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A theoretical probe of a German experiment on stationary moving traffic jams
Transportation Research Part B 37 (2003) 251–261 www.elsevier.com/locate/trb
A theoretical probe of a German experiment on stationary moving traffic jams Wei-Hua Lin
a,*
, Hong K. Lo
b,1
a
b
Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721-0020, USA Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Received 15 November 2000; received in revised form 10 July 2001; accepted 4 March 2002
Abstract Kerner and Rehborn (Phys. Rev. E 53 (2) (1996)) reported on the observation of two stationary moving jams that lasted for about an hour on a 13 km long German highway section. They attributed the phenomenon to intrinsic characteristics of traffic flow, something that would arise spontaneously within the traffic stream due to drivers’ driving behavior. We show in this paper that these moving jams are not particularly peculiar but can be explained with the hydrodynamic theory of traffic flow, or the Lighthill– Whitham–Richards model, and the merge and diverge models in the cell transmission model. In fact, we demonstrate that this stationary jam phenomenon can be replicated with a simple two-wave velocity (or triangular) flow–density relationship in conjunction with the hydrodynamic theory. This finding provides some evidence to support that a triangular flow–density relationship is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Kerner and Rehborn (1996) reported on the observation of two stationary moving traffic jams on a 13 km long German highway section as shown in Fig. 1(a). In the figure, traffic is moving in the upward direction. The two regions with light shades shown on the time–space plane correspond to the reduced flow within the moving traffic jams. Fig. 1(b) shows the speed profile across the two moving traffic jams. The phenomenon lasts for about 50 min. According to their *
Corresponding author. Tel.: +1-520-621-6553; fax: +1-520-621-6555. E-mail addresses: [email protected] (W.-H. Lin), [email protected] (H.K. Lo). 1 Tel.: +852-2358-8742; fax: +852-2358-1534. 0191-2615/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 0 2 ) 0 0 0 1 2 - 7
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Fig. 1. The propagation of the moving traffic jams through highway A5 between Frankfurt and Bad Homburg in Germany ðsource: Kerner and Rehborn (1996)Þ: (a) a time–space plot of the flow data; (b) the observed speed profile.
observations, the basic parameters for the moving jams, such as the flow, speed, and density, stay constant over time and space despite the existence of three intersections along the section (I1, I2, and I3) with vehicles entering or leaving the highway. Kerner and Rehborn attributed the phenomenon to intrinsic characteristics of traffic flow, something that would arise spontaneously within the traffic stream due to drivers’ driving behavior.
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253
We show in this paper that moving traffic jams are not particularly peculiar but can be explained with the hydrodynamic theory of traffic flow, or the Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956) and the merge and diverge model in the cell transmission model (Daganzo, 1994, 1995a, 1996). The LWR model is a continuum traffic flow model that is especially powerful in explaining the spatial and temporal effect of queues on a long crowded highway. The basic cell transmission model is a discrete version of the LWR model. The network cell transmission model extends the theory so as to capture the traffic dynamics in merge and diverge areas. It is interesting to see that by putting these two models together the observed parallel moving jams can be reproduced. Furthermore, if one uses a simple two-wave velocity flow–density relationship (or triangular relationship) for which there is no diffusion, the fit is very good. This observation is significant since it is an additional validation of the simple two-wave velocity model, consistent with the results for validating the two-wave velocity model obtained by Cassidy and Windover (1995) and Windover (1998). This reveals that a triangular flow–density relationship is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon. In addition, our results provide an alternative explanation to the cause of the observed stationary moving traffic jams. The stationary moving jams could be triggered by some capacity-reducing events downstream, such as incidents, competing flows entering the highway section of concern, or other exogenous events, instead of some intrinsic characteristics of traffic flow. In Section 2, an overview of the LWR model and the cell transmission model will be presented. In particular, we will discuss the role of the functional form of the flow–density relationship in capturing the temporal and spatial effect of the queuing process. In Section 3, we will construct graphically the exact solution for the stationary moving jam problem when arrivals and departures are assumed to be deterministic. Finally, the deterministic assumption will be relaxed. A numerical solution of the LWR model for non-deterministic arrivals and departures will be given in Section 4 based on the ordinary cell transmission model (Daganzo, 1994, 1995a) and the lagged cell transmission model (Daganzo, 1999).
2. Model description The central component of the hydrodynamic theory of traffic flow or the LWR model is the classic continuity equation for flow conservation that defines the relationship between flow (q) and density (k) over time and space in the following form: ok oq þ ¼ 0: ot ox
ð1aÞ
The fundamental hypothesis of the theory is that traffic flow at location x at time t is a function of the traffic density only, i.e. q ¼ Qðk; x; tÞ:
ð1bÞ
The relationship between flow and density expressed in (1b) is usually termed the fundamental relationship of traffic flow. For homogeneous roads and a time-independent flow–density relationship, the equation can be further simplified as q ¼ QðkÞ. The assumed flow–density
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relationship can generally be calibrated with three parameters: free flow speed, capacity, and jam density. These parameters are functions of the roadway configuration. The point on the flow– density curve that corresponds to the capacity flow and optimal density separates the curve into two regimes. The left regime represents uncongested traffic and the right regime represents congested traffic. The Eqs. (1a) and (1b) imply that a slight change in flow would propagate along a kinematic wave, or a characteristic line. Big changes in flow or density could arise in two situations. In the situation when an obstruction is suddenly removed from the roadway, it is assumed in the model that the lead vehicle would accelerate to the maximum allowed speed instantly with a wave velocity equal to the free flow speed. At the same time a wave propagates back passing through the waiting vehicles. The densities between the two waves are fully defined by the functional form of the flow density relationship. This treatment eliminates the possible multivalued solutions for the process of queuing dissipation. In the situation where a sudden drop in speed occurs in the traffic stream, such as the case when vehicles join a queue, there is a discontinuity in flow and density. The velocity of the discontinuity that separates two traffic regions can be represented by the following equation: qu qd ; ð1cÞ u¼ ku kd where (qu , ku ) and (qd , kd ) represent the traffic states upstream and downstream of the interface, respectively. In the case of traffic breakdown, u is negative, indicating that the shock wave travels in the direction against the traffic stream. Eqs. (1a)–(1c) can be solved exactly. However, the solutions are often tedious, especially when the flow–density relationship is non-linear, a functional form which is widely adopted in practice. The cell transmission model is a discrete version of the LWR model. In the cell transmission model, the roadway is partitioned into discrete segments or ‘‘cells’’ labeled 1; 2; 3; . . . ; i, and the time is partitioned into discrete steps labeled 1; 2; 3; . . . ; t. The partition is done in such a way that it takes a single time step (Dt ¼ 1) to traverse a cell at free-flow travel speed. When the functional relationship between flow and density is assumed to be piece-wise linear, Eqs. (1a)–(1c) can be substituted by a set of difference equations as follows: zi ðtÞ ¼ minfni ðtÞ; Qi ðtÞ; Qiþ1 ðtÞ; aðNiþ1 ðtÞ niþ1 ðtÞÞg;
ð2aÞ
yiþ1 ðtÞ ¼ zi ðtÞ;
ð2bÞ
ni ðt þ 1Þ ¼ ni ðtÞ zi ðtÞ þ yi ðtÞ;
ð2cÞ
where zi ðtÞ is the number of vehicles leaving cell i at time [t, t þ 1), yiþ1 ðtÞ the number of vehicles entering cell i þ 1 at time [t, t þ 1), and ni ðtÞ the number of vehicles inside cell i at time [t, t þ 1). Qi is the maximum number of vehicles allowed to leave cell i, an entity equivalent to capacity. Ni is the maximum number of vehicles that can reside within cell i, an entity equivalent to jam density. The model can be proven to be convergent to the continuum model when the discretized time and space elements approach zero (Daganzo, 1995b). Eq. (2a) determines the outflow for cell i. Eq. (2b) ensures flow conservation such that the inflow of a cell is equal to the outflow of its upstream cell. Eq. (2c) is a state function for flow conservation. In the network model, boundary conditions were derived for merges and diverges, consistent with traffic rules and the first-in-first-out (FIFO) sequence in the merges and diverges ðfor details, see Daganzo (1995a, 1996)Þ.
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255
The functional form of the flow–density relationship adopted in the LWR model is directly relevant to the spatial pattern of queues. In the past, many forms of flow–density relationship have been proposed (Greenshields, 1935; Greenberg, 1959; Drake et al., 1967; Pipes, 1967). Though the functional forms of the flow density relationship can vary, they should preserve some common properties as discussed in Castillo and Benitez (1995). Different forms of the flow– density relationship may yield different queuing patterns, especially in the course of queuing dissipation, even though they are calibrated with the same parameters. Stationary moving jams can only exist when the congested regime of the flow–density relationship is linear. If the right regime of the flow–density relationship were non-linear, then a fan of waves of spreading densities would emerge at the front or at the end of the jam in the queue dissipation process on the time– space plane depending on whether the right regime is concave or convex (see Fig. 2, for example, for a case with a concave flow–density relationship), which would result in a significant diffusion in the region between the two moving jams. Consequently, the acceleration wave in the process of queue dissipation cannot possibly be parallel to the deceleration wave. On the contrary, the density profile generated from a triangular flow–density relationship would not diffuse in any direction. The presence of two parallel interfaces and the absence of measurable diffusion in the German data strongly suggest the plausibility of a linear right regime of the flow–density relationship. This is also in good agreement with recent experiments (Cassidy and Windover, 1995; Windover, 1998) which showed that small disturbances in traffic propagate without spreading. Another necessary condition for the parallel interfaces of the two stationary moving jams to arise is that the initial traffic state on the roadway must be either at capacity or congested regime
Fig. 2. Multi-valued densities in the queue dissipation process resulting from a concave flow–density relationship.
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of the flow density relationship. Furthermore, for the moving jam to sustain, intersection 3 (I3) must have a net inflow while intersection 2 (I2) a net outflow. Note that the exact values used to calibrate the parameters are omitted in the discussion below, since the essence of the result––that the two resultant moving jams are parallel––remains for a range of parameter values.
3. An exact solution Since the data for flows entering or exiting the highway from I2 and I3 were not provided in Kerner and Rehborn (1996), we consider that they are proportional to the mainline flows when traffic is congested. Before the onset of the jams, the flow level on the mainline highway is slightly under capacity and traffic in the region is moderately congested. The time–space diagram for an exact solution to the German site with all the boundary conditions, based on the LWR model, is displayed in Fig. 3. The relationship between flow and density is of triangular shape, defined by three constants: the free-flow speed (mf ), the optimum density (ko ), and the jam density (kj ), shown in the upper right area of the figure.
Fig. 3. Time–space solution to the moving jam problem (assuming that the moving traffic jam is induced by an incident and an unknown capacity-reducing event further downstream). The interfaces between traffic states correspond to acceleration or deceleration waves.
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The solid lines represent time–space interfaces between the traffic states in various regions of the time–space plane. The traffic states, labeled ‘‘A, A1, A2, C, C1, C2, F, O,’’ correspond to the points on the flow–density relationship shown in the upper right area of the figure. The area inside the dashed box corresponds to the time–space plane for the highway section shown in Fig. 1. Initially, traffic is at state A, A1, and A2 for the sections upstream of I2, between I2 and I3, and downstream of I3, respectively. In the occurrence of two downstream capacity-reducing events as shown in Fig. 3, the hydrodynamic theory in conjunction with the triangular flow–density relationship predict the formation of the two moving jams. When capacity is reduced downstream, the stationary moving traffic jams, represented by labels F, travel in the upstream direction. The two moving jams are separated by a region with near capacity flows that last only for a short period of time. When the second capacity-reducing event is cleared, traffic at the downstream end returns to the capacity level. The traffic states in the entire section of concern are shown by labels C, C1, and C2.
Fig. 4. Time–space solution to the moving jam problem (assuming that the moving traffic jam is induced by two incidents with same durations and magnitudes). The interfaces between traffic states correspond to acceleration or deceleration waves.
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Since information is unavailable regarding the cause of the moving traffic jams, we assume that the first moving traffic jam was induced by an incident and the second moving traffic jam by some unknown capacity-reducing event further downstream, outside the time–space plane shown in Fig. 3. The same moving traffic jams could also arise, according to the hydrodynamic theory, by two accidents as shown in Fig. 4. Note that under the occurrence of either of these two events (i.e., as shown in Fig. 3 or Fig. 4), the structure of the two moving jams shown in the area within the dashed box remains unchanged. Therefore, capturing the exact location of the source of the jams is irrelevant to the issue of concern. The cause of these two moving jams can simply be generalized as two capacity-reducing events that happened downstream. In the simulation that follows, the two moving jams are generated by two capacity-reducing events.
4. Numerical solution for non-deterministic arrivals and departures If the arrivals and departures are non-deterministic, it is difficult to construct a solution using the method shown in the previous section. In this section we apply Eqs. (2a)–(2c) in Section 2 and the network model in the cell transmission model (Daganzo, 1994, 1995a) to generate the flow and speed profiles for the moving traffic jam problem. The numerical solution of the LWR model to the moving jam problem with non-deterministic arrivals and departures is shown in Fig. 5 with the ordinary cell transmission model. A total of 360 cells are used to represent the roadway section of concern. A small perturbation term is added to the flow entering and exiting the intersections. The shading intensity in the figure corresponds to the magnitude of the predicted traffic flow. The stationary moving traffic jams are shown in the figure by two areas with light shading. The plot confirms that the randomness in arrivals and
Fig. 5. Shading plot to the moving jam problem (numerical solution with non-deterministic arrivals and departures based on the ordinary cell transmission model using 360 cells).
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259
Fig. 6. Comparison of observed and predicted speed profile: (a) the observed speed profile ðsource: Kerner and Rehborn (1996)Þ; (b) the predicted speed profile based on the cell transmission model.
departures at the boundary will not alter the qualitative behavior of the solution. The observed speed profile and the predicted speed profile are plotted in Fig. 6(a) and (b). The two exhibit the same trend and the similar level. Note that minor discrepancies are expected since the LWR model is a macroscopic model in which flows and speeds are aggregated across all the lanes. Besides, there is also numerical diffusion at the edges of the moving jam as shown in Fig. 5. The numerical diffusion can be attenuated with the lagged cell transmission model (Daganzo, 1999). Two plots are displayed in Fig. 7 in which the lagged cell transmission model is applied. In Fig. 7(a), the solution is improved with the lagged cell transmission model even though the total number of cells used to represent this section is reduced by one-fourth. The edges of the moving jam become much sharper when 360 cells are used as shown in Fig. 7(b).
5. Conclusions In this paper, we provided an explanation to the formation of the observed stationary moving traffic jams in a German experiment with the hydrodynamic theory of traffic flow, or the LWR model. The findings of the paper are summarized as follows:
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Fig. 7. Shading plot to the moving jam problem (numerical solution with non-deterministic arrivals and departures based on the lagged cell transmission model): (a) 90 cells; (b) 360 cells.
(1) The stationary moving traffic jam observed in a German highway can be reproduced with the hydrodynamic theory of traffic flow and the merge and diverge models in the cell transmission model. (2) To reproduce the stationary moving jams, the functional form of the flow–density relationship has to be linear in the regime representing congested traffic. (3) Moving traffic jams can be caused by downstream capacity-reducing events, such as incidents, competing flow from a downstream intersection, etc. The exact location where the capacityreducing event happens would not alter the qualitative behavior of the stationary moving jam.
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(4) The shock wave and the acceleration wave of the moving traffic jams remain parallel on a time–space plane for a range of parameters, even when arrivals and departures at all intersections are non-deterministic. These findings provide some evidence to support that a triangular flow–density relationship, especially a linear flow–density relationship for the congested regime, is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon.
References Cassidy, M.J., Windover, J.R., 1995. Methodology for assessing dynamics of freeway traffic flow. Transportation research record, No. 1484. Castillo, J.M., Benitez, F.G., 1995. On the functional form of speed–density relationship, Part 1: General theory, Part II: Empirical investigation. Transport. Res. B 29 (5), 373–406. Daganzo, C.F., 1994. The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transport. Res. B 28 (4), 269–287. Daganzo, C.F., 1995a. The cell transmission model: network traffic simulation. Transport. Res. B 29, 79–93. Daganzo, C.F., 1995b. A finite difference approximation of the kinematic wave model of traffic flow. Transport. Res. B 29 (4), 261–276. Daganzo, C.F., 1996. The nature of freeway gridlock and how to prevent it. In: Lesort, J.B. (Ed.), Transportation and Traffic Theory, Proc. 13th Int. Symp. Trans. Traffic Theory. Pergamon-Elsevier, Tarrytown, NY, pp. 629–646. Daganzo, C.F., 1999. The lagged cell-transmission model. In: Ceder, A. (Ed.), Transportation and Traffic Theory, Proc. 14th Int. Symp. Trans. Traffic Theory, Elsevier, New York, NY, 81–104. Drake, J.S., Schofer, J.L., May, A.D., 1967. A statistical analysis of speed–density hypotheses. Highway Res. Rec. 156, 53–87. Greenberg, H., 1959. An analysis of traffic flow. Operat. Res. 7 (1), 79–85. Greenshields, B.D., 1935. A study in highway capacity. Highway Res. Board Proc. 14, 448–477. Kerner, B.S., Rehborn, H., 1996. Experimental features and characteristics of traffic jams. Phys. Rev. E 53 (2), 1996. Lighthill, M.J., Whitham, G.B., 1955. On kinematic waves, II. A theory of traffic flow on long crowded roads. Proc. R. Soc. London, Ser. A 229, 317–345. Pipes, L.A., 1967. Car-following models and the fundamental diagram of road traffic. Transport. Res. 1, 21–29. Richards, P.I., 1956. Shockwaves on the highway. Operat. Res. 4, 42–51. Windover, J.R., 1998. Empirical studies of the dynamic features of freeway traffic. Ph.D. thesis, Department of Civil and Environmental Engineering, Institute of Transportation Studies, University of California, Berkeley, CA.
A theoretical probe of a German experiment on stationary moving traffic jams Wei-Hua Lin
a,*
, Hong K. Lo
b,1
a
b
Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721-0020, USA Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Received 15 November 2000; received in revised form 10 July 2001; accepted 4 March 2002
Abstract Kerner and Rehborn (Phys. Rev. E 53 (2) (1996)) reported on the observation of two stationary moving jams that lasted for about an hour on a 13 km long German highway section. They attributed the phenomenon to intrinsic characteristics of traffic flow, something that would arise spontaneously within the traffic stream due to drivers’ driving behavior. We show in this paper that these moving jams are not particularly peculiar but can be explained with the hydrodynamic theory of traffic flow, or the Lighthill– Whitham–Richards model, and the merge and diverge models in the cell transmission model. In fact, we demonstrate that this stationary jam phenomenon can be replicated with a simple two-wave velocity (or triangular) flow–density relationship in conjunction with the hydrodynamic theory. This finding provides some evidence to support that a triangular flow–density relationship is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Kerner and Rehborn (1996) reported on the observation of two stationary moving traffic jams on a 13 km long German highway section as shown in Fig. 1(a). In the figure, traffic is moving in the upward direction. The two regions with light shades shown on the time–space plane correspond to the reduced flow within the moving traffic jams. Fig. 1(b) shows the speed profile across the two moving traffic jams. The phenomenon lasts for about 50 min. According to their *
Corresponding author. Tel.: +1-520-621-6553; fax: +1-520-621-6555. E-mail addresses: [email protected] (W.-H. Lin), [email protected] (H.K. Lo). 1 Tel.: +852-2358-8742; fax: +852-2358-1534. 0191-2615/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 0 2 ) 0 0 0 1 2 - 7
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Fig. 1. The propagation of the moving traffic jams through highway A5 between Frankfurt and Bad Homburg in Germany ðsource: Kerner and Rehborn (1996)Þ: (a) a time–space plot of the flow data; (b) the observed speed profile.
observations, the basic parameters for the moving jams, such as the flow, speed, and density, stay constant over time and space despite the existence of three intersections along the section (I1, I2, and I3) with vehicles entering or leaving the highway. Kerner and Rehborn attributed the phenomenon to intrinsic characteristics of traffic flow, something that would arise spontaneously within the traffic stream due to drivers’ driving behavior.
W.-H. Lin, H.K. Lo / Transportation Research Part B 37 (2003) 251–261
253
We show in this paper that moving traffic jams are not particularly peculiar but can be explained with the hydrodynamic theory of traffic flow, or the Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956) and the merge and diverge model in the cell transmission model (Daganzo, 1994, 1995a, 1996). The LWR model is a continuum traffic flow model that is especially powerful in explaining the spatial and temporal effect of queues on a long crowded highway. The basic cell transmission model is a discrete version of the LWR model. The network cell transmission model extends the theory so as to capture the traffic dynamics in merge and diverge areas. It is interesting to see that by putting these two models together the observed parallel moving jams can be reproduced. Furthermore, if one uses a simple two-wave velocity flow–density relationship (or triangular relationship) for which there is no diffusion, the fit is very good. This observation is significant since it is an additional validation of the simple two-wave velocity model, consistent with the results for validating the two-wave velocity model obtained by Cassidy and Windover (1995) and Windover (1998). This reveals that a triangular flow–density relationship is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon. In addition, our results provide an alternative explanation to the cause of the observed stationary moving traffic jams. The stationary moving jams could be triggered by some capacity-reducing events downstream, such as incidents, competing flows entering the highway section of concern, or other exogenous events, instead of some intrinsic characteristics of traffic flow. In Section 2, an overview of the LWR model and the cell transmission model will be presented. In particular, we will discuss the role of the functional form of the flow–density relationship in capturing the temporal and spatial effect of the queuing process. In Section 3, we will construct graphically the exact solution for the stationary moving jam problem when arrivals and departures are assumed to be deterministic. Finally, the deterministic assumption will be relaxed. A numerical solution of the LWR model for non-deterministic arrivals and departures will be given in Section 4 based on the ordinary cell transmission model (Daganzo, 1994, 1995a) and the lagged cell transmission model (Daganzo, 1999).
2. Model description The central component of the hydrodynamic theory of traffic flow or the LWR model is the classic continuity equation for flow conservation that defines the relationship between flow (q) and density (k) over time and space in the following form: ok oq þ ¼ 0: ot ox
ð1aÞ
The fundamental hypothesis of the theory is that traffic flow at location x at time t is a function of the traffic density only, i.e. q ¼ Qðk; x; tÞ:
ð1bÞ
The relationship between flow and density expressed in (1b) is usually termed the fundamental relationship of traffic flow. For homogeneous roads and a time-independent flow–density relationship, the equation can be further simplified as q ¼ QðkÞ. The assumed flow–density
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relationship can generally be calibrated with three parameters: free flow speed, capacity, and jam density. These parameters are functions of the roadway configuration. The point on the flow– density curve that corresponds to the capacity flow and optimal density separates the curve into two regimes. The left regime represents uncongested traffic and the right regime represents congested traffic. The Eqs. (1a) and (1b) imply that a slight change in flow would propagate along a kinematic wave, or a characteristic line. Big changes in flow or density could arise in two situations. In the situation when an obstruction is suddenly removed from the roadway, it is assumed in the model that the lead vehicle would accelerate to the maximum allowed speed instantly with a wave velocity equal to the free flow speed. At the same time a wave propagates back passing through the waiting vehicles. The densities between the two waves are fully defined by the functional form of the flow density relationship. This treatment eliminates the possible multivalued solutions for the process of queuing dissipation. In the situation where a sudden drop in speed occurs in the traffic stream, such as the case when vehicles join a queue, there is a discontinuity in flow and density. The velocity of the discontinuity that separates two traffic regions can be represented by the following equation: qu qd ; ð1cÞ u¼ ku kd where (qu , ku ) and (qd , kd ) represent the traffic states upstream and downstream of the interface, respectively. In the case of traffic breakdown, u is negative, indicating that the shock wave travels in the direction against the traffic stream. Eqs. (1a)–(1c) can be solved exactly. However, the solutions are often tedious, especially when the flow–density relationship is non-linear, a functional form which is widely adopted in practice. The cell transmission model is a discrete version of the LWR model. In the cell transmission model, the roadway is partitioned into discrete segments or ‘‘cells’’ labeled 1; 2; 3; . . . ; i, and the time is partitioned into discrete steps labeled 1; 2; 3; . . . ; t. The partition is done in such a way that it takes a single time step (Dt ¼ 1) to traverse a cell at free-flow travel speed. When the functional relationship between flow and density is assumed to be piece-wise linear, Eqs. (1a)–(1c) can be substituted by a set of difference equations as follows: zi ðtÞ ¼ minfni ðtÞ; Qi ðtÞ; Qiþ1 ðtÞ; aðNiþ1 ðtÞ niþ1 ðtÞÞg;
ð2aÞ
yiþ1 ðtÞ ¼ zi ðtÞ;
ð2bÞ
ni ðt þ 1Þ ¼ ni ðtÞ zi ðtÞ þ yi ðtÞ;
ð2cÞ
where zi ðtÞ is the number of vehicles leaving cell i at time [t, t þ 1), yiþ1 ðtÞ the number of vehicles entering cell i þ 1 at time [t, t þ 1), and ni ðtÞ the number of vehicles inside cell i at time [t, t þ 1). Qi is the maximum number of vehicles allowed to leave cell i, an entity equivalent to capacity. Ni is the maximum number of vehicles that can reside within cell i, an entity equivalent to jam density. The model can be proven to be convergent to the continuum model when the discretized time and space elements approach zero (Daganzo, 1995b). Eq. (2a) determines the outflow for cell i. Eq. (2b) ensures flow conservation such that the inflow of a cell is equal to the outflow of its upstream cell. Eq. (2c) is a state function for flow conservation. In the network model, boundary conditions were derived for merges and diverges, consistent with traffic rules and the first-in-first-out (FIFO) sequence in the merges and diverges ðfor details, see Daganzo (1995a, 1996)Þ.
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255
The functional form of the flow–density relationship adopted in the LWR model is directly relevant to the spatial pattern of queues. In the past, many forms of flow–density relationship have been proposed (Greenshields, 1935; Greenberg, 1959; Drake et al., 1967; Pipes, 1967). Though the functional forms of the flow density relationship can vary, they should preserve some common properties as discussed in Castillo and Benitez (1995). Different forms of the flow– density relationship may yield different queuing patterns, especially in the course of queuing dissipation, even though they are calibrated with the same parameters. Stationary moving jams can only exist when the congested regime of the flow–density relationship is linear. If the right regime of the flow–density relationship were non-linear, then a fan of waves of spreading densities would emerge at the front or at the end of the jam in the queue dissipation process on the time– space plane depending on whether the right regime is concave or convex (see Fig. 2, for example, for a case with a concave flow–density relationship), which would result in a significant diffusion in the region between the two moving jams. Consequently, the acceleration wave in the process of queue dissipation cannot possibly be parallel to the deceleration wave. On the contrary, the density profile generated from a triangular flow–density relationship would not diffuse in any direction. The presence of two parallel interfaces and the absence of measurable diffusion in the German data strongly suggest the plausibility of a linear right regime of the flow–density relationship. This is also in good agreement with recent experiments (Cassidy and Windover, 1995; Windover, 1998) which showed that small disturbances in traffic propagate without spreading. Another necessary condition for the parallel interfaces of the two stationary moving jams to arise is that the initial traffic state on the roadway must be either at capacity or congested regime
Fig. 2. Multi-valued densities in the queue dissipation process resulting from a concave flow–density relationship.
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of the flow density relationship. Furthermore, for the moving jam to sustain, intersection 3 (I3) must have a net inflow while intersection 2 (I2) a net outflow. Note that the exact values used to calibrate the parameters are omitted in the discussion below, since the essence of the result––that the two resultant moving jams are parallel––remains for a range of parameter values.
3. An exact solution Since the data for flows entering or exiting the highway from I2 and I3 were not provided in Kerner and Rehborn (1996), we consider that they are proportional to the mainline flows when traffic is congested. Before the onset of the jams, the flow level on the mainline highway is slightly under capacity and traffic in the region is moderately congested. The time–space diagram for an exact solution to the German site with all the boundary conditions, based on the LWR model, is displayed in Fig. 3. The relationship between flow and density is of triangular shape, defined by three constants: the free-flow speed (mf ), the optimum density (ko ), and the jam density (kj ), shown in the upper right area of the figure.
Fig. 3. Time–space solution to the moving jam problem (assuming that the moving traffic jam is induced by an incident and an unknown capacity-reducing event further downstream). The interfaces between traffic states correspond to acceleration or deceleration waves.
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The solid lines represent time–space interfaces between the traffic states in various regions of the time–space plane. The traffic states, labeled ‘‘A, A1, A2, C, C1, C2, F, O,’’ correspond to the points on the flow–density relationship shown in the upper right area of the figure. The area inside the dashed box corresponds to the time–space plane for the highway section shown in Fig. 1. Initially, traffic is at state A, A1, and A2 for the sections upstream of I2, between I2 and I3, and downstream of I3, respectively. In the occurrence of two downstream capacity-reducing events as shown in Fig. 3, the hydrodynamic theory in conjunction with the triangular flow–density relationship predict the formation of the two moving jams. When capacity is reduced downstream, the stationary moving traffic jams, represented by labels F, travel in the upstream direction. The two moving jams are separated by a region with near capacity flows that last only for a short period of time. When the second capacity-reducing event is cleared, traffic at the downstream end returns to the capacity level. The traffic states in the entire section of concern are shown by labels C, C1, and C2.
Fig. 4. Time–space solution to the moving jam problem (assuming that the moving traffic jam is induced by two incidents with same durations and magnitudes). The interfaces between traffic states correspond to acceleration or deceleration waves.
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Since information is unavailable regarding the cause of the moving traffic jams, we assume that the first moving traffic jam was induced by an incident and the second moving traffic jam by some unknown capacity-reducing event further downstream, outside the time–space plane shown in Fig. 3. The same moving traffic jams could also arise, according to the hydrodynamic theory, by two accidents as shown in Fig. 4. Note that under the occurrence of either of these two events (i.e., as shown in Fig. 3 or Fig. 4), the structure of the two moving jams shown in the area within the dashed box remains unchanged. Therefore, capturing the exact location of the source of the jams is irrelevant to the issue of concern. The cause of these two moving jams can simply be generalized as two capacity-reducing events that happened downstream. In the simulation that follows, the two moving jams are generated by two capacity-reducing events.
4. Numerical solution for non-deterministic arrivals and departures If the arrivals and departures are non-deterministic, it is difficult to construct a solution using the method shown in the previous section. In this section we apply Eqs. (2a)–(2c) in Section 2 and the network model in the cell transmission model (Daganzo, 1994, 1995a) to generate the flow and speed profiles for the moving traffic jam problem. The numerical solution of the LWR model to the moving jam problem with non-deterministic arrivals and departures is shown in Fig. 5 with the ordinary cell transmission model. A total of 360 cells are used to represent the roadway section of concern. A small perturbation term is added to the flow entering and exiting the intersections. The shading intensity in the figure corresponds to the magnitude of the predicted traffic flow. The stationary moving traffic jams are shown in the figure by two areas with light shading. The plot confirms that the randomness in arrivals and
Fig. 5. Shading plot to the moving jam problem (numerical solution with non-deterministic arrivals and departures based on the ordinary cell transmission model using 360 cells).
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Fig. 6. Comparison of observed and predicted speed profile: (a) the observed speed profile ðsource: Kerner and Rehborn (1996)Þ; (b) the predicted speed profile based on the cell transmission model.
departures at the boundary will not alter the qualitative behavior of the solution. The observed speed profile and the predicted speed profile are plotted in Fig. 6(a) and (b). The two exhibit the same trend and the similar level. Note that minor discrepancies are expected since the LWR model is a macroscopic model in which flows and speeds are aggregated across all the lanes. Besides, there is also numerical diffusion at the edges of the moving jam as shown in Fig. 5. The numerical diffusion can be attenuated with the lagged cell transmission model (Daganzo, 1999). Two plots are displayed in Fig. 7 in which the lagged cell transmission model is applied. In Fig. 7(a), the solution is improved with the lagged cell transmission model even though the total number of cells used to represent this section is reduced by one-fourth. The edges of the moving jam become much sharper when 360 cells are used as shown in Fig. 7(b).
5. Conclusions In this paper, we provided an explanation to the formation of the observed stationary moving traffic jams in a German experiment with the hydrodynamic theory of traffic flow, or the LWR model. The findings of the paper are summarized as follows:
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Fig. 7. Shading plot to the moving jam problem (numerical solution with non-deterministic arrivals and departures based on the lagged cell transmission model): (a) 90 cells; (b) 360 cells.
(1) The stationary moving traffic jam observed in a German highway can be reproduced with the hydrodynamic theory of traffic flow and the merge and diverge models in the cell transmission model. (2) To reproduce the stationary moving jams, the functional form of the flow–density relationship has to be linear in the regime representing congested traffic. (3) Moving traffic jams can be caused by downstream capacity-reducing events, such as incidents, competing flow from a downstream intersection, etc. The exact location where the capacityreducing event happens would not alter the qualitative behavior of the stationary moving jam.
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(4) The shock wave and the acceleration wave of the moving traffic jams remain parallel on a time–space plane for a range of parameters, even when arrivals and departures at all intersections are non-deterministic. These findings provide some evidence to support that a triangular flow–density relationship, especially a linear flow–density relationship for the congested regime, is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon.
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