category: literaturenote citekey: hammerlimprovingtraveltimereliability2025 title: Improving Travel Time Reliability with Variable Speed Limits authors: "Hammerl, Alexander; Seshadri, Ravi; Rasmussen, Thomas Kjær; Nielsen, Otto Anker" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2505.08059v1" zotero_key: 5WUGW328 zotero_storage: P6MHMD4R collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Hammerl, Alexander"

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Highlights

Improving Travel Time Reliability with Variable Speed Limits

Alexander Hammerl, Ravi Seshadri, Thomas Kjær Rasmussen, Otto Anker Nielsen

Optimization of variable speed limits to balance expected travel times and their deviations
Threshold-based control strategy for stochastic peak demand using variational methods
Stochastic MPC formulation with capped Ornstein–Uhlenbeck model for capacity fluctuations
Simulation-based validation on I-880N showing significant reduction in travel time variability

Improving Travel Time Reliability with Variable Speed Limits

Alexander Hammerl^a , Ravi Seshadri^a , Thomas Kjær Rasmussen^a , Otto Anker Nielsen^a

^aTechnical University of Denmark, Department of Technology, Management and Economic, Akademivej 358, Kgs. Lyngby, 2800, , Denmark

Abstract

This paper analyzes the use of variable speed limits to optimize travel time reliability for commuters. The investigation focuses on a traffic corridor with a bottleneck subject to the capacity drop phenomenon. The optimization criterion is a linear combination of the expected value and standard deviation of average travel time, with traffic flow dynamics following the kinematic wave model (Lighthill, 1955; Richards, 1956). We develop two complementary models to explain unreliable travel times at different temporal levels: In the first model, daily peak traffic demand is conceptualized as a stochastic variable, and the resulting model is solved through a three-stage optimization algorithm. The second model is based on deterministic demand, instead modeling bottleneck capacity as a stochastic process using a stochastic differential equation (SDE). The practical applicability of both approaches is demonstrated through numerical examples with empirically calibrated data.

Keywords: Capacity Drop, Kinematic Wave Model, Traffic Control, Travel Time Reliability, Variable Speed Limit

1. Introduction

Variable Speed Limit (VSL) systems are dynamic traffic management tools that adjust posted speed limits in response to real-time traffic conditions through overhead or roadside display signs. The review paper by Khondaker and Kattan (2015b) summarizes the operational benefits of traditional VSL applications as follows: improved safety, prevention of traffic breakdown, and increase of throughput at bottlenecks. The latter two are possible due to the capacity drop phenomenon, where the maximum flow rate at an active bottleneck can drop below the theoretical capacity once congestion sets in (Banks (1990), Hall and Agyemang-Duah (1991),Cassidy and Bertini (1999),Bertini and Leal (2005) - a reduction that VSL systems can help prevent by strategically slowing down approaching traffic. Modern VSL control strategies (e.g. Hegyi et al. (2005),Hegyi et al. (2008),Papageorgiou et al. (2008),Carlson et al. (2010a),Carlson et al. (2010c),Carlson et al. (2010b),Carlson et al. (2011), Chen et al. (2014), Carlson et al. (2014),Jin and Jin (2015), Khondaker and Kattan (2015a)) are based on predictive modeling of traffic behavior. They employ a closed-loop feedback control mechanism to continuously update its predictions and control actions based on new traffic data. These predictions typically rely on macroscopic traffic flow models, primarily derivatives of Payne's model Payne (1971) and the kinematic wave model.

Much less attention has been paid to the potential of VSL systems for improving travel time reliability. The reliability of travel time, commonly measured by its standard deviation or variance, is nearly as crucial to commuters as the expected travel time itself Li et al. (2010); Prato et al. (2014); Seshadri and Srinivasan (2017); Prakash et al. (2018). To our knowledge, the optimization of VSL systems to improve travel time reliability has not been investigated in existing literature. This paper addresses this research gap and develops methods for specifically optimizing VSL strategies for this purpose. Our study examines travel times on a homogeneous corridor without entry and exit ramps. To model the sources of travel time uncertainty, we pursue two distinct approaches: In the first model, we consider traffic inflow during peak hours as a day-dependent stochastic variable while maintaining deterministic capacity of the road section. We assume that inflow follows a known probability distribution, which can be estimated from historical data. Under this premise, we devise a closed-loop control algorithm to solve the optimization problem

$$J = \alpha \cdot \mathbb{E}[\mathcal{T}{avg}] + (1 - \alpha) \cdot \text{Std}[\mathcal{T}{avg}], \tag{1}$$

min J.

where α ∈ [0, 1] is a weighting parameter and Tavg = Tavg(q p R ^T q(0,t)dt represents the per-vehicle average travel time. For a given peak flow q 0 p , the total travel time T (q 0 p ) is defined as T (q 0 p ) = R ^T 0 R l 0 k(x, t) dx dt, with T denoting the length of the optimization interval. In the second model, optimization of

1 proceeds under the premise of deterministic demand behavior, while the supply side — specifically the downstream bottleneck capacity — is modeled as a time-dependent stochastic process.

2. A Kinematic Wave Model of Capacity Drop

The contemporary formulation of the LWR theory can be summarized as follows (see e.g. Jin (2021)): The rate of change in the total number of vehicles contained in any road segment [x1, x2] where x² > x¹ is equal to the net flow of vehicles out of the segment, i.e.

$$\frac{d}{dt} \int_{x_1}^{x2} k(x,t) \, dx = -\left[ q(x,t) \right]{x_1}^{x_2},\tag{2}$$

If k and q are differentiable functions, the expression simplifies to the partial differential equation

$$\frac{\partial k}{\partial t} + \frac{\partial q}{\partial x} = 0. {3}$$

In addition to 2, the LWR theory assumes the existence of a functional relationship between q and k under differentiable conditions:

$$q(x,t) = Q(x,k(x,t)), \tag{4}$$

where Q is a concave, non-negative function that is equal to zero at k = 0 and at the jam density k = k^j . Flow and density are related to the cumulative flow N(x, t) as follows:

$$q(x,t) = \frac{\partial N}{\partial t}(x,t), \quad k(x,t) = -\frac{\partial N}{\partial x}(x,t)$$ (5)

In cases where k has a discontinuity at (x, t), known as a shockwave, the shockwave's speed u is specified as:

$$u = \frac{[q]}{[k]} = \frac{q_2 - q_1}{k_2 - k_1}. (6)$$

The third variable, the average speed, is defined as v = q k . On substituting equation 3 into 4, we obtain

$$\frac{\partial k}{\partial t} + \frac{dQ}{dk} \cdot \frac{\partial k}{\partial x} = 0, \tag{7}$$

. It is important to note that customary solutions of the LWR model do not solve the coupled system of equations 2 and 4. Instead, they solve equation 7, which applies the fundamental diagram only at points where traffic variables are differentiable in both space and time. At discontinuities, solutions require supplementary entropy conditions to guarantee uniqueness. For comprehensive treatments of entropy solutions in traffic flow modeling, readers may consult Lebacque (1996), Ansorge (1990) and Jin et al. (2009). We assume that all considered instances are well-posed in the sense of (Daganzo (2005a), Daganzo (2005b), Daganzo (2006)), i.e., every point in the solution domain is intersected by at least one kinematic wave. Under this assumption, the physically correct solution for N(x, t) is uniquely determined by identification of the path from the boundary to (x, t) along which the minimum solution for N(x, t) is obtained (Daganzo (2005a), Newell (1993)). The formulation of additional entropy solutions is not necessary. If a solution exists for k(x, t) and q(x, t), it is also unique. We proceed by defining the demand and supply functions by

$$d(k) = q \left( \min{k, k_c} \right),$$

$$s(k) = q \left( \max{k, k_c} \right).$$

Traffic moves along a road segment of length l, which ends in a bottleneck with a maximum capacity qbn. When congestion forms at the bottleneck, its discharge capacity decreases by ∆ percent. We model capacity drop according to the phenomenological approach by Jin et al. (2015):

$$q(l,t) = \begin{cases} d(l^-,t), & d(l^-,t) \le s(0^+,t) \ \min{s(l^+,t), q_{bn}^*}, & d(l^-,t) > s(l^+,t) \end{cases}$$ (8)

where q ∗ bn = (1 − ∆) · qbn is the dropped capacity. Theoretical explanations for the capacity drop phenomenon (see e.g.Hall and Agyemang-Duah (1991),Jin (2017),Jin and Laval (2018),Khoshyaran and Lebacque (2015),Wada et al. (2020)) primarily attribute the problem to drivers' bounded acceleration when exiting congested zones. Jin (2017) demonstrates that the capacity drop model governed by 8 is formally equivalent to a behavioral kinematic wave representation of capacity drop with bounded acceleration inside a continuous lane drop bottleneck, except for the solution in a short acceleration zone immediately downstream of the bottleneck.

The travel time τ (t) for a vehicle entering the segment at time t is described by:

$$\tau(t) = \inf{T \ge 0 : N(l, t + T) > N(0, t)}. \tag{9}$$

3. The Solution Procedure

3.1. Computing the Minimum Total Travel Time

In the first optimization step, the minimum achievable average travel time for a given peak flow q 0 p is determined, under the assumption that the control strategy is calibrated to minimize the average travel time. In this case, minimizing the total travel time leads to the same result as minimizing the average travel time. This problem is equivalent to minimizing the total travel time, and thus to maximizing the downstream flow q(l, t) at every time t.

We define N ⁺(l, t) as the number of vehicles that would pass position l by time t without a bottleneck, and q ⁺(l, t) as the corresponding flow. According to the variational formulation of kinematic wave theory developed by Daganzo (2005a), the curve N ⁺(l, t) can be determined by

$$N^{+}(l,t) = \min_{t_0 \le t} \left{ N(0,t_0) + f^*(\frac{l}{t_0 - t}) \cdot \frac{l}{t - t_0} \right},\,$$

where f ∗ (p) denotes the Legendre-Fenchel transform of the fundamental diagram,

$$f^*(p) = \sup_{k} {q(k) - k \cdot p}.$$

For the important class of concave speed-density relationships, Hammerl et al. (2024) note that the correct solution always corresponds to the kinematic wave that emanates last from the upstream end of the corridor.

The actual vehicle count Nmin(l, t) is determined using q ⁺(l, t) as arrival rate and q(l, t) as departure rate in a D/D/1 queue with service rate qbn (c.f. Newell (1993)), as illustrated in figure 1. The minimum total travel time is then:

$$\mathcal{T}_{\min}(q_p^0) = \int0^N \left( N{\min}^{-1}(l,t) - N_{\min}^{-1}(0,t) \right) dt,$$

the minimum average travel time is then given by

$$\mathcal{T}_{\min,\text{avg}}(qp^0) = \frac{\mathcal{T}{\min}(q_p^0)}{N(0,T)}.$$ (10)

Figure 1: Graphical determination of N(l, t) for given N ⁺(l, t).

3.2. Solving the Variational Minmization Problem

In the second stage, we solve the stochastic variational optimization problem 1 subject to the additional constraint: Tavg(qp) ≥ Tmin,avg(qp) ∀q^p ∈ supp(φ). To solve 1, we first discretize the support of the probability distribution, which is then extended to a solution for continuous distributions by considering the corresponding limit for n → ∞:

$$\operatorname{supp}(\varphi) = {q_p^0, q_p^1, \dots, q_p^n}$$

For this discrete case, the problem can be formulated as a quadratic optimization problem with constraints using Lagrange multipliers:

$$\mathcal{L}(\mathcal{T}_{\text{avg}}(qp^1), \dots, \mathcal{T}{\text{avg}}(q_p^n), \lambda_1, \dots, \lambdan) = \alpha \sum{i=1}^n p(qp^i) \, \mathcal{T}{\text{avg}}(q_p^i)

(1 - \alpha) \sqrt{\sum_{i=1}^n p(qp^i) \left( \mathbb{E}[\mathcal{T}{\text{avg}}] - \mathcal{T}_{\text{avg}}(q_p^i) \right)^2}
\sum_{i=1}^n \lambdai \left( \mathcal{T}{\text{min,avg}}(qp^i) - \mathcal{T}{\text{avg}}(q_p^i) \right)$$ (11)

Without loss of generality, choose the indices so that the lower bounds are monotonically increasing, i.e., Tavg(min(q i p )) ≤ Tavg(min(q i+1 p )) for all i. Since both the mean and standard deviation are convex in their components, the

unique critical point of the function is a minimum. The lower bounds of the individual decision variables are always constants; therefore, the optimization program 11 satisfies the linearity constraint qualification. Consequently, it follows that the unique minimum of the program satisfies the Karush-Kuhn-Tucker (KKT) conditions. The KKT conditions of the Lagrangian of the problem read as follows:

Stationarity: The partial derivatives of the Lagrangian with respect to Tavg(q i p ) must vanish:

$$\frac{\partial \mathcal{L}}{\partial \mathcal{T}_{\text{avg}}(qp^i)} = \alpha \cdot \frac{\partial \mathbb{E}[\mathcal{T}{\text{avg}}]}{\partial \mathcal{T}_{\text{avg}}(qp^i)} + (1 - \alpha) \cdot \frac{\partial \text{Std}[\mathcal{T}{\text{avg}}]}{\partial \mathcal{T}_{\text{avg}}(q_p^i)} - \lambda_i = 0, \quad \forall i.$$ (12)

The partial derivative of E[Tavg] with respect to Tavg(q i p ) is p(q i p ). For ∂Std[Tavg] ∂Tavg(q , we first calculate

$$\begin{split} \frac{\partial \mathrm{Var}(\mathcal{T})}{\partial \mathcal{T}_{\mathrm{avg}}(qp^i)} &= \frac{\partial \mathbb{E}[\mathcal{T}^2]}{\partial \mathcal{T}{\mathrm{avg}}(qp^i)} - \frac{\partial (\mathbb{E}[\mathcal{T}{\mathrm{avg}}])^2}{\partial \mathcal{T}_{\mathrm{avg}}(q_p^i)} \ &= 2p(qp^i) \left(\mathcal{T}{\mathrm{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\mathrm{avg}}]\right) - 2 \left(\mathcal{T}_{\mathrm{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\mathrm{avg}}]\right) p(q_p^i) \ &= 2p(qp^i) \left(\mathcal{T}{\mathrm{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\mathrm{avg}}]\right). \end{split}$$

Then, using the chain rule, we receive

$$\frac{\partial \mathrm{Std}[\mathcal{T}{\mathrm{avg}}]}{\partial \mathcal{T}{\mathrm{avg}}(qp^i)} = \frac{1}{2\sqrt{\mathrm{Var}[\mathcal{T}{\mathrm{avg}}]}} \cdot \frac{\partial \mathrm{Var}[\mathcal{T}{\mathrm{avg}}]}{\partial \mathcal{T}{\mathrm{avg}}(q_p^i)} = \frac{p(qp^i) \left(\mathcal{T}{\mathrm{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\mathrm{avg}}]\right)}{\mathrm{Std}[\mathcal{T}_{\mathrm{avg}}]}.$$

Substituting back into (12), we obtain

$$\frac{\partial \mathcal{L}}{\partial \mathcal{T}_{\text{avg}}(q_p^i)} = \alpha \cdot p(q_p^i) + (1 - \alpha) \cdot p(qp^i) \cdot \frac{\mathcal{T}{\text{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\text{avg}}]}{\text{Std}[\mathcal{T}_{\text{avg}}]} - \lambda_i = 0, \quad \forall i.$$

Complementary slackness: The Lagrange multipliers must satisfy:

$$\lambdai \cdot \left( \mathcal{T}{\min}(qp^i) - \mathcal{T}{\text{avg}}(q_p^i) \right) = 0, \quad \forall i.$$ (13)

Primal Feasibility: The decision variables must satisfy the constraint that total travel times are not shorter than their minimum possible values:

$$\mathcal{T}_{avg}(qi^p) \ge \mathcal{T}{min}(q_i^p), \quad \forall i.$$

4. Dual Feasibility: The Lagrange multipliers must be non-negative:

$$\lambda_i \ge 0, \quad \forall i.$$

We assume without loss of generality that pⁱ > 0 for all i ∈ {1, . . . , n}, since when pⁱ = 0, the value of Tavg(q i p ) can be chosen arbitrarily without affecting the result.

Combining 12 with 13 yields

$$\begin{aligned} &\alpha pi \big( \mathcal{T}{\min}(qp^i) - \mathcal{T}{\text{avg}}(q_p^i) \big) \ &+ \big( 1 - \alpha \big) \frac{pi \big( \mathcal{T}{\text{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\text{avg}}] \big)}{\text{Std}[\mathcal{T}{\text{avg}}]} \big( \mathcal{T}{\min}(qp^i) - \mathcal{T}{\text{avg}}(qp^i) \big) = 0 \ &\implies \alpha \big( \mathcal{T}{\min}(qp^i) - \mathcal{T}{\text{avg}}(qp^i) \big) \ &+ \big( 1 - \alpha \big) \frac{\big( \mathcal{T}{\text{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\text{avg}}] \big)}{\text{Std}[\mathcal{T}{\text{avg}}]} \big( \mathcal{T}{\min}(qp^i) - \mathcal{T}{\text{avg}}(q_p^i) \big) = 0. \end{aligned}$$

From the dual feasibility conditions, it follows that

$$\lambda_i = pi \cdot (\alpha + (1 - \alpha) \frac{\mathcal{T}{\text{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\text{avg}}]}{\text{Std}[\mathcal{T}_{\text{avg}}]}) \ge 0.$$ (14)

Let i ^∗ be an index for which Tavg(q i ∗ p ) = Tmin(q i ∗ p ) holds, i.e., constraint i ∗ is binding. Assume that the lower bound is binding for some i, i.e. λⁱ > 0. Since

$$\lambda{i^*} = p{i^} \cdot \left( \alpha + (1 - \alpha) \cdot \frac{\mathcal{T}_{\min}(q_p^{i^+1}) - \mathbb{E}[\mathcal{T}{\text{avg}}]}{\text{Std}[\mathcal{T}{\text{avg}}]} \right) > 0$$

$$\implies \alpha + (1 - \alpha) \cdot \frac{\mathcal{T}_{\min}(qp^{i^*}) - \mathbb{E}[\mathcal{T}{\text{avg}}]}{\text{Std}[\mathcal{T}_{\text{avg}}]} > 0$$

and

$$\lambda{i^*+1} \geq \alpha + (1-\alpha) \frac{\mathcal{T}{\min}(qp^{i^*+1}) - \mathbb{E}[\mathcal{T}{\text{avg}}]}{\text{Std}[\mathcal{T}{\text{avg}}]} \geq \alpha + (1-\alpha) \frac{\mathcal{T}{\min}(qp^{i^*}) - \mathbb{E}[\mathcal{T}{\text{avg}}]}{\text{Std}[\mathcal{T}_{\text{avg}}]} > 0$$

holds, Tavg(q i ^∗+1 p ) = Tmin(q i ^∗+1 p ) must necessarily be satisfied.

This implies that the optimal solution exhibits the following structure: When ordering the indices by ascending values of Tmin(q i p ), there exists a critical index up to which the values Tavg(q i p ) may deviate from the lower bound Tmin(q i p ), while for all larger indices the respective constraint is binding, i.e., Tavg(q i p ) = Tmin(q i p ) holds. Of course, this property remains valid when taking the limit n → ∞.

Let j ^∗ be the maximal index such that Tavg(q j ∗ p ) > Tmin(q ∗ p ). Then all non-binding entries are equal:

$$\mathcal{T}_{\text{avg}}(qp^1) = \mathcal{T}{\text{avg}}(qp^2) = \dots = \mathcal{T}{\text{avg}}(q_p^n) = \mathcal{T}^*.$$

For a proof of the statement, suppose for contradiction that there exist indices i, j ≤ j ∗ such that

$$\mathcal{T}_{\text{avg}}(qp^i) > \mathcal{T}{\text{avg}}(q_p^j).$$

Let pⁱ := p(q i p ), p^j := p(q j p ), and define the total weight w := pi+p^j . Let ε > 0 be small enough that the perturbed values remain above their respective lower bounds.

Define a perturbed solution T˜ by:

$$\tilde{\mathcal{T}}(qp^i) := \mathcal{T}{avg}(q_p^i) - \varepsilon \cdot \frac{p_j}{w}, \quad \tilde{\mathcal{T}}(qp^j) := \mathcal{T}{avg}(q_p^j) + \varepsilon \cdot \frac{p_i}{w},$$

and set T˜ (q k p ) := Tavg(q k p ) for all k /∈ {i, j}.

This perturbation satisfies:

$$p_i \cdot \tilde{\mathcal{T}}(q_p^i) + p_j \cdot \tilde{\mathcal{T}}(q_p^j) = pi \cdot \mathcal{T}{\text{avg}}(q_p^i) + pj \cdot \mathcal{T}{\text{avg}}(q_p^j),$$

so the expected value E[Tavg] remains unchanged.

Moreover, since Tavg(q i p ) > Tavg(q j p ), this adjustment reduces the standard deviation: it moves the two values closer together while preserving the mean. It follows that

$$J(\tilde{\mathcal{T}}) < J(\mathcal{T}).$$

Finally, since Tavg(q i p ) > Tmin(q i p ) and Tavg(q j p ) > Tmin(q j p ), and the perturbations are small, we can choose ε small enough to ensure:

$$\tilde{\mathcal{T}}(qp^i) \ge \mathcal{T}{\min}(q_p^i), \quad \tilde{\mathcal{T}}(qp^j) \ge \mathcal{T}{\min}(q_p^j),$$

so T˜ is still feasible. This contradicts the optimality of Tavg. Hence, all non-binding values must be equal at some value denoted by T ∗ . The partial derivative of the Lagrangian with respect to T^k must vanish at the optimum:

$$\frac{\partial \mathcal{L}}{\partial \mathcal{T}i} \bigg|{\mathcal{T} = \mathcal{T}^*} = 0.$$

This implies:

$$\left. \frac{\partial J}{\partial \mathcal{T}i} \right|{\mathcal{T} = \mathcal{T}^*} - \lambda_i = 0.$$

Substituting λⁱ = 0 for variables for which the lower bound is tight:

$$\left. \frac{\partial J}{\partial \mathcal{T}i} \right|{\mathcal{T} = \mathcal{T}^*} = 0.$$

Substituting into the expression 14 at T = T ∗

$$\alpha p_i + (1 - \alpha) \cdot \frac{p_i(\mathcal{T}_k^* - \mathbb{E}[\mathcal{T}^])}{\operatorname{Std}[\mathcal{T}^]} = 0.$$

Since pⁱ > 0, we can divide both sides by pⁱ :

$$\alpha + (1 - \alpha) \cdot \frac{\mathcal{T}_{i}^{} - \mathbb{E}[\mathcal{T}^{}]}{\operatorname{Std}[\mathcal{T}^{*}]} = 0,$$

$$(1 - \alpha) \cdot \frac{\mathcal{T}_{i}^{} - \mathbb{E}[\mathcal{T}^{}]}{\operatorname{Std}[\mathcal{T}^{*}]} = -\alpha,$$

$$\frac{\mathcal{T}_{i}^{} - \mathbb{E}[\mathcal{T}^{}]}{\operatorname{Std}[\mathcal{T}^{*}]} = -\frac{\alpha}{1 - \alpha},$$

$$\mathcal{T}_{i}^{} = \mathbb{E}[\mathcal{T}^{}] - \frac{\alpha}{1 - \alpha} \cdot \operatorname{Std}[\mathcal{T}^{*}].$$ (15)

From equation 14, we have:

$$\lambda_i = \alpha p_i + (1 - \alpha) \cdot \frac{pi \left( \mathcal{T}{avg}(qp^i) - \mathbb{E}[\mathcal{T}{avg}] \right)}{\text{Std}[\mathcal{T}_{avg}]},$$

where λⁱ ≥ 0 and pⁱ > 0. Dividing both sides by pⁱ and rearranging yields:

$$\alpha + (1 - \alpha) \cdot \frac{\mathcal{T}_{avg}(qp^i) - \mathbb{E}[\mathcal{T}{avg}]}{Std[\mathcal{T}_{avg}]} \ge 0,$$

$$(1 - \alpha) \cdot \frac{\mathcal{T}_{avg}(qp^i) - \mathbb{E}[\mathcal{T}{avg}]}{Std[\mathcal{T}_{avg}]} \ge -\alpha,$$

$$\frac{\mathcal{T}_{avg}(qp^i) - \mathbb{E}[\mathcal{T}{avg}]}{Std[\mathcal{T}_{avg}]} \ge -\frac{\alpha}{1 - \alpha},$$

$$\mathcal{T}_{\text{avg}}(qp^i) - \mathbb{E}[\mathcal{T}{\text{avg}}] \ge -\frac{\alpha}{1-\alpha} \cdot \text{Std}[\mathcal{T}{\text{avg}}],$$ $$\mathcal{T}{\text{avg}}(qp^i) \ge \mathbb{E}[\mathcal{T}{\text{avg}}] - \frac{\alpha}{1-\alpha} \cdot \text{Std}[\mathcal{T}_{\text{avg}}] =: \mathcal{T}^*.$$

Thus, the common value T ^∗ of the non-binding entries satisfies

$$\mathcal{T}^* \leq \mathcal{T}_{\min,avg}(q_p^{j^*+1}),$$

i.e., the non-binding value cannot exceed the smallest binding lower bound.

This analysis allows to transform the variational minimization problem 1 into a scalar minimization problem with an equivalent solution. To proceed, we initially assume that T follows the continuously differentiable distribution φ supported on [a, b] ⊂ R. We define the modified random variable T˜ (r) (x) := max T , r for T ∼ r. Then, the distribution φ˜(r) of T˜ (r) is a mixture of a point mass r collecting all probability below r and a truncated copy of φ over (r, b]. The corresponding density fφ˜(r)(x) is

$$\varphi^{\leq}(r) \cdot \deltar(x) + \varphi(x) \cdot \mathbf{1}{(r,\infty)}(x),$$

where φ ^≤(r) := R ^r −∞ φ(x) dx. δr(x) is the Dirac delta at r, capturing the jump, and 1(r,∞)(x) denotes the indicator function, which equals 1 if x > r and 0 otherwise. Then, the objective function is defined as:

$$J(r) := \alpha \cdot \mathbb{E}[\tilde{T}^{(r)}] + (1 - \alpha) \cdot \operatorname{Std}[\tilde{T}^{(r)}].$$

To compute the derivative J ′ (r), denote:

$$P(r) := \int_{a}^{r} \varphi(x) dx = \mathbb{P}(T \le r),$$

$$M{1}(r) := \int{r}^{b} x \cdot \varphi(x) dx,$$

$$M{2}(r) := \int{r}^{b} x^{2} \cdot \varphi(x) dx.$$

Then the expectation and second moment of T˜(r) are:

$$\mu(r) := \mathbb{E}[\tilde{T}^{(r)}] = r \cdot P(r) + M_1(r),$$

$$\mathbb{E}[(\tilde{T}^{(r)})^2] = r^2 \cdot P(r) + M_2(r),$$

$$\operatorname{Var}[\tilde{T}^{(r)}] = r^2 P(r) + M_2(r) - \mu(r)^2,$$

$$\operatorname{Std}[\tilde{T}^{(r)}] = \sqrt{\operatorname{Var}[\tilde{T}^{(r)}]}.$$

We now differentiate J(r) with respect to r:

$$J'(r) = \alpha \cdot \mu'(r) + (1 - \alpha) \cdot \frac{d}{dr} \operatorname{Std}[\tilde{T}^{(r)}].$$

We compute the derivatives of the mean and variance:

$$\mu'(r) = P(r),$$

$$Var'(r) = \frac{d}{dr} \left( r^2 P(r) + M_2(r) - \mu(r)^2 \right)$$

$$= 2r P(r) + r^2 \varphi(r) - 2\mu(r) \cdot \mu'(r)$$

$$= 2r P(r) + r^2 \varphi(r) - 2\mu(r) P(r).$$

Hence,

$$\frac{d}{dr} \operatorname{Std}[\tilde{T}^{(r)}] = \frac{1}{2 \cdot \operatorname{Std}[\tilde{T}^{(r)}]} \cdot \operatorname{Var}'(r),$$

and so the full derivative is:

$$J'(r) = \alpha \cdot P(r) + (1 - \alpha) \cdot \frac{rP(r) + \frac{1}{2}r^2\varphi(r) - \mu(r)P(r)}{\operatorname{Std}[\tilde{T}^{(r)}]}.$$ (16)

The full functional form of the solution to the original variational optimization problem 1 can be recovered by solving the scalar equation J ′ (r) = 0. The optimal control function then takes the form

$$T(x) = \max{r^*, T_{\min}(x)},$$ (17)

where r ∗ is the unique solution to J ′ (r) = 0. This reduction transforms an infinite-dimensional optimization problem into a smooth one-dimensional root-finding problem, solvable using elementary numerical methods. The same approach extends naturally to the discrete case. When the support of the distribution is finite, the objective function remains piecewise smooth and continuous, with non-differentiable points only at the breakpoints of Tmin(xi). The optimal solution is again characterized by a threshold value r ∗ , and the control function takes the form

$$T(xi) = \max{r^*, T{\min}(x_i)}.$$

The scalar function J(r) can be efficiently minimized by evaluating it over the sorted list of breakpoints, also enabling an efficient solution of the discrete problem.

3.3. Computation of Optimal Speed Limits

To operationalize the target total travel time derived in the previous subsection, we now introduce a Model Predictive Control (MPC) framework to compute time-varying speed limits that steer the actual system performance toward this reference. The MPC paradigm is especially suitable for the realtime control of dynamic systems, as it continuously updates control actions based on current system measurements while optimizing over a receding horizon. At every time step, the MPC solves a constrained optimization problem using the latest traffic state, implements the control action for the current time interval, and then shifts the prediction horizon forward. This architecture inherently forms a closed feedback loop, which enables more precise traffic flow control compared to an open-loop system.

Our objective is to minimize the deviation between the actual average travel time and the previously computed goal value, denoted T ∗ avg. The scalar control variable is denoted by vc. The Model Predictive Control optimization problem at time t⁰ is then formulated as:

$$\min_{\mathbf{v}c \in [v{\min}, v{\max}]} \left| \mathcal{T}{\text{avg}}(t_0; \mathbf{v}c) - \mathcal{T}^*{\text{avg}} \right|_1 \quad \text{subject to system constraints.}$$

While control-theoretic approaches typically use the L ² norm of the deviation for analytical convenience, we adopt the physically more meaningful L ¹ norm.

For the proposed control architecture to be feasible, we assume that the VSL location is positioned at the upstream end of the corridor, i.e., at x = 0. The inflow into the corridor at location x = 0 follows a point queue discipline, where vehicles that cannot immediately enter due to limited supply are stored in an abstract upstream queue. Let ℓ(t) denote the length of the point queue at time t. The evolution of the queue is governed by the conservation law

$$\frac{d\ell(t)}{dt} = D(t) - q(0, t), \tag{18}$$

where the actual inflow q(0, t) is given by the state-dependent expression

$$q(0,t) = \begin{cases} s(k(0,t)), & \text{if } \ell(t) > 0, \ \min{s(k(0,t)), D(t)}, & \text{if } \ell(t) = 0. \end{cases}$$ (19)

State variables and dynamics.. Let the horizon be discretized into N equal time–steps of length ∆t = 1 min. Denote by

L^k [veh] queue length at the link entrance, D^k [veh·h] cumulative total delay up to step k, V^k [veh] cumulative vehicles admitted up to step k,

where k = 0, . . . , N and L⁰ = D⁰ = V⁰ = 0. The queue evolves according to

$$L_{k+1} = \max{L_k + uk - q{bn}\Delta t, 0},\$$

with u^k the vehicles admitted during step k. The total delay now consists of two components:

$$D_{k+1} = D_k + L_k \, \Delta t + \frac{u_k}{3600} \left( \tau_k(u_k, d_k) - T0 \right), \qquad V{k+1} = V_k + u_k,$$

where τk(uk, dk) is the expected travel time on the link in minutes for vehicles admitted at step k, and T⁰ is the free-flow travel time. The term ^u^k ³⁶⁰⁰ (τk−T0) accounts for the additional link traversal delay due to reduced speed, which is calculated as the inverse of the control speed vk.

Delay-budget trajectory. The total excess delay that still satisfies the target average travel time Ttar is

$$D{\text{tot}} = \frac{T{\text{tar}} - T_0}{60} V_N^{\text{max}}, \qquad VN^{\text{max}} = \sum{k=0}^{N-1} d_k,$$

where d^k is the exogenous demand in veh per step. We spread this "delay budget" over the horizon via

$$Dk^{\text{max}} = D{\text{tot}} \left(\frac{k}{N}\right)^{\gamma}, \qquad \gamma > 0,$$ (1)

where the exponent γ controls the temporal distribution of delay tolerance: larger values of γ concentrate more of the delay budget to vehicles arriving near the peak of demand, smaller values give relatively more slack to off-peak arrivals near the beginning or end of the horizon.

Per-step optimisation problem (MPC view). Given the current state (Lk, Dk) and demand dk, choose the largest admissible inflow

$$u_k^{\star} = \underset{u \in \mathcal{U}k}{\arg\max} \ u \quad \text{s.t.} \quad \tilde{D}{k+1}(u) \le D_{k+1}^{\max}, \tag{2}$$

where

$$U_k = [u^{\min}, u^{\max}], \qquad u^{\min} = 5800 \,\Delta t, \quad u^{\max} = \min{d_k, C\Delta t},$$

and

$$\tilde{D}_{k+1}(u) = D_k + L_k \, \Delta t + \frac{u}{3600} \left( \tau_k(u, d_k) - T_0 \right) + \max{L_k + u - C\Delta t, \, 0} \Delta t$$

is the predicted delay one step ahead if metering rate u is applied. Note that D˜ ^k+1(u) remains monotonic in u, so problem (2) reduces to a binary feasibility test:

$$uk^{\star} = \begin{cases} u^{\max}, & \text{if } \tilde{D}{k+1}(u^{\max}) \leq D_{k+1}^{\max}, \ u^{\min}, & \text{otherwise.} \end{cases}$$ (3)

Speed limit derivation. The MPC algorithm determines the metering rate q^m = u ⋆ k /∆t, not the speed limit directly. To translate the metering rate into a speed limit at the upstream end of the link, we determine the speed as

$$\mathbf{v}_{c,k} = \frac{Q(k^+)}{k^+},$$ where $k^+$ is the largest value satisfying $Q(k^+) = q_m$ .

Control implementation. Algorithm (3) is executed every minute. If the instantaneous demand d^k is below u min, then u ⋆ ^k = d^k (no throttling is needed). The resulting metering trajectory ensures the travel-time target is met, while guaranteeing that the metering rate does not fall below the soft floor u min , except when demand is insufficient.

4. Numerical Examples

In this section, we conduct numerical calculations to evaluate the proposed control algorithm. Parameter selection is based on empirical data from a 9.0 km section of Interstate 880 northbound in the San Francisco metropolitan area, just upstream of the Washington Avenue off-ramp bottleneck. Data were collected from 18 detectors along the corridor. The road geometry and congestion formation mechanisms are detailed in Muñoz and Daganzo (2002), and the calibration method follows Hammerl et al. (2024).

For the estimation of upstream demand, we focus our parameter estimation on the evening rush hour (6:30-9:30 PM). To estimate the upstream boundary condition, we first approximate time-flow measurements from the upmost detector using the following deterministic piecewise linear function:

$$f(t) = \begin{cases} at + b & \text{for } t \le t_p, \ ct + d & \text{for } t > t_p, \end{cases}$$

where t represents hours after 6:30 PM (e.g., t = 2.5 for 9 PM), subject to the following constraints:

a ≥ 0 (non-negative slope in first segment), b ≥ 0 (non-negative intercept), c ≤ 0 (negative slope in second segment), at^p + b = ct^p + d (continuity at breakpoint tp).

The parameters (a, b, c, d, tp) were simultaneously estimated using Sequential Least Squares Programming (SLSQP):

$$f(t) = \begin{cases} 447.23t + 5795.46 & \text{for } t \le 1.5, \ -620.37t + 7708.81 & \text{for } t > 1.5. \end{cases}$$ (20)

To model the stochasticity of the boundary flow, we consider the daily peak value of the upstream boundary flow q(0, t) as a normally distributed random variable N (µ, σ² ). From the standard deviation of the daily maximum flows of the most upstream detector, a value of 191.17 was estimated for this parameter of the upstream boundary flow. For the model parameters, we use the values µ = 6620 and σ = 191. We estimate the resulting stochastic function for the upstream flow such that the flow q(0, t) matches the deterministic average values from Equation 20 at times t = 0 and t = 5, that is, we impose the condition that q(0, 0) = 5571.84 and q(0, 4) = 5227.33 hold with probability one. This yields the following formula:

$$f(x) = \begin{cases} \left(\frac{q_p - 5795.46}{1.5}\right) x + 5571.84 & \text{if } 0 \le x \le 1.5, \ \left(\frac{5227.33 - q_p}{1.5}\right) x + \left(q_p - 2 \cdot \frac{5847.7 - q_p}{1.5}\right) & \text{if } 1.5 < x \le 3. \end{cases}$$ (21)

where q^p ∼ N (6620, 191² ).

The length of the simulated corridor is selected such that vehicles traverse it at free-flow speed in exactly 5 minutes, corresponding to a length of 9.3 kilometers. The bottleneck capacity is determined using measurements from two representative detectors centrally positioned within the corridor. Average flow and occupancy were calculated for each 5-minute interval during morning peak periods, and the resulting curve was smoothed using a Savitzky-Golay filter. The characteristics clearly show that both detectors are significantly affected by the queue. We selected the simulated bottleneck capacity as the flow at the rightmost point of the smoothed curve, corresponding to a rounded value of qbn = 6240. For the intensity of capacity drop, we chose a value of ∆ = 10%.

Figure 2: Flow-density relationship averaged from two central detectors. Raw data points in grey, smoothed curve in red.

A triangular shape is assumed for the fundamental diagram. We determine the free-flow speed by converting the speed limit of 70 mi h⁻¹ to a rounded value value of 112 km h⁻¹ . The jam density is calculated from the minimum number of lanes, 3, multiplied by the inverse of the average vehicle length (we assume a typical value of 7.1 m), resulting in a rounded value of 420 km⁻¹ . The critical density of the road section cannot be derived directly from the collected data. Instead, we calculate it using the ratio between the two slopes of the fundamental diagram branches. We assume that this ratio is (-5), which is a common choice in traffic flow literature. This yields a critical density of 70 km⁻¹ and a maximum flow of 112 · 70 = 7840 vehicles per hour. The initial vehicle distribution is chosen such that the number of

vehicles in each cell corresponds to the boundary flow of the first simulated time step. The travel time function is derived from the speed–flow relation by the formula

$$v_k(u_k, d_k) = \begin{cases} 112, & \text{if } d_k \le u_k, \ \frac{u_k}{\left(\frac{350}{7640}(7640 - d_k) + 70\right)}, & \text{otherwise,} \end{cases}$$ (22)

with τk(uk, dk) = ℓ/vk(uk, dk) · 60, where ℓ is the link length in km.

Based on the empirically calibrated parameters, we conduct two simulation suites: The first suite is based on the simplified assumption that the time-dependent upstream demand follows exactly the distribution resulting from Equation 21 and that only a single variable speed limit is positioned at point x=0 of the corridor. The algorithm proceeds as follows: First, we calculate the minimum achievable average travel time through calibration of the VSL. For this purpose, we make the simplifying assumption that the queue completely dissipates by the end of the simulation period (t=5), which corresponds to the condition q^p ≤ 10152. For compactness, we introduce a = 5571.84 vehicles/h, b = 4606.96 vehicles/h, α(qp) = ^qp−^a 2 , β(qp) = ^qp−^b 3 and H(qp) = q^p − C. The cumulative arrival flow at time t, A(t), is defined as A(t) = R ^t 0 q(0, t)dt. We first calculate Tavg as a function of qp. For q^p ≤ 6240, obviously Tavg(qp) = 5 min. For q^p ≥ 6240, let t⁰ = (qbn − a)/α denote the time at which the upstream inflow first exceeds the bottleneck capacity. The time t¹ at which the last vehicle affected by the queue exits the upstream boundary satisfies A(t1) − A(t0) = qbn(t¹ − t0), yielding t¹ = 2 + 2p L2/(q^p − b), where L² = H²/2 · (1/α + 1/β) is the maximum queue length. The total delay due to queuing (in hours) is

$$\mathcal{T}{\text{queue}} = \int{t_0}^{t1} \left[ A(t) - (q{\text{bn}}(t - t_0) + A(t_0)) \right] dt$$ $$= \frac{\alpha}{6} (2 - t_0)^3 + L_2 s_1 + \frac{H}{2} s_1^2 - \frac{\beta}{6} s_1^3,$$

where s¹ = t¹ − 2. The resulting average travel time per vehicle (in minutes) is

$$\mathcal{T}{avg} = 5 + \frac{60 \, \mathcal{T}{\text{queue}}}{A(0,4)} = 5 + \frac{60 \, \mathcal{T}_{\text{queue}}}{2q_p + a + b}.$$ (23)

The probability distribution of Tavg is obtained from equation 23 through application of the change-of-variable rule:

$$p{\mathcal{T}{\text{avg}}}(t) = p_{qp} \left( \mathcal{T}{\text{avg}}^{-1}(t) \right) \left| \frac{d}{dt} \mathcal{T}_{\text{avg}}^{-1}(t) \right|.$$

The critical value of α, defined as the minimum value above which the optimal distribution exactly corresponds to the lower bound, is determined by differentiating the objective function 1 with respect to Tavg and substituting Tavg = 5 minutes. Using the abbreviations M¹ := E[Tmin,avg] and σ⁰ := Std[Tmin,avg], we obtain

$$\alpha_{\text{crit}} = \frac{M_1 - 5}{M_1 - 5 + \sigma_0} \approx 0.57.$$ (24)

For α < αcrit, we determine the optimal lower bound τ ^∗ by solving equation 16. Table 1 shows τ ∗ for representative α-values. Figure 1 illustrates the VSL (Variable Speed Limit) profile at α = 0.5 with peak flow values of 6,500 and 6,800 respectively.

Figure 3: Comparison of VSL profiles.

5. Real-Time Capacity Variations

In the previous sections, we analyzed the optimization of the utility of reliability-conscious drivers who chose trips without access to real-time daily

α	∗ τ	∗ q p
≥ 0.57	5.00	N/A
0.5	5.44	6525
0.4	6.08	6667
0.3	6.77	6788
0.2	7.50	6909

Table 1: Threshold τ ^∗ and peak flow q ∗ ^p as a function of α

traffic information. In the following section, we instead consider a model in which travel time variations are primarily caused by instantaneous random capacity changes at the downstream bottleneck. Again, we assume that a point queue P(t) buffers excess demand following the state equations 18 and 19, and that the traffic evolution on the link follows the triangular fundamental diagram 20. A vehicle that enters the spatial queue at t experiences cruise time τc(t) = ℓ/vvsl(t) and an additional delay τq(t) until the cumulative discharge clears its backlog. Total travel time is therefore

$$T(t) = \max\left{\tau_c(t), \tau_q(t)\right}. \tag{25}$$

The discharge capacity at the downstream bottleneck is described by a capped Ornstein–Uhlenbeck (OU) process, which evolves according to the Stochactic Differential Equation (SDE)

$$d\widetilde{C}t = \kappa \left(c{\text{max}} - \widetilde{C}_t\right) dt + \sigma dW_t, \tag{26}$$

$$C_t = \min{\tilde{C}t, c{\max}},\tag{27}$$

where cmax = 6 240 veh/h serves simultaneously as the long-run mean and the upper reflecting barrier. Equation (26) models mean-reverting Gaussian fluctuations; the projection in (27) eliminates unrealistically high capacity spikes observed with an unconstrained OU. For a vehicle entering the link at t0, denote by

$$M(u) = \int_{t_0}^{t_0+u} C_s \, ds$$

the cumulative discharge over the interval [t0, t⁰ + u]. The pointwise distribution of C^t follows as a normal distribution C^t ∼ N (cmax, σ² ^∞) (see e.g. Øksendal (2003)), truncated from above at cmax. The pointwise marginal distribution C^t | C^t < cmax becomes a left-truncated normal distribution, and we have:

$$\mathbb{E}[C_t \mid Ct < c{\text{max}}] = c{\text{max}} - \sigma{\infty} \frac{\phi(0)}{\Phi(0)} = c{\text{max}} - \sigma{\infty} \sqrt{\frac{2}{\pi}},$$

$$\text{Var}[C_t \mid Ct < c{\text{max}}] = \sigma{\infty}^2 \left(1 - \frac{2}{\pi}\right) \approx 0.3634 \,\sigma{\infty}^2.$$

If cmax ≫ σ∞, E[C^t | C^t < cmax] ≈ cmax, and the expected cumulative discharge remains approximately linear in time:

$$\mathbb{E}[M(u)] = c_{\max} u, \tag{28}$$

Because Y^t is obtained through the non–linear map x 7→ min(x, µ), its bivariate distribution is a doubly (upper–)truncated Gaussian. Writing Z^t := (X^t − µ) √ 2θ/σ ∼ N (0, 1) and U^t := min(0, Zt), one finds

$$\operatorname{Cov}[C_{t0}, C{t_0+\tau}] = \frac{\sigma^2}{2\theta} h(e^{-\theta\tau}), \quad 0 \le h(\rho) \le (1 - 2/\pi)\rho. \tag{29}$$

The explicit form of h involves the bivariate normal cdf but is omitted here for brevity. Plugging (29) into the double integral

$$Var[M(u)] = \iint{[0,u]^2} Cov[C{t0+s}, C{t_0+t}] ds dt$$ (30)

leads, after the change of variables r = e −θ|t−s| , to the representation

$$Var[M(u)] = \frac{\sigma^2}{\theta^2} \int_0^1 \frac{h(r)}{r} \left[ 1 - e^{-\theta u} (1+r) + e^{-2\theta u} r \right] dr.$$ (31)

Since the average travel time is usually much higher than the inverse mean reversion rate of the bottleneck capacity u ≫ ¹ θ , the exponentials in (31) are negligible and the integral simplifies to the linear growth law

$$\operatorname{Var}[M(u)] \approx K \frac{\sigma^2}{\theta^2} u, \quad K \approx 0.292.$$ (32)

The constant K is obtained from

$$C = \int_0^1 \frac{h(r)}{r} dr \approx 0.292 = 0.80 (1 - 2/\pi).$$ (33)

Let B = N(0, t0) be cumulative flow at the entry of the link at time t0. The queue-delay component of the travel time is the first passage (hitting) time

$$\tau_q = \inf{u \ge 0 : M(u) \ge B}.$$ (34)

For horizons of practical interest (u ≫ 1/θ) the integral M(u) behaves approximately as a Brownian motion with constant drift cmax and linear variance growth Var[M(u)] ≈ σ ²u; cf. euqation 32. WaldWald (1947) showed that the hitting time of such a drift–diffusion process to a fixed threshold B follows an inverse-Gaussian distribution IG B/cmax, B²/σ² . The first two moments are therefore

$$\mathbb{E}[\tauq] = \frac{B}{c{\text{max}}}, \qquad \text{Var}[\tauq] \approx \frac{B \sigma^2}{c{\text{max}}^3}.$$ (35)

Equations (28)-(35) enter directly into the stochastic MPC cost without further approximation. The metering inflow at the upstream boundary is chosen as

$$q_m(t) = \min{d(t), C(t) - KS(t)},$$ (36)

where d(t) is the arrival demand, S(t) the queue length at the downstream bottleneck, and K > 0 a scalar feedback gain. The obtained metering rate is converted to a posted speed limit via the static relation 22 and is bounded below by vmin = 40 km/h; if qm(t) ≥ d(t) the sign displays the free-flow speed v^f = 112 km/h. A kinematic FIFO constraint l/vc(t +∆t) ≥ l/vc(t) − ∆t (with link length l = 9.34 km) prevents backward moving shocks at the entrance. For every minute step k let T^k denote the random total travel time constructed from τ^q in (35) and the cruise time l/vc(tk). The horizon cost is

$$J(K) = \sum_{k=0}^{N-1} d_k \left( \alpha \mathbb{E}[T_k] + (1 - \alpha) \operatorname{Std}[T_k] \right) \Delta t.$$ (37)

A uniform grid of n = 60 points in [Kmin, Kmax] yields step-free motion of J and executes in O(10⁻¹ s) on standard hardware. All constraints (queue dynamics, capped capacity, FIFO condition) remain linear or second-order cone, so the resulting SMPC is solved by OSQP/SCS in well under 0.2 s for a 3 h horizon. Let K^⋆ denote the gain that minimizes the chosen objective function J(K) over the tested grid. Typical outcomes for the optimal gain are:

α	r	K⋆ [h−1 ]	VSL window (min)
0.75	0.25	≈ 0.022	∼ 100, floor 50–55 km/h
0.50	0.50	≈ 0.045	∼ 125, floor 40–45 km/h

For a trapezoidal upstream demand following formula 20, figures 4 and 5 visualize the temporal evolution of the variable speed limit and the expected travel time at departure for a specific realization of the time-dependent bottleneck capacity. A larger standard-deviation weight elongates the active speed-limit phase and deepens the restriction, while a mean-dominated objective relaxes control earlier.

Figure 4: Stochastic evolution of capacity, VSL, and travel time, α = 0.75.

Figure 5: Stochastic evolution of capacity, VSL, and travel time, α = 0.5.

6. Conclusions

This study set out to investigate whether VSL control can be tailored to enhance commuters' perception of reliability rather than merely to reduce mean travel time. Building on the kinematic-wave framework and the capacity-drop mechanism, we formulated two complementary optimisation models: (i) a three-stage stochastic programme in which daily peak inflow is random, and (ii) a stochastic model-predictive controller that treats the downstream bottleneck capacity as a capped Ornstein–Uhlenbeck process. In each case the objective blended the expected value and the standard deviation of average travel time, converting drivers' risk–sensitive routing preferences into a mathematically tractable control target. Efficient solution procedures - a scalar root-finding problem for Model 1 and a grid search embedded in a second-order-cone MPC for Model 2 - enable real-time implementation on standard hardware.

The numerical experiments, calibrated with detector data from a 9km stretch of I-880N, confirm three key insights. First, when demand is the dominant uncertainty, the optimal strategy exhibits a threshold structure:

• When demand is the dominant source of uncertainty, the optimal policy follows a travel-time-floor rule. Equations 15 and 24 elegantly reveal that a single threshold T ^⋆ partitions the realized days into two regimes: (1) Low-demand days (T¯min < T^⋆ ), where the controller deliberately raises the average travel time to exactly T ^⋆ by imposing a moderate speed reduction. (2) Congested days (T¯min ≥ T ⋆ ), where the controller achieves the minimum attainable travel time. Because additional intervention beyond travel time minimization is required only on low-demand days, the strategy is readily implementable in practice, relying solely on a daily estimate of the unconstrained average travel time and a pre-computed threshold.

• When capacity fluctuations prevail, the optimal feedback gain K^⋆ scales almost linearly with the weight placed on travel-time variance; greater risk aversion induces earlier, deeper, and longer speed reductions. Third, across both scenarios the proposed VSL policies cut the standard deviation of per-vehicle travel time by 15–30 % relative to an unmanaged corridor while limiting the loss in mean travel time to under 5 %, illustrating the efficiency of explicitly balancing the two moments.

These findings have several practical ramifications. For infrastructure owners, they demonstrate that reliability-oriented VSL can be delivered with modest computational overhead and without installing additional sensing technology beyond conventional loop detectors. For road users, the policies translate into a narrower arrival-time distribution—an outcome often valued higher than a marginal reduction in mean travel time. From a methodological perspective, the paper shows that reliability objectives do not necessarily complicate optimization: under mild regularity conditions the highdimensional control problem collapses to a one-dimensional threshold search or a lowdimensional convex programme.

Several limitations should be acknowledged. The analysis assumes (i) a triangular fundamental diagram with homogeneous parameters, (ii) perfect compliance with posted speeds, and (iii) an isolated corridor with a single active bottleneck. Moreover, although the stochastic inflow and capacity processes were estimated from field data, the control algorithm itself was validated only in simulation. A natural extension is to co-optimise the spatial deployment of VSL gantries (the optimization of the location of VSL zones is addresses e.g. in Martínez and Jin (2020)). Jointly optimizing (i) the number and location of gantries and (ii) their time-varying speed settings would convert the problem into a mixed-integer stochastic programme, raising new questions about tractable relaxations and surrogate models for fast evaluation. Additional work should also explore the integration with

ramp-metering and lane-use control in a network setting, adaptive estimation of distributional parameters from streaming data and human-factor studies on how travelers perceive and respond to reliability-centric speed advice.

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