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Measuring capacities in multimodal maritime port systems with anchorage queues

This paper introduces a framework to distinguish between and estimate a port's sustainable operating capacity versus its absolute ultimate capacity using queueing theory and differential equations, demonstrated through a case study of the Port of Houston that identifies shifting bottlenecks between liquid-bulk terminals and pilot availability under different conditions.

Original authors: Debojjal Bagchi, Kyle Bathgate, Kenneth N. Mitchell, Magdalena I. Asborno, Marin M. Kress, Stephen D. Boyles

Published 2026-03-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Debojjal Bagchi, Kyle Bathgate, Kenneth N. Mitchell, Magdalena I. Asborno, Marin M. Kress, Stephen D. Boyles

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a busy seaport not as a static building, but as a giant, living waterway highway system. Ships are the cars, the ocean is the on-ramp, the narrow channel is a single-lane tunnel, and the terminals are the parking garages where they unload their cargo.

This paper tackles a very specific problem: How do we measure how many cars this highway can actually handle?

The authors argue that we've been asking the wrong question. We usually ask, "How many cars can this road handle?" But the answer depends entirely on when you ask and how you ask. They propose that every port has two different speed limits:

1. The "Daily Commuter" Limit (Operating Capacity)

Think of this as the sustainable speed limit for a normal day.

The Analogy: Imagine a highway that can comfortably handle 60 cars per hour. If you try to push 65 cars through every hour, traffic will eventually jam, and the line of cars waiting to get on the highway (the "anchorage queue") will grow forever.
What it means: This is the maximum number of ships a port can process stably over a long period (months or years). If you stay at this speed, the line of waiting ships stays manageable.
How they measured it: They used a mathematical "queueing theory" model (like the math used to figure out how many cashiers a grocery store needs). They looked at historical data of ships waiting and how long they waited to calculate the "safe" speed limit.

2. The "Emergency Evacuation" Limit (Ultimate Capacity)

Think of this as the absolute maximum speed the road can physically handle for a short burst, even if it causes chaos.

The Analogy: Imagine a hurricane hits, and the highway is closed. When it reopens, everyone tries to get on at once. For a few hours, the road might squeeze through 90 cars per hour. The line of waiting cars will get huge, and the drivers will be furious, but the road can physically move that many cars for a short time.
What it means: This is the absolute highest number of ships the port can process if you ignore the fact that the waiting line is getting out of control. It's useful for planning disaster recovery (e.g., "How fast can we clear the backlog after a storm?").
How they measured it: They built a giant computer simulation of the Port of Houston, cranked up the number of incoming ships to extreme levels, and watched what happened. Then, they used a special math formula (an Ordinary Differential Equation) to predict the "ceiling" of that performance.

The Case Study: The Port of Houston

The authors tested their ideas on the Port of Houston, one of the busiest ports in the US. Here is what they found:

The "Safe" Speed: The port can comfortably handle about 0.9 ships per hour (roughly 21 ships a day) without the waiting line growing out of control.
The "Max" Speed: If they really pushed it, they could process about 1.4 ships per hour (roughly 33 ships a day), but the waiting line would balloon, and ships would wait much longer.

The Big Surprise: Bottlenecks Change

The most interesting part of the paper is that the traffic jam changes depending on the situation.

On a Normal Day (Operating Capacity): The bottleneck is the Liquid-Bulk Terminals (where oil and chemicals are stored). It's like a narrow parking garage exit. Even if the road is clear, the garage is full, so ships can't get in.
After a Disaster (Ultimate Capacity): The bottleneck shifts to Pilots.
- Who are pilots? They are the expert boat captains who guide ships through the narrow, tricky channels.
- The Analogy: Imagine the highway is wide open, but you only have enough traffic cops (pilots) to direct cars through the tunnel. If you have 100 cars waiting but only 2 cops, the cars can't move fast, no matter how wide the road is.
- After a storm, when everyone is rushing to get in, the lack of pilots becomes the biggest problem, not the storage space.

Why Does This Matter?

This framework helps port planners make smarter decisions:

Long-Term Planning: If you want to increase the daily flow of goods, you should build more storage terminals (fix the "Daily Commuter" bottleneck).
Disaster Planning: If you want to recover faster after a hurricane, you need more pilots (fix the "Emergency Evacuation" bottleneck).

The Takeaway

You can't just say "The port is full." You have to ask, "Full for a normal day, or full for an emergency?"

Operating Capacity tells you how to build a sustainable, efficient system for the future.
Ultimate Capacity tells you how resilient the system is when things go wrong and you need to clear a massive backlog.

By understanding the difference, port officials can stop guessing and start investing in the right things to keep the global supply chain moving.

1. Problem Statement

Measuring the capacity of multimodal maritime port systems is critical for infrastructure planning, bottleneck identification, and recovery planning following disruptions. However, existing literature lacks standardized methods to distinguish between two fundamentally different types of capacity:

Operating Capacity ( $C_o$ ): The maximum sustainable vessel throughput over an extended period under stable operating conditions (where anchorage queues remain bounded).
Ultimate Capacity ( $C_u$ ): The absolute maximum vessel throughput achievable over a finite time horizon, regardless of stability (where queues may grow unbounded, often seen during post-disruption recovery or demand surges).

Current models typically rely on component-based analysis (which ignores system interdependencies) or simulation-based approaches that use proxy indicators (like wait-time ratios) without a theoretical foundation to differentiate between stable and unstable throughput. Furthermore, many existing methods require extensive, high-fidelity data that is often unavailable.

2. Methodology

The authors propose a dual-framework approach to estimate both capacity types, utilizing different modeling techniques suited to their specific definitions and data requirements.

A. Operating Capacity Model (Queueing Theory)

Approach: A parsimonious queueing-theoretic model based on the G/M/1 queue assumption.
System Structure: The port is modeled as a network where vessels queue at an anchorage before passing through a shared waterway channel (modeled as a multi-class G/M/1 queue) and then to specific terminals (modeled as independent G/M/1 queues).
Key Mechanism:
- Uses Kingman's formula to approximate waiting times based on arrival rates, service rates, and coefficients of variation.
- Uses Little's Law to link observed queue lengths and waiting times to service rates.
- Estimates the operating capacity of the channel ( $\mu_C$ ) and each terminal class ( $\mu^T_i$ ) using archival Automatic Identification System (AIS) data and port logs.
- The overall operating capacity is defined as the minimum of the channel capacity and the sum of terminal capacities: $C_o = \min(\mu_C, \sum \mu^T_i)$ .
Data Requirement: Only requires archival AIS vessel tracking data and terminal records; does not require complex simulation.

B. Ultimate Capacity Model (Simulation + ODE)

Approach: A hybrid method combining a detailed Discrete-Event Simulation (DES) with an Ordinary Differential Equation (ODE) model.
Simulation: A high-fidelity DES of the Port of Houston (including waterside, terminals, and landside) is run under varying vessel arrival rates ( $\lambda$ ).
ODE Formulation: The relationship between arrival rates and the number of vessels exiting the anchorage ( $N_{exits}$ $N_{e x i t s}$ ) is modeled using a piecewise ODE:
- Growth Phase ( $\lambda \le \lambda_c$ ): Modeled to show throughput increasing linearly then plateauing as it approaches ultimate capacity ( $C_u$ ).
- Decline Phase ( $\lambda > \lambda_c$ ): Models the "congested regime" where excessive arrivals cause resource competition (e.g., pilots, tugs), reducing throughput toward a saturated capacity ( $C_s$ ).
Parameter Estimation: The parameters of the ODE ( $C_u, C_s, \theta, \beta, \lambda_c$ ) are estimated by minimizing the Mean Squared Error (MSE) between the ODE predictions and simulation outputs using a Differential Evolution algorithm.

3. Key Contributions

Theoretical Distinction: Formally defines and differentiates between operating capacity (sustainable/stable) and ultimate capacity (maximum/short-term), proving mathematically that $C_o < C_u$ for any finite system.
Queueing-Based Operating Capacity: Develops a closed-form queueing model that estimates sustainable capacity using only historical AIS data, avoiding the need for resource-intensive simulations for long-term planning.
Simulation-Driven Ultimate Capacity: Introduces an ODE-based framework to fit simulation data, providing a theoretical basis for defining ultimate capacity and predicting system behavior under extreme congestion.
Empirical Validation: Applies the framework to the Port of Houston, a complex multimodal system, validating the models against real-world data and demonstrating their utility in identifying shifting bottlenecks.

4. Results (Case Study: Port of Houston)

The study utilized 2022–2024 AIS data and a calibrated DES for the Port of Houston.

Capacity Estimates:
- Operating Capacity ( $C_o$ ): Approximately 0.9 vessels per hour (vph).
- Ultimate Capacity ( $C_u$ ): Approximately 1.4 vph.
- This indicates the port can sustain a ~14% increase in arrivals long-term, but can process up to ~78% more vessels in the short term (before queues become unmanageable).
Bottleneck Analysis:
- Under Stable Conditions: Liquid-bulk terminals are the primary bottleneck. The waterway channel itself is not the limiting factor (channel capacity $\approx$ 1.5 vph).
- Under Disrupted/High-Demand Conditions: Pilot availability becomes the dominant bottleneck. A 10% reduction in pilot availability reduces ultimate capacity by ~7.1%, whereas similar reductions in terminal berths have negligible effects on ultimate capacity.
Sensitivity:
- Reducing liquid-bulk terminal resources significantly impacts operating capacity.
- Reducing pilot resources significantly impacts ultimate capacity (resilience).

5. Significance and Implications

Strategic Planning: The framework allows port authorities to distinguish between investments needed for long-term throughput (e.g., expanding terminal berths) versus investments needed for resilience and recovery (e.g., increasing pilot availability).
Disruption Management: By quantifying ultimate capacity, planners can better estimate recovery rates following events like fog, storms, or closures, understanding that the port can temporarily process vessels faster than its sustainable rate.
Data Efficiency: The operating capacity model offers a low-cost, data-light alternative to full simulation for initial screening and long-term trend analysis.
Theoretical Advancement: The paper bridges the gap between queueing theory (stable systems) and dynamic simulation (unstable systems), providing a unified mathematical relationship between the two capacity metrics.

In conclusion, the paper provides a robust, dual-metric framework for port capacity that moves beyond static definitions, enabling more nuanced decision-making for both infrastructure investment and emergency response planning.