Capacity drop: a comparison between stop-and-go wave and standing queue at lane-drop bottleneck

ABSTRACT In freeways, the capacity drop means that the maximum traffic flow is higher than congestion discharge rates there. Various capacity drop magnitudes have been empirically observed before. But the mechanism behind this wide capacity drop range is not yet found. This contribution fills in the gap by relating the congestion discharge rates to different congestions in empirical observations. Two days’ data show that the outflows of stop-and-go waves are always lower than those of standing queues. Different discharge rates, ranging from 5220 to 6040 veh/h at the same site, always accompany different congestion states. Moreover, the different observations show that a higher discharge rate means a higher density in the free-flow branch in the fundamental diagram. This contribution shows that discharging rates probably could be controlled by transforming the congestion states. For instance, transforming a stop-and-go wave into a standing queue at a bottleneck might increase the bottleneck throughput.


Introduction
Generally congestion is observed as a form of vehicular queueing, which can be categorized into stopand-go waves and standing queues. In the stop-and-go waves, two congestion fronts move upstream along a freeway. However, in the standing queue, the head of the queue is fixed at a bottleneck. An active bottleneck is a bottleneck with free-flow situation downstream and a traffic jam upstream. The activation of a bottleneck signals the onset of a standing queue. Theoretically, downstream of an active bottleneck the outflow of the standing queue should be the maximum flow on the road or capacity. However, the queue discharging rate of congestion is often lower than the maximum flow on a road without congestion. This phenomenon is called the bottleneck capacity drop (Banks 1991;Hall and Agyemang-Duah 1991;Cassidy and Bertini 1999;Bertini and Leal 2005).
Researchers have observed the capacity drop phenomenon for decades at bottlenecks. Those observations point out that the range of capacity drop, difference between the bottleneck capacity and the queue discharging rate can vary in a wide range. The capacity of the road and the queue discharging flow is essential for the total delay on the road. Hall and Agyemang-Duah (1991) report a drop of around 6% on empirical data analysis at an on-ramp bottleneck. Cassidy and Bertini (1999) place the drop ranging from 8% to 10% from bottlenecks formed by a lane-drop or a horizontal curve. Srivastava and Geroliminis (2013) observe that the capacity falls by approximately 15% at an on-ramp bottleneck. Chung, Rudjanakanoknad, and Cassidy (2007) present a few empirical observations of capacity drop from 3% to 18% at three active bottlenecks. The three bottlenecks are formed by on-ramp merge, lanedrop and a horizontal curve. Excluding the influences of light rain, they show at the same location the capacity drop can range from 8% to 18%. Cassidy and Rudjanakanoknad (2005) observe capacity drop ranging from 8.3% to 14.7% from on-ramp bottlenecks. Oh and Yeo (2012) collect empirical observations of capacity drop in nearly all previous research before 2008. The drop ranges from 3% up to 18%.
Even though a large amount of research effort has been put into the capacity drop, some significant macroscopic features on capacity drop are still unclear. For example, it is not clear to what extent the capacity reduces when a different congestion occurs upstream. Moreover, it is not clear what is the amount of traffic on each lane (flow distribution over lanes), especially at the downstream of a bottleneck with compulsory merging behaviors upstream. Hence, this paper tries to show more empirical observations to forward traffic research to reveal more empirical features. These findings can contribute to a better understanding of the traffic processes, possibly leading to control principles mitigating congestion. Moreover, it also gives an indication of the lane change behavior at the bottleneck locations.
The question answered in this paper is: what are the differences between traffic states downstream of stop-and-go waves compared to downstream of standing queues at the same site. In answering this question, we use the following four subquestions. First, to what extent does the capacity reduce downstream of a stop-and-go wave? Most of previous researches observe the capacity drop phenomenon at active bottlenecks. Few of those studies reveal features of capacity drop downstream of a stop-and-go wave. Kerner (2002) observes that the outflow of a wide moving jam can be higher than minimum outflow of synchronized flow, and lower than the maximum outflow of synchronized flow. We categorize congestion into stop-and-go waves and standing queues, showing the present empirical observations of capacity drop in stop-and-go waves. Second, to what extent does the outflow of congestion, that is, the capacity with congestion upstream, vary at the same road section without other disturbances such as weather and road layouts? In short, this subquestion thus discusses the stochasticity of the outflow of the queue. Previous research shows that discharging flows of standing queues at one bottleneck only exhibit small deviations (Cassidy and Bertini 1999). But those researches only target standing queues at an active bottleneck. In contrast to the standing queue, whose traffic states are limited to a narrow range because the road layout dictates the congested traffic state upstream, different stopand-go waves can result in different congestion states. The study of stop-and-go waves can enlarge the observation samples. Third, what is the flow in each lane in queue discharge conditions? This might shed light on the capacity drop as well. Four, what is the traffic flow distribution over lanes downstream of a bottleneck with compulsory merging behaviors upstream, especially locations near bottlenecks? The study of the flow distribution can show the utilization of lanes when the capacity drop is observed, which can benefit increasing queue discharge rates with multi-lane dynamic management.
To answer those questions, this paper studies a traffic scenario where a standing queue forms immediately after a stop-and-go wave passes. It seems that the standing queue is induced by the stop-and-go wave. In this scenario, there can be at least two congestion states and two outflow states observed at the same road section at the same day.
The remainder of the paper is set up as follows. Section 2 describes methodologies applied in this paper. This section applies shock wave analysis to recognize those different congestions. Section 3 shows the study site and the study data. In Section 4, empirical observations are presented, including various traffic states and flow distributions in each lane. Finally, Section 5 presents the conclusions.

Methodologies
This paper targets a homogeneous freeway section with a lane-drop bottleneck upstream. In the expected scenario, a standing queue forms immediately after the passing of a stop-and-go wave. It seems like the bottleneck is activated by the stop-and-go wave. In this way, we can compare the outflows of congestion at that location and possible location-specific influences are excluded from the analysis.
Since the differences in the capacity drop (in standing queues) between any two days at the same bottleneck lie in a small range among days (Cassidy and Bertini 1999), it is difficult to observe standing queues in distinctly different congestion states at the same bottleneck. However, the congestion level in a stop-and-go wave is considerably different from the congestion in a standing queue. Congestion level is represented by vehicle speed in the congestion and density. Previous research (Laval and Daganzo 2006;Chung, Rudjanakanoknad, and Cassidy 2007) shows that the capacity drop is strongly related to the congestion level; hence it is expected that downstream of a stop-and-go wave traffic states differ from those downstream of a standing queue. In this way, several state points at the same road stretch can be observed empirically, including free flow and congestion states. Shock wave analysis is applied to identify those congestion states qualitatively.
By comparing the outflows downstream of congestion, this paper shows the capacity drop corresponding to the two different congestion types, stop-and-go wave and standing queue. The key of the traffic state analysis is to identify those traffic states. To avoid unnecessary deviations, this paper applies slanted cumulative counts to calculate flow. The slanted cumulative curve, also known as oblique cumulative curves, is drawn by subtracting a reference flow from the cumulative number of passing vehicles. The slanted cumulative curve can promote the visual identification of changing flows (Cassidy and Bertini 1999).
Both of these two outflows are detected downstream of the congestion. There are repetitive observations. For the duration of congestion until the congestion is dissolved, there are no other influences from downstream. The outflow of a stop-and-go wave can be detected at some location where the speed returns to the free-flow speed after the breakdown phenomenon, and the discharging flow can be detected at each location downstream of an active bottleneck.

Shock wave analysis
The states which occur are determined using shock wave analysis. Figure 1 shows the resulting traffic states, including the regions in space-time where the outflows can be measured. For the sake of simplicity, we choose triangular fundamental diagrams. Figure 1(a) shows these fundamental diagrams, the smaller one for a three-lane section and the larger one for a four-lane section. The outflow of a stop-and-go wave, shown as state 5, and discharging flow of standing queue, shown as state 6, both lie in the free-flow branch, see Figure 1. The flows in both of these states are lower than the capacity shown as state 1 to represent the capacity drop. A stop-and-go wave, state 2 in Figure 1, propagates upstream to the bottleneck and this triggers a standing queue, state 4. Figure 1(b) shows that once the bottleneck has been activated both states 5 and 6 can be observed in the downstream of the bottleneck. Further away from the bottleneck, longer time of state 5 can be observed. Note that because states 5 and 6 are always located in the free-flow branch, the shock wave between these two states are always a positive line parallel to the free-flow branch. Therefore, in Figure 1(b) the shock waves between states 5 and 6 are always the same in the x-t plot, no matter which state shows a higher flow. All those states are predicted theoretically by shock wave analysis, which should be observed in empirical observations. Hence, for measuring the outflow, observations at locations far away from the bottleneck are preferred. In that case, the outflow of a stop-and-go wave can be measured for a long enough time and compared clearly there.
With the same methodology, different outflow features in different lanes are analyzed. This shows the performance of each lane during the transition from the outflow of a stop-and-go wave to queue discharging flow. This paper applies slanted cumulative counts to calculate the outflow in each lane. Note that in the Netherlands the rule is Keep Right Unless Overtaking. This asymmetric rule might lead to a different lane choice, for instance for slugs and rabbits (Daganzo 2002), as well as leading to different traffic operations.

Data handling
This paper reveals the flow distribution in each lane as a function of average density over lanes in Section 4.4. The density (ρ) is estimated through dividing flow (q) by space mean speed (ν S ).
In the Netherlands, loop detector data are time mean speed (ν T ) and flow (q). Knoop, Hoogendoorn, and van Zuylen (2009)  and space mean speed ν S , especially during speed in congestion. Yuan et al. (2010) present a correction algorithm based on flow-density relations to calculate space mean speed. This method requires that traffic states should lie on the linear congested branch of the fundamental diagram. However, this paper considers acceleration states downstream of a bottleneck, so we need another method. Knoop, Hoogendoorn, and van Zuylen (2009) show an empirical relation between space mean speed and time mean speed, see Figure 2. The space mean speed actually is estimated as harmonic speed. This relation is applied to space mean speed calculation in Ou (2011). This paper also applies the relation to calculate the space mean speed and the density.

Data
The data analyzed are one minute aggregated, collected around a lane-drop bottleneck on the freeway A4 in the Netherlands. This paper considers the north-bound direction just around Exit 8 (The Hague) in A4 shown in Figure 3. The layout of the study site is shown in the right part of Figure 3. The targeted bottleneck is a lane-drop bottleneck which is circled in Figure 3. Downstream of this bottleneck, there is another lane-drop bottleneck next to Exit 7. Drivers in the targeted road section are driving from a four-lane section to a three-lane section (the upward direction in Figure 3), so a lane-drop bottleneck occurs. The data are collected from 10 locations with approximately 500 m spacing between them, giving a total length of around 5 km. There are two detectors in the four-lane section, followed by eight in the three-lane section. This paper does not consider detectors further downstream because vehicles will change into a shoulder lane to leave the freeway through Exit 7, possibly leading to external disturbances, for instance lane changing near the off-ramp.
Data for analysis are collected on two days, Monday 18 May 2009 and Thursday 28 May 2009. Figure 4 shows the speed contour plots in the study section on two days. There are two similar traffic situations during both days. The first event is a stop-and-go wave. On 18 May the stop-and-go wave originated from the lane-drop bottleneck near Exit 7 at about 16:45. On 28 May the stop-and-go wave enters the selected stretch from further downstream at around 16:55. At 17:40 and 17:50 (18 and 28 May, respectively), the next stop-and-go wave reaches the lane-drop bottleneck. Downstream of the second stop-and-go wave there is congestion. When calculating the outflows, this study analyzes the data before the entering of the second stop-and-go wave in order to avoid influences of this congestion. When analyzing the flow distribution, we analyze the data collected from 16:00 to 19:00.  During the targeted period, there is no other influence from downstream, that is, the bottleneck is active.

Results
This section first presents the different states, then the capacity estimates and then in Sections 4.3 and 4.4 the lane-specific features are discussed.

State identification
This section describes empirical observations. Figure 5 shows empirical slanted cumulative counts across three lanes at eight locations downstream of the bottleneck on two study days. The arrow in each figure shows the shock wave which propagates downstream from the bottleneck. This means the traffic is in a free-flow state, and not influenced by the off-ramp downstream. The outflow of the stop-and-go wave and the discharging flow of the standing queue are clearly distinguishable with the shock wave between these two states, see the upward arrows in Figure 5. Generally, the empirical observations are in line with the expectations presented in section 2. This shock wave separates the outflow of the stop-and-go wave from the discharging flow of the standing queue. This shock wave has been expected in section 2 (see Figure 1(b)). At one location, we first observe the outflow of the stop-and-go wave and then observe the discharging flow of the standing queue. First, we find the outflow of the stop-and-go wave only directly downstream of the stop-and go wave. The wave travels upstream, from location 1 to location 8. Once it reaches location 8, the traffic state will change, with a wave propagating downstream, which takes some time before it reaches location 8. During that whole time, at location 1 the outflow of the stop-and-go wave can be detected.
The discharging flows found for the two days are constant for each day, at 6040 veh/h (18 May) and 5700 veh/h (28 May), see Figure 4. Although they are different for both days, the flows are remarkably constant over time. There is also a difference between the flows downstream of the standing queues on 18 and 28 May. This holds for all locations downstream of the bottleneck, including the acceleration phase. The flow is different but constant during both days. During the acceleration process, the density continuously decreases. Since the flows differ for the two days, the speeds must differ for the two days for situations with an equal density. This means that drivers leave a larger gap than necessary in the day with the lower flow (28 May), since apparently -given the speed-density relationship for the other day -they can drive with lower speeds given the spacing.
Moreover, the downstream direction of the shock wave implies that the off-ramp (Exit 7 in Figure 3) does not influence the discharging flow. Oh and Yeo (2012) implies that the off-ramp at the downstream location mitigates the capacity drop. In our study site, the off-ramp which is located far away has no effects. The shock waves propagating downstream indicate no influence from downstream. Figure 6 shows the capacities (with congestion upstream) which are the outflow of congestion at a homogeneous three-lane freeway section. In Figure 6, all red dashed lines show the slanted cumulative curves at the downstream locations and the blue bold lines represent speed evolution there. All figures in Figure 6 show firstly a decrease of flow (during the time the stop-and-go wave is present), indicated by a cumulative flow line with a negative slope. Afterwards, at location 1 the flow is constant for about 20 minutes, at approximately 5400 veh/h on 18 May and 5220 veh/h on 28 May. Figure 6(c) and 6(d) shows the slanted cumulative curves for location 8, just downstream of the bottleneck. After the stop-and-go wave reaches location 8, the jam soon transforms into a standing queue and the outflow increases up to 6040 and 5700 veh/h, respectively. These two discharging  flows propagate downstream from the bottleneck and reach location 1. In Figure 6, we label the moment when the higher discharge rate reached as 'A'. The higher outflow (6040 and 5700 veh/h) is not temporary and remains for at least 15 minutes at each location. The solid black line in each of the figures indicates a flow to which the slanted cumulative curve can be compared. In each figure, the increasing slope of black lines shows that the outflow of the stop-and-go wave is lower than the discharging flow of the standing queue. Typically, we find that the outflow of the stop-and-go wave lies in the range of 5220-5400 veh/h and the outflow of the standing queue is in the range of 5700-6040 veh/h. All data points are presented in Table 1. The number of states corresponds to Figure 1. States 2, 4, 5 and 6 in Figure 1(a) are identified quantitatively. States 2 and 4 stand for congestion states. States 5 and 6 represent states of capacities. We thus find a correlation between the type of congestion and its outflow. In fact, the outflow of a stop-and-go wave is lower than the outflow of a standing queue at the same location.

Outflows in each lane
When congestion occurs, each lane presents different features regarding outflows. In Figure 7, slanted cumulative counts and speed in each lane are presented, shown as a red dashed line and a blue bold line, respectively. Slow vehicles and trucks usually drive in the shoulder lane due to the Keep Right Unless Overtaking rule. Therefore, the flow and speed detected in each lane at the same location differ from each other. In both Figure 6(a) and 6(b), aggregated data over three lanes show an increase of outflow at the moment the wave separates the outflow from the stop-and-go wave and the outflow from the standing queue reaches the detector. In Figure 7 28 May this increase is found in all lanes. The lack of change in flow in the shoulder lane is remarkable, but at the moment is it unclear what could be the reason.

Flow distribution over lanes
When the bottleneck has been active, there are several different traffic states in the downstream of the bottleneck. Along the distance, the density decreases. Therefore, in the targeted scenario, a large range of density can be detected, which can reveal the flow distribution as a function of density across lanes. The flow distributions are shown in Figure Table 1 and Figure 7). Those circles and triangles stand for the state of the outflow in each lane at location 1, that is, state 5 and state 6 (see Figure 1), respectively. Note that at location 1 on 18 May 2009 there is no distinction between state 5 and state 6. Therefore, when calculating the flow distribution in these two states (states 5 and 6), we use the same flow, that is 1437 veh/h as shown in Figure 7(e). Note that, the lower flow in state 5 (compared to state 6) in the center lane (see Figure 7(d)) does not mean the flow distribution in state 5 should be lower than that in state 6. That explains why in the center lane the flow distribution in state 5 is higher than that in state 6 (see Figure 8 and 8(d)) represent the flow distributions at each location. The lines with five-point stars stand for the distribution at location 8. Figure 8(a) and 8(b) shows flow distributions on two different days. Both figures show a common feature. When the density lies within the range 22-60 veh/km, the flow in the center lane is higher than in both other lanes, although it keeps decreasing as density grows. When the density is around 60 veh/km, the fraction of the flow at the shoulder lane reaches the minimum at around 23%. For the shoulder lane the decrease of the fraction of the flow was sharp, but afterwards the increase is only marginal. Meanwhile, from 60 veh/km the fraction of the flow in the median lane stops increasing with density and begins to stabilize at around 38%. Note that the density of 60 veh/km corresponds to a typical critical density, that is 20 veh/km/lane (Treiber and Kesting 2013).
When the density exceeds 132 veh/km (18 May) and 95 veh/km (28 May), the fraction of the flow in the median is almost equal to the fraction of the flow in the center lane, at around 35% for each while the flow percentage at shoulder lane is around 30%. So even in states with a very high density, flows in the shoulder lane are still lower than in the other lanes. When density reaches up to 220 veh/km, the flow begins to be distributed evenly over three lanes on 18 May while the flow distribution is more unstable on 28 May. It is not surprising because in an extremely high-density situation standing vehicles can lead to some detection problems. We explain this by the following. Vehicles force themselves into the traffic stream and it takes some time -and hence distance -before equilibrium distribution sets in again. Therefore, it is believed that a high percentage of vehicles choose to leave the median lane by changing lanes between location 8 and location 7. This situation is only visible when the density reaches up to 130 veh/km. In the future research, more empirical data (especially trajectory data set) are needed for justifying the behavioral explanation on the different flow distributions at different locations.
Among three lanes, due to the Keep Right Unless Overtaking rule in the Netherlands, we can assume that the shoulder lane (slow lane) is the first choice for drivers when the density is extremely low. As the density increases to around 20 veh/km, the occupation of the center lane begins to be higher than that in the shoulder lane. The use of the median lane (fast lane) is the least at that time. As the density increases, in contrast to the shoulder lane whose flow fraction reduces considerably, the use of the median lane sharply grows. Finally, the median lane and center lane are highly made use of while the shoulder lane is being underutilized. Figure 9 shows the speed in each lane at the same average density over three lanes. Circles, triangular and dots indicate the speed in the median lane, center lane and shoulder lane, respectively. When the density is lower than around 70 veh/km, the speed decreases from the median lane towards the shoulder lane, that is due to the Keep Right Unless Overtaking rule. The median lane is the fastest lane. In Figure 9, when the average density is higher than 70 veh/km, circles, triangles and dots greatly overlap. That means the speed is becoming more equal among the lanes. Because in congestion the speeds are almost equal in all lanes (shown as the highly overlapped area among circles, triangles and dots), the low flow in the shoulder lane must be due to a low density or large spacing. That means that microscopically in congestion the spacing between successive vehicles in the shoulder lane is the largest among the three lanes. Figure 10 shows the flow distributions in the four-lane freeway section upstream of the lane-drop bottleneck. Note that the outflow of the upstream four-lane freeway section is the inflow of the downstream three-lane freeway section. There are two locations for the data collection, location 9 and location 10 in Figure 3. Traffic flow moves from location 10 to location 9. The figure only shows the data for 18 May; the data for 28 May are similar. In fact, we can distinguish two pairs of lanes. First,  lanes 1 and 2 are the median and shoulder lanes of one of the upstream branches of the road. The flow distributions at lanes 3 and 4 are similar to those of lanes 1 and 2, respectively, also originating from a two-lane road upstream. The flow distribution at the two locations differs considerably. On one hand, in contrast to location 10 which is in the upstream of location 9, location 9 shows a lower flow in the median lane, especially for low densities. On the other hand, at location 9 the flow in the shoulder lane is higher for low densities. The non-compensated amount of lane changes can be estimated by the difference in flow per lane between the two detectors for a certain density (e.g. one can see how much lower the flow is). Compensation is possible by other vehicles making opposite movements (e.g. vehicles moving into the lane). In lane 3, the right center lane, the flow is higher at location 9. Down

Conclusions
This paper compares the downstream states of a stop-and-go wave with those of a standing queue. The standing queue in this paper is induced at a lane-drop bottleneck by a stop-and-go wave. Therefore, at one bottleneck there are two different congestion states observed. In the downstream of the congestion there are free-flow states, that means the two outflows detected downstream of congestion are the capacities of the road section. This paper applies shock wave analysis to find those two outflows at the same road section, which is well traceable in the real data. The most important finding is that the outflow of stop-and-go waves is much lower than that of a standing queue. Therefore, the capacity with congestion upstream can vary in a rather wide range, for example, from 5220 to 6040 veh/h at a three-lane road section. The various capacities could be related to congestion states, which means promising traffic control strategies could increase the queue discharge rate and minimize traffic delays.
There are two other findings. First, different features of outflow from congestion in different lanes can be found. Strong fluctuations occasionally can be observed in the shoulder lane, which might even trigger stop-and-go waves later on, for instance near a next bottleneck. Second, the flow distribution over three lanes is presented. This shows that particularly near the head of a standing queue more vehicles can merge into the lane adjacent to the ending lane, thereby locally increasing the capacity of that lane. The capacity of the shoulder lane is markedly wasted when in congestion. The reason for the low flow distribution in the shoulder lane is the large spacing between successive vehicles.
Future research should show the mechanisms behind these features, from a behavioral perspective (whether people behave differently), from a vehicle perspective (what the influences of different acceleration profiles are) or from a flow perspective (what for instance the influence of voids is). In the future, a promising control strategy, based on our empirical research, should be proposed to minimize queue discharge rates and traffic delays.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work is financially supported by China Scholarship Council (CSC); the NWO grant 'There is plenty of room in the other lane'; and the visiting scholar grant from the Delft University of Technology Transport Institute.