in the loop, s required by most trffic regultions. Our results show tht the gridlock cnnot be voided even if ll vehicles perfectly follow the well-intended trffic regultions. β α Figure. System configurtion, where the rrow indictes the trffic direction. T- shped intersection, where the shded cell indictes the exit junction. 3. Results If ll vehicles go round the loop just one time, i.e. γ =, there is no trffic interweve. The model reduces to ASEP on stright rodwy. The simultion results for the density ρ on the loop re shown in Fig. 2. When α < β, trffic is free-flowing nd ρ depends on α only. When α > β, trffic is congested nd ρ depends on β only. An brupt trnsition long α = β seprtes the free flow phse nd the congestion phse. When vehicles go round the loop more thn one times, trffic in the loop will interweve with the incoming trffic. Two new trffic phses cn be observed. At γ =.5, vehicles go round the loop one nd hlf times on verge, where 6% of vehicles go round the loop one time, 3% two times, 8% three times, nd % four times. The results re shown in Fig. 2. Four different phses cn be observed. Besides the free flow nd congestion, the plteu indictes bottleneck phse nd the nrrow stripe with mximum density ρ = indictes gridlock phse. When both α nd β re lrger thn the criticl vlue, the plteu emerges nd ρ is independent of both α nd β. When β becomes 2
4th Interntionl Conference on Mthemticl Modeling in Physicl Sciences (IC-MSqure25) IOP Publishing Journl of Physics: Conference Series 633 (25) 227 doi:.88/742-6596/633//227 very smll, the severe congestion leds to the gridlock. The density reches the mximum ρ = on the loop nd lso on the incoming rodwy; while the outgoing rodwy becomes empty. When the trffic interweve further increses, the gridlock phse becomes dominnt. At γ =, vehicles go round the loop two times on verge. The rtio of vehicles trvelling round the loop one time decreses to 37% nd the rtios of multi-time trveller increse ccordingly. The results re shown in Fig. 3(c), where we observe only free flow nd gridlock. The free flow phse is limited to α < β nd smll α..5 r.5 r.5 r.5.5.5.5.5.5 (c) Figure 2. on the loop s function of inflow nd outflow ρ(α, β): γ = ; γ =.5; (c) γ =..8.6.4.8.6.4.2 G C F.2.4.6.8.2 G C B.2.4.6.8.8 (c).8 (d).6.4.6.4.2 F B G.2 C B G.2.4.6.8 Interweve γ.2.4.6.8 Interweve γ Figure 3. evolution in the consecutive trnsitions: ρ(β) with fixed α =.3 nd γ =.5; ρ(β) with fixed α =.6 nd γ =.5; (c) ρ(γ) with fixed α =.4 nd β =.6; (d) ρ(γ) with fixed α =.6 nd β =.4; This model presents four stedy phses: Free flow (F ), Congestion (C), Bottleneck (B), nd Gridlock (G). Simultion results in Fig. 2 indicte vrious kind of phse trnsitions s the three control prmeters (α, β, γ) vry. Bsiclly the trffic conditions become worse nd the free 3
4th Interntionl Conference on Mthemticl Modeling in Physicl Sciences (IC-MSqure25) IOP Publishing Journl of Physics: Conference Series 633 (25) 227 doi:.88/742-6596/633//227 flow cn no longer be mintined when either α increses, β decreses, or γ increses. First we consider the simple trnsition, which involves only two phses. As α increses, we observe three brupt trnsitions (F C, F B, F G). As β decreses, we observe two brupt trnsitions (F C, F G) nd one smooth trnsition (B G). As γ increses, we observe one smooth trnsition (C G). Then we consider the consecutive trnsition, which involves three phses. As α increses, there is no consecutive trnsition. As β decreses, we observe two consecutive trnsitions (F C G, B C G). As γ increses, we observe two consecutive trnsitions (F B G, C B G). Typicl results re shown in Fig. 3. As α increses, ll three types of congestion cn be the finl phse. As β decreses or γ increses, gridlock becomes the dominte finl phse. Bottleneck cnnot be the finl phse. Congestion cn be the finl phse only if there is no interweve (γ = ). 4. Discussions This model is ble to differentite the four trffic phses. The free flow is chrcterized by the inflow (α). The conventionl congestion is cused by direct comprison between inflow (α) nd outflow (β), i.e. α > β, nd chrcterized by the outflow (β). The bottleneck is cused minly by the interweve (γ), i.e. independent of both α nd β. The model lso presents the gridlock, which often occurs in rel trffic but is totlly bsent in ASEP. We observe two kinds of gridlock: one s nrrow stripe nd the other s plteu in Fig. 2. The stripe phse is gridlock trnsited from the conventionl congestion s shown in Figs. 3 nd 3; while the plteu phse is gridlock trnsited from the bottleneck s shown in Figs. 3(c) nd 3(d). The stripe phse ppers only t the boundry of the prmeter spce. In the limit β, the conventionl congestion is the finl phse provided there is no interweve γ =. With interweve γ, the conventionl congestion becomes unstble. The gridlock cn be triggered when the density on the loop is high. As result, the high density in the outgoing rodwy tkes drstic turn to reduce to zero in the limit β, s shown in Figs. 3 nd 3. In contrst, the plteu phse occupies finite region of the prmeter spce. All densities evolve grdully from bottleneck to gridlock, s shown in Figs. 3(c) nd 3(d). Unlike the other three trffic phses, the gridlock implies the loss of mobility which cnnot be restored. Thus the observtion of gridlock will depend on the time scle. Typicl results re shown in Figs. 4 nd 4..8.6.4 T = 3 T = 4 T = 5 T = 6 T = 7.8.6.4 T = 3 T = 4 T = 5 T = 6 T = 7.2 C B.2.4.6.8.2 C B.2.4.6.8 Interweve γ Figure 4. on the loop t vrious time scles T, where the simultion runs period of T before tking the verge: the sme s Fig. 3; the sme s Fig. 3(d). 5. References [] Chowdhury D, Snten L nd Schdschneider A 2 Phys. Rep. 329 99 [2] Hung D W 25 Comput. Phys. Comm. 89 72 [3] Derrid B 998 Phys. Rep. 3 65 [4] Hung D W 2 Internt. J. Modern Phys. C2 89 4