Algorithms of evaluation of the waiting time and the modelling of the terminal activity

This paper approaches the application of the waiting model with Poisson inputs and priorities in the port activity. The arrival of ships in the maritime terminal is numerically modelled, and specific parameters for the distribution functions of service and of inputs are determined, in order to establish the waiting time of ships in the seaport and a stationary process. The modelling is based on waiting times and on the traffic coefficient.

💡 Research Summary

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The paper presents a comprehensive analytical and numerical study of waiting‑time performance in a maritime terminal where ship arrivals follow a Poisson process and are classified into five priority levels. The authors adopt the classic single‑server queue notation 1‖‖rr G M, extending it to incorporate arbitrary service‑time distributions for each priority class and to model four distinct interruption policies: (1) continuation of an interrupted service, (2) loss of the current service and immediate start of a new one, (3) restarting the interrupted service from the beginning, and (4) a hybrid “restart‑after‑loss” rule.

Key theoretical tools include the Laplace‑Stieltjes transform βk(s) of the service‑time distribution Bk(x) and its first moment βk, which together allow the derivation of class‑specific mean waiting times Wk and the full waiting‑time distribution Wk(x) via numerical inversion (e.g., Gaver‑Stehfest). The traffic coefficient ρ, defined as the ratio of the effective arrival load to service capacity, is expressed for each policy; stability of the system requires ρ < 1.

To obtain concrete numerical results, the authors develop a C++ program that (i) computes βk(s) for chosen distributions, (ii) performs numerical inversion to recover waiting‑time distributions, and (iii) estimates model parameters from real data using the method of moments (Pearson’s approach). Four families of service‑time distributions are examined: uniform, exponential, Erlang (order 2), and Gamma (shape α = 3).

The empirical part uses ship‑movement logs from the Constanţa Seaport for February 2016. From these logs the authors extract arrival rates (λk) ranging roughly from 0.3 to 0.8 ships per hour and fit service‑time parameters for each distribution. Tables 4.1‑4.5 present the resulting traffic coefficients and average waiting times under each policy and distribution. The main findings are:

1. Exponential service times consistently yield the smallest mean waiting times (≈0.2 h) and the lowest traffic coefficients (ρ as low as 0.04), indicating the most efficient processing regime.
2. Policy 1 (continuation of interrupted service) and Policy 4 (restart from the beginning) produce the most favorable ρ values, while Policies 2 (loss) and 3 (restart) often push ρ toward the instability threshold (ρ ≈ 1).
3. Higher‑priority classes experience markedly reduced waiting times, confirming the effectiveness of the priority‑based scheduling in alleviating congestion for critical vessels.
4. Model validation against observed waiting times shows an average prediction error of 10‑20 %, demonstrating that the analytical framework captures the essential dynamics of the terminal.

Based on these results, the authors recommend operational guidelines to keep ρ < 1: (i) reduce variability in service times through automation and standardized handling procedures, (ii) prioritize high‑priority ships to minimize overall queue length, and (iii) adopt a continuation‑of‑service policy whenever feasible.

In summary, the paper contributes a unified queueing‑theoretic model that integrates Poisson arrivals, multi‑class priorities, arbitrary service‑time laws, and interruption policies. By coupling Laplace‑Stieltjes analysis with a robust C++ implementation, it provides port managers with a practical tool for quantifying waiting‑time performance, assessing stability via the traffic coefficient, and making data‑driven decisions to optimize terminal throughput.