- Email: [email protected]
Some errors in macroscopic traffic models based on car following models
Tmnspn Rcs..B Vol. I4B. pp. 241-242 0 Pergamon Press Ltd..1980.Printed in Great Britain
SOME ERRORS IN MACROSCOPIC TRAFFIC MODELS BASED ON CAR FOLLOWING MODELS BERNARD F. BYRNE Department of Civil Engineering,
West Virginia University,
(Received
Morgantown,
WV 26506, U.S.A.
26 December 1978)
Abstract-The macroscopic traffic flow models developed from the car following models of Gazis et al. (1961) are shown to have a flaw in that they do not meet certain of the boundary conditions that researchers have said that they do. This does not affect many existing models but, nevertheless, should be cleared up.
In general, there have been two approaches to the development of models of traffic flow. These are the macroscopic models, which seek to describe traffic flow in terms of average variables such as density, flow and average speeds; and the microscopic, or car following models, which describe traffic flow in terms of the speed and spacing of individual vehicles. These two approaches were shown to be interrelated in the work of Gazis et al. (l%l), later further developed by May and Keller (1967). Gazis et al. (l%l) proposed, as a general car following model, the following: [see Gazis et al. (l%l), eqn (9), or May and Keller (1967), eqn (7)]
.il,+l(t + T) = aK+dt
+ ~)[kl(~)-~n+d~)l [x”(t)-x”+l(t)l’
(1)
.
If we assume that eqn (1) holds in steady-state condition, as well as in transient conditions, then several simplifications occur. The time lag disappears, and the spacing becomes the steady-state spacing (s). If the speed of individual car is taken to be the speed of the stream (u), then eqn (1) simplifies to du _=Urn The solution to this equation
where the meaning of
ads
S’ .
is Gazis et al. (l%l),
eqn (11)
fp(p = m or 1) is fp(x)=x’_P, (p# 1)
and fp(x) = In x. To find the constant
of integration,
(p = 1).
c’, either of two boundary I
u=u,atk=O
or II
u = 0 at k = kh 241
conditions
may be applied.
242
B. F. BYRNE
In the above, ur is the “free speed” and ki is the jam density and is related to k by
The only situation in which the first boundary condition can be applied is when I> 1. For I = 1, filn(s)=ln(l/k) which is undefined for k = 0. The only situation in which the second boundary condition can be applied is when m < 1. For m = 1,
fm=ln(U) which is undefined for u = 0. Also, for m > 1,
which is undefined for u = 0. Therefore, the constant of integration can be evaluated as
c’=fn&) only when I > 1, contrary to the statement by Gazis et al. (1961, p. 548), that it can be evaluated for m > 1, If 1, and m# 1, I > 1. Also, the constant of integration can be evaluated as
c’ = -cf,(sJ only when m < 1, contrary to Gazis et al. (1961, p. 548), who state that In can be evaluated for all other combinations of m and 1 except m = 1, I < 1. This implies that, in addition to the stated case of m = 1, I < 1, three other cases must be added to the list of those for which boundary conditions cannot be satisfied, namely I< 1, m > 1; I = 1, m = 1; I = 1, m > 1. This also implies that the matrix developed by May and Keller (1%7, p. 22) cannot be filled for these four cases. However, the matrix developed by May and Keller (1967, p. 23) for existing macroscopic models is satisfactory, as is the non-integer mode1 developed by May and Keller (1967, p. 29). The technique developed by May and Keller (1%7, pp. 26-28) to compare various models and determine suitable values for m and 1 shows the unsatisfactory region quite well. This technique consists of graphing parameters of various models on m, 1 diagrams. In examining these m, 1 diagrams, there are very few points in the region where neither boundary condition is satisfied. CONCLUSIONS
It has been shown that the macroscopic models developed from the car following models have some flaws in them, namely, that the boundary conditions cannot be satisfied in some cases in which previous researchers have stated that they can be satisfied. The conditions thus described generally do not affect existing models.
REFERENCES Gazis D. C., Herman R. and Rothery R. W. (l%l) Nonlinear follow-the-leader models of traffic flow. Ops Res. 9, 545-567. May A. D. Jr. and Keller H. E. M. (1%7) Non-integer car following models. Highway Res. Rec. 199,19-32.
SOME ERRORS IN MACROSCOPIC TRAFFIC MODELS BASED ON CAR FOLLOWING MODELS BERNARD F. BYRNE Department of Civil Engineering,
West Virginia University,
(Received
Morgantown,
WV 26506, U.S.A.
26 December 1978)
Abstract-The macroscopic traffic flow models developed from the car following models of Gazis et al. (1961) are shown to have a flaw in that they do not meet certain of the boundary conditions that researchers have said that they do. This does not affect many existing models but, nevertheless, should be cleared up.
In general, there have been two approaches to the development of models of traffic flow. These are the macroscopic models, which seek to describe traffic flow in terms of average variables such as density, flow and average speeds; and the microscopic, or car following models, which describe traffic flow in terms of the speed and spacing of individual vehicles. These two approaches were shown to be interrelated in the work of Gazis et al. (l%l), later further developed by May and Keller (1967). Gazis et al. (l%l) proposed, as a general car following model, the following: [see Gazis et al. (l%l), eqn (9), or May and Keller (1967), eqn (7)]
.il,+l(t + T) = aK+dt
+ ~)[kl(~)-~n+d~)l [x”(t)-x”+l(t)l’
(1)
.
If we assume that eqn (1) holds in steady-state condition, as well as in transient conditions, then several simplifications occur. The time lag disappears, and the spacing becomes the steady-state spacing (s). If the speed of individual car is taken to be the speed of the stream (u), then eqn (1) simplifies to du _=Urn The solution to this equation
where the meaning of
ads
S’ .
is Gazis et al. (l%l),
eqn (11)
fp(p = m or 1) is fp(x)=x’_P, (p# 1)
and fp(x) = In x. To find the constant
of integration,
(p = 1).
c’, either of two boundary I
u=u,atk=O
or II
u = 0 at k = kh 241
conditions
may be applied.
242
B. F. BYRNE
In the above, ur is the “free speed” and ki is the jam density and is related to k by
The only situation in which the first boundary condition can be applied is when I> 1. For I = 1, filn(s)=ln(l/k) which is undefined for k = 0. The only situation in which the second boundary condition can be applied is when m < 1. For m = 1,
fm=ln(U) which is undefined for u = 0. Also, for m > 1,
which is undefined for u = 0. Therefore, the constant of integration can be evaluated as
c’=fn&) only when I > 1, contrary to the statement by Gazis et al. (1961, p. 548), that it can be evaluated for m > 1, If 1, and m# 1, I > 1. Also, the constant of integration can be evaluated as
c’ = -cf,(sJ only when m < 1, contrary to Gazis et al. (1961, p. 548), who state that In can be evaluated for all other combinations of m and 1 except m = 1, I < 1. This implies that, in addition to the stated case of m = 1, I < 1, three other cases must be added to the list of those for which boundary conditions cannot be satisfied, namely I< 1, m > 1; I = 1, m = 1; I = 1, m > 1. This also implies that the matrix developed by May and Keller (1%7, p. 22) cannot be filled for these four cases. However, the matrix developed by May and Keller (1967, p. 23) for existing macroscopic models is satisfactory, as is the non-integer mode1 developed by May and Keller (1967, p. 29). The technique developed by May and Keller (1%7, pp. 26-28) to compare various models and determine suitable values for m and 1 shows the unsatisfactory region quite well. This technique consists of graphing parameters of various models on m, 1 diagrams. In examining these m, 1 diagrams, there are very few points in the region where neither boundary condition is satisfied. CONCLUSIONS
It has been shown that the macroscopic models developed from the car following models have some flaws in them, namely, that the boundary conditions cannot be satisfied in some cases in which previous researchers have stated that they can be satisfied. The conditions thus described generally do not affect existing models.
REFERENCES Gazis D. C., Herman R. and Rothery R. W. (l%l) Nonlinear follow-the-leader models of traffic flow. Ops Res. 9, 545-567. May A. D. Jr. and Keller H. E. M. (1%7) Non-integer car following models. Highway Res. Rec. 199,19-32.