# Vertical and Horizontal Queue Models for Oversaturated Signal Intersections with Quasi-Real-Time Reconstruction of Deterministic and Shockwave Queueing Profiles Using Limited Mobile Sensing Data

Authors: Yongyang Liu; Jingqiu Guo; Yibing Wang
Journal: Journal of Advanced Transportation
 (2018)

---

## Abstract

Deterministic/point/vertical and shockwave/physical/horizontal queueing models are widely used in traffic operation to estimate vehicular queue length and delays at bottlenecks such as signalized intersections. The consistency between the two types of queueing model in terms of their estimation performance has been a subject of debate for decades. This paper reexamines the issue, typically with respect to oversaturated signal intersections, and demonstrates the consistency based on analytical studies and microscopic simulations. While fixed-location sensor data was dominating, it was hardly possible to construct the deterministic or shockwave queueing profile using real data. For this reason, either profile had significance only at a conceptual level and could not be put into practical usage. With the quick spread of mobile sensing data, however, the situation has drastically changed. In this context, this paper also intends to develop an efficient approach to the reconstruction of the deterministic and shockwave queueing profiles in a quasi-real-time manner using very limited mobile sensing data. Microscopic simulations with AIMSUN have demonstrated the efficiency of the approach as well as the analytical results obtained in this paper.


---

## Body

## 1. Introduction

This paper studies the consistency of the vertical and horizontal queue models as well as the quasi-real-time reconstruction of the deterministic and shockwave queueing profiles using very limited mobile sensing data for oversaturated signal intersections.The vertical and horizontal queue models are two distinctive queueing models widely used in transport and traffic engineering. The vertical queue model provides a physically impossible but conceptually valid description of queueing processes. It postulates that any vehicle that has to queue before passing a bottleneck is stacked in a vertical pile at the bottleneck, on top of any other such vehicles that arrived earlier. When the queue is released, departing vehicles exit from the bottom of the queue. A vertical queue does not occupy any road space and has no influence on upstream approaching vehicles. It is therefore also called a point queue. In contrast, the horizontal queue model formulates the formation and dissipation of a physical queue. Figure1 illustrates the two types of queue with respect to a signalized intersection.Figure 1
The horizontal and vertical queue models for a signalized intersection (x and kjam are given in Figure 5).The deterministic queueing profile is commonly used to describe the vertical queueing process. It is based on cumulative diagrams, which were initially introduced to transport by Moskowitz and Newman [1] and Gazis and Potts [2] and later highlighted by Newell for the potential as a traffic analysis tool (Daganzo [3]). On the other hand, the shockwave queueing profile is based on traffic flow theory [4, 5] to describe the spatiotemporal dynamics of the horizontal queues. Both queueing profiles are depicted in Figure 2 for an oversaturated signal intersection.Figure 2
Deterministic and shockwave queueing profiles for an oversaturated signal intersection (traffic conditionsⒸ, Ⓢ, and Ⓤ are defined with Figure 5).The consistency of the vertical and horizontal queue models or equivalently of the deterministic and shockwave queueing profiles (in terms of their modeling performance) is of much interest and has been a subject of debate for decades. Several textbooks and handbooks have described the two models or two queueing analysis methods as separate tools for treating bottleneck problems (see, e.g., McShane and Roess [6]). Makigami et al. [7] studied the two analysis methods using a three-dimensional (space-time-queue) model and demonstrated their consistency. Wirasinghe [8] proved that the two methods delivered identical results in the total travel delay. Michalopoulos and Pisharody [9] however commented that the deterministic profile overestimated travel delays, while Daganzo [10] pointed out a flaw in that work and emphasized that, with the flaw removed, the two methods were still consistent. More than a decade later, the inconsistency issue was again raised by Chin [11], McShane and Roess [6], and Nam and Drew [12], whereas Lawson et al. [13], Erera et al. [14], Daganzo [3], and Lovell and Windover [15] underlined the consistency once more. More recently, Rakha and Zhang [16] and Yi et al. [17] concurred with the consistency but tried to improve Nam and Drew’s approach [12].The vertical and horizontal queue models are quite different in their operational mechanisms, which is very much the reason for the consistency issue to be raised from time to time. With the literature review, we see that thorough and analytical investigations on the issue still lack. This paper intends to close the debates via a systematic study, particularly for the case of oversaturated signal intersections. This is actually the first motivation of the current work.Traffic signals are one of the crucial components of traffic networks. Queue length [8, 18–20] and total delay [6, 8, 20, 21] of vehicles due to signal operation are among the most important performance measures for traffic signal control. Many relevant studies were conducted using the deterministic queueing profile [8, 10, 12, 15, 20] or shockwave queueing profile [8, 18, 19] with fixed-location sensing data (e.g., loop detector data). As a matter of fact, both deterministic and shockwave queueing profiles are individual-vehicle-trajectory oriented. Fixed-location sensors can hardly provide sufficient information for constructing either profile for the purpose of traffic signal control. For this reason, while fixed-location sensing technologies were dominating, both queueing profiles only had significance at a conceptual level and could not offer strong operability in practice. In recent years, however, the emergence of mobile sensing technologies based on connected vehicles (CVs) has been attracting increasingly more attention. Equipped with GPS (or other tracking devices) and wireless communication systems, CVs are able to provide real-time information (of their positions, speeds, and acceleration), thus leading in total to better spatial coverage of traffic conditions. This has also offered opportunities of reexamining the two queueing models/profiles for traffic modeling and signal control.A number of studies have been conducted using mobile sensing data, to address shockwave profile reconstruction for the purpose of queue length estimation [22–26]. The focus of those works is on the identification of some critical points on vehicle trajectories based on the shockwave model. For instance, [22] tried to figure out the critical points that fall on the congestion forming shockwave curves (CFSC) to determine vehicle trajectories. The work in [24] fitted vehicle trajectories with piecewise linear lines and took critical points as the intersection of the lines. The work in [25] fixed the critical points using both speed and acceleration information. The work in [26] formulated the construction of CFSC as a convex optimization problem. In addition, [27–29] studied queueing delay patterns using mobile-sensing-based travel time data and estimated queue length for signalized intersections. So far, still missing is a simple and efficient approach to the reconstruction of shockwave queueing profiles using mobile sensing data, and this is the second motivation of the current work.It should be noted that only the shockwave queueing profiles can be directly reconstructed using mobile sensing data. Due to its abstract nature, the deterministic queueing profile can only be built upon the corresponding shockwave queueing profile in consideration of the consistency between the two profiles.To sum up, the objective of this work is twofold. First, it aims to close the debates on the consistency of the two queueing models via some systematic and analytical studies. Second, it intends to develop an efficient approach to the reconstruction of the shockwave and deterministic queueing profiles in a quasi-real-time manner using very limited mobile sensing data. The work was done with regard to oversaturated signal intersections. Microscopic simulation tool AIMSUN was used to deliver emulated mobile sensing data and evaluate the accuracy of queueing profile reconstruction, with various penetration rates of connected vehicles taken into account.The remainder of the paper is organized as follows. Section2 introduces the vertical and horizontal queue models as well as the deterministic and shockwave queueing profiles with respect to an oversaturated signal intersection. The consistency properties of the two queueing models/profiles are presented in Section 3. Section 4 focuses on the quasi-real-time reconstruction of the two queueing profiles using very limited mobile sensing data. Section 5 reports on simulation studies, with satisfactory results delivered. Section 6 concludes the paper.


## 2. Queueing Models for Oversaturated Signal Intersections

### 2.1. Vertical and Horizontal Queue Models

With reference to Figure1, either queueing model takes as an input-output system the displayed road link between the upstream entry and the downstream intersection. Consider one vehicle and its counterpart enter the road link at the same time and experience the horizontal and vertical queueing, separately. The two vehicles should pass the stop line at the intersection at the same time. This is actually a minimum requirement for the two queueing models; i.e., given the same input, the model outputs should be the same, despite quite different internal mechanism of queueing.The two queueing models can be further examined with Figures3 and 4. Figure 3 depicts a queue forming process over a red signal phase at the intersection and focuses on three typical time instants: (a) both vertical and horizontal queues exist with three vehicles A, B, and C, while D is approaching either queue; (b) D just joins the horizontal queue while it is still running towards the vertical queue at the stop line; (c) D reaches the stop line and becomes the new tail of the vertical queue. Focusing on vehicle D, we see(i)
it makes no difference to the two types of queue before time instant (b);(ii)
it joins the horizontal queue earlier than the vertical queue;(iii)
after joining the horizontal queue it keeps its original speed until joining the vertical queue;(iv)
the process with which it joins the vertical queue as depicted in Figure2(c) is completed instantaneously.Figure 3
Vehicles arrive to join a vertical queue (upper) and a horizontal queue (lower) over a red signal phase: (a) when D is approaching either queue; (b) when D just joins the horizontal queue; (c) when D just joins the vertical queue.
(a) (b) (c)Figure 4
Vehicles depart from a vertical queue (upper) and a horizontal queue (lower): (a) vehicles are all queueing before the end of a red phase; (b) C has departed from the horizontal queue but is still waiting in the vertical queue; (c) C is passing the stop line from both queues.
(a) (b) (c)Figure 5
A triangular fundamental diagram.Figure4 displays a queue dissolving process and focuses on three typical time instants: (a) vehicles are queueing right before the end of a red signal phase; (b) vehicles A and B have already passed the stop line, and vehicle C is waiting in the vertical queue at the stop line while it has departed from the horizontal queue and is running towards the stop line; (c) vehicle C is passing the stop line from both queues. The process that C departs from the vertical queue to pass the stop line is completed instantaneously. In summary,(i)
a horizontal queue makes a distinction between “departing from the queue” and “passing the stop line”, while a vertical queue does not;(ii)
a horizontal queue vehicle and its counterpart in the vertical queue pass the stop line at the same time.


### 2.2. Deterministic and Shockwave Queueing Profiles

With reference to Figure1, traffic conditions along the road link and at the intersection are defined with the fundamental diagram in Figure 5. More specifically U(qu,ku,v), Sqs,ks,v, and C0,kjam,0 address the upstream arrival flow, the capacity flow at the intersection, and the queueing condition on the road link, with x and w denoting the queue forming and dissolving shockwaves.Without loss of generality, Figure2 plots the deterministic and shockwave queueing profiles for the intersection under signal oversaturation over two cycles. A number of definitions and notations are first given. On the time axes, TR1, TG1, TR2, and TG2 represent the start and end times of red phases. The superscripts “d” and “s” of symbols on the time axes refer to the “deterministic” and “shockwave” queueing profiles, respectively. The two profiles are connected via individual vehicle trajectories; see, e.g., vehicles i, p, and m. As a consequence, a number of time instants appear in pairs, e.g., TEd,1=TEs,1, TEd,2=TEs,2, Tu,φd=Tu,φs, Ts,φd=Ts,φs (φ=i,p,m). TEd,2 (or TEs,2) denotes the time instant by which vehicles that are delayed over the two cycles have all been served, while TEd,1 (or TEs,1) is a virtual time instant by which all vehicles delayed over the first cycle would have been served without signal oversaturation. Tu,φd or Tu,φs denotes the time vehicle φ joins the vertical queue, and Ts,φd or Ts,φs denotes the time vehicle φ leaves the vertical queue (or equivalently passes the stop line with both queues). The timewise consistency between the two profiles is discussed later with Theorems 1 and 2. Moreover, Tmax1 and Tmax2, defined only for the shockwave profile, are the time instants the physical queues spill back to reach their far ends.On theN axis of the deterministic profile, NC1 refers to the actual number of vehicles served by the end of the first cycle, Nmax2, corresponding to TEd,2 (or TEs,2), refers to the total number of vehicles that are delayed but eventually served over the two cycles, and Nmax1, corresponding to TEd,1 (or TEs,1), is a virtual quantity addressing the number of vehicles that would have been served by the end of the first cycle without signal oversaturation. In addition, Ni, Np, and Nm denote the accumulative numbers for vehicles i, p, and m in the vertical queue.On theL axis of the shockwave profile, Lφθ denotes the distance between vehicle φ and the stop line when it joins the horizontal queue in cycle θ. Lmax1 and Lmax2, corresponding to Tmax1 and Tmax2, respectively, address the far ends that the physical queues can reach.In the case of signal oversaturation over two cycles, some vehicles are involved with the horizontal queue once, while some others may experience queueing twice. With “single” queueing, a vehicleφ joins and leaves the vertical queue at Tu,φd and Ts,φd (e.g., vehicles i and m). With “double” queueing, a vehicle φ (e.g., vehicle p) joins the vertical queue at Tu,φd and would leave it at Ts,φd,1 if there was no oversaturation but factually leaves it at Ts,φd,2. Accordingly, the timewise consistency between the two profiles reads Tu,φd=Tu,φs; Ts,φd=Ts,φs or Ts,φd,1=Ts,φs,1 and Ts,φd,2=Ts,φs,2.When a vehicleφ joins a horizontal queue, its distance from the stop line follows a different way of definition. If φ joins the horizontal queue only in cycle 1, then only Lφ1 is defined (e.g., φ=i); if it joins the queue only in cycle 2, only Lφ2 is defined (e.g., φ=m); if it joins the queue twice, both Lφ1 and Lφ2 (e.g., φ=p) make sense.


## 2.1. Vertical and Horizontal Queue Models

With reference to Figure1, either queueing model takes as an input-output system the displayed road link between the upstream entry and the downstream intersection. Consider one vehicle and its counterpart enter the road link at the same time and experience the horizontal and vertical queueing, separately. The two vehicles should pass the stop line at the intersection at the same time. This is actually a minimum requirement for the two queueing models; i.e., given the same input, the model outputs should be the same, despite quite different internal mechanism of queueing.The two queueing models can be further examined with Figures3 and 4. Figure 3 depicts a queue forming process over a red signal phase at the intersection and focuses on three typical time instants: (a) both vertical and horizontal queues exist with three vehicles A, B, and C, while D is approaching either queue; (b) D just joins the horizontal queue while it is still running towards the vertical queue at the stop line; (c) D reaches the stop line and becomes the new tail of the vertical queue. Focusing on vehicle D, we see(i)
it makes no difference to the two types of queue before time instant (b);(ii)
it joins the horizontal queue earlier than the vertical queue;(iii)
after joining the horizontal queue it keeps its original speed until joining the vertical queue;(iv)
the process with which it joins the vertical queue as depicted in Figure2(c) is completed instantaneously.Figure 3
Vehicles arrive to join a vertical queue (upper) and a horizontal queue (lower) over a red signal phase: (a) when D is approaching either queue; (b) when D just joins the horizontal queue; (c) when D just joins the vertical queue.
(a) (b) (c)Figure 4
Vehicles depart from a vertical queue (upper) and a horizontal queue (lower): (a) vehicles are all queueing before the end of a red phase; (b) C has departed from the horizontal queue but is still waiting in the vertical queue; (c) C is passing the stop line from both queues.
(a) (b) (c)Figure 5
A triangular fundamental diagram.Figure4 displays a queue dissolving process and focuses on three typical time instants: (a) vehicles are queueing right before the end of a red signal phase; (b) vehicles A and B have already passed the stop line, and vehicle C is waiting in the vertical queue at the stop line while it has departed from the horizontal queue and is running towards the stop line; (c) vehicle C is passing the stop line from both queues. The process that C departs from the vertical queue to pass the stop line is completed instantaneously. In summary,(i)
a horizontal queue makes a distinction between “departing from the queue” and “passing the stop line”, while a vertical queue does not;(ii)
a horizontal queue vehicle and its counterpart in the vertical queue pass the stop line at the same time.


## 2.2. Deterministic and Shockwave Queueing Profiles

With reference to Figure1, traffic conditions along the road link and at the intersection are defined with the fundamental diagram in Figure 5. More specifically U(qu,ku,v), Sqs,ks,v, and C0,kjam,0 address the upstream arrival flow, the capacity flow at the intersection, and the queueing condition on the road link, with x and w denoting the queue forming and dissolving shockwaves.Without loss of generality, Figure2 plots the deterministic and shockwave queueing profiles for the intersection under signal oversaturation over two cycles. A number of definitions and notations are first given. On the time axes, TR1, TG1, TR2, and TG2 represent the start and end times of red phases. The superscripts “d” and “s” of symbols on the time axes refer to the “deterministic” and “shockwave” queueing profiles, respectively. The two profiles are connected via individual vehicle trajectories; see, e.g., vehicles i, p, and m. As a consequence, a number of time instants appear in pairs, e.g., TEd,1=TEs,1, TEd,2=TEs,2, Tu,φd=Tu,φs, Ts,φd=Ts,φs (φ=i,p,m). TEd,2 (or TEs,2) denotes the time instant by which vehicles that are delayed over the two cycles have all been served, while TEd,1 (or TEs,1) is a virtual time instant by which all vehicles delayed over the first cycle would have been served without signal oversaturation. Tu,φd or Tu,φs denotes the time vehicle φ joins the vertical queue, and Ts,φd or Ts,φs denotes the time vehicle φ leaves the vertical queue (or equivalently passes the stop line with both queues). The timewise consistency between the two profiles is discussed later with Theorems 1 and 2. Moreover, Tmax1 and Tmax2, defined only for the shockwave profile, are the time instants the physical queues spill back to reach their far ends.On theN axis of the deterministic profile, NC1 refers to the actual number of vehicles served by the end of the first cycle, Nmax2, corresponding to TEd,2 (or TEs,2), refers to the total number of vehicles that are delayed but eventually served over the two cycles, and Nmax1, corresponding to TEd,1 (or TEs,1), is a virtual quantity addressing the number of vehicles that would have been served by the end of the first cycle without signal oversaturation. In addition, Ni, Np, and Nm denote the accumulative numbers for vehicles i, p, and m in the vertical queue.On theL axis of the shockwave profile, Lφθ denotes the distance between vehicle φ and the stop line when it joins the horizontal queue in cycle θ. Lmax1 and Lmax2, corresponding to Tmax1 and Tmax2, respectively, address the far ends that the physical queues can reach.In the case of signal oversaturation over two cycles, some vehicles are involved with the horizontal queue once, while some others may experience queueing twice. With “single” queueing, a vehicleφ joins and leaves the vertical queue at Tu,φd and Ts,φd (e.g., vehicles i and m). With “double” queueing, a vehicle φ (e.g., vehicle p) joins the vertical queue at Tu,φd and would leave it at Ts,φd,1 if there was no oversaturation but factually leaves it at Ts,φd,2. Accordingly, the timewise consistency between the two profiles reads Tu,φd=Tu,φs; Ts,φd=Ts,φs or Ts,φd,1=Ts,φs,1 and Ts,φd,2=Ts,φs,2.When a vehicleφ joins a horizontal queue, its distance from the stop line follows a different way of definition. If φ joins the horizontal queue only in cycle 1, then only Lφ1 is defined (e.g., φ=i); if it joins the queue only in cycle 2, only Lφ2 is defined (e.g., φ=m); if it joins the queue twice, both Lφ1 and Lφ2 (e.g., φ=p) make sense.


## 3. Consistency Properties of the Queueing Profiles

The consistency properties of the two queueing profiles are described via four theorems and one corollary. The mathematical proof is found in AppendicesA and B. To the best of our knowledge, these properties were not explicitly discussed, especially for oversaturated signal intersections. The theorems and corollary are presented on the basis of Figure 2, with respect to signal oversaturation over two cycles, but also apply to more general cases.Theorem 1.
For a signal cycleθ(θ=1,2), TEs,θ=TEd,θ.Proof.
SeeAppendix A.With reference to Figure2, TEs,1 and TEd,1 are two virtual quantities while TEs,2 and TEd,2 are well defined. The theorem says that the two types of queue are cleared at the same time in the case of either undersaturation (θ=1) or oversaturation (θ=2).Theorem1 sets a basis for the proof of the other theorems and corollary. Following Theorem 1, TEd,θ and TEs,θ(θ=1,2) are replaced with TEθ(θ=1,2) in the sequel.Theorem 2.
Consider a vehicleφ and its counterpart, which go through the vertical and horizontal queues separately. Then,
(1) the vehicles do not join the two queues at the same time, butTu,φs=Tu,φd;
(2) the vehicles do not depart from the queues at the same time but pass the stop line at the same time, i.e.,Ts,φs=Ts,φd or Ts,φs,θ=Ts,φd,θ(θ=1,2).Proof.
See Appendix A.Regarding the property demonstrated with Theorem2-(2), see also Section 2.1.Corollary 3.
Given a vehicleφ, its total horizontal queue delay dφs is equal to its vertical queue delay dφd.Proof.
See Appendix  A.This may also be checked graphically with, e.g., vehiclei in Figure 2. The trajectory of vehicle i in the shockwave profile is ②③⑤, while ②④⑤ is its vertical queue trajectory in time. Clearly, vehicle i spends the same period of time in both profiles, i.e., between ④ and ⑤ for the vertical queue and between ② and ③ for the horizontal queue.Theorem 4.
For any vehicleφ in a signal cycle θ(θ=1,2),(1)Nφ-NCθ-1=Lφθ·kjamTGθ≤Ts,φs,θ≤TEθ,where NC0=0, NC1=qs·(TR2-TG1).Proof.
See Appendix A.Any vehicleφ and its counterpart do not join the horizontal queue and the corresponding vertical queue at the same time, but their positions in the queues (Nφ and Lφθ) are interrelated as stated above. With reference to Figure 2, only Li1 is defined for vehicle i; then Ni=Li1·kjam; both Lp1 and Lp2 are defined for vehicle p; then Np=Lp1·kjam and Np-NC1=Lp2·kjam; only Lm2 is defined for vehicle m; then Nm-NC1=Lm2·kjam.It is noted that Theorem4 also applies to more general cases with θ>2, where NCθ=qs·(TRθ+1-TGθ). The proof is omitted.Corollary 5.
For any vehicleφ that experiences queueing more than once, (2)Lφ2=Lφ1-TR2-TG1·qskjamProof.
See Appendix A.Given a vehicleφ that experiences queueing more than once over multiple signal cycles, the corollary formulates the relation of this vehicle’s queue length in two neighboring cycles.It is noted that Corollary5 also applies to more general cases with θ>2:(3)Lφθ+N=Lφθ-∑i=1NTRθ+i-TGθ+i-1·qskjamThe proof is omitted.Theorem 6.
The total delays created by the vertical and horizontal queueing models are equal, i.e.,SV=SH·kjam, where SV is the total area of the deterministic queueing profile over the whole period [TR1,TEd,2] in Figure 2, while SH is the total area of the two shockwave polygons over periods [TR1,Tmax1] and [TR2,Tmax2] in Figure 2.Proof.
See Appendix  A.So far what has been discussed with Figure2 is the case of a constant traffic demand. The theorems and corollary can certainly be extended to the case of a varying traffic demand (Figure 6), and the corresponding proof is found in Appendix B.Figure 6
Deterministic and shockwave profiles under varying traffic demands.


## 4. Quasi-Real-Time Reconstruction of Queueing Profiles Using Limited Mobile Sensing Data

Conventional fixed-location detectors such as loops, radars, and cameras cannot provide sufficient or suitable measurements to construct the deterministic and shockwave queueing profiles. Albeit well-known in traffic engineering, the vertical/horizontal queue models and the deterministic/shockwave queueing profiles have long been considered only at a conceptual level. With the emergence of the mobile sensing technologies, however, the increasingly wider spread of mobile sensing data has made it possible to construct both queueing profiles in quasi-real time. In fact, what can be directly created using mobile sensing data is the shockwave queueing profile, but with the consistency properties presented in Section3, especially Theorems 1, 2, and 4, the deterministic profile can also be constructed. As an illustrative example, Figure 7(a) presents the shockwave profiles constructed using mobile sensing data emulated with the microscopic simulation tool AIMSUN. The shockwave profiles over the five most congested cycles are highlighted, over which the corresponding signal intersection is oversaturated. Figure 7(b) displays the associated deterministic profile. The approaches employed to create both profiles are described in this section, and more simulation details are found in Section 5.Figure 7
Queueing profiles based on microscopic simulation with 100% penetration rate of CVs: (a) shockwave profile (with the signal phases at the upstream intersection); (b) the corresponding deterministic profile.
(a) (b)


### 4.1. Basic Ideas

Given an intersection as shown in Figure8, its arriving flow may consist of several components, according to the signal phase setting at the intersection right upstream: the through movement in phase (a); the right-turn movement during phase (c); the left-turn movement within phase (d). As an illustrative example, Figure 7(a) presents the correspondence of flows reaching the downstream intersection to the upstream signal phases as given in Figure 8.Figure 8
Arrival flow from upstream intersection.Based on vehicle to vehicle/infrastructure communication, assume with reference to Figure8 that the start and end times of each signal phase of the two intersections are known, and each movement towards the downstream intersection as shown in Figure 8 (i.e., each flow in Figure 7(a)) contains at least two connected vehicles (CVs). Then, each linear section of the congestion forming shockwave curve (CFSC) as shown in Figure 9 contains at least two measured points (e.g., y0 and y1), and the whole CFSC can be basically determined in quasi-real time. Similarly, the congestion dissolving shockwave w can also be determined. w is supposed to be constant by Figure 5, which is also confirmed with Figure 7(a) in simulation. Clearly, the more the CVs are available, the more accurately the shockwave profile is identified. This way of determining the shockwave profile was also considered in [12, 13, 15] via linear optimization. Once the shockwave profile is fixed, the deterministic profile can be readily determined as well.Figure 9
Two queueing profiles with variant upstream arrival flow.Consider a CVi in Figure 9. The information of its positions and speeds as well as the corresponding time instants is available along its trajectory, e.g., at points y1, y3, and y4. By Theorem 2 and the definition of the vertical queue, the time it joins the deterministic queueing profile at y5 is equal to the time instant it reaches virtually y2. y2 can be approximately determined as 1/n·∑k=1ntk+L(tk)/v(tk), where n denotes a given number of data points collected right before it joins the horizontal queue at y1, L(tk) denotes its distance from the stop line at each time instant tk, and v(tk) denotes the instantaneous speed. By Theorem 2, y6 and y4 are consistent in time. Moreover, based on Theorem 4, it is straightforward to get Ni for the deterministic profile with measured Li in the shockwave profile. Thus, the full coordinates of y5 and y6 are fixed. As such, as long as two CVs are available for each portion of the arrival flow, both shockwave and deterministic queueing profiles can basically be determined; see, e.g., Figure 9.


### 4.2. Creating a Shockwave Profile

#### 4.2.1. The Case of Two Connected Vehicles in Each Subflow

As previously mentioned, traffic arriving at an intersection can be decomposed into several flow portions as regulated by the signal phases at the intersection right upstream. There may also be a number of periods over which no vehicle arrives at the intersection. Such periods may appear in the middle of a CFSC, e.g., between flows ① and ② in Figure7(a), or at the end of a CFSC, e.g., between flows ② and ③ in Figure 7(a). Note that there are two possibilities concerning no vehicle arrival at a subject intersection: (1) the signal phase of the upstream intersection is prohibitive (e.g., phase (b) in Figure 8 and Figure 7(a)); (2) the upstream signal phase is permissive but by coincidence there is no demand; see in Figure 7(a), e.g., phase (d) in relation to flow ① and phase (c) to flow ⑥. Either case may occur in the middle of or at the end of a CFSC. It is stipulated in this work that if no vehicle arrival happens in the middle of a CFSC, the portion of CFSC concerning (1) is plotted as a horizontal line, e.g., between flows ⑤ and ⑥ in Figure 7(a), and the portion concerning (2) is plotted as a line, e.g., between ① and ② in Figure 7(a). However, if either (1) or (2) happens at the end of a CFSC, the corresponding portion is naturally plotted as a horizontal line, e.g., between ② and ③ and between ⑥ and ⑦ in Figure 7(a).It is noted that Figure7(a) plots the case of 100% penetration rate of CVs, where every trajectory displayed corresponds to one specific CV. However, the market penetration rate of CVs will be low in the foreseeable future, which means, in the context of Figure 7(a), only a very small number of vehicle trajectories displayed are actually observable. It is then a question how to construct the shockwave profile and accordingly the deterministic profile using very limited vehicle trajectory data. To be realistic, we assume in this work that each flow as shown in Figure 7(a) contains at most two CVs.While constructing the shockwave profile using very limited mobile sensing data, it is common to encounter the situation of missing data. Six possible cases are presented in Figure10. Figures 10(a) and 10(b) address the occurrence of missing data in the middle of CFSCs, while Figures 10(c)–10(f) address the cases at the end of CFSCs. Note that on top of “low penetration of CVs”, “no vehicle arrival” as previously discussed also contributes to the result of missing data.Figure 10
An illustration of handling missing data while constructing the shockwave queueing profiles using very limited mobile sensing data.
(a) (b) (c) (d) (e) (f)In Figure10, CVs r and s correspond to one green phase of the upstream intersection, while CVs e and f correspond to the next green phase. Each of subfigures (b), (d), and (f) involves a prohibitive phase (e.g., phase (b) in Figure 8 and period OD in Figure 10(b)), while none of subfigures (a), (c), and (e) involves such a phase. Moreover, m represents a virtual vehicle that leaves the upstream intersection at the start of the next green phase, and virtual vehicle n leaves at the end time of the current green phase, which is followed by a prohibitive phase. Taking Figure 7(a) as the ground truth (with the 100% market penetration), all cases in Figure 10 except (e) and (f) are encountered while estimating both profiles using very limited mobile sensing data as sampled from Figure 7(a). More details are given in Section 5.In what follows, we discuss how to treat the cases shown in Figure10, with various situations of missing data taken into account, so as to reconstruct the shockwave profiles. In Figure 10(a), CVs r and s are from one green phase of the upstream intersection while CVs e and f correspond to another. It is not difficult to distinguish this since the information of all signal phases and of CVs are available. As such, y2 is related to the last CV of that green phase, and y3 is to the first CV of the next green phase. Then, two linear portions of CFSC can be determined, respectively, with CV data points y1, y2, y3, and y4. Note that the start time of the upstream green phase for e and f is known. Let a virtual vehicle m leave the upstream intersection at this time instant; the trajectory of m can be drawn with its slope equal to the mean speed of e and f. As such, points D and C can be fixed. It is noted that the portion of the congestion forming shockwave curve (CFSC) between y1 and D belongs to the previous green phase. As such, there exist two extreme but possible cases: (1) y2CD, i.e., section y1C is with a constant traffic volume while no vehicle arrives over period CD; (2) y2D, i.e., sections y1y2 and y2D are with different traffic volumes. Moreover, between cases (1) and (2), there are a number of intermediate possibilities as displayed. Note that the ground truth can be any of the above cases. Based on our simulation results in Section 5, we believe case y2D is more likely to happen than y2CD.Figure10(b) refers to the case that a no-flow phase OD is involved. In this case, let virtual vehicle n leave the upstream intersection at the end time of the green phase associated with vehicles r and s; then the trajectory of n is fixed with its speed equal to the mean speed of vehicles r and s. As such, point O can be determined. The treatment of the portion between y2 and O in Figure 10(b) is the same as that between y2 and D in Figure 10(a).Figures10(c)–10(f) address the case that the data is missing at the end of a CFSC. First, we consider Figure 10(c), where CVs e and f from the next green phase of the upstream intersection do not join the queue in the cycle of the subject intersection for vehicles r and s, and this is also assumed to be the case for all vehicles from the next green phase (among them only e and f are observable). Let virtual vehicle m leave the upstream intersection at the start of the next green phase, and then points C and D can be fixed with the information of y1, y2, y3, and y4. Figure 10(d) refers to a similar case but the signal phase next to that for CVs r and s is a prohibitive phase. Let virtual vehicle n leave the upstream intersection at the end of the current green phase, and points C and D can be fixed. Note that CD in Figure 10(c) or 11(d) represents an extreme case, while the real case can be any one between y2y4 and CD. Moreover, Figures 10(e) and 10(f) present two special but possible cases of Figures 10(c) and 10(d).Figure 11
Further consideration for the queueing profile reconstruction.
(a) (b)The combination of the cases in Figure10 could very likely happen in practice. For instance, with Figures 10(a) and 10(d) in mind, Figure 11(a) illustrates the reconstruction of the shockwave profiles, under the assumption that each subflow contains only two CVs, e.g., r  and  s and c  and  d. More specifically, the shockwave forming curve y2Dy3 and the queue end EF can be fixed. Albeit not fully accurate, our simulation studies in Section 5 demonstrate that the inaccuracy is not essential as far as the total delay estimate is concerned (Theorem 6).


#### 4.2.2. The Case of One Connected Vehicle in Each Subflow

At an early stage of the CV deployment, it may not be even practical to assume two CVs available in each flow from the upstream. So, it is necessary to also consider the case that only one CV is involved in some flows. Let us focus on Figure11(b), where vehicle s and f are the only CVs within their respective flows. We again consider a virtual car n that may leave the upstream intersection at the end of the green phase that is associated with vehicle f. Then, we can fix point E with the shockwave dissolving speed w determined with y6 and y8. Furthermore, with the information of y4 and the trajectory of virtual vehicle m that leaves at the start of the upstream green phase, we can determine point D. With D and y2, we can determine point B.Once the shockwave profile is fixed, the deterministic profile can also be plotted on the basis of Theorems1, 2, and 4; see the illustrated correspondence between points A–E with points A′–E′ in Figure 11.It should be emphasized that this work only requires the following information of the application site: densitykjam of the concerned road link, the start and end times of the signal phases at the upstream intersection and the subject intersection, and the position and speed information of CVs.


## 4.1. Basic Ideas

Given an intersection as shown in Figure8, its arriving flow may consist of several components, according to the signal phase setting at the intersection right upstream: the through movement in phase (a); the right-turn movement during phase (c); the left-turn movement within phase (d). As an illustrative example, Figure 7(a) presents the correspondence of flows reaching the downstream intersection to the upstream signal phases as given in Figure 8.Figure 8
Arrival flow from upstream intersection.Based on vehicle to vehicle/infrastructure communication, assume with reference to Figure8 that the start and end times of each signal phase of the two intersections are known, and each movement towards the downstream intersection as shown in Figure 8 (i.e., each flow in Figure 7(a)) contains at least two connected vehicles (CVs). Then, each linear section of the congestion forming shockwave curve (CFSC) as shown in Figure 9 contains at least two measured points (e.g., y0 and y1), and the whole CFSC can be basically determined in quasi-real time. Similarly, the congestion dissolving shockwave w can also be determined. w is supposed to be constant by Figure 5, which is also confirmed with Figure 7(a) in simulation. Clearly, the more the CVs are available, the more accurately the shockwave profile is identified. This way of determining the shockwave profile was also considered in [12, 13, 15] via linear optimization. Once the shockwave profile is fixed, the deterministic profile can be readily determined as well.Figure 9
Two queueing profiles with variant upstream arrival flow.Consider a CVi in Figure 9. The information of its positions and speeds as well as the corresponding time instants is available along its trajectory, e.g., at points y1, y3, and y4. By Theorem 2 and the definition of the vertical queue, the time it joins the deterministic queueing profile at y5 is equal to the time instant it reaches virtually y2. y2 can be approximately determined as 1/n·∑k=1ntk+L(tk)/v(tk), where n denotes a given number of data points collected right before it joins the horizontal queue at y1, L(tk) denotes its distance from the stop line at each time instant tk, and v(tk) denotes the instantaneous speed. By Theorem 2, y6 and y4 are consistent in time. Moreover, based on Theorem 4, it is straightforward to get Ni for the deterministic profile with measured Li in the shockwave profile. Thus, the full coordinates of y5 and y6 are fixed. As such, as long as two CVs are available for each portion of the arrival flow, both shockwave and deterministic queueing profiles can basically be determined; see, e.g., Figure 9.


## 4.2. Creating a Shockwave Profile

### 4.2.1. The Case of Two Connected Vehicles in Each Subflow

As previously mentioned, traffic arriving at an intersection can be decomposed into several flow portions as regulated by the signal phases at the intersection right upstream. There may also be a number of periods over which no vehicle arrives at the intersection. Such periods may appear in the middle of a CFSC, e.g., between flows ① and ② in Figure7(a), or at the end of a CFSC, e.g., between flows ② and ③ in Figure 7(a). Note that there are two possibilities concerning no vehicle arrival at a subject intersection: (1) the signal phase of the upstream intersection is prohibitive (e.g., phase (b) in Figure 8 and Figure 7(a)); (2) the upstream signal phase is permissive but by coincidence there is no demand; see in Figure 7(a), e.g., phase (d) in relation to flow ① and phase (c) to flow ⑥. Either case may occur in the middle of or at the end of a CFSC. It is stipulated in this work that if no vehicle arrival happens in the middle of a CFSC, the portion of CFSC concerning (1) is plotted as a horizontal line, e.g., between flows ⑤ and ⑥ in Figure 7(a), and the portion concerning (2) is plotted as a line, e.g., between ① and ② in Figure 7(a). However, if either (1) or (2) happens at the end of a CFSC, the corresponding portion is naturally plotted as a horizontal line, e.g., between ② and ③ and between ⑥ and ⑦ in Figure 7(a).It is noted that Figure7(a) plots the case of 100% penetration rate of CVs, where every trajectory displayed corresponds to one specific CV. However, the market penetration rate of CVs will be low in the foreseeable future, which means, in the context of Figure 7(a), only a very small number of vehicle trajectories displayed are actually observable. It is then a question how to construct the shockwave profile and accordingly the deterministic profile using very limited vehicle trajectory data. To be realistic, we assume in this work that each flow as shown in Figure 7(a) contains at most two CVs.While constructing the shockwave profile using very limited mobile sensing data, it is common to encounter the situation of missing data. Six possible cases are presented in Figure10. Figures 10(a) and 10(b) address the occurrence of missing data in the middle of CFSCs, while Figures 10(c)–10(f) address the cases at the end of CFSCs. Note that on top of “low penetration of CVs”, “no vehicle arrival” as previously discussed also contributes to the result of missing data.Figure 10
An illustration of handling missing data while constructing the shockwave queueing profiles using very limited mobile sensing data.
(a) (b) (c) (d) (e) (f)In Figure10, CVs r and s correspond to one green phase of the upstream intersection, while CVs e and f correspond to the next green phase. Each of subfigures (b), (d), and (f) involves a prohibitive phase (e.g., phase (b) in Figure 8 and period OD in Figure 10(b)), while none of subfigures (a), (c), and (e) involves such a phase. Moreover, m represents a virtual vehicle that leaves the upstream intersection at the start of the next green phase, and virtual vehicle n leaves at the end time of the current green phase, which is followed by a prohibitive phase. Taking Figure 7(a) as the ground truth (with the 100% market penetration), all cases in Figure 10 except (e) and (f) are encountered while estimating both profiles using very limited mobile sensing data as sampled from Figure 7(a). More details are given in Section 5.In what follows, we discuss how to treat the cases shown in Figure10, with various situations of missing data taken into account, so as to reconstruct the shockwave profiles. In Figure 10(a), CVs r and s are from one green phase of the upstream intersection while CVs e and f correspond to another. It is not difficult to distinguish this since the information of all signal phases and of CVs are available. As such, y2 is related to the last CV of that green phase, and y3 is to the first CV of the next green phase. Then, two linear portions of CFSC can be determined, respectively, with CV data points y1, y2, y3, and y4. Note that the start time of the upstream green phase for e and f is known. Let a virtual vehicle m leave the upstream intersection at this time instant; the trajectory of m can be drawn with its slope equal to the mean speed of e and f. As such, points D and C can be fixed. It is noted that the portion of the congestion forming shockwave curve (CFSC) between y1 and D belongs to the previous green phase. As such, there exist two extreme but possible cases: (1) y2CD, i.e., section y1C is with a constant traffic volume while no vehicle arrives over period CD; (2) y2D, i.e., sections y1y2 and y2D are with different traffic volumes. Moreover, between cases (1) and (2), there are a number of intermediate possibilities as displayed. Note that the ground truth can be any of the above cases. Based on our simulation results in Section 5, we believe case y2D is more likely to happen than y2CD.Figure10(b) refers to the case that a no-flow phase OD is involved. In this case, let virtual vehicle n leave the upstream intersection at the end time of the green phase associated with vehicles r and s; then the trajectory of n is fixed with its speed equal to the mean speed of vehicles r and s. As such, point O can be determined. The treatment of the portion between y2 and O in Figure 10(b) is the same as that between y2 and D in Figure 10(a).Figures10(c)–10(f) address the case that the data is missing at the end of a CFSC. First, we consider Figure 10(c), where CVs e and f from the next green phase of the upstream intersection do not join the queue in the cycle of the subject intersection for vehicles r and s, and this is also assumed to be the case for all vehicles from the next green phase (among them only e and f are observable). Let virtual vehicle m leave the upstream intersection at the start of the next green phase, and then points C and D can be fixed with the information of y1, y2, y3, and y4. Figure 10(d) refers to a similar case but the signal phase next to that for CVs r and s is a prohibitive phase. Let virtual vehicle n leave the upstream intersection at the end of the current green phase, and points C and D can be fixed. Note that CD in Figure 10(c) or 11(d) represents an extreme case, while the real case can be any one between y2y4 and CD. Moreover, Figures 10(e) and 10(f) present two special but possible cases of Figures 10(c) and 10(d).Figure 11
Further consideration for the queueing profile reconstruction.
(a) (b)The combination of the cases in Figure10 could very likely happen in practice. For instance, with Figures 10(a) and 10(d) in mind, Figure 11(a) illustrates the reconstruction of the shockwave profiles, under the assumption that each subflow contains only two CVs, e.g., r  and  s and c  and  d. More specifically, the shockwave forming curve y2Dy3 and the queue end EF can be fixed. Albeit not fully accurate, our simulation studies in Section 5 demonstrate that the inaccuracy is not essential as far as the total delay estimate is concerned (Theorem 6).


### 4.2.2. The Case of One Connected Vehicle in Each Subflow

At an early stage of the CV deployment, it may not be even practical to assume two CVs available in each flow from the upstream. So, it is necessary to also consider the case that only one CV is involved in some flows. Let us focus on Figure11(b), where vehicle s and f are the only CVs within their respective flows. We again consider a virtual car n that may leave the upstream intersection at the end of the green phase that is associated with vehicle f. Then, we can fix point E with the shockwave dissolving speed w determined with y6 and y8. Furthermore, with the information of y4 and the trajectory of virtual vehicle m that leaves at the start of the upstream green phase, we can determine point D. With D and y2, we can determine point B.Once the shockwave profile is fixed, the deterministic profile can also be plotted on the basis of Theorems1, 2, and 4; see the illustrated correspondence between points A–E with points A′–E′ in Figure 11.It should be emphasized that this work only requires the following information of the application site: densitykjam of the concerned road link, the start and end times of the signal phases at the upstream intersection and the subject intersection, and the position and speed information of CVs.


## 4.2.1. The Case of Two Connected Vehicles in Each Subflow

As previously mentioned, traffic arriving at an intersection can be decomposed into several flow portions as regulated by the signal phases at the intersection right upstream. There may also be a number of periods over which no vehicle arrives at the intersection. Such periods may appear in the middle of a CFSC, e.g., between flows ① and ② in Figure7(a), or at the end of a CFSC, e.g., between flows ② and ③ in Figure 7(a). Note that there are two possibilities concerning no vehicle arrival at a subject intersection: (1) the signal phase of the upstream intersection is prohibitive (e.g., phase (b) in Figure 8 and Figure 7(a)); (2) the upstream signal phase is permissive but by coincidence there is no demand; see in Figure 7(a), e.g., phase (d) in relation to flow ① and phase (c) to flow ⑥. Either case may occur in the middle of or at the end of a CFSC. It is stipulated in this work that if no vehicle arrival happens in the middle of a CFSC, the portion of CFSC concerning (1) is plotted as a horizontal line, e.g., between flows ⑤ and ⑥ in Figure 7(a), and the portion concerning (2) is plotted as a line, e.g., between ① and ② in Figure 7(a). However, if either (1) or (2) happens at the end of a CFSC, the corresponding portion is naturally plotted as a horizontal line, e.g., between ② and ③ and between ⑥ and ⑦ in Figure 7(a).It is noted that Figure7(a) plots the case of 100% penetration rate of CVs, where every trajectory displayed corresponds to one specific CV. However, the market penetration rate of CVs will be low in the foreseeable future, which means, in the context of Figure 7(a), only a very small number of vehicle trajectories displayed are actually observable. It is then a question how to construct the shockwave profile and accordingly the deterministic profile using very limited vehicle trajectory data. To be realistic, we assume in this work that each flow as shown in Figure 7(a) contains at most two CVs.While constructing the shockwave profile using very limited mobile sensing data, it is common to encounter the situation of missing data. Six possible cases are presented in Figure10. Figures 10(a) and 10(b) address the occurrence of missing data in the middle of CFSCs, while Figures 10(c)–10(f) address the cases at the end of CFSCs. Note that on top of “low penetration of CVs”, “no vehicle arrival” as previously discussed also contributes to the result of missing data.Figure 10
An illustration of handling missing data while constructing the shockwave queueing profiles using very limited mobile sensing data.
(a) (b) (c) (d) (e) (f)In Figure10, CVs r and s correspond to one green phase of the upstream intersection, while CVs e and f correspond to the next green phase. Each of subfigures (b), (d), and (f) involves a prohibitive phase (e.g., phase (b) in Figure 8 and period OD in Figure 10(b)), while none of subfigures (a), (c), and (e) involves such a phase. Moreover, m represents a virtual vehicle that leaves the upstream intersection at the start of the next green phase, and virtual vehicle n leaves at the end time of the current green phase, which is followed by a prohibitive phase. Taking Figure 7(a) as the ground truth (with the 100% market penetration), all cases in Figure 10 except (e) and (f) are encountered while estimating both profiles using very limited mobile sensing data as sampled from Figure 7(a). More details are given in Section 5.In what follows, we discuss how to treat the cases shown in Figure10, with various situations of missing data taken into account, so as to reconstruct the shockwave profiles. In Figure 10(a), CVs r and s are from one green phase of the upstream intersection while CVs e and f correspond to another. It is not difficult to distinguish this since the information of all signal phases and of CVs are available. As such, y2 is related to the last CV of that green phase, and y3 is to the first CV of the next green phase. Then, two linear portions of CFSC can be determined, respectively, with CV data points y1, y2, y3, and y4. Note that the start time of the upstream green phase for e and f is known. Let a virtual vehicle m leave the upstream intersection at this time instant; the trajectory of m can be drawn with its slope equal to the mean speed of e and f. As such, points D and C can be fixed. It is noted that the portion of the congestion forming shockwave curve (CFSC) between y1 and D belongs to the previous green phase. As such, there exist two extreme but possible cases: (1) y2CD, i.e., section y1C is with a constant traffic volume while no vehicle arrives over period CD; (2) y2D, i.e., sections y1y2 and y2D are with different traffic volumes. Moreover, between cases (1) and (2), there are a number of intermediate possibilities as displayed. Note that the ground truth can be any of the above cases. Based on our simulation results in Section 5, we believe case y2D is more likely to happen than y2CD.Figure10(b) refers to the case that a no-flow phase OD is involved. In this case, let virtual vehicle n leave the upstream intersection at the end time of the green phase associated with vehicles r and s; then the trajectory of n is fixed with its speed equal to the mean speed of vehicles r and s. As such, point O can be determined. The treatment of the portion between y2 and O in Figure 10(b) is the same as that between y2 and D in Figure 10(a).Figures10(c)–10(f) address the case that the data is missing at the end of a CFSC. First, we consider Figure 10(c), where CVs e and f from the next green phase of the upstream intersection do not join the queue in the cycle of the subject intersection for vehicles r and s, and this is also assumed to be the case for all vehicles from the next green phase (among them only e and f are observable). Let virtual vehicle m leave the upstream intersection at the start of the next green phase, and then points C and D can be fixed with the information of y1, y2, y3, and y4. Figure 10(d) refers to a similar case but the signal phase next to that for CVs r and s is a prohibitive phase. Let virtual vehicle n leave the upstream intersection at the end of the current green phase, and points C and D can be fixed. Note that CD in Figure 10(c) or 11(d) represents an extreme case, while the real case can be any one between y2y4 and CD. Moreover, Figures 10(e) and 10(f) present two special but possible cases of Figures 10(c) and 10(d).Figure 11
Further consideration for the queueing profile reconstruction.
(a) (b)The combination of the cases in Figure10 could very likely happen in practice. For instance, with Figures 10(a) and 10(d) in mind, Figure 11(a) illustrates the reconstruction of the shockwave profiles, under the assumption that each subflow contains only two CVs, e.g., r  and  s and c  and  d. More specifically, the shockwave forming curve y2Dy3 and the queue end EF can be fixed. Albeit not fully accurate, our simulation studies in Section 5 demonstrate that the inaccuracy is not essential as far as the total delay estimate is concerned (Theorem 6).


## 4.2.2. The Case of One Connected Vehicle in Each Subflow

At an early stage of the CV deployment, it may not be even practical to assume two CVs available in each flow from the upstream. So, it is necessary to also consider the case that only one CV is involved in some flows. Let us focus on Figure11(b), where vehicle s and f are the only CVs within their respective flows. We again consider a virtual car n that may leave the upstream intersection at the end of the green phase that is associated with vehicle f. Then, we can fix point E with the shockwave dissolving speed w determined with y6 and y8. Furthermore, with the information of y4 and the trajectory of virtual vehicle m that leaves at the start of the upstream green phase, we can determine point D. With D and y2, we can determine point B.Once the shockwave profile is fixed, the deterministic profile can also be plotted on the basis of Theorems1, 2, and 4; see the illustrated correspondence between points A–E with points A′–E′ in Figure 11.It should be emphasized that this work only requires the following information of the application site: densitykjam of the concerned road link, the start and end times of the signal phases at the upstream intersection and the subject intersection, and the position and speed information of CVs.


## 5. Microscopic Simulation Evaluation

So far, we have discussed how to construct the shockwave and deterministic queueing profiles with very limited mobile sensing data. This section aims to evaluate the accuracy of the profile construction using the microscopic traffic simulator AIMSUN. Note that it makes little sense to do the evaluation on any performance index concerning individual cars, as the CV penetration rate is very low. Rather, we handle the evaluation on the basis of the total vertical and horizontal queue delays, which have been discussed with Theorem6. More specifically, the ground truth is the sum of delays of all emulated individual vehicles over the simulation time horizon. The two intersections considered are 300 meters apart. The phase plan of the upstream intersection is given in Figure 12 with reference to Figure 8. The subject intersection at the downstream follows the same four-phase signal plan, with equal green and red phases of 30 s for the traffic flow from the concerned approach. The demands for the through, left-turn, and right-turn traffic at the upstream intersection (Figure 8) are 600 veh/h, 250 veh/h, and 150 veh/h. The jam density kjam of the considered road link is determined with simulation to be 0.202 veh/m/lane. The simulation time horizon is 20 min. The simulation results are shown in Figure 7, but we focus on the period of the five most congested cycles from 270 s to 570 s.Figure 12
The phase plan for the upstream intersection.By Theorem6, we evaluate the accuracy of the profile construction based on the estimated total delay using both deterministic and shockwave profiles in comparison to the ground truth. The corresponding results are presented in Table 1. Clearly, the total delays determined with the two queueing profiles are quite close to the ground truth.Table 1
Comparison of total delay between the two queueing profiles in the case of 100% penetration rate.
Delay of 100% CVs Ratio by the ground truth Deterministic 2405.00 98.52% Shockwave 2341.72 95.93% Ground truth 2441.20 100.00%As shown in Figure7(a), 10 flows are involved. Except flow ⑨, each of those flows contains more than two vehicles. We assume each flow other than ⑨ contributes at most two CVs, and the only vehicle in flow ⑨ is a CV. We have then considered 10 different cases as displayed in Table 2. In each case, the blank space means the corresponding flow has two CVs while the cross means the corresponding flow has only one CV. From top to bottom in Table 2, the number of CVs is reduced by 1 each case. Finally, in case ⑩, every flow has only one CV. The 10 flows are of different volumes. The sequence of adding the cross signs is determined as follows: the less the volume that one flow has, the higher the possibility that it has only one CV.Table 2
Ten cases.
① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ case 1         × case 2   ×      × case 3   ×      × × case 4   ×   ×   × × case 5 ×  ×   ×   × × case 6 ×  ×   × ×  × × case 7 ×  × ×  × ×  × × case 8 ×  × × × × ×  × × case 9 × × × × × × ×  × × case 10 × × × × × × × × × ×Focusing on the five most congested cycles in Figure7(a), Figures 13–15 present the queueing profiles constructed for all 10 cases aforementioned. The blue curves address the cases of 100% penetration rate, while the red curves correspond to the cases with at most two CVs in each subflow. It is noted that each horizontal pair of star signs represents a CV queueing. The corresponding penetration rate and estimation performance for each case are presented in Table 3.Table 3
Total delay comparison for ten cases.
Case Number of CVs Penetration rate Total Delay (s) Ratio by 100% CV data Shockwave Deterministic Shockwave Deterministic 1 19 22.62% 2412.16 2464.28 103.01% 102.46% 2 18 21.43% 2447.29 2508.28 104.51% 104.29% 3 17 20.24% 2429.25 2491.90 103.74% 103.61% 4 16 19.05% 2394.53 2477.85 102.26% 103.03% 5 15 17.86% 2415.03 2467.55 103.13% 102.60% 6 14 16.67% 2430.02 2497.02 103.77% 103.83% 7 13 15.48% 2419.52 2483.11 103.32% 103.25% 8 12 14.29% 2413.91 2409.39 103.08% 100.18% 9 11 13.10% 2405.28 2421.46 102.71% 100.68% 10 10 11.90% 2205.30 2204.71 94.17% 91.67%Figure 13
Reconstruction of queueing profiles for cases 1-2 in Table2.
(a) (b)Figure 14
Reconstruction of queueing profiles for cases 3–6 in Table2.
(a) (b) (c) (d)Figure 15
Reconstruction of queueing profiles for cases 7–10 in Table2.
(a) (b) (c) (d)Even in the extreme case ⑩, the performance is still acceptable. This indicates that the new technologies of mobile sensing have the potential of enabling even with a very low market penetration rate of connected vehicles a number of tasks that would otherwise be unpractical.


## 6. Conclusion

This paper addresses via analytical studies and microscopic simulations the lasting debates on the consistency between the deterministic/point/vertical queue model and the shockwave/physical/horizontal queue model, particularly with regard to oversaturated signal intersections. This paper also develops an efficient approach to the quasi-real-time reconstruction of the deterministic and shockwave queueing profiles using very limited mobile sensing data. Microscopic simulations with AIMSUN have demonstrated the efficiency of the approach as well as the analytical results obtained.



---

*Source: 6986198-2018-06-10.xml*