A Hybrid Queuing Model for Coordinated Vehicle Platooning on Mixed-Autonomy Highways: Training and Validation

Platooning of connected and autonomous vehicles (CAVs) is an emerging highway operation with a strong potential for throughput improvement and fuel savings [1, 2, 3]. Platooning technology has been progressing fast recently [4, 5, 6, 7]. The mixed-autonomy traffic flow stability and throughput are extensively studied [8, 9, 10, 11]. However, the macroscopic interaction between CAV platoons and non-CAV background traffic has not been well modeled or understood. In particular, platoons require much larger spaces for lane-changing than non-CAVs, and platoons may hinder non-CAVs’ lane-changing; see Fig. 1.

Consequently, platoons may disrupt local traffic flow, which may prevent platooning from attaining its maximal benefits; see [12] for quantification of such compromise.

In this paper, we develop a hybrid queuing model as well as an easy-to-use training algorithm specifically for highways with a mixture of CAV platoons and non-CAV flow. The model predicts the CAV and non-CAV states based on the traffic demand and highway parameters. We develop a simulation-based gradient descent algorithm to train the model parameters in an online manner. We show that the proposed training algorithm can attain a prediction error less than 15% (resp. 25%) in a stationary (non-stationary) setting with respect to results given by the Simulation of Urban Mobility (SUMO [13]). We also show that the online trained model can be used for platoon headway regulation.

Our hybrid queuing model (HQM) consists of two traffic classes: CAVs and non-CAVs. CAVs arrive at the system as discrete batches (platoons), and non-CAVs arrive as a continuous flow. CAV platoons are “condensed” traffic: a platoon of $n$ CAVs is treated equivalently as $\eta n$ non-CAVs, where $\eta\in(0,1)$ captures the impact of reduced spacing between CAVs. A bottleneck restricts the discharge to downstream. Traffic arrives at the bottleneck after traversing the distance from the highway entrance to the bottleneck, and a constant, nominal traverse time is assumed. The vehicle discharge rate is upper-bounded by the highway capacity. The HQM can be viewed as a simplification of a class of more sophisticated models for mixed-autonomy traffic, which are designed for more detailed analysis or control purposes [10, 14, 15, 16]. The HQM can be further abstracted and used for higher-level decisions such as routing [17]. This model is a significant refinement of a preliminary version proposed some of us in [18].

Our training algorithm determines model parameters by minimizing the square error for the vehicle states on the highway section. The parameters to be determined include the nominal traverse time, the nominal capacity, and the non-CAV’s priority ratio. The algorithm improves the parameters heuristically with constant step sizes. The HQM is simulated to compute the square error with respect to input data. The algorithm terminates when the reduction in square error is less than a threshold. Since the fluid queuing model is simple and the space for parameter search is not very large, the computational efficiency is acceptable.

We show via simulation that the proposed HQM and the training algorithm are valid for control design. We consider the regulation of between-platoon headways in response to real-time traffic condition. The HQM can be used to determine the minimal allowable headway between two platoons, which is critical for the headway regulation policy. We show that the policy based on the HQM is very close to the simulated optimal policy.

The rest of this paper is organized as follows. Section 2 defines the HQM and formulates the model training problem. Section 3 presents the training algorithms in both stationary and non-stationary settings. Section 4 validates the model and evaluates the control algorithm in SUMO. Section 5 gives concluding remarks.

In this section, we introduce the hybrid queuing model (HQM) and formulate the training problem.

We model the generic highway section in Fig. 1 using the HQM in Fig. 2. Cell 1 models the bottleneck. Cells 2 through $T$ are virtual cells to model the traverse time from the entrance to the bottleneck; the nominal traverse time is $T$. Every vehicle entering the highway first goes to cell $T$ and then switch from one cell to the immediately downstream one. No queuing dynamics exists in cells 2 through $T$. Non-CAVs enter the highway as a continuous traffic flow with rate $a(t)\in\mathbb{R}_{\geq 0}$. CAVs enter the highway as discrete platoons with size $l$. The number of CAV arrivals during the $t$th time step is $b(t)\in\mathcal{B}$, where $\mathcal{B}=\{0,l,2l,\ldots\}$. The mixture of continuous and discrete arrivals leads to the hybrid nature of our model.

The state of the model includes (i) the non-CAV states $x_{k}(t)$ in cells $k=1,2,\ldots,T$ and (ii) the CAV states $y_{k}(t)$ in cells $k=1,2,\ldots,T$. Hence, the state space is $x(0)\in\mathbb{R}_{\geq 0}^{T}$. To define the state space, let $\gamma>1$ be a scaling factor for CAVs in a platoon.

That is, a platoon of $n$ CAVs is equivalent to $n/\gamma$ non-CAVs in terms of its queuing effect; see Fig. 3. A typical value for $\gamma$ is 2 [2, 19]. Hence, the set of $x(t)$ is $\mathbb{R}_{\geq 0}^{T}$, while the set of $y(t)$ is $\mathcal{Y}^{T}$, where $\mathcal{Y}=\{0,l/\gamma,2l/\gamma,\ldots\}$.

Consider an arbitrary initial condition $(x(0),y(0))\in\mathbb{R}_{\geq 0}^{T}\times\mathcal{Y}^{T}$. At the $t$th time step, the state is updated as follows.

1. 1.

   For $k=T$,

   	$\displaystyle x_{T}(t+1)=a(t),$
   	$\displaystyle y_{T}(t+1)=b(t)/\gamma.$

2. 2.

   For $k=2,3,\ldots,T-1$,

   	$\displaystyle x_{k}(t+1)=x_{k+1}(t),$
   	$\displaystyle y_{k}(t+1)=y_{k+1}(t).$

3. 3.

   For $k=1$, the discharged flows are given by

   	$\displaystyle f(t)=\min\Big{\{}x_{1}(t),\rho F\Big{\}},$		(1a)
   	$\displaystyle g(t)=\Big{\lfloor}\frac{1}{l}\min\Big{\{}y_{1}(t),F-f(t)\Big{\}}\Big{\rfloor}l,$		(1b)

   where $F\in\mathbb{R}_{\geq 0}$ is the bottleneck’s capacity and $\rho\in[0,1]$ specifies the priority of non-CAV traffic with respect to CAVs; $\rho=1$ means that non-CAVs have full priority over CAVs. The traffic state in cell 1 is updated as follows.

   	$\displaystyle x_{1}(t+1)=x_{1}(t)+x_{2}(t)-f(t),$
   	$\displaystyle y_{1}(t+1)=y_{1}(t)+y_{2}(t)-g(t).$

The above queuing dynamics can be interpreted as follows. CAVs and non-CAVs first enter cell $T$ and then enter cell 1 after $(T-1)$ time steps. No queuing occurs in cells $T,T-1,\ldots,2$, while a two-class traffic queue may exist in cell 1. The parameters that define the model are the nominal traverse time $T$, the priority $\rho$, and the capacity $F$. The observable quantities are the inflows $a(t),b(t)$ and the traffic states

where $\hat{m}(t)$ and $\hat{n}(t)$ stand for number of non-CAV and CAV based on the HQM respectively.

Thus, the estimation problem can be formulated as follows:
Given $a(s),b(s),m(s),n(s)$, $s=0,1,2,\ldots,t$,
Find $T,\rho,F$
such that some cost is minimized.
The objective of the HQM is to accurately predict the traffic states $m(t),n(t)$ over the highway section. Therefore, the cost function penalizes the error between the SUMO-simulated values $\{m(s),n(s);s=0,1,2,\ldots,t\}$ and the predicted values $\{\hat{m}(s),\hat{n}(s);s=0,1,2,\ldots,t\}$. The observed values are obtained from SUMO, while the predicted values are from the HQM. The specific form of the cost function depends on the specific setting, which we discuss in the next section.

In this section, we consider the model training algorithm in two typical settings, viz. the stationary and the non-stationary settings. We use

to denote the vector of parameters of the HQM. These parameters are the decision variables of the training problem. Since the state $\hat{x}(t)$ and $\hat{y}(t)$ (and thus $\hat{m}(t)$ and $\hat{n}(t)$) depends on $\theta$, we write $\hat{x}(t;\theta)$, $\hat{y}(t;\theta)$, $\hat{m}(t;\theta)$, $\hat{n}(t;\theta)$ to emphasize such dependency.

Fig. 4 illustrates the mechanism of the training algorithm. SUMO data (vehicle trajectories) are aggregated as CAV and non-CAV counts on the highway section. The HQM uses its internal dynamics to predict the vehicle counts (model prediction), which is supposed to best match the counts (experimental data) given by SUMO. The training algorithm aims to minimizes the prediction error.

In the stationary setting, all parameters are time-invariant, i.e., $\theta(t)=\vartheta$ for all $t\geq 0$, where $\vartheta$ is a constant vector. In this setting, SUMO first generates the experimental data first, and then the training algorithm matches the model prediction and the experimental data by varying the model parameters. In the non-stationary setting, $\theta(t)$ is a non-constant function of $t$. In this setting, as SUMO keeps generating data, the training algorithm computes model prediction and adjusts model parameters simultaneously.

In this paper, we considered the training and validation of a hybrid queuing model for highway bottlenecks with CAV platoons. This model describes CAV platoons as discrete bulk arrivals and non-CAV traffic as continuous flow. We studied model training algorithms in both stationary and non-stationary settings. We validate the hybrid queuing model with respect to SUMO results. In the stationary setting, our model gives a prediction error of about 15%. In the non-stationary setting, the model parameters are able to track the change in the environment, and a 25% prediction error is attained. The above results provide the basis for analysis and design of learning-based platoon coordination algorithms, which are promising directions for future work.