22:
3519:) methods that can be used to do this, depending on the accuracy to which the user would prefer. Using higher order methods to calculate a vehicle's position in the next timestep will yield a result with higher accuracy (if each method uses the same timestep). Numerical methods can also be used to find positions of vehicles in other car following models, such as the
3507:
speed, the speed at which the driver intends to travel, and the acceleration of the vehicle. Analyzing the variables in these two equations, it becomes apparent that as the gap between two vehicles decreases (i.e. a following vehicle approaches a leading vehicle) the velocity given by the congested branch of the equation will decrease and is more likely to prevail.
3631:
In 2000, Wilson used Gipps' model for simulating driver behavior on a ring road. In this case, every vehicle in the system is following another vehicle – the leader follows the last vehicle. The results of the experiment showed that the cars followed a free-flow time-space trajectory when the density
712:
to adjust the model to real-life conditions. Gipps states that it is desirable for the interval between successive recalculations of acceleration, speed and location to be a fraction of the reaction time which necessitates the storage of a considerable quantity of historical data if the model is to
185:
and movement of gases (Lighthill and
Whitman (1955) and Richards (1956) postulated the density of traffic to be a function of position; Newell (1955) makes an analogy between vehicle motion along a sparsely populated roadway and the movement of gases). First mention of simulating traffic with “high
180:
The method of modeling individual cars along a continuous space originates with
Chandler et al. (1958), Gazis et al. (1961), Lee (1966) and Bender and Fenton (1972), though many other papers proceeded and have since followed. In turn, these papers have bases in several works from the mid-1950s. Of
805:
Gipps sets limitations on the model through safety considerations and assuming a driver would estimate his or her speed based on the vehicle in front to be able to come to a full and safe stop if needed (1981). Pipes (1953) and many others have defined following characteristics placed into models
3529:(first order, and perhaps the simplest of the numerical methods) can be used to obtain accurate results, but the timestep would have to be very small, resulting in a greater amount of computation. Also, as a vehicle comes to a stop and the following vehicle approaches it, the term underneath the
3506:
These two equations used to determine the velocity of a vehicle in the next timestep represent free-flow and congested conditions, respectively. If the vehicle is in free-flow, the free-flow branch of the equation indicates that the speed of the vehicle will increase as a function of its current
1234:
Gipps defines the model by a set of limitations. The following vehicle is limited by two constraints: that it will not exceed its driver's desired speed and its free acceleration should first increase with speed as engine torque increases then decrease to zero as the desired speed is reached.
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where the first argument of the minimization regimes describes an uncongested roadway and headways are large, and the second argument describes congested conditions where headways are small and speeds are limited by followed vehicles.
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Gipps' model is based directly on driver behavior and expectancy for vehicles in a stream of traffic. Limitations on driver and vehicle parameters for safety purposes mimic the traits of vehicles following vehicles in the front of the
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The impetus for modeling vehicles in a stream of traffic and their subsequent actions and reactions comes from the need to analyze changes to roadway parameters. Indeed, many factors (to include driver,
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For the purpose of simulation, it is important to make sure the velocity and position of every vehicle has been calculated for a timestep before determining the moving along to the next timestep.
3012:
If the final assumption is true, that is, the driver travels as fast and safely as possible, the new speed of the driver's vehicle is given by the final equation being Gipps' model:
3537:
equation could potentially fall below zero if Euler's method is being used and the timestep is too large. The position of the vehicle in the next timestep is given by the equation:
1895:
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2022:
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2737:, and the driver is willing to brake hard, a model system can continue without disruption to flow. Thus, the previous equation can be rewritten with this in mind to yield
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Higher order methods not only use the velocity in the current timestep, but velocities from the previous timestep to generate a more accurate result. For instance,
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Using higher-order methods reduces the probability that the term under the square root in the congested branch of the velocity equation will fall below zero.
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After determining the velocity of the vehicle at the next timestep, its position at the next timestep should be calculated. There are several numerical (
806:
based on various driver department codes defining safe following speeds, known informally as a “2 second rule,” though is formally defined through code.
3223:{\displaystyle v_{n}(t+\tau )={\mbox{min}}\left\{v_{n}(t)+2.5a_{n}\tau (1-v_{n}(t)/V_{n})\left(0.025+v_{n}\left(t\right)/V_{n}\right)^{1/2}\right.,}
392:
which is defined primarily by one vehicle (noted by subscript n) following another (noted by subscript n-1); reaction time of the following vehicle
199:
and roadway conditions, to name a few) affect how traffic behaves. Gipps (1981) describes models current to that time to be in the general form of:
1655:
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The model is named after Peter G. Gipps who developed it in the late-1970s under S.R.C. grants at the
Transport Operations Research Group at the
3002:{\displaystyle v_{n}(t+\tau )\leq b_{n}\tau +{\sqrt {b_{n}^{2}\tau ^{2}-b_{n}\left(2\left-v_{n}(t)\tau -v_{n-1}(t)^{2}/{\hat {b}}\right)}}}
1428:
205:
2317:{\displaystyle x_{n-1}(t)-v_{n-1}(t)^{2}/2b_{n-1}-s_{n-1}\geq x_{n}(t)+\left\tau /2+v_{n}(t+\tau )\theta -v_{n}(t+\tau )^{2}/2b_{n}}
105:
1859:
For safety, the driver of vehicle n (the following vehicle) must ensure that the difference between point where vehicle n-1 stops (
3815:
2679:{\displaystyle -v_{n}(t+\tau )^{2}/2b_{n}+v_{n}(t+\tau )(\tau /2+\theta )-\left+v_{n}(t)\tau /2+v_{n-1}(t)^{2}/2{\hat {b}}\leq 0}
39:
3655:
3645:
3482:{\displaystyle \left.b_{n}\tau +{\sqrt {b_{n}^{2}\tau ^{2}-b_{n}\left-v_{n}(t)\tau -v_{n-1}(t)^{2}/{\hat {b}}\right]}}\right\}}
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has no obvious connection with identifiable characteristics of driver or vehicle. So, he introduces a new and improved model.
122:
86:
43:
998:, that is, the physical length plus a margin into which the following vehicle is not willing to intrude, even when at rest,
58:
3549:(second order) averages the velocity from the current and previous timestep to determine the next position of a vehicle:
1962:). However, Gipps finds the driver of vehicle n allows for an additional buffer and introduces a safety margin, of delay
3757:
Gipps, P. G. 1981 A behavioural car-following model for computer simulation. Transportation
Research Part B, 15, 105-111
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Computer
Program MULTSIM for Simulating Output from Vehicle Detectors on a Multi-Lane Signal Controlled Road
32:
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on the ring road was low. However, as the number of vehicles on the road increases (density increases),
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Model parameters should correspond to observable driver characteristics without undue calculation, and,
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Gazis, D. C., Herman R. and
Rothery R. W. (1961) Non-linear follow the leader models of traffic flow.
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1985:
1412:{\displaystyle v_{n}(t+\tau )\leq v_{n}(t)+2.5a_{n}\tau (1-v_{n}/V_{n})(0.025+v_{n}(t)/V_{n})^{1/2}}
574:
1933:
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Spyropoulou, Ioanna (2007). "Simulation Using Gipps' Car-Following Model—An In-Depth
Analysis".
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3788:. Transport Operations Research Group Working Paper No. 20, University of Newcastle-Upon-Tyne.
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Butchers Method (fifth order) uses an even more elegant solution to solve the same problem:
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between successive recalculations of speed and position is the same as driver reaction time.
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waves begin to form as the congested part of the Gipps’ Model velocity equation prevails.
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Seddon, P. A. (1972) Program for simulating the dispersion of platoons in road traffic.
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Wilson, R. E. (2001). "An analysis of Gipps' car-following model of highway traffic".
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1823:{\displaystyle x_{n}^{\ast }=x_{n}(t)+\left\tau /2-v_{n}(t+\tau )^{2}/2b_{n}}
186:
speed computers” is given by
Gerlough and Mathewson (1956) and Goode (1956).
3770:
Bender, J. C. and Fendon R. E. (1972) On vehicles longitudinal dynamics. In
3534:
144:. Gipps' model is differentiated by other models in that Gipps uses a
1543:{\displaystyle x_{n-1}^{\ast }=x_{n-1}(t)-v_{n-1}(t)^{2}/2b_{n-1}}
382:{\displaystyle a_{n}(t+\tau )=l_{n}{\frac {\left^{k}}{\left^{m}}}}
3791:
Lee, G. (1966) A generalization of linear car-following theory.
15:
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is the apparent reaction time, a constant for all vehicles.
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is the maximum acceleration which the driver of vehicle
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Using numerical methods to generate time-space diagrams
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is the most severe braking that the driver of vehicle
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Gipps’ model should reflect the following properties:
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special importance are a few that have analogies to
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717:program. He also points out that the parameters
125:behaviour by motorists in the United Kingdom.
797:The model should behave as expected when the
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1422:The third constraint, braking, is given by
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106:Learn how and when to remove this message
1085:is the location of the front of vehicle
568:of the following and preceding vehicle;
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2360:, it is replaced by an estimated value
168:to reduce the computation required for
132:and the Transport Studies Group at the
121:is a mathematical model for describing
7:
1982:when driver n is traveling at speed
44:adding citations to reliable sources
3722:IMA Journal of Applied Mathematics
1230:Constraints leading to development
14:
978:is the effective size of vehicle
613:of the following vehicle at time
130:University of Newcastle-Upon-Tyne
20:
3772:Traffic Flow and Transportation
3612:= v(t-Ď„) + (3/4)(v(t) - v(t-Ď„))
3603:= v(t-Ď„) + (1/2)(v(t) - v(t-Ď„))
3594:= v(t-Ď„) + (1/4)(v(t) - v(t-Ď„))
1890:{\displaystyle x_{n-1}^{\ast }}
1610:{\displaystyle x_{n-1}^{\ast }}
645:; and finally, model constants
31:needs additional citations for
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2017:{\displaystyle v_{n}(t+\tau )}
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606:{\displaystyle a_{n}(t+\tau )}
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1955:{\displaystyle x_{n}^{\ast }}
1642:{\displaystyle x_{n}^{\ast }}
148:within the function equal to
3774:, 19–32. Elsevier, New York.
3646:Newell's car-following model
942:{\displaystyle (b_{n}<0)}
3656:List of Runge–Kutta methods
1649:(for vehicle n is given by
3832:
2382:{\displaystyle {\hat {b}}}
561:{\displaystyle v_{n-1}(t)}
483:{\displaystyle x_{n-1}(t)}
3699:10.1080/18128600708685675
3556:x(t+Ď„) = x(t) + (1/90)(7k
2689:If the introduced delay,
638:{\displaystyle (t+\tau )}
134:University College London
3651:Intelligent driver model
3521:intelligent driver model
1163:is the speed of vehicle
1156:{\displaystyle v_{n}(t)}
1078:{\displaystyle x_{n}(t)}
519:{\displaystyle v_{n}(t)}
441:{\displaystyle x_{n}(t)}
3816:Road traffic management
3742:10.1093/imamat/66.5.509
2730:{\displaystyle \tau /2}
2702:{\displaystyle \theta }
2353:{\displaystyle b_{n-1}}
1975:{\displaystyle \theta }
1923:{\displaystyle s_{n-1}}
1849:{\displaystyle t+\tau }
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3784:Gipps, P. G. (1976)
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40:improve this article
3734:2001JApMa..66..509W
3541:x(t+Ď„)= x(t) +v(t)Ď„
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1572:{\displaystyle n-1}
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3666:Traffic simulation
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1196:{\displaystyle t}
1176:{\displaystyle n}
1118:{\displaystyle t}
1098:{\displaystyle n}
1047:wishes to travel,
1040:{\displaystyle n}
991:{\displaystyle n}
903:{\displaystyle n}
854:{\displaystyle n}
777:{\displaystyle m}
757:{\displaystyle k}
705:{\displaystyle m}
685:{\displaystyle k}
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2554:
2553:
2526:
2525:
2490:
2464:
2463:
2451:
2450:
2438:
2433:
2432:
2411:
2410:
2388:
2386:
2385:
2380:
2378:
2377:
2369:
2359:
2357:
2356:
2351:
2349:
2348:
2323:
2321:
2320:
2315:
2313:
2312:
2300:
2295:
2294:
2273:
2272:
2242:
2241:
2226:
2218:
2214:
2198:
2197:
2176:
2175:
2149:
2148:
2136:
2135:
2117:
2116:
2098:
2093:
2092:
2077:
2076:
2049:
2048:
2023:
2021:
2020:
2015:
1998:
1997:
1981:
1979:
1978:
1973:
1961:
1959:
1958:
1953:
1950:
1945:
1929:
1927:
1926:
1921:
1919:
1918:
1896:
1894:
1893:
1888:
1885:
1880:
1855:
1853:
1852:
1847:
1829:
1827:
1826:
1821:
1819:
1818:
1806:
1801:
1800:
1779:
1778:
1763:
1755:
1751:
1735:
1734:
1713:
1712:
1686:
1685:
1672:
1667:
1648:
1646:
1645:
1640:
1637:
1632:
1616:
1614:
1613:
1608:
1605:
1600:
1578:
1576:
1575:
1570:
1549:
1547:
1546:
1541:
1539:
1538:
1520:
1515:
1514:
1499:
1498:
1471:
1470:
1451:
1446:
1418:
1416:
1415:
1410:
1408:
1407:
1403:
1390:
1389:
1380:
1366:
1365:
1344:
1343:
1334:
1329:
1328:
1307:
1306:
1282:
1281:
1254:
1253:
1224:
1222:
1221:
1216:
1202:
1200:
1199:
1194:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1143:
1142:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1065:
1064:
1046:
1044:
1043:
1038:
1026:
1024:
1023:
1018:
1016:
1015:
997:
995:
994:
989:
977:
975:
974:
969:
967:
966:
948:
946:
945:
940:
929:
928:
909:
907:
906:
901:
889:
887:
886:
881:
879:
878:
860:
858:
857:
852:
840:
838:
837:
832:
830:
829:
783:
781:
780:
775:
763:
761:
760:
755:
743:
741:
740:
735:
733:
732:
711:
709:
708:
703:
691:
689:
688:
683:
671:
669:
668:
663:
661:
660:
644:
642:
641:
636:
612:
610:
609:
604:
587:
586:
567:
565:
564:
559:
548:
547:
525:
523:
522:
517:
506:
505:
489:
487:
486:
481:
470:
469:
447:
445:
444:
439:
428:
427:
412:; the locations
411:
409:
408:
403:
388:
386:
385:
380:
378:
376:
375:
370:
366:
356:
355:
334:
333:
312:
311:
306:
302:
292:
291:
270:
269:
248:
246:
245:
218:
217:
167:
165:
164:
159:
111:
104:
100:
97:
91:
89:
48:
24:
16:
3831:
3830:
3826:
3825:
3824:
3822:
3821:
3820:
3806:
3805:
3802:Vol. 18, 81–90.
3767:
3765:Further reading
3762:
3761:
3756:
3749:
3719:
3718:
3714:
3684:
3683:
3679:
3674:
3642:
3620:
3611:
3602:
3593:
3584:
3575:
3571:
3567:
3563:
3559:
3513:
3440:
3418:
3393:
3366:
3347:
3319:
3318:
3314:
3310:
3306:
3296:
3283:
3250:
3249:
3246:
3241:
3240:
3184:
3168:
3158:
3151:
3147:
3146:
3133:
3109:
3087:
3062:
3061:
3057:
3022:
3017:
3016:
2965:
2943:
2918:
2891:
2872:
2844:
2843:
2839:
2835:
2831:
2821:
2808:
2775:
2747:
2742:
2741:
2711:
2710:
2691:
2690:
2640:
2618:
2585:
2558:
2539:
2511:
2510:
2506:
2455:
2442:
2424:
2402:
2394:
2393:
2362:
2361:
2334:
2329:
2328:
2304:
2286:
2264:
2233:
2189:
2167:
2166:
2162:
2140:
2121:
2102:
2084:
2062:
2034:
2029:
2028:
1989:
1984:
1983:
1964:
1963:
1932:
1931:
1904:
1899:
1898:
1861:
1860:
1832:
1831:
1810:
1792:
1770:
1726:
1704:
1703:
1699:
1677:
1654:
1653:
1619:
1618:
1581:
1580:
1555:
1554:
1524:
1506:
1484:
1456:
1427:
1426:
1391:
1381:
1357:
1335:
1320:
1298:
1273:
1245:
1240:
1239:
1207:
1206:
1185:
1184:
1165:
1164:
1134:
1129:
1128:
1107:
1106:
1087:
1086:
1056:
1051:
1050:
1029:
1028:
1007:
1002:
1001:
980:
979:
958:
953:
952:
920:
912:
911:
892:
891:
870:
865:
864:
843:
842:
821:
816:
815:
766:
765:
746:
745:
724:
719:
718:
694:
693:
674:
673:
652:
647:
646:
615:
614:
578:
573:
572:
533:
528:
527:
497:
492:
491:
455:
450:
449:
419:
414:
413:
394:
393:
347:
319:
318:
314:
313:
283:
255:
254:
250:
249:
237:
209:
204:
203:
192:
178:
150:
149:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
3829:
3827:
3819:
3818:
3808:
3807:
3804:
3803:
3796:
3789:
3782:
3775:
3766:
3763:
3760:
3759:
3747:
3728:(5): 509–537.
3712:
3693:(3): 231–245.
3676:
3675:
3673:
3670:
3669:
3668:
3663:
3658:
3653:
3648:
3641:
3638:
3618:
3609:
3600:
3591:
3582:
3573:
3569:
3565:
3561:
3557:
3512:
3509:
3500:
3499:
3498:
3497:
3496:
3495:
3494:
3493:
3492:
3491:
3490:
3489:
3477:
3470:
3463:
3460:
3453:
3447:
3443:
3439:
3436:
3431:
3428:
3425:
3421:
3417:
3414:
3411:
3408:
3405:
3400:
3396:
3392:
3388:
3384:
3381:
3378:
3373:
3369:
3365:
3360:
3357:
3354:
3350:
3346:
3343:
3340:
3337:
3332:
3329:
3326:
3322:
3317:
3313:
3309:
3303:
3299:
3295:
3290:
3286:
3280:
3275:
3271:
3265:
3262:
3257:
3253:
3248:
3219:
3215:
3210:
3206:
3202:
3197:
3191:
3187:
3182:
3177:
3174:
3171:
3165:
3161:
3157:
3154:
3150:
3145:
3140:
3136:
3131:
3127:
3124:
3121:
3116:
3112:
3108:
3105:
3102:
3099:
3094:
3090:
3086:
3083:
3080:
3077:
3074:
3069:
3065:
3060:
3049:
3046:
3043:
3040:
3037:
3034:
3029:
3025:
3010:
3009:
2995:
2988:
2985:
2978:
2972:
2968:
2964:
2961:
2956:
2953:
2950:
2946:
2942:
2939:
2936:
2933:
2930:
2925:
2921:
2917:
2913:
2909:
2906:
2903:
2898:
2894:
2890:
2885:
2882:
2879:
2875:
2871:
2868:
2865:
2862:
2857:
2854:
2851:
2847:
2842:
2838:
2834:
2828:
2824:
2820:
2815:
2811:
2805:
2800:
2796:
2790:
2787:
2782:
2778:
2774:
2771:
2768:
2765:
2762:
2759:
2754:
2750:
2726:
2722:
2718:
2698:
2687:
2686:
2675:
2672:
2666:
2663:
2657:
2653:
2647:
2643:
2639:
2636:
2631:
2628:
2625:
2621:
2617:
2614:
2610:
2606:
2603:
2600:
2597:
2592:
2588:
2584:
2580:
2576:
2573:
2570:
2565:
2561:
2557:
2552:
2549:
2546:
2542:
2538:
2535:
2532:
2529:
2524:
2521:
2518:
2514:
2509:
2505:
2502:
2499:
2496:
2493:
2489:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2462:
2458:
2454:
2449:
2445:
2441:
2437:
2431:
2427:
2423:
2420:
2417:
2414:
2409:
2405:
2401:
2375:
2372:
2347:
2344:
2341:
2337:
2325:
2324:
2311:
2307:
2303:
2299:
2293:
2289:
2285:
2282:
2279:
2276:
2271:
2267:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2240:
2236:
2232:
2229:
2225:
2221:
2217:
2213:
2210:
2207:
2204:
2201:
2196:
2192:
2188:
2185:
2182:
2179:
2174:
2170:
2165:
2161:
2158:
2155:
2152:
2147:
2143:
2139:
2134:
2131:
2128:
2124:
2120:
2115:
2112:
2109:
2105:
2101:
2097:
2091:
2087:
2083:
2080:
2075:
2072:
2069:
2065:
2061:
2058:
2055:
2052:
2047:
2044:
2041:
2037:
2013:
2010:
2007:
2004:
2001:
1996:
1992:
1971:
1949:
1944:
1940:
1917:
1914:
1911:
1907:
1884:
1879:
1876:
1873:
1869:
1857:
1856:
1845:
1842:
1839:
1817:
1813:
1809:
1805:
1799:
1795:
1791:
1788:
1785:
1782:
1777:
1773:
1769:
1766:
1762:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1733:
1729:
1725:
1722:
1719:
1716:
1711:
1707:
1702:
1698:
1695:
1692:
1689:
1684:
1680:
1676:
1671:
1666:
1662:
1636:
1631:
1627:
1604:
1599:
1596:
1593:
1589:
1568:
1565:
1562:
1551:
1550:
1537:
1534:
1531:
1527:
1523:
1519:
1513:
1509:
1505:
1502:
1497:
1494:
1491:
1487:
1483:
1480:
1477:
1474:
1469:
1466:
1463:
1459:
1455:
1450:
1445:
1442:
1439:
1435:
1420:
1419:
1406:
1402:
1398:
1394:
1388:
1384:
1379:
1375:
1372:
1369:
1364:
1360:
1356:
1353:
1350:
1347:
1342:
1338:
1333:
1327:
1323:
1319:
1316:
1313:
1310:
1305:
1301:
1297:
1294:
1291:
1288:
1285:
1280:
1276:
1272:
1269:
1266:
1263:
1260:
1257:
1252:
1248:
1232:
1231:
1227:
1226:
1214:
1204:
1192:
1172:
1152:
1149:
1146:
1141:
1137:
1126:
1114:
1094:
1074:
1071:
1068:
1063:
1059:
1048:
1036:
1014:
1010:
999:
987:
965:
961:
950:
938:
935:
932:
927:
923:
919:
899:
877:
873:
862:
850:
828:
824:
812:
811:
810:Model notation
803:
802:
795:
792:
773:
753:
731:
727:
701:
681:
659:
655:
634:
631:
628:
625:
622:
602:
599:
596:
593:
590:
585:
581:
557:
554:
551:
546:
543:
540:
536:
515:
512:
509:
504:
500:
479:
476:
473:
468:
465:
462:
458:
437:
434:
431:
426:
422:
401:
390:
389:
374:
369:
365:
362:
359:
354:
350:
346:
343:
340:
337:
332:
329:
326:
322:
317:
310:
305:
301:
298:
295:
290:
286:
282:
279:
276:
273:
268:
265:
262:
258:
253:
244:
240:
236:
233:
230:
227:
224:
221:
216:
212:
191:
188:
183:fluid dynamics
177:
174:
157:
142:traffic stream
114:
113:
55:"Gipps' model"
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3828:
3817:
3814:
3813:
3811:
3801:
3797:
3794:
3790:
3787:
3783:
3780:
3776:
3773:
3769:
3768:
3764:
3754:
3752:
3748:
3743:
3739:
3735:
3731:
3727:
3723:
3716:
3713:
3708:
3704:
3700:
3696:
3692:
3688:
3681:
3678:
3671:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3643:
3639:
3637:
3635:
3629:
3626:
3623:
3622:
3614:
3613:
3605:
3604:
3596:
3595:
3587:
3586:
3578:
3577:
3553:
3550:
3548:
3547:Heun's Method
3543:
3542:
3538:
3536:
3532:
3528:
3527:Eulers Method
3524:
3522:
3518:
3510:
3508:
3504:
3475:
3468:
3458:
3451:
3445:
3437:
3429:
3426:
3423:
3419:
3415:
3412:
3406:
3398:
3394:
3390:
3386:
3379:
3371:
3367:
3363:
3358:
3355:
3352:
3348:
3344:
3338:
3330:
3327:
3324:
3320:
3315:
3311:
3307:
3301:
3297:
3293:
3288:
3284:
3278:
3273:
3269:
3263:
3260:
3255:
3251:
3239:
3238:
3237:
3236:
3235:
3234:
3233:
3232:
3231:
3230:
3217:
3208:
3204:
3200:
3195:
3189:
3185:
3180:
3175:
3172:
3169:
3163:
3159:
3155:
3152:
3148:
3138:
3134:
3129:
3122:
3114:
3110:
3106:
3103:
3097:
3092:
3088:
3084:
3081:
3075:
3067:
3063:
3058:
3047:
3041:
3038:
3035:
3027:
3023:
3015:
3014:
3013:
2993:
2983:
2976:
2970:
2962:
2954:
2951:
2948:
2944:
2940:
2937:
2931:
2923:
2919:
2915:
2911:
2904:
2896:
2892:
2888:
2883:
2880:
2877:
2873:
2869:
2863:
2855:
2852:
2849:
2845:
2840:
2836:
2832:
2826:
2822:
2818:
2813:
2809:
2803:
2798:
2794:
2788:
2785:
2780:
2776:
2772:
2766:
2763:
2760:
2752:
2748:
2740:
2739:
2738:
2724:
2720:
2716:
2696:
2673:
2670:
2661:
2655:
2651:
2645:
2637:
2629:
2626:
2623:
2619:
2615:
2612:
2608:
2604:
2598:
2590:
2586:
2582:
2578:
2571:
2563:
2559:
2555:
2550:
2547:
2544:
2540:
2536:
2530:
2522:
2519:
2516:
2512:
2507:
2503:
2497:
2494:
2491:
2487:
2483:
2474:
2471:
2468:
2460:
2456:
2452:
2447:
2443:
2439:
2435:
2429:
2421:
2418:
2415:
2407:
2403:
2399:
2392:
2391:
2390:
2370:
2345:
2342:
2339:
2335:
2309:
2305:
2301:
2297:
2291:
2283:
2280:
2277:
2269:
2265:
2261:
2258:
2252:
2249:
2246:
2238:
2234:
2230:
2227:
2223:
2219:
2215:
2208:
2205:
2202:
2194:
2190:
2186:
2180:
2172:
2168:
2163:
2159:
2153:
2145:
2141:
2137:
2132:
2129:
2126:
2122:
2118:
2113:
2110:
2107:
2103:
2099:
2095:
2089:
2081:
2073:
2070:
2067:
2063:
2059:
2053:
2045:
2042:
2039:
2035:
2027:
2026:
2025:
2008:
2005:
2002:
1994:
1990:
1969:
1947:
1942:
1938:
1915:
1912:
1909:
1905:
1882:
1877:
1874:
1871:
1867:
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713:be used in a
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123:car-following
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57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
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3011:
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1553:for vehicle
1552:
1421:
1233:
804:
786:
570:acceleration
490:and speeds
391:
197:traffic flow
193:
179:
176:Introduction
138:
127:
119:Gipps' model
118:
117:
102:
96:January 2014
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
3531:square root
3517:Runge–Kutta
3800:Simulation
3672:References
3661:Simulation
715:simulation
190:Definition
66:newspapers
3793:Ops. Res.
3779:Ops. Res.
3707:111305074
3634:kinematic
3462:^
3427:−
3416:−
3413:τ
3391:−
3364:−
3356:−
3345:−
3328:−
3294:−
3285:τ
3261:τ
3107:−
3098:τ
3042:τ
2987:^
2952:−
2941:−
2938:τ
2916:−
2889:−
2881:−
2870:−
2853:−
2819:−
2810:τ
2786:τ
2773:≤
2767:τ
2717:τ
2697:θ
2671:≤
2665:^
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2605:τ
2556:−
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2520:−
2504:−
2498:θ
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2475:τ
2422:τ
2400:−
2374:^
2343:−
2284:τ
2262:−
2259:θ
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2220:τ
2209:τ
2138:≥
2130:−
2119:−
2111:−
2071:−
2060:−
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2009:τ
1970:θ
1948:∗
1913:−
1883:∗
1875:−
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1790:τ
1768:−
1757:τ
1746:τ
1670:∗
1635:∗
1603:∗
1595:−
1579:at point
1564:−
1533:−
1493:−
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1465:−
1449:∗
1441:−
1318:−
1309:τ
1271:≤
1265:τ
1213:τ
1105:at time *
630:τ
598:τ
542:−
464:−
400:τ
345:−
328:−
281:−
264:−
229:τ
156:τ
3810:Category
3640:See also
3585:= v(t-Ď„)
3535:velocity
1830:at time
1617:, where
1183:at time
799:interval
146:timestep
3730:Bibcode
80:scholar
3705:
3621:= v(t)
82:
75:
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61:
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3703:S2CID
3568:+ 32k
3564:+ 12k
3560:+ 32k
3153:0.025
1352:0.025
1203:, and
87:JSTOR
73:books
3572:+ 7k
931:<
764:and
59:news
3738:doi
3695:doi
3085:2.5
3053:min
1296:2.5
136:.
42:by
3812::
3750:^
3736:.
3726:66
3724:.
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3576:)Ď„
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36:.
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