Dwell-Travel Time Quantiles

• Dwell-travel time quantiles are defined as the conditional quantiles of dwell or travel times, computed via inverse CDF methods under statistical regularity conditions.
• They incorporate methodologies such as empirical process theory, phase-type distributions with ODE embeddings, and root-finding algorithms to ensure precise quantile estimation.
• Advanced frameworks integrate deep learning and Markov models to produce calibrated uncertainty estimates for enhanced traffic management and stochastic modeling.

Dwell-travel time quantiles describe the conditional quantiles of random variables representing the time spent (dwell time) in a state or the total travel time along a specified path. Accurate and operational estimation of dwell or travel time quantiles is central in applications ranging from traffic management to stochastic compartmental modeling. A rigorous foundation for these quantiles involves the statistical properties of their empirical estimators, parametric representations via phase-type (PH) distributions or Markov models, and contemporary deep learning-based regressor architectures. This entry details formal definitions, computational methodologies, and modeling frameworks for dwell and travel time quantiles as established in the technical literature.

1. Formal Definitions and Empirical Quantile Properties

For a stochastic process $\{X_t: t \in E\}$, where $X_t$ denotes the dwell or travel time at index $t$ (with $E$ typically a time or spatial index set), the marginal cumulative distribution function (CDF) is

$$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$

The population $\alpha$-quantile at index $t$ is

$$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$

for $\alpha \in I \subset (0,1)$.

Given an i.i.d. sample $\{X_t^i\}_{i=1}^n$ at each $X_t$0, the empirical CDF is $X_t$1, and the empirical quantile is

$X_t$2

Under suitable regularity conditions—including sample-path continuity, tightness, and density bounds near the target quantile—Kuelbs and Zinn established a uniform functional central limit theorem (CLT): $X_t$3 where $X_t$4 is a mean-zero Gaussian process with explicit covariance: $X_t$5 This provides the asymptotic distributional law for empirical dwell/travel-time quantile processes, crucial for inference and confidence bands (Kuelbs et al., 2011).

2. Phase-Type Distributions and ODE Embeddings

Dwell and travel time random variables in many Markovian or compartmental models are naturally represented as phase-type (PH) distributions. For a continuous-time Markov chain with $X_t$6 transient states and absorbing state $X_t$7, let

• $X_t$8 ($X_t$9 row vector) denote the initial probability on the transient states,
• $t$0 ($t$1 matrix) be the subgenerator matrix, with negative diagonals and nonnegative off-diagonals.

Then the time to absorption $t$2 is $t$3. The CDF and associated survival and density functions are as follows: $t$4 The $t$5-th raw moment is $t$6.

To bypass explicit computation of $t$7, an equivalent embedding as a linear ODE is used: $t$8 This approach underpins the Generalized Linear Chain Trick (GLCT), which generalizes embedding dwell/travel time PH-distributions into ODE models beyond exponential/Erlang assumptions (Hurtado et al., 2020).

3. Computational Methods for Quantile Extraction

Given a dwell/travel time variable with CDF $t$9, the $E$0-th quantile $E$1 is

$E$2

For PH models, $E$3 is available either via direct computation of the matrix exponential or by solving the corresponding ODE. Numerical inversion is accomplished as follows:

1. Define $E$4.
2. Choose a bracket $E$5 such that $E$6.
3. Apply a scalar root-finder (bisection, secant, Brent's method) to solve $E$7; at each trial $E$8, compute $E$9 either by matrix exponentiation or ODE integration.
4. Optionally, use Newton-Raphson updates: $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$0 using the closed-form derivative $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$1 (Hurtado et al., 2020, Levering et al., 2022).

This methodology extends to any fitted PH-distribution (e.g., Coxian, hypo/hyper-exponential mixtures). For classical Erlang($$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$2) forms: $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$3 and quantiles are obtained by scalar root-finding in $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$4.

4. Markovian Velocity and Path-Based Travel Time Quantile Modeling

For travel times across transportation networks, especially under stochastic or schedule-dependent regimes, the Markovian Velocity Model (MVM) generalizes the PH approach. The MVM constructs a large CTMC, representing both recurrent (e.g., daily rush periods, scheduled roadworks) and nonrecurrent (e.g., accidents) velocity regimes along a path. Each path-specific travel time $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$5 can be shown to be PH-distributed, where the generator $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$6 results from a Kronecker-sum assembly of link-specific subgenerators and velocity-dependent "rewards".

Formally, for initial CTMC state $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$7, the CDF is

$$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$8

where $$F(t, x) = \Pr\{X_t \le x\}, \qquad x \in \mathbb{R}.$$9 selects the starting state and $\alpha$0 projects onto non-absorbing states.

Quantiles are extracted as for standard PH models by root-finding. Full algorithmic workflow includes data preprocessing (incident/loop detector logs), fitting inter-incident/incident-duration PH distributions, block generator construction, speed assignment, and CDF/quantile computation by matrix exponentiation (Levering et al., 2022).

5. Deep Learning and Quantile Regression for Dwell/Travel Times

Recent approaches parametrize the conditional dwell/travel time quantile function directly using neural networks. The Quantile Graph Wavenet predicts the $\alpha$1-th conditional quantile $\alpha$2, for feature vector $\alpha$3, by minimizing the pinball loss: $\alpha$4 The network is conditioned on the target quantile $\alpha$5, embedded as an input channel.

Once trained, the set $\alpha$6 acts as a (possibly implicit) parametrization of the conditional distribution. Quantile inference requires only forward passes for each desired $\alpha$7. This allows for direct extraction of prediction intervals: $\alpha$8 without the need for sampling or explicit density estimation (Maas et al., 2020).

6. Path Prediction and Calibrated Uncertainty with DutyTTE

In frameworks such as DutyTTE for origin-destination travel time quantile estimation, path prediction is formalized as a finite-horizon Markov decision process (MDP) with a reinforcement learning policy: $\alpha$9 Travel time quantile estimation along the decoded path employs a Mixture-of-Experts (MoE) model, aggregating segment-level uncertainty representation into an overall travel time prediction, $t$0, and one-sided uncertainty estimates $t$1. The predictive interval is

$t$2

Training is supervised via the Mean Interval Score (MiS) to promote both sharpness and empirical coverage. Post-training, statistical calibration using Hoeffding's upper-confidence bound ensures that the empirical miscoverage rate does not exceed the nominal level $t$3 with high probability: $t$4 The smallest scaling $t$5 guaranteeing $t$6 for all $t$7 is selected, and the final output interval is $t$8 (Mao et al., 2024).

7. Concluding Remarks and Model Comparison

Dwell and travel time quantile estimation spans a rigorous spectrum from classical empirical process theory (CLT and functional uniform convergence), parametric Markovian modeling (PH-distributions), to nonparametric and deep learning-based quantile regression. The table below summarizes key methodologies and their representative features:

Approach	Key Formula for $t$9-th Quantile	Reference
PH-distribution/ODE	Solve $$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$0	(Hurtado et al., 2020)
Markovian Velocity MVM	$$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$1, $$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$2 via $$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$3	(Levering et al., 2022)
Empirical Process	$$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$4	(Kuelbs et al., 2011)
Quantile Graph Wavenet	$$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$5 (network output)	(Maas et al., 2020)
DutyTTE (MoE + UQ)	$$Q(t, \alpha) = F(t, \cdot)^{-1}(\alpha) = \inf\{x: F(t, x) \ge \alpha\},$$6	(Mao et al., 2024)

A plausible implication is that, depending on application scale, feature availability, and coverage guarantees required, different approaches may be preferential: ODE/PH-model-based quantiles for mechanistic models, empirical process theory for inference, and deep neural models for high-dimensional observation-based settings. All rigorous quantile workflows (including neural models) require careful calibration or statistical guarantees to ensure correct nominal coverage.