MULTIPLE TRAFFIC JAMS IN FULL VELOCITY DIFFERENCE MODEL WITH REACTION-TIME DELAY

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1 ISSN It j simul model 14 (2015) 2, Origial scietific paper MULTIPLE TRAFFIC JAMS IN FULL VELOCITY DIFFERENCE MODEL WITH REACTION-TIME DELAY Xu, R. G. School of Aerospace Egieerig ad Applied Mechaics, Togji Uiversity, Shaghai , Chia xurog.gai@163.com Abstract A full velocity differece model for traffic flow, icludig driver s reactio time delay is cosidered. The uiform flow ad traffic jams are iterpreted through stability ad bifurcatio aalysis. Specifically, the uiform flow is represeted by the equilibrium of the model. Liear stability reveals that whe the equilibrium loses its stability, local bifurcatio tur up through Hopf bifurcatios. To aalyse the behaviour of the model after bifurcatig, umerical cotiuatio techiques are employed. Braches of oscillatig solutios ad the correspodig stabilities are obtaied. It is show that bifurcatig oscillatios ca coexist ad correspod to differet traffic patters. To visualize the spatial patters, umerical simulatios are performed, which are preseted by velocity time histories ad spatio-temporal diagrams. Aalysig the characteristic features, these oscillatig solutios are classified ito three types, ad further correspod to three types of traffic jams: almost traffic jams, width-equal traffic jam ad width-alterated traffic jam. The obtaied results provide a explaatio of how multiple jams iduced by driver s reactio time delay occur. (Received, processed ad accepted by the Chiese Represetative Office.) Key Words: Full Velocity Differece Model, Reactio-Time Delay, Stability, Bifurcatio, Numerical Cotiuatio, Traffic Patters 1. INTRODUCTION I recet years, traffic flow problems have received cosiderable iterest from various disciplies [1]. Oe of the reasos is that traffic jams are becomig more ad more serious problems. However, although part of our daily lives, traffic dyamics ad especially, traffic jams, have ot bee well uderstood. To study the traffic dyamics ad to uderstad the mechaism of traffic jams, modellig of the traffic flow is the first step. I cotiuum models, istead of cosiderig idividual vehicles, cotiuous desity distributio ad velocity distributio (as fuctios of space ad time) are used to describe the traffic flow, where partial differetial equatios (PDEs) are employed to describe the traffic dyamics [2]. I cellular automata models, space, time ad velocity are cosidered to be discrete, ad updated rules are used to describe the time developmet of traffic flow [3]. I car-followig models, discrete etities move i cotiuous space ad time, ad the vehicles motios are described by ordiary differetial equatios (ODEs) [4] or by delay differetial equatios (DDEs) if time delay is itroduced [5]. The optimal velocity (OV) model [4] is the most widely used car-followig model. Geerally, the quality of these developed models is evaluated by fittig the models to empirical data [6]. The data-fittig method easily leads to research capturig. However, some essetial characteristics of traffic dyamics, e.g., the coexistece of multiple stable traffic flow, caot be fully revealed through such data-fittig method. Hece, to comprehesively uderstad ad classify the characteristic features of traffic flow, especially, the hidde ustable motio, geeral models with varyig traffic flow parameters eed to be studied [7]. Amog the previous research, bifurcatio theory is a useful ad effective way to study the dyamical behaviour of the traffic flow models. Igarashi et al. [8] first cosidered the bifurcatio pheomea i a car-followig model ad poited out that the subcritical Hopf bifurcatio was the dyamical origi of the evolutio betwee uiform flow ad cogested DOI: /IJSIMM14(2)CO7 325

2 traffic flow. Hopf bifurcatio was ivestigated through umerical cotiuatio techiques [9] by Gasser et al. [10]. Notice that durig drivig, due to huma physiological factors, drivers require a fiite period of time to process stimuli from the precedig vehicle, which iheretly itroduces the drivers reactio time delay ito the car-followig models. I 1936, driver s reactio time delay was first recogized i traffic dyamics [11]. However, it was first itroduced ito the OV model i 1998, where drivers were supposed to react to their curret velocity ad headway with the same time delay [12]. Davis [13] proposed a modified time-delayed OV model, which assumed that drivers could recogize their speed without time delay, but eed some time to react to their headway. This model was used by Yu et al. [14] to ivestigate the desity waves, ad by Orosz et al. [15] to perform global bifurcatio aalysis through umerical cotiuatio techiques [16]. A ew car-followig model cosiderig the effects of two differet time delays was aalysed i [17]. I the above OV models, drivers reactio does ot deped o the velocity differece of the vehicles. However, drivers reactio should deped o ot oly the chages i headway, but more importatly, the velocity differece betwee vehicles [18]. Shamoto [19] poited out that the acceleratio of a vehicle had a positive correlatio with the velocity differece. The preseted research is motivated by two aspects. O oe had, a appropriate car-followig model should iclude the headway, curret ow velocity ad velocity differece. Moreover, drivers reactio time delay with respect to these factors should be cotaied, because a recet study [20] has poited out that drivers reactio time may lead to ustable traffic flow. O the other had, explorig all the possible dyamic behaviour of the model with respect to varyig model parameters, especially the driver s reactio time delay, is of great sigificace to the comprehesive uderstadig of the mechaism of traffic jams. To do so, i this paper, a full velocity differece model with drivers reactio time delay is built. Through liear aalysis, uiform flow stability is studied ad the stability boudary of the uiform flow is obtaied. The, based o umerical cotiuatio techiques, the bifurcatio behaviour of the model whe the uiform flow loses stability is ivestigated with respect to drivers reactio time delay. It is foud that multiple oscillatig solutios with distict characteristics ca coexist. Through umerical simulatios, these oscillatig solutios ad their correspodig traffic patters are studied ad classified ito three types. 2. TIME-DELAYED FULL VELOCITY DIFFERENCE MODEL I this sectio, a time-delayed full velocity differece model is built. Suppose that idetical vehicles drive by idetical drivers are placed o a circular road, where overtakig ad lae-chagig effects are igored. Cosider the situatio that the i th vehicle follows the i+1 st vehicle, ad the th vehicle follows the first oe. Deote the positio of each vehicle by x i (i = 1, 2,, ), the velocity by v i. Hece, the relative displacemets (called headways) h i (t) ca be expressed as h i (t) = x i+1 (t) x i (t). Takig the time derivative of h i (t) yields the relative velocity of two adjacet vehicles (called velocities differece): hi ( t) v i1( t) v i( t), i 1,2,..., -1 (1) Here the dot deotes the derivatio with respect to time. Assume that the legth of the circular road is L ad the legth of each vehicle ca be eglected. Sice the th car follows the first car, it is reasoable to get 1 h L h, such that: i1 i h ( t) v ( t) v ( t) (2) The origial OV model was developed by Bado et al. [4], i.e.: 1 v ( t) [ V( h ( t)) v ( t)] (3) i i i 326

3 Itroducig the velocity differece ito the eq. (3) yields the full-velocity deferece model [21], i.e.: v ( t) [ V( h ( t)) v ( t)] h ( t) (4) i i i i I eqs. (3) ad (4), the parameters α > 0 ad ß > 0 are the sesitivity, ad V(h i ) is kow as the optimal velocity fuctio. Models cosiderig the time delay i reactio to the headway ad to its ow velocity are developed i [12] ad [13], respectively. The preseted model specifies that the acceleratio of each vehicle is a fuctio of the three stimuli that the driver receives: the headways, the curret velocities ad the velocity differece. Besides, it is assumed that the drivers reactio time delay with respect to all these three stimuli was ot zero. Deote τ 1, τ 2 ad τ 3 as driver s reactio time delay with respect to the stimuli of headway, its ow velocity ad velocity differece, respectively. Thus the acceleratio of the i th vehicle ca be expressed as: vi ( t) [ V( hi ( t 1)) v i( t 2)] hi ( t 3) (5) Owig to the o-zero time delay, such model is amed as time-delayed full velocity differece model. Combiig eqs. (1), (2) ad (5) ad usig the huma driver set-up i [7] (τ 1 = τ 3 = τ, τ 2 = 0) yields our dyamic model expressed by delay differetial equatios: hi ( t) vi+1( t) vi ( t) vi( t) ( V ( hi( t )) vi ( t)) ( vi+1( t ) vi ( t )) i 1,..., 1 (6) 1 v ( t) ( V ( L hi ( t )) v ( t)) ( v1( t ) v ( t )) i1 Orosz preseted a o-dimesioalized optimal velocity fuctio [15], which will be used i this study, i.e.: 0, if h[0, hstop ] 3 V( h) [( h h 0 stop )] (7) v, if h [ h 3 3 stop, ) hstop ( h hstop ) where h stop > 0 stads for a jam headway, ad v 0 > 0 is the desired speed. 3. LINEAR STABILITY ANALYSIS Uiform traffic flow is such a situatio that equidistat cars move with the same time-idepedet velocity ad zero velocity differece with respect to their earby vehicle. Notice that uiform flow exists i the stable regio of the liearized system. Hece, i what follows liear stability aalysis will first be performed. It is obvious that eq. (6) possesses uiform flow equilibrium: L hi ( t) h, vi ( t) v V ( h ), hi ( t) 0, i 1,..., (8) I order to study the stability of this equilibrium, eq. (6) ca be liearized as: h ( t) v i1( t) v i( t) hi ( t) v i( t) V ( h )( hi ( t ) h ) ( vi +1( t ) vi ( t )) i 1,..., 1 1 v ( t) v ( t) V ( h )( L hi ( t ) h ) ( v1( t ) v( t )) i1 (9) 327

4 Usig the trial solutio e λt with λ C, oe obtais the characteristic equatio i the form: 1 2k i 2 ( ) e ( V ( h ) ) e ( V ( h ) )e 0 (10) k1 Substitutig the critical eigevalue λ = i ω, ω > 0 ito eq. (10), separatig the real ad imagiary parts ad usig some trigoometric idetities, oe obtais the Hopf stability curves i the (τ, ß) plae: 2 2 k ( ( V ( h ) ) ( V ( h ) )cot ) 1 arccos ( V ( h ) ) (11) k 4V( h ) csc 2 Each curve of eq. (11) belogs to each spatial wave umber k of oscillatios alog the rig ad is parameterized by the frequecy ω. Notig that k = 1, 2,, 1 is itroduced by the th of uity. For the followig parameters: = 33, α = 0.9 s -1, v 0 = 11 m/s, h stop = 14 m, L = 1122 m (12) The Hopf stability curves correspodig to k = 1 to 1 are show i Fig. 1. The stability curves for k 11 ad k 12 is deoted by blue ad gree curves, respectively. A solid arrow idicates the icrease of the wave umber. I the grey area of Fig. 1, the uiform flow is stable. Crossig ay stability curve, a pair of complex-cojugate eigevalues cross the imagiary axis, i.e., a Hopf bifurcatio takes place. For small sesitivity ß (ß < 0.4), the curve correspodig to k = 1 costitutes the stability boudary betwee the uiform flow ad oscillatios. With the icrease of time delay, curves correspodig to k > 1 result i further oscillatio modes aroud the already ustable equilibrium. However, for large ß (ß > 0.4), k = 11 ad k = 12 curves costitute the stability boudary, ad oscillatios correspodig to other values of k show up alterately with the icrease of time delay. Thus, oe otices that eve at the liear level the time delay ca iduce differet spatial dyamics, which will be further aalysed i sectios 4 ad 5. To demostrate the complexity i the ustable uiform flow domai with varyig time delay, further oliear aalysis is ecessary. We eed to use the umerical cotiuatio methods [16] to describe the global dyamics k 12-1 (s ) k 11 k 0.2 k (s) Figure 1: Liear stability diagrams for = 33 cars. 328

5 4. NUMERICAL BIFURCATION ANALYSIS I this sectio, umerical cotiuatio techiques [16] are employed to demostrate the complexity arise at the oliear level whe the drivers reactio time delay varies. Bifurcatio aalysis of the time-delayed full velocity differece model is performed. The basic idea is to fid a bifurcatio with the chage of time delay τ (the dyamics chage qualitatively at a critical value) ad the cotiue the bifurcatio solutio as τ is chaged. To this ed, the amplitude of the velocity oscillatios is used as a fuctio of reactio-time delay, which is defied as: vi,amp max vi ( t) mi vi ( t), i 1,..., (13) t Sice it is assumed that each vehicle ad each driver possess the same characteristics, which lead to a Z symmetry of the system, a result is: T T vi ( t) vi 1( t ), hi ( t) hi 1( t ) (14) where T is the period of the velocity oscillatios. I other words, it is sufficiet to plot the profile of the first car, ad the the profiles for all other cars are simply shifted copies. Sice the studied system is described by ifiite dimesioal DDEs eq. (6), the software package DDE-BIFTOOL is adopted to fid ad cotiue both equilibrium ad oscillatig solutios of eq. (6), eve if they are ustable. Cosiderig the real traffic istace, oe takes τ [0, 1.5] as the cotiuatio rage [22]. With parameters i eq. (12) ad ß = 0.1, umerical cotiuatios are carried out. At the same time, to determie the stability of oscillatig solutios, the Floquet multipliers are calculated. Fold bifurcatio of oscillatig solutios are detected, where two oscillatig solutios merge ad disappear. Neimark-Sacker bifurcatio occurs where the oscillatig solutio chages its stability ad quasiperiodic oscillatio arises (the detail of which is ot studied i this paper). A period-doublig bifurcatio occurs where the oscillatig solutio chages its stability ad eve yields more complicate oscillatio. (m/ s) v 1,amp ( a) * * * * * * k a) (s) b) (s) Figure 2: a) Solid ad dashed curves idicate stable ad ustable solutios, respectively; b) Local zoom of the rectagle regio i part a). (m/ s) v 1,amp Fig. 2 shows the cotiuatio results of the first car for differet values of wave umber k. The horizotal axis is the reactio-time delay τ, ad the vertical axis displays the amplitude of velocity oscillatio of the first car for a differet wave umber. To clearly observe the bifurcatios, the rectagular regio i Fig. 2 a is zoomed i Fig. 2 b. Stable ad ustable solutios are deoted by solid curves ad dashed curves respectively. Fold, Neimark-Sacker, ad period-doublig bifurcatios of the periodic oscillatios are deoted by crosses (), stars t ( b) * * * * * * ** * 329

6 () ad diamods () respectively. The solid circles o the v 1,amp = 0 lie are the subcritical Hopf bifurcatio poits of uiform flow. Note that the brach correspodig to k = 1 belogs to the stability boudary of the uiform flow equilibrium, while the curves correspodig to k > 1 result i further oscillatio modes aroud the already ustable equilibrium. These braches of oscillatios will udergo further bifurcatios. By calculatig the Floquet multipliers with the DDE-BIFTOOL, it is foud that fold bifurcatio () occurs o every brach k where braches fold back. While period doublig bifurcatios () oly occur o the braches k = 2 where the stability chages. Except the braches k = 1, 2, Neimark-Sacker bifurcatios () occur o all the other braches (k = 3, 4, 5, ), where the stability also chages. Oe should otice that for ß = 0.1, τ C = 0.6 (see the first vertical lie show i Fig. 2) is the first period-doublig bifurcatios poit correspodig to the wave umber k = 2, where two stable periodic oscillatios correspodig to wave umber k = 1 ad k = 2 coexist. Fig. 2 also shows that for ay value of time delay that locates betwee (τ C, 1.5), several stable ad ustable periodic oscillatios correspodig to differet wave umbers k ca coexist. Here we say that the periodic oscillatio is stable i this umerical cotiuatio techique if the maximum Floquet multiplier is smaller tha 1.09, i.e. max μ It should be oted that to meet the requiremets of precise umerical cotiuatio, criteria max μ 1.09 is employed, istead of the theoretical criteria max μ 1. To clearly describe the coexistece of stable ad ustable periodic oscillatios τ = 1 s is take as a example (see the secod vertical lie show i Fig. 2). The results obtaied from the DDE-BIFTOOL reveal that for τ = 1, the periodic oscillatios are stable for the braches correspodig to k = 1,, 5 but are ustable for the braches k = 6,, 9. To uderstad the evolutio of each oscillatio correspodig to a differet wave umber k, iitial value simulatios of the eq. (6) from suitable iitial data are carried out (τ = 1), results of which are show i Fig. 3. It is foud that for the wave umbers k = 1,, 5 the velocity of the first car performs stable periodic oscillatio. However, the oscillatios related to wave umbers k = 6,, 9 are ustable durig the trasiet stage, as is predicted, but evetually evolve ito stable oscillatios, but of differet types. Ustable periodic oscillatios for k = 7 ad 8 become stable period-doublig oscillatios, while ustable periodic oscillatios for k = 6 ad 9 evetually become stable periodic oscillatios. As a result, all the periodic oscillatios correspodig to wave umber k = 1,, 9 fially evolve ito stable forms, which ca be further classified ito two types: stable periodic oscillatios for k = 1, 2, 3, 4, 5, 6, 9 ad stable period-doublig oscillatio for k = 7, 8. I the ext sectio, detailed characteristics of these oscillatios ad their correspodig traffic patters will be discussed. 5. MULTIPLE TRAFFIC JAMS WITH NUMERICAL SIMULATIONS To uderstad the traffic patters correspodig to differet wave umbers, i this sectio umerical simulatios are performed for each case (k = 1,, 9) to visualize the fial stable oscillatio i time history diagrams (Fig. 3) ad the resultig spatial patters i spatiotemporal diagrams (Fig. 4). I our model, we say that a traffic jam occurs if there is a regio of the rig alog which the cars remai statioary (velocity is less tha 0.01 m/s i our defiitio). We deote T jam as the cost time that the vehicles stay i the jams regio. We first look at the time history of the velocity of the first car for various wave umbers k i Fig. 3. It is foud that for k = 1, 2, 3, 6, 9 there exist fiite time itervals where the velocity v 1 is below 0.01 m/s; i.e., the vehicles remai statioary durig these fiite time itervals, which meas T jam > 0. As a result, vehicles i such a mode will advace for a momet ad stop for a momet. Such velocity oscillatios are hece amed as stop ad go periodic 330

7 oscillatio. For k = 4, 5, mi v1 ( t) is very close but up to 0.01 m/s, ad max v1 ( t) t[0,1000] but very close to the desired speed v 0. t[0,1000] is below Figure 3: The time histories of the velocity of the first car for wave umbers k = 1,, 9. I these cases, vehicles will ot stop ad stay statioary (T jam = 0), but will move at speeds with wide fluctuatios. Such velocity oscillatios are amed as large amplitude periodic oscillatios. For k = 7, 8, similar stop ad go pheomea are observed. Additioally, as stated i Sectio 4, the oscillatios fially evolve ito period-doublig oscillatios. Therefore, they are amed as stop ad go period-doublig oscillatios. Cosequetly, accordig to the characteristics of velocity distributio i Fig. 3 ad based o our defiitio of statioary, three types of stable oscillatio exist whe τ = 1, amely, stop ad go periodic oscillatios for k = 1, 2, 3, 6, 9, large amplitude periodic oscillatios for k = 4, 5, ad stop ad go period-doublig oscillatios for k = 7, 8. I order to visualize traffic patters correspodig to these three types of oscillatio, we choose the wave umbers k = 1, k = 5 ad k = 8 as examples to illustrate the stop ad go periodic oscillatio, large amplitude periodic oscillatio, ad stop ad go period-doublig oscillatio respectively. With costat fuctio alog the iterval [ τ, 0] as iitial coditios, the spatio-temporal diagrams (i.e., positio of each car at each time) for these three types of oscillatios are show i Figs. 4 a, b, ad c respectively. 331

8 Notice that i the spatio-temporal diagrams 4 a ad 4 c, curves have some horizotal sectios. Such horizotal curves mea that the vehicles do ot go forward for some time; i.e., they are jammed. Hece, the horizotal sectios i the spatio-temporal diagram are amed as cogested cluster. Oce vehicles have etered the cogested clusters, they have to wait some time. As a result, cogested cluster is a idicator of traffic jams. a) b) c) Figure 4: Spatiotemporal diagram for differet wave umbers: a) k = 1, b) k = 5 ad c) k = 8; the trajectory of the first vehicle is emphasized as blue curves. I Fig. 4 a, it is observed that each vehicle has horizotal sectios i positio, idicatig that the vehicles will be jammed for some time. Moreover, it reads from Fig. 4 a that each vehicle wait exactly the same time at each cogested cluster; i.e., T jam = 33 s for k = 1. Such a traffic patter is hece referred to as width-equal traffic jam. I this traffic patter, drivers eed to decrease or icrease their velocities more tha oce durig oe circle, but have to face oe or two traffic jams durig oe circle ad wait the same time i every cogested cluster oce havig etered. I Fig. 4 b, oe ca see that there is o horizotal sectio i positio for each vehicle; i.e., T jam = 0 for k = 5, which meas that the velocities are always positive (vehicles without beig completely stopped) alog the rig for oe circle. However, i such a case, drivers will frequetly apply the brake to decelerate ad the press the gas pedal to accelerate, seve or eight times durig oe circle. Accordig to our defiitio, o traffic jam will occur eve though the miimum velocity is very small (but NOT zero). Such traffic a patter is referred to as almost traffic jam. I Fig. 4 c, after eterig a steady state, each vehicle also has several horizotal segmets i positio, with total waitig time of T jam = 20 s, but is discotiuously divided ito a wide traffic jam with T 1 jam = 15 s ad a arrow jam with T 2 jam = 5 s, as marked i Fig. 4 c. I this traffic patter, oce eterig the cogested clusters, vehicles will meet oe after aother traffic jam. This patter is referred to as the wide-alterated traffic jam. 332

9 k 1,2,3 6,9 4,5 7,8 trasiet oscillatio stable ustable stable ustable Table I: The classificatio of traffic jams. period period period-doublig steady oscillatio stop ad go large amplitude stop ad go traffic patter width-equal almost traffic jam width-alterated T T jam (s) 33 0 = 15, T = jam jam Table I lists the characteristics of the three types of oscillatio ad their correspodig traffic patters. These three differet traffic patters ca coexist, ad the iitial coditios determie which oe is approached. 6. CONCLUSION I this paper, a full velocity differece model for traffic flow with drivers reactio time delay is developed, which comprehesively icludes the drivers reactio with respect to the stimuli of headway, its ow velocity ad velocity differece. I order to study the effects of time delay o traffic dyamics, stability aalysis ad bifurcatio aalysis are carried out. The local stability of the uiform flow equilibrium is ivestigated whe drivers reactio time delay is varied, which reveals that the traffic system will udergo Hopf bifurcatios at critical poits ad uiform flow becomes ustable, iducig oscillatig solutios. After that, umerical cotiuatio techiques are employed to ivestigate the bifurcatio whe reactio-time delay is far from the critical poit. Followig the braches of oscillatios, fold, Neimark-Sacker ad period-doublig bifurcatios are detected, which cause qualitative chages of the oscillatig solutios. It is show that several stable ad ustable oscillatig solutios ca coexist whe time delay is larger tha the critical value. Numerical simulatios are performed to show that all periodic oscillatios will evetually evolve ito stable periodic or stable period-doublig oscillatios. These two stable oscillatios ca be further classified ito three types accordig to their velocity characteristic, which are large amplitude periodic oscillatio, stop ad go periodic oscillatio ad stop ad go period-doublig oscillatio. Fially, spatial patters correspodig to these oscillatios are show i a spatio-temporal diagram. We show that multiple types of traffic jams, amely, almost traffic jams, width-equal traffic jams ad width-alterated traffic jams, coexist. The preseted paper establishes a full velocity differece model with drivers reactio time delay, the dyamics of which are studied i terms of bifurcatio. The obtaied results provide a classificatio of traffic jam, which sigificatly improves our uderstadig of the dyamics of traffic flow as well as the mechaism of traffic jams iduced by driver s reactio time delay. I further work, more complicate oscillatios ad traffic patters will be studied which have arise ear the bifurcatig oscillatios. REFERENCES [1] Helbig, D. (2001). Traffic ad related self-drive may-particle systems, Reviews of Moder Physics, Vol. 73, No. 4, , doi: /revmodphys [2] Lighthill, M. J.; Whitham, G. B. (1955). O kiematic waves. II. A theory of traffic flow o log crowded roads, Proceedigs of the Royal Society A: Mathematical, Physical ad Egieerig Scieces, Vol. 229, No. 1178, , doi: /rspa [3] Nagel, K.; Schreckeberg, M. (1992). A cellular automato model for freeway traffic, Joural de Physique I, Vol. 2, No. 12, , doi: /jp1: [4] Bado, M.; Hasebe, K.; Nakayama, A.; Shibata, A.; Sugiyama, Y. (1995). Dyamical model of traffic cogestio ad umerical simulatio, Physical Review E, Vol. 51, No. 2, , doi: /physreve

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