INSTITUTE OF TRANSPORTATION STUDIES
UNIVERSITY OF CALIFORNIA, BERKELEY
A Capacity- Increasing Mechanism in Freeway Traffic and
On- ramp Metering Strategies to Affect It
Kwangho Kim and Michael J. Cassidy
Research Report
UCB- ITS- RR- 2011- 2
April 2011
A Capacity- Increasing Mechanism in Freeway Traffic and
On- ramp Metering Strategies to Affect It
By
Kwangho Kim and Michael J. Cassidy*
Department of Civil and Environmental Engineering and the Institute of Transportation Studies,
University of California, Berkeley, CA 94720, U. S. A.
Abstract
A reason is unveiled for the time- varying pattern in discharge flow that is commonly observed at freeway
bottlenecks. We hypothesize that four known effects in freeway traffic can interact upstream of a
bottleneck in ways that trigger periodic bursts in its discharge flow. Repeated observations of a 3- km
freeway stretch support the hypothesis. Controlled experiments show that the capacity- increasing
mechanism can be favorably modulated by metering the site’s on- ramps in an unconventional manner.
The unconventional strategy repeatedly produced higher average discharge flows and shorter on- ramp
queues than did a more traditional metering policy.
1. Introduction
The onset of queuing at a freeway bottleneck is commonly accompanied by a reduction in its discharge
flow ( e. g. Banks, 1991; Hall and Agyemang- Duah, 1991; Persaud, et al, 1998; Cassidy and Bertini, 1999;
Bertini et al, 2005). Various explanations have been offered for this so- called “ capacity drop” ( Banks,
1991; Cassidy and Rudjanakanoknad, 2005; Laval and Daganzo, 2006; Chung et al, 2007). The cause
seems to be linked to flow disruptions, which are created by vehicle lane- changing maneuvers made just
upstream of the bottlenecks.
Curiously, the discharge flow following a capacity drop will usually not persist at a single low rate for
the entire rush. Instead, this flow often recovers to a higher rate soon after the initial drop. The recovery
can then persist for some minutes before returning to a lower rate; and the bottleneck’s active period is
often characterized by multiple sequences of high and low discharge flow ( e. g. see Figures 5 and 10 of
Cassidy and Bertini, 1999).
* Author for correspondence; e- mail: cassidy@ ce. berkeley. edu
1
Understanding the mechanism behind the discharge- flow recoveries can shed light on better ways of
managing freeway traffic. If control strategies can favorably affect the mechanism, perhaps by prolonging
the recoveries, then commute delays would diminish ( see, for example, Wattleworth and Berry, 1965;
Banks, 1991; Newell, 1993; Cassidy, 2003).
We hypothesize that the mechanism entails the interactions of four known effects in freeway traffic
( sect 2). Detailed and repeated observations of a 3- km freeway stretch support the hypothesis ( sect 3). An
unconventional on- ramp metering strategy is proposed in the attempt to favorably influence the
mechanism; and repeated field tests of this strategy yield favorable results ( sect 4). Practical implications
are discussed ( sect 5).
2. Hypothesis
We shall consider four freeway traffic effects. Each has already been reported in the literature. Our
hypothesis concerns how these effects may interact upstream of a bottleneck. The traffic conditions and
freeway geometries needed to induce the interactions are typical of those on urban freeways. We
conjecture that, under these conditions, the four effects can arise in sequence, with one effect triggering
the next; and that this will ultimately cause the periodic recoveries often observed in bottleneck discharge
flow. We describe the hypothesis using the idealized freeway stretch in Figure 1.
Consider a queue that forms at a bottleneck and gradually expands. If the queue eventually reaches an
upstream interchange ( e. g. Merging Area 1 in Figure 1a), the queue would then disrupt the vehicle
merging or diverging maneuvers there. This can create on- ramp queuing at the interchange, and has been
known to cause a dense cluster of slow- moving traffic ( termed a jam) to form within the freeway queue,
often near its tail; see Figure 1b. The jam’s formation has been termed the “ pinch effect” ( Kerner, 2002;
Schönhof and Helbing, 2007).
The jam’s upstream and downstream fronts can propagate intact, in the upstream direction. If the jam
later arrives to another interchange further upstream ( e. g. Merging Area 2 in Figure 1a), and if merging or
diverging maneuvers occur there in sufficient numbers, the resulting disruptions may cause a new and
very restrictive bottleneck to form at that upstream interchange. This phenomenon has been termed the
“ catch effect” ( Kerner, 2002; Laval, 2005), presumably because a bottleneck is “ caught” at a new
location.
The jam may continue to propagate upstream. The new bottleneck left behind starves the downstream
queue of flow. The queue therefore decouples. The tail of the downstream queue will propagate forward,
as in Figure 1c, leaving a pocket of free flow conditions in its wake. Kerner ( 2002) has reported that the
2
pinch effect can trigger the catch effect, much as we have just described. The more novel aspects of our
hypothesis are as follows.
3
Figure 1 A hypothetical freeway and the four effects
We conjecture that within this free flow pocket, traffic will flow at high rates, like those that invariably
precede capacity drops. If the free flow pocket engulfs an on- ramp junction that resides upstream of the
original bottleneck ( Merging Area 1), then the high flows in that pocket may be further fueled by
increased inflows from the on- ramp. ( Note that vehicles discharging from the on- ramp’s queue would no
longer be constrained by a freeway queue.) The result has been termed the pumping effect: freeway
drivers finding themselves in freely flowing but dense traffic become especially motivated to accept small
headways as they approach the merge ( Daganzo, 2002; Cassidy and Rudjanakanoknad, 2005). This new-found
motivation on the part of drivers may reflect their attempts to ward- off lane- changing maneuvers
into their paths.
As higher flows are now “ pumped” from the merge, a fourth and final effect may arise: motivated
drivers would likely remember and retain their preference for small headways as they approach and travel
through the original bottleneck downstream. Previous study suggests that a “ driver memory effect” of this
kind could continue for a kilometer or so, even if these drivers were to encounter a local queue at the
downstream bottleneck ( Cassidy and Windover, 1998); see Figure 1d.
4
We know from observation ( e. g. Cassidy and Bertini, 1999) that the resulting increase in bottleneck
discharge flow will usually continue for only a few minutes. The high rate probably subsides for reasons
akin to those that cause the initial capacity drop: disruptive vehicle lane- changing maneuvers made just
upstream. Once the recovery subsides, the above process can begin anew.
Our hypothesis seems reasonable, given that it rests on four effects that are known to occur. We next
test the hypothesis against real data.
3. Observations
The freeway site and traffic data used for the study are described first ( sect 3.1). The site’s downstream-most
bottleneck and its time- varying pattern in discharge flow are examined next ( sect 3.2). We then
confirm how the pinch effect triggers the catch effect ( sect 3.3), and unveil how these interactions trigger,
in turn, the pumping effect ( sect 3.4). The evidence of driver memory and the resulting recovery in
bottleneck discharge flow are presented last ( sect 3.5).
3.1 Site and Data
The study site is the 3- km stretch of northbound Interstate 5 shown in Figure 2. Its features include: a
downstream bottleneck formed by a changing curvature in the site’s horizontal alignment; an upstream
merge area formed by the on- ramp from Seamas Avenue ( with rush- hour ramp demand of 600~ 700 vph);
and a 0.5- km- long weaving section still further upstream ( with on- ramp demand of 1400~ 1600 vph, and
off- ramp demand of only 100~ 300 vph during the rush).
Figure 2 Study Site, Northbound I- 5, Sacramento, California
5
The traffic data include vehicle arrival times that were extracted from videos at the locations labeled
X0~ X3 in the figure. ( Note how our numeric labels increase in the upstream direction.) Vehicle entry and
exit times at the on- and off- ramps were also extracted from videos. The site’s loop detector stations, also
shown in the figure, measure vehicle counts and occupancies ( a dimensionless measure of density) over
30- sec intervals.
6
All data were collected during morning rush periods. These include observations taken over three days
in 2006 and 2008 when the ramps were not metered. * Additional observations were taken over another
six days as part of metering experiments ( to be described in sect 4). Attempts to collect data on several
other days were foiled due to queues from downstream that spilled- over to the site early in the rush
periods.
3.2 Downstream Bottleneck and Freeway Queues
A contour map of detector occupancy is shown for the site during part of a typical morning rush ( on
October 18, 2006, when ramps were not metered) in Figure 3a. Note that in this figure, traffic moves in
the downward direction, and that a sketch of the site is included on the left- hand- side for the reader’s
convenience. Occupancies of 17% or less ( unshaded regions in the figure) denote free flow traffic
conditions, where flow = demand; and occupancies greater than 17% ( shaded regions) roughly denote
queues, the darker the shade the denser the queue. The figure reveals that conditions remained freely
flowing at locations downstream of the horizontal curve, just beyond kilometer- post 2.6. Thus we see that
for the period shown in Figure 3a, the discharge flows from the downstream curve were not constrained
by other queues from further downstream. This state of affairs continued until a queue of this kind spilled
over at 7: 50 hrs. The day’s data were not analyzed beyond that time.
Queues arose upstream of the curve. Note that prior to 7: 18 hrs, local queues periodically formed near
the curve’s entrance. Beyond 7: 18, a queue existed somewhere on the site through the remainder of the
rush.
The queuing pattern unveiled in Figure 3a means that from 7: 18 until the curve bottleneck deactivated
at 7: 50, vehicles discharged from the curve at maximum rates. These rates varied with time, however.
This variation is evident in Figure 3b. It presents a curve of cumulative vehicle count plotted with an
oblique axis for time ( O- curve). Its counts were measured at location X0, downstream of the curve
bottleneck for a duration that spanned most of the bottleneck’s active period. Note that the slopes of an O-
* Two of the days were in 2006 when meters had yet to be installed at the site. On the third day ( in 2008), meters
were present at the ramps, but rates were relaxed ( i. e. raised) so that ramp vehicles were served as they arrived, and
thus ramp queues were nearly non- existent.
7
Figure 3a Occupancy contour map ( Oct 18, 2006)
Figure 3b Discharge flow from the curve bottleneck ( Oct 18, 2006)
8
curve are flows in excess of a background rate ( Cassidy and Windover, 1995; Cassidy and Bertini, 1999;
Muñoz and Daganzo, 2003), which was 8000 vph in the present case. Visual inspection of the O- curve in
Figure 3b shows that following a capacity drop, the bottleneck’s discharge flow temporarily recovered:
first at 7: 24 hrs; then at 7: 34 hrs. Recall that the same pattern has been reported at other freeway
bottlenecks ( e. g. Cassidy and Bertini, 1999).
The mechanism behind these recoveries is much like the one hypothesized. The evidence follows.
3.3 Pinch and Catch Effects
The first two effects, and their sequence, are visible in Figure 3a, and the figure has been annotated to
highlight this. Note first the darkly shaded region that traces the upstream and downstream fronts of a
backward- moving jam. Recall that this jam is the product of the pinch effect. Further visual inspection of
the figure shows that the jam initially formed near the Seamas merge, which suggests that the jam formed
when the queue from the curve bottleneck disrupted merging traffic. The jam then propagated into the
weaving section upstream.
Note too the larger region of dark shading that arose at the upstream end of the weaving section on the
heels of the jam. This shading reveals a bottleneck that formed due to the catch effect. We further see how
the queue downstream of the “ caught” bottleneck gradually became less dense over time, to the point
where a free flow pocket eventually emerged.
A reader interested in further and more detailed evidence of the pinch and catch effects, and their
interaction, can refer to Appendix I. We omit the additional evidence in this paper in the interest of
brevity, and because our findings of the pinch and catch effects are much like those already reported ( e. g.
Kerner, 2002).
3.4 Pumping Effect
Figure 3a shows that the free flow pocket eventually enveloped the Seamas merge. As the pocket was
forming, flow rose within it, and the pumping effect soon ensued. Part of the evidence is visible in the
upper, boldly- drawn O- curve in Figure 4a. Its counts were measured at location X1, just downstream of
the Seamas merge.
Visual inspection of that bold O- curve shows that a high flow of 9200 vph persisted at X1 prior to about
7: 18 hrs. We further see that flow at X1 diminished soon thereafter, when it became constrained by the
arrival of the queue from the curve bottleneck downstream. The flow diminished further still with the
arrival of the backward- moving jam just after 7: 20 hrs.
9
Figure 4a O- curves at X1 and X0 ( Oct 18, 2006)
Figure 4b O- curve of Seamas Avenue on- ramp vehicles ( Oct 18, 2006)
10
When the jam’s downstream front passed over X1, the flow there began to increase. And from the bold
O- curve we see that by about 7: 23: 30, flow at X1 rose to a rate of 9300 vph.
This high flow seems to have been spurred- on by a rise in on- ramp inflow that began very soon
thereafter. This is evident in Figure 4b, which displays the O- curve of inflows from the Seamas on- ramp.
That O- curve shows that a low inflow of 350 vph persisted for 5 minutes starting at 7: 13: 30. This low
flow may have merely been due to a temporary reduction in demand. When ramp flow returned to a high
rate at 7: 18: 30 hrs, the queue from the downstream curve bottleneck had just arrived to the merge ( as was
revealed in Figures 3a and 4a). Note as a result how the on- ramp’s O- curve in Figure 4b displays
pronounced wiggles immediately thereafter. These wiggles reveal how small platoons of ramp vehicles
were often impeded by the freeway queue before forcing their ways onto the freeway.
Things changed once the free flow pocket began forming at the merge. The O- curve in Figure 4b shows
that by 7: 24 hrs, on- ramp inflow increased slightly ( to 650 vph) and became smoother ( the wiggles
diminished in amplitude). These features of on- ramp inflow beyond 7: 24 hrs, combined with the rise in
outflow from the Seamas merge that began a short time earlier ( bold O- curve in Figure 4a), unveil the
pumping effect.
3.5 Driver memory and Discharge Flow Recovery
The lower, thinly- drawn O- curve in Figure 4a was constructed from the counts at location X0 located
downstream of the curve bottleneck. ( This O- curve was previously displayed for a longer duration in
Figure 3b.) That lower O- curve in Figure 4a shows that bottleneck discharge flow was only 8100 vph
following the capacity drop at 7: 18 hrs. Recall, however, how discharge flow recovered to 9500 vph at
around 7: 24 hrs. From this we see how the high flows pumped out of the free flow pocket upstream
continued through the curve bottleneck. Thus, drivers remembered their preference for small headways,
even as they passed through the localized queue that persisted for much of the time at the curve
bottleneck’s entrance ( see again Figure 3a). Visual comparison of the two O- curves in Figure 4a shows
that the recovery flow ( thin O- curve) was slightly higher than the flow pumped from upstream ( bold O-curve).
The thin O- curve shows that the recovery lasted for only about 4 minutes. When the recovery subsided,
the mechanism began anew: Figure 3b previously showed that the short- run discharge flows were
characterized by sequences of reductions and recoveries until the bottleneck deactivated.
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4. Controlled Experiments
The findings of the previous section were reproduced across days. This reproducibility will now be
explored in the context of field experiments. The experiments will also verify that the capacity- increasing
mechanism can be affected to favorable ends.
A simple but unconventional on- ramp metering logic was designed to modulate the mechanism ( sect
4.1). The controlled experiments, which were performed over multiple days, show that this logic
consistently produced higher long- run average discharge flows from the curve bottleneck, whether
compared against an absence of ramp control, or against a more conventional metering scheme ( sect 4.2).
The reasons for this outcome are confirmed with data ( sect 4.3).
4.1 Proposed Metering Logic
The logic underlying our unconventional metering strategy is summarized below. Full details on the
algorithm are furnished in Appendix II. The logic was designed for use during the later portion of each
rush, after the queue from the curve bottleneck first spilled- over to the Seamas merge ( see again Figure
2).
Immediately after detector occupancies infer this first spill- over, the metering rate at the upstream- most
on- ramp from 43rd Avenue is to be relaxed ( i. e. set at a high rate). High ramp inflows would therefore
flood the site’s weaving area. The intent is to create dense, slow- moving traffic there. This, in turn, is
aimed at making restrictive the weaving section’s bottleneck that is eventually triggered ( i. e. “ caught”) by
the arrival of a backward jam. Once the queue decouples as a result, the receding portion of queue
downstream would be fueled by low input flow ( from the restrictive weave bottleneck). The tail of the
receding queue should therefore propagate forward at high velocity.
Once detector occupancies infer that free flow conditions are restored to the Seamas merge as a result,
the metering rate at its on- ramp is to be relaxed. The intent is to modulate the capacity- increasing
mechanism; i. e. we seek ramp inflows that are sufficiently high to produce the pumping effect, but low
enough to reduce the likelihood that disruptive vehicle lane- changing will terminate the discharge- flow
recovery. In this way, we seek to prolong the durations of the recoveries.
Relaxed metering is to persist at the Seamas on- ramp until the queue from the curve bottleneck again
spills over to that merge area. ( The metering rate at Seamas would be relaxed yet again whenever a free
flow pocket is returned to the merge). In an effort to maintain a restrictive bottleneck in the weaving
section, the 43rd on- ramp is to be metered at relaxed rates until the curve bottleneck downstream is no
longer active. The relaxed metering can have the additional benefit of reducing the on- ramp queues.
12
4.2 Outcomes
The above strategy was tested for three days at the site. In each day’s test, the logic was deployed only
after the queue from the curve bottleneck first expanded to the Seamas merge. Prior to that event, the
meters were operated in a more conventional fashion: restrictive metering rates were deployed at each on-ramp
whenever freeway queues were detected nearby ( based on detector occupancies). This conventional
logic sought to retard the growth of freeway queues; see Appendix III for a full description of this
algorithm. The logic is compatible with so- called “ demand- capacity” metering strategies that are
prominent in the literature ( e. g. Wattleworth and Berry, 1965; Payne et al, 1973; Papageorgiou el al,
1998).
For comparison, experiments were performed for three additional days, during which times the
conventional metering algorithm was used throughout the rush periods. Outcomes in the absence of ramp
metering were also compared for good measure.
The outcomes are summarized in Table 1. Its third column displays the duration over which each day’s
measurement was taken. Each of these periods spanned the time from the queue’s initial expansion to the
Seamas merge, until the curve bottleneck was deactivated.*
Table 1 Outcomes of Experiments
Average Experiment Metering Rates ( vph)
Group
Date
Measurement
Duration ( min)
Seamas 43rd
Average
Discharge Flow ( vph)
Oct 18, 2006 18 620 1590 8560
Nov 2, 2006 64 610 1570 8550
Feb 13, 2008 21 740 1550 8440
No
Metering
Average 34 660 1570 8520
May 20, 2009 29 510 1100 8660
Sep 16, 2009 42 450 1040 8480
Sep 17, 2009 18 450 1050 8420
Conventional
Metering
Average 30 470 1060 8520
Apr 15, 2009 54 570 1440 8790
Apr 29, 2009 34 640 1470 8730
Proposed
Metering
May 14, 2009 37 610 1520 8870
* On most days, the bottleneck was de- activated at the end of the rush. On three days ( Oct 18, 2006; Feb 13, 2008;
Sep 17, 2009), the bottleneck was deactivated late in each rush by the arrival of a queue from downstream.
13
Average 42 610 1480 8800
14
The table’s fourth and fifth columns present each day’s average inflows from the site’s two on- ramps.
As expected, these were highest in the absence of metering, and lowest under conventional metering.
Very importantly, the sixth column presents ( in bold) each day’s long- run average discharge flow from
the curve bottleneck. Within a single experiment group, these flows did not vary much across days: daily
differences were always less than 2%. Interestingly, we see that under conventional metering, the average
discharge flow taken over the three days of experiments was identical to the 3- day average in the absence
of metering: 8520 vph. Tellingly, the proposed logic produced a long- run discharge flow each day that
was always larger than any day’s discharge flow from the other experiment groups. The likelihood that
this outcome occurred purely by chance is about 1 percent.* The 3- day average under the proposed logic
( 8800 vph) represents a 3% gain over the average of the other two groups.
It seems that the proposed logic performed as intended. This favorable performance is confirmed by
looking closely at the data.
4.3 Closer Look
The data indicate that the proposed logic modulated the mechanism, such that the recoveries in discharge
flow were less pronounced, but longer- lived, than those measured in the absence of ramp metering. As an
example, Figure 5a displays O- curves measured on a day when the proposed logic was deployed ( May 14,
2009). They are typical of the O- curves for that experiment group. The middle, boldly- drawn O- curve
was constructed from vehicle counts at X1, located just downstream of the Seamas merge. That O- curve
shows how flows from the merge recovered on the heels of a jam’s passage, i. e. the flow rose to 8510 vph
starting at around 7: 34 hrs. The lower, thinly- drawn O- curve shows how the discharge flows ( measured at
X0) began to rise very soon thereafter. The recovery was small at first ( 8760 vph). Yet as the free flow
pocket expanded, vehicles eventually began discharging the merge at a high rate of 9250 vph.
This high rate was soon fueled by increased on- ramp flows. At 7: 39 hrs, the metering rate for the
Seamas on- ramp was relaxed ( the rate went from 530 vph to 750 vph). Note the evidence of the pumping
effect: the bold O- curve in Figure 5a shows that the flow discharging from the merge increased to 9350
vph. The lower, thin O- curve shows how the discharge flow further increased as a result: discharge flow
rose to 9380 vph. Note that the recovery subsided at 7: 43: 30. The average discharge rate during the 9- min
recovery period was 9200 vph.
* That the outcome could be due to random chance is akin to randomly choosing nine numbers and discovering that
the first three of these are the largest of the nine.
15
Figure 5a O- curves under proposed logic ( May 14, 2009)
Figure 5b O- curves under conventional logic ( May 20, 2009)
16
Contrast this with the recovery previously observed in the absence of metering. The thin O- curve in
Figure 4a showed that, with no metering, the recovery was slightly more pronounced ( discharge flow rose
to 9500 vph), but much shorter- lived ( 4 minutes). This is part of the reason why the proposed logic
produced higher long- run average discharge flows, as compared against the “ no metering” case.
Of further interest, we find that the discharge flows measured outside of the recovery periods were
higher when ramps were metered. As an example, consider the discharge flow following capacity drop in
the unmetered case ( 8100 vph, see the thin O- curve in Figure 4a); and compare it against the flow that
followed a capacity drop when our proposed logic was used ( 8430 vph; see the lower O- curve in Figure
5a). We suspect that by restricting inflows from the Seamas ramp, metering diminished flow disruptions
caused by lane- changing. Diminished lane- changing has been shown to “ smooth” and increase the
discharge flows through freeway bottlenecks ( see Cassidy et al, 2010).
The conventional logic, with its low metering rates, produced this latter benefit as well. Save for this,
the conventional approach was counterproductive for two reasons. First, metering the 43rd Avenue on-ramp
at low rates impeded the formations of free flow pockets; i. e., by preventing the on- ramp vehicles
from swamping the weaving section, the bottleneck “ caught” there was never restrictive. Second, the
restrictive metering rates at the Seamas on- ramp never fueled the pumping effect. Evidence follows.
For comparison, consider again the O- curves in Figure 5a. Recall that these were measured while the
proposed logic was in use. The thin, upper- most O- curve in the figure was measured just upstream of the
Seamas merge ( and downstream of the weaving section) at a location that we labeled accumulations ( i. e.
the numbers occupying a freeway segment at an instant) were extracted from videos in the three 200 m-long
zones shown in Figure A1. These accumulations were measured every 10 seconds in each of the
three zones labeled 1~ 3 in the figure.
Figure A1. Study Site, Northbound I- 5, Sacramento, California
17
1. Pinch Effect
Telltale signs of the pinch effect are visible in Figures A2 and A3. Figure A2 presents O- curves measured
at X1
+, X2 and X3. These O- curves were constructed so that the vertical displacements between any two of
them are the vehicle accumulations between their measurement locations ( Newell, 1993). Note how the
flow at X1
+ ( near the Seamas merge area) diminished at around 7: 19 hrs, as evident by the slope reduction
in the lowest and thinly- drawn O- curve in the figure. Comparable slope changes in the O- curves at X2 and
X3 occurred at successively later times. These reveal how flows were constrained by an expanding queue
from the site’s curve bottleneck ( Cassidy and Windover, 1995; Cassidy and Bertini, 1999).
We can further see that a jam ( attendant to the pinch effect) formed near the queue’s tail. Soon after
the queue arrived at each Xi , the flow there was temporarily constrained even further; e. g. note from
Figure A2 how flow at X1
+ dropped to 6170 vph from 7: 20: 30 to 7: 22 hrs. The boldly- drawn dashed
arrow in the figure highlights how the jam propagated upstream intact.
Figure A2 O- curves at
18
t= t 0
K 0 ( t ), K 1 ( t)
K0( t) ! Kcr?
K1( t ) ! Kcr?
Rseamas( t)
= Rseamas, restrict - _
R43rd ( t)
= R43rd, restrict - _
Rseamas( t)
= Rseamas, resrict - _
R43rd( t)
= R43rd, relax
Rseamas( t)
= Rseamas, relax
R43rd ( t)
= R43rd , restrict - _
Rseamas( t)
= Rseamas, relax
R43rd ( t)
= R43rd , relax
Yes No
Yes No
t= t+ !
K1( t) ! Kcr?
Yes No
Figure A5 Flow chart of the conventional metering logic
References
Newell, G. F., 1993. A simplified theory of kinematic waves in highway traffic, Part I: General theory.
Transportation Research B 27, 281- 287.
Cassidy, M. J., Windover, J. R., 1995. Methodology for assessing dynamics of freeway traffic flow. Transpn Res.
Rec. 1484, 73- 79.
Cassidy, M. J., Bertini, R. L., 1999. Observations at a freeway bottleneck, In: Ceder, A. ( Ed.), Proceedings of the 14th
International Symposium on Transportation and Traffic Theory. Pergamon, Newyork, pp. 107- 146.