🔬 physics

Bursty Arrivals, Smooth Sojourns: Non-Poissonian Temporal Dynamics in a Logistics Warehouse

This paper analyzes high-resolution pallet-level data from a Spanish warehouse to reveal that while arrival and departure processes exhibit heavy-tailed, non-Poissonian burstiness, outbound sojourn times follow a log-normal distribution, collectively uncovering distinct operational regimes and weekly cycles that remain invisible to traditional aggregate statistics.

Original authors: Alfonso de Miguel-Arribas, Yamir Moreno

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alfonso de Miguel-Arribas, Yamir Moreno

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 massive warehouse not as a static building full of boxes, but as a living, breathing organism with a heartbeat. For decades, experts have tried to understand how this organism works by looking at the "average" heartbeat—counting how many trucks arrive per hour or how long a package sits on average. They assumed the traffic was like rain falling on a roof: steady, random, and predictable.

This paper argues that the warehouse is actually more like a city during rush hour. It doesn't just "rain" events; it has storms (bursts of activity) followed by calm spells. By looking at the data with a fresh set of "physics glasses," the authors discovered that the timing of events in this Spanish warehouse is far more chaotic and structured than the old models suggested.

Here is the breakdown of their findings in everyday terms:

1. The "Bursty" Heartbeat (Arrivals and Departures)

The old way of thinking assumed that pallets (the wooden platforms holding goods) arrive and leave at a steady, random pace, like people walking into a park at a constant rate.

The authors found the opposite. The warehouse is bursty.

The Analogy: Imagine a subway station. Instead of one person arriving every minute, you get a sudden rush of 50 people all at once (a burst), followed by 20 minutes of silence.
The Finding: The time between pallets arriving or leaving isn't random; it's "heavy-tailed." This means you get huge gaps followed by intense clusters. The warehouse doesn't run on a smooth timer; it runs on a stop-and-go rhythm.

2. The "Log-Normal" Sojourn (How Long Things Stay)

While the arrival of goods is chaotic, the time a pallet spends waiting inside the warehouse (its "sojourn") is surprisingly smooth and predictable.

The Analogy: Think of a coffee shop line. The arrival of customers is chaotic (a burst at 8:00 AM, quiet at 10:00 AM). But the time it takes to get your coffee follows a specific pattern: most people get it in 5 minutes, some in 10, and very few in 30. It rarely takes 2 hours.
The Finding: The time pallets spend in the outbound buffer zone follows a "log-normal" curve. This suggests that while the start of the process is chaotic, the service itself is constrained by human rules, schedules, and bottlenecks that keep most items moving within a specific window (usually around 24 hours).

3. The "Weekend Hangover" (The Weekly Cycle)

The most striking discovery is how the warehouse reacts to the calendar, specifically the weekend.

The Analogy: Imagine a river that flows fast during the week but hits a dam on Saturday and Sunday. The water (pallets) piles up behind the dam. When the dam opens on Monday, the water rushes out all at once.
The Finding: The authors tracked the "age" of the cargo. They saw a clear weekly cycle:
- Monday Morning: The warehouse is full of "old" cargo that arrived before the weekend and is finally leaving, mixed with "fresh" cargo arriving that day.
- Mid-Week: The system clears out, and the cargo is young.
- Weekend: Activity slows down, and the "age" of the remaining cargo grows.
- This proves the warehouse isn't a steady machine; it's a rhythm driven by the human work week.

4. Two Different "Speeds" of Waiting

The authors created a clever way to separate "real" delays from "calendar" delays.

The Analogy: Imagine you are waiting for a bus.
- Scenario A (Regular): The bus is late because the traffic is heavy. You wait 20 minutes, and 20 other people got on the bus while you waited. Your wait time matches the activity.
- Scenario B (The Weekend Effect): You wait 20 minutes, but the bus driver is on a break, and nobody else got on the bus. You waited a long time, but the "activity" was zero.
The Finding: Most pallets follow Scenario A (Regular). However, a specific group of pallets—those that arrived before the weekend and left on Monday—follow Scenario B. They sit there for a long time not because of traffic, but because the "clock" of the warehouse stopped ticking over the weekend.

The Big Picture

The paper concludes that if you only look at the "average" number of trucks moving through a warehouse, you miss the whole story. You miss the bursts, the weekly rhythms, and the hidden delays caused by the calendar.

By treating the warehouse like a complex system (using tools from statistical physics), the authors showed that limited but high-quality data can reveal the "personality" of the warehouse. It's not just a storage room; it's a dynamic system with its own unique, non-random heartbeat that changes depending on the day of the week and the specific route the goods are taking.

Technical Summary: Bursty Arrivals, Smooth Sojourns

Problem Statement
Warehouse operations are critical nodes in logistics networks, yet their temporal dynamics are frequently analyzed through aggregate performance indicators or queueing models that assume stationary, memoryless (Poissonian) event timing. While statistical physics has extensively characterized burstiness and heavy-tailed inter-event times in human and socio-technical systems, these concepts remain underexplored in warehouse contexts. Standard models often fail to capture hidden bottlenecks, feedback loops, or temporal correlations that manifest as deviations from exponential distributions. This paper addresses the gap by investigating whether real-world warehouse operations, specifically at the pallet level, exhibit non-Poissonian temporal structures and if these structures reveal operational regimes invisible to aggregate statistics.

Methodology
The study utilizes one month of high-resolution, pallet-level data from a large logistics warehouse in Spain (May 2023). The dataset distinguishes between inbound operations (arrival to inbound buffer zone, shelving) and outbound operations (arrival to outbound buffer zone, departure). Due to identifier resets during shelving, inbound and outbound subsystems are analyzed separately.

The analytical framework applies tools from statistical physics and renewal theory:

Distributional Analysis: Inter-arrival, inter-departure, and sojourn (residence) times are fitted to power-law tails ( $P(\tau) \sim \tau^{-\gamma}$ ) using maximum likelihood estimation, with cutoffs determined by Kolmogorov–Smirnov distance. Sojourn times are also tested against log-normal distributions.
Burstiness and Memory: The study computes the burstiness coefficient ( $B$ ) and memory coefficient ( $M$ ) to characterize the temporal organization of event sequences. $B$ measures the dispersion of inter-event times (ranging from regular to bursty), while $M$ quantifies correlations between consecutive intervals.
Flow Disaggregation: Dynamics are analyzed globally and disaggregated by seven distinct logistic flows (e.g., FactoryPickup, DirectShip, CustomerPickup) to identify route-specific temporal signatures. Jensen–Shannon (J–S) divergence is used to quantify distributional similarity between flows.
Renewal and Aging Analysis: The "age" of cargo in the outbound buffer zone is tracked over calendar time to observe how the population of active pallets evolves, specifically looking for weekly accumulation and clearance cycles.
Activity–Sojourn Scaling: Inspired by Little's Law, the study plots the sojourn time of individual pallets against the number of arrivals ( $A_i$ ) that occurred during their stay. This establishes a null benchmark ( $\tau \approx A/\lambda$ ) to distinguish between delays caused by high traffic volume and those caused by calendar-driven service interruptions.

Key Contributions

Empirical Characterization: The paper provides a proof-of-concept for applying statistical-physics diagnostics to warehouse operations, moving beyond aggregate throughput to analyze event-level temporal signatures.
Methodological Integration: It combines distributional fitting, burstiness–memory coefficients, and J–S divergence to create a multi-faceted view of temporal heterogeneity across different logistics flows.
Novel Diagnostics: The authors introduce two specific diagnostic tools for the outbound buffer: a renewal-theoretic aging analysis to visualize weekly cycles and an activity–sojourn scaling plot to separate regular turnover from calendar-induced stagnation.

Results

Non-Poissonian Dynamics: Inter-arrival and inter-departure times are strongly heterogeneous and heavy-tailed, consistent with non-Poissonian behavior. Fitted power-law exponents ( $\gamma$ ) range from approximately 1.86 to 2.93 depending on the process. In contrast, outbound sojourn times are unimodal and well-described by a log-normal distribution ( $\mu \approx 10.76, \sigma \approx 1.10$ ), suggesting constrained service mechanisms with multiplicative micro-delays.
Burstiness vs. Smoothness: The Memory-Burstiness diagram reveals a structural dichotomy. Arrival and departure sequences are highly bursty ( $B \approx 0.8$ ) with near-zero memory, indicating irregular but weakly correlated intervals. Sojourn times, however, exhibit low burstiness ( $B \approx 0.08$ ) and moderate positive memory ( $M \approx 0.49$ ), indicating smoother, temporally persistent residence times.
Flow Heterogeneity: Disaggregation by logistics flow shows systematic differences. Routes involving CustomerPickup and InboundDrag exhibit distinct temporal signatures and higher distributional divergence compared to high-volume preparation routes.
Weekly Aging Cycles: The aging analysis reveals a recurrent weekly pattern. Cargo accumulates during weekends (shifting the age distribution to longer timescales), and the buffer is cleared upon the resumption of activity on Mondays. This creates a bimodal age distribution at the start of the week.
Operational Regimes: The activity–sojourn scaling identifies two regimes. Most pallets follow a near-linear baseline where residence time scales with experienced activity. A reproducible "off-baseline" branch exists for pallets arriving before weekends and departing on Mondays; these pallets experience long residence times despite low concurrent activity, indicating a calendar-driven stagnation effect rather than congestion.

Significance and Claims
The paper claims that warehouse timing is not adequately described by a single memoryless process. Instead, it operates through two coupled layers: external, schedule-driven activation of events (arrivals/departures) and internal service rules regulating residence times. The significance of the work lies in demonstrating that limited but high-resolution industrial data can reveal operational structures—such as weekly accumulation cycles and flow-specific temporal fingerprints—that are invisible to aggregate throughput statistics.

The authors frame these findings as a "compact diagnostic framework" for temporal complexity. They explicitly state that their claims are diagnostic and mechanistic rather than universal, noting limitations such as the short observation window (one month) and the lack of spatial data. The study does not propose new optimization algorithms but rather provides a method to interpret operational rhythms, distinguishing between delays caused by traffic volume and those caused by scheduled service interruptions.