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

Abstract: Warehouses are central nodes in logistics networks: they buffer material flows, synchronize heterogeneous actors, and absorb temporal mismatches between inbound and outbound operations. Yet most warehouse analyses still rely on aggregate performance indicators or on queueing assumptions in which event timing is stationary and approximately memoryless. Here we use one month of high-resolution pallet-level data from a large Spanish warehouse to characterize arrivals, departures, and outbound residence times from a statistical-physics perspective. Inter-arrival and inter-departure times are strongly heterogeneous and compatible with heavy-tailed, non-Poissonian behavior, whereas outbound sojourn times are more naturally described by a log-normal distribution, suggesting constrained service mechanisms with a characteristic operational scale. Disaggregation by logistics flow reveals systematic differences in burstiness, memory, and distributional similarity. A renewal-based aging analysis uncovers recurrent weekly accumulation and clearance cycles in the outbound buffer zone. Finally, a Little's-Law-inspired activity--sojourn scaling identifies two operational regimes: a near-linear baseline under regular turnover and a reproducible off-baseline branch associated with weekend accumulation and Monday dispatches. These results provide a compact diagnostic framework for temporal complexity in warehouse operations and show how limited but high-resolution industrial data can reveal operational structure invisible to aggregate throughput statistics.

• The paper demonstrates that warehouse arrivals and departures are highly bursty but weakly correlated, while sojourns are smoother, log-normally distributed, and moderately persistent, revealing dynamics that memoryless queueing models miss.
• The analysis combines heavy-tail fitting, burstiness–memory metrics, renewal aging, Jensen–Shannon divergence, and activity–sojourn scaling to identify flow-specific timing patterns and distinguish traffic-driven delays from calendar-driven weekend stagnation.
• The findings show a recurring weekly cycle in outbound cargo age, with roughly 70% of completed sojourns ending within 24 hours and 1.2%–26.5% of flow-specific pallets classified as off-baseline delays, supporting schedule coordination and pre-weekend clearance policies.

Overview and motivation

This paper analyzes one month of pallet-level event data from a large Spanish warehouse operated by Carreras Grupo Logístico, applying tools from statistical physics—burstiness, memory coefficients, renewal theory, and Jensen–Shannon divergence—to characterize temporal dynamics that standard queueing models typically assume away. The authors position the work against a literature in which warehouse operations are dominated by optimization frameworks that treat event timing as stationary or memoryless. Prior empirical work in logistics had already reported deviations from Poissonian behavior: Wang and Guo found unimodal inter-event distributions with power-law tails in inbound aerial logistics [1004.xxxx-style Physica A work], and Yao et al. observed bursty outbound processes with exponents near $\gamma \approx 2.5$ and Hurst exponents above 0.5. The present study extends this small body of evidence to a single-facility, high-resolution setting, explicitly framed as a proof of concept rather than a universal characterization.

The dataset comprises two event tables—inbound arrivals and shelvings at the inbound buffer zone (IBZ), and outbound arrivals, sojourns, and departures at the outbound buffer zone (OBZ)—recorded during May 2023. Because pallet identifiers are reset between shelving and outbound registration, inbound and outbound subsystems cannot be linked at the individual-pallet level; the analysis treats them separately. The outbound table is annotated with seven logistics flows spanning four inbound origins (FactoryPickup, InboundDrag, PickPrep, FullLoadPrep) and three outbound destinations (CustomerPickup, DirectShip, OutboundDrag).

Non-Poissonian inter-event statistics

Inter-arrival and inter-departure time distributions span several orders of magnitude, from seconds to days, and are compatible with heavy-tailed scaling over part of their support. Maximum-likelihood tail fits (with lower cutoffs selected by minimizing Kolmogorov–Smirnov distance) yield $\gamma_{\mathrm{I,arr}} = 2.19$, $\gamma_{\mathrm{shelf}} = 1.86$, $\gamma_{\mathrm{O,arr}} = 2.89$, and $\gamma_{\mathrm{O,dep}} = 2.93$. The authors are careful to state that these should be read as evidence for heavy-tailed, non-Poissonian timing rather than definitive proof of asymptotic power-law behavior, given the short observation window and strong calendar dependence—a caveat that also applies to prior claims of scale-free timing in human dynamics, where cascading Poisson processes modulated by circadian or weekly activation windows can mimic apparent scale-free behavior.

Outbound sojourn times behave differently: they are unimodal and well described by a log-normal distribution with $\mu = 10.76$ and $\sigma = 1.10$. Most pallets remain in the OBZ between $10^3$ and $10^5$ seconds; approximately 70% of completed outbound sojourns end within 24 hours, while 28.7% exceed it. The mechanistic interpretation offered is multiplicative accumulation of proportional micro-delays—handling variation, dock availability, order consolidation, transport scheduling—with bounded adjustment around soft service targets producing a log-normal stationary solution via a Fokker–Planck description. This contrast between bursty point-process arrivals and constrained, log-normal service times suggests two coupled layers of dynamics: schedule-driven external activation and internally regulated residence.

Disaggregation by flow reveals systematic heterogeneity. Inter-arrival times show the largest variability across flows: high-volume routes such as FullLoadPrep→DirectShip and PickPrep→OutboundDrag lie below the global mean, while FactoryPickup→CustomerPickup has a mean inter-arrival time exceeding 24 hours. Jensen–Shannon divergence analysis identifies FactoryPickup→CustomerPickup and InboundDrag-originating routes as systematically dissimilar from the rest, indicating distinct replenishment logic for transfer-driven flows and additional timing constraints introduced by customer-controlled collection.

Memory–burstiness structure

On the Goh–Barabási $M$–$\gamma_{\mathrm{I,arr}} = 2.19$0 phase diagram, global inter-arrival and inter-departure sequences occupy a region of high burstiness ($\gamma_{\mathrm{I,arr}} = 2.19$1, $\gamma_{\mathrm{I,arr}} = 2.19$2) with near-zero memory ($\gamma_{\mathrm{I,arr}} = 2.19$3, $\gamma_{\mathrm{I,arr}} = 2.19$4): highly irregular but only weakly correlated at the level of consecutive intervals. Sojourn sequences invert this signature, with low burstiness ($\gamma_{\mathrm{I,arr}} = 2.19$5) and moderate positive memory ($\gamma_{\mathrm{I,arr}} = 2.19$6), indicating smooth residence times with temporal persistence—runs of similarly delayed or similarly fast cargo within particular flows. Flow-level points split into two groups, one near the origin and another with moderate-to-high memory but weak burstiness. This compression of distributional heterogeneity into an interpretable two-dimensional map constitutes the paper's central empirical claim: warehouse timing is not well described by any single memoryless process.

Renewal aging and the weekly cycle

Framing the OBZ as a renewal system, the authors track the time-resolved age distribution $\gamma_{\mathrm{I,arr}} = 2.19$7 of active cargo at 60-second resolution. The distribution exhibits a recurrent weekly cycle: early-week populations are dominated by recently arrived pallets; ages stretch through the week as slow-moving cargo accumulates; weekend slowdown concentrates the distribution above the 24-hour scale; and Monday restarts produce a bimodal pattern combining aged pre-weekend cargo with fresh arrivals, which Tuesday operations then clear. The mean and median age separate when a small fraction of long-residence pallets pulls the mean upward while the median tracks fast turnover—a population-level manifestation of the waiting-time paradox.

Under stationarity with i.i.d. sojourns, renewal theory predicts $\gamma_{\mathrm{I,arr}} = 2.19$8, which for the fitted log-normal exceeds both the mean sojourn and the exponential benchmark $\gamma_{\mathrm{I,arr}} = 2.19$9. The empirical mean age matches neither benchmark, instead oscillating with the weekly operating cycle. This mismatch is presented as informative rather than problematic: the warehouse is a non-stationary renewal system whose apparent equilibrium depends on calendar time, implying that static queueing equilibria are poor descriptors of its internal state.

Activity–sojourn scaling

Using Little's Law as a macroscopic reference rather than a microscopic identity, the authors define a null expectation $\gamma_{\mathrm{shelf}} = 1.86$0, where $\gamma_{\mathrm{shelf}} = 1.86$1 counts OBZ arrivals experienced during pallet $\gamma_{\mathrm{shelf}} = 1.86$2's stay. In log–log plots of sojourn time versus activity, most pallets form a near-linear band close to this baseline, consistent with quasi-stationary turnover. A second, reproducible branch appears above the baseline in nearly every sufficiently populated flow, with residence times of $\gamma_{\mathrm{shelf}} = 1.86$3–$\gamma_{\mathrm{shelf}} = 1.86$4 seconds and overwhelmingly associated with pallets entering before a weekend and departing on Monday. Crucially, these pallets do not have exceptionally high activity footprints—the deviation reflects calendar-driven stagnation rather than congestion.

A residual-based classification ($\gamma_{\mathrm{shelf}} = 1.86$5 and $\gamma_{\mathrm{shelf}} = 1.86$6 h), which does not use day-of-departure information, independently confirms the calendar interpretation: off-baseline pallets concentrate among Monday departures, often approaching or reaching 100% in several flows, with positive mean residuals across all reported flows. Off-baseline fractions range from 1.2% (FactoryPickup→OutboundDrag) to 26.5% (InboundDrag→CustomerPickup). Operationally, this separates delays caused by high traffic—which call for capacity or staffing adjustments—from delays caused by scheduled interruptions, which call for schedule coordination or pre-weekend clearance rules.

Limitations and open questions

The authors state their limitations plainly. The one-month window prevents claims about universality, cross-season stability, or asymptotic scaling exponents; several flow-specific subsets are small (e.g., 11 pallets for FactoryPickup→CustomerPickup arrivals, 27 for its sojourns), so their fitted parameters and divergence values are descriptive only. The absence of spatial annotation leaves open whether temporal signatures connect to storage location, travel distance, dock assignment, or picker routing. The raw data cannot be released for privacy reasons, though code and curated derived datasets are available. Three specific questions remain open: whether the reported signatures—bursty activation, log-normal residence times, flow-specific fingerprints, weekly aging cycles—are stable across warehouses, firms, sectors, and seasons; how non-stationary arrival processes combine with flow-specific service rules and calendar-dependent capacity in generative models; and whether spatially annotated data can link temporal diagnostics to layout and storage policy.

Conclusion

The paper demonstrates that even a limited, single-site industrial dataset contains measurable temporal structure invisible to aggregate throughput statistics. Its principal results are the sharp dichotomy between bursty, weakly correlated arrival/departure processes and smooth, memory-bearing log-normal sojourns; the identification of flow-specific temporal fingerprints via burstiness–memory and divergence analyses; the empirical documentation of a weekly aging–clearance cycle that violates stationary renewal expectations; and a Little's-Law-inspired scaling diagnostic that cleanly separates traffic-driven from calendar-driven delay. The framework is deliberately diagnostic rather than universal, and its value lies in providing reproducible, interpretable temporal signatures that bridge statistical physics and warehouse operations research.