A Divide-and-Conquer Approach for Modeling Arrival Times in Business Process Simulation

Han van der Aa; Heiner Stuckenschmidt; Konrad \"Ozdemir; Lukas Kirchdorfer; Stjepan Kusenic

REVIEW 3 major objections 5 minor 23 references

A Divide-and-Conquer Approach for Modeling Arrival Times in Business Process Simulation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read AT-KDE claims that splitting arrival data by global trend, weekday, and intraday bin and fitting a kernel density estimate to each subset is more accurate and robust than static or learned arrival models in business process simulation…

desk verdict Solid empirical arrival-modeling pipeline for BPS; the 'robust' claim needs tempering and variance reporting, but it deserves review. read the letter →

arxiv 2505.22381 v1 pith:ILD6QIJS submitted 2025-05-28 cs.LG

Lukas Kirchdorfer , Konrad Özdemir , Stjepan Kusenic , Han van der Aa , Heiner Stuckenschmidt This is my paper

classification cs.LG

keywords businessprocesssimulationcasearrivalmodelingkerneldensityestimationtimeseriessegmentationchangepointdetectionweekdayclusteringintradaydynamicseventlogs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Business process simulation is only as trustworthy as its case-arrival model, yet most data-driven simulators assume a static distribution of inter-arrival times, ignoring seasonal, weekly, and within-day variation. This paper claims that those static models systematically misrepresent workload, which distorts cycle-time and waiting-time estimates. It proposes AT-KDE, which divides the arrival dataset into subsets that share global trends, weekday patterns, and intraday behavior, then fits a separate kernel density estimate to each subset and samples arrivals from the ensemble. In experiments on 20 event logs, the paper reports that AT-KDE has the lowest average square-root CADD in 12 of them, with small gaps on most of the rest, and that this accuracy is achieved in 13 seconds of training and 0.03 seconds of simulation. The contribution is a general way to make arrival modeling non-stationary without abandoning simple, interpretable density estimation.

What carries the argument

The load-bearing mechanism is the three-level partition of the arrival dataset into disjoint subsets, followed by a dedicated Gaussian KDE per subset, where KDE stands for kernel density estimation, a non-parametric density estimator that smooths observed samples with a kernel. The partition is found by (1) detecting change points in sliding-window differences of moving-average daily arrival counts and clustering the resulting segments with DBSCAN, (2) clustering weekdays within each global segment using Ward's method, and (3) splitting each day into equal time bins. Each subset's inter-arrival times are then modeled by a KDE with bandwidth chosen by Silverman's rule and refined on a validation split. The partition converts a globally non-stationary arrival process into locally stationary pieces, so that a simple static density estimator suffices inside each piece; the ensemble of KDEs is what generates new timestamps, and the segment-label sequence is what decides which piece applies to a simulated day.

What would settle it

Take an event log where daily arrivals drop sharply immediately after the training period, fit AT-KDE with its default settings, and compare its square-root CADD on the test period with that of a trend-fitted baseline. If the baseline's CADD is substantially lower and the difference vanishes when the most-recent-segment fallback is replaced with a trend extrapolation, the paper's central robustness claim is falsified for that scenario.

Watch

Extended reading notes

Core claim

The central discovery the paper defends is that a non-parametric, divide-and-conquer arrival model can capture the dynamic structure of real organizational arrivals that static distributions miss, while remaining computationally cheap. AT-KDE works in three partitioning steps: change-point detection on moving averages of daily arrival counts plus DBSCAN clustering groups periods into global segment clusters; Ward clustering groups days of the week within each global cluster; and each day within a weekday cluster is cut into equal-length bins. A KDE over inter-arrival times is fitted to each subset, and simulation samples from the corresponding KDE for each simulated day, bin by bin, replicating recurring global patterns or falling back to the most recent segment when no recurrence is found. The paper's evidence is experimental: across 20 logs spanning finance, government, and healthcare, AT-KDE produced the best average square-root CADD in 12 logs and near-best results in most others, with a notable miss on a log where arrivals drop abruptly in the test period. The claim is therefore that the partition-plus-KDE recipe is both more accurate and more robust than static distribution fitting, decomposition-based forecasting, and learned forecasting models.

Load-bearing premise

If no repeating global pattern can be seen in the training data, AT-KDE assumes the most recent segment's behavior continues into the future, and that assumption is what fails when the simulation period behaves differently.

Editorial extensions

If this is right

If AT-KDE's accuracy carries over into full simulation, simulators that use it should produce more trustworthy cycle-time, waiting-time, and resource-utilization estimates, because arrival load directly drives congestion.
The approach needs only case start timestamps, so any event log with start times can feed it without extra data collection.
On logs with little global drift, AT-KDE performs on par with static baselines, meaning it degrades gracefully instead of hurting accuracy.
For logs with abrupt test-period changes, AT-KDE is not the best choice; a trend-focused forecaster can catch a drop that segmentation does not see.
The empirical win on 12 of 20 logs with near-best gaps on most others supports the paper's robustness claim, but the single large miss shows the fallback rule is the limiting failure mode.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same segment-cluster-bin-fit-KDE recipe could be applied to other time-dependent simulation inputs, such as activity durations or inter-event delays; the paper names this as future work, and a testable extension would compare event-level distribution distances.
The paper's segmentation features are coarse daily counts and inter-arrival percentiles; using richer features, such as spectral or quantile summaries, might reduce reliance on the most-recent-segment fallback on logs with subtle trends.
Because the method samples from KDEs of inter-arrival times within fixed bins, it cannot produce arrivals outside the observed working hours; if a simulation needs to test schedule changes, the bin bounds would need to be learned or relaxed.
A full-process evaluation that propagates these arrivals through a resource-aware simulator would likely show larger gains on cycle-time metrics than on arrival CADD alone, since correct load is the main driver of queuing delay.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the take: AT-KDE is a genuine empirical contribution for arrival modeling in business process simulation. It's not a fundamental advance, but it's a well-built pipeline that outperforms static and most dynamic baselines on 12 of 20 logs, with code and data shipped. The main problem is that the phrase 'far more accurate and robust' is not supported by the evidence as presented.

What's new: the divide-and-conquer structure — global segmentation via IQR-based change point detection, DBSCAN segment clustering, weekday clustering, intraday binning, and per-cell KDE — applied specifically to BPS arrivals. The evaluation is more thorough than typical BPS papers: 20 diverse logs, temporal hold-out, comparisons to Prophet, LSTM, XGBoost, Chronos, Hawkes, and static baselines. Runtime is measured and sensible. The P2P case study nicely illustrates why ignoring a drift in training data hurts static methods.

The soft spots are the ones the stress-test note identifies, but I want to keep them in proportion. Algorithm 3's fallback — if no recurring global pattern is found in the training labels, just reuse the most recent cluster for the whole test period — is indeed a zero-order forecast. It cannot anticipate a trend that appears only after training. Hospital is the visible failure, and the paper's Limitations section candidly admits this. That is honest, but it means 'robust' is too strong. The claim should be something like 'accurate on average across diverse logs, with a known failure mode for new test-period trends.'

Also, Table 2 gives averages over 10 runs without standard deviations in the main text. The repo apparently has them, but the paper should show them. The hyperparameter settings (omega=7, L=3, sensitivity range Z) are reported, but there is no sensitivity analysis, which matters for a method that is supposed to be 'auto.'

None of this is fatal. The central empirical claim largely holds: static baselines are clearly worse, and AT-KDE is at least competitive with dynamic methods. The method is a reasonable default for BPS arrival modeling. The authors just need to temper the abstract and add the variance/sensitivity information.

This paper deserves serious peer review. It's a solid applied ML contribution to process mining. I would send it to a BPM or process mining venue, and I'd accept it after minor revision addressing the overclaim and missing variance. I'd also put it on the reading list for anyone working on simulation from event logs.

Referee Report

3 major / 5 minor

Summary. The paper proposes AT-KDE, a divide-and-conquer method for modeling case-arrival times in business process simulation. AT-KDE segments the arrival time series globally, clusters weekdays, bins each day into intraday intervals, fits one kernel density estimate per cell of this partition, and then generates new arrival timestamps by sampling from the appropriate KDE models. The authors evaluate AT-KDE on 20 event logs against seven baselines (Mean, Best Distribution, Prophet, LSTM, Chronos, XGBoost, and a Hawkes-process-based NPP), reporting the square-root Case Arrival Distribution Distance (CADD) on a temporal hold-out split. They report that AT-KDE achieves the best score on 12 of 20 logs and is substantially faster at simulation time than the dynamic baselines, and they conclude that AT-KDE is more accurate and robust than existing approaches.

Significance. If the empirical claims hold, the paper makes a useful contribution: it provides a non-parametric, computationally efficient arrival model that captures multiple temporal scales, and it evaluates the method on a broad, publicly available benchmark with a temporal split and several strong baselines. The release of code and results is a concrete strength. The main limitation is that the headline claims of accuracy and, especially, robustness are stronger than the evidence reported in the paper: no error bars or significance tests are shown for the 10-run averages, and the one large failure (Hospital) plus the acknowledged heuristic for future segment assignment leave the robustness claim under-supported. The central method is sound, but the evidence base needs strengthening before the claims can be accepted at face value.

major comments (3)

[Section 5.2, Table 2] Table 2 reports only the mean square-root CADD over 10 runs and does not report standard deviations, confidence intervals, or significance tests, although the text says the repository contains the full overview including standard deviations. The claim that AT-KDE is "far more accurate and robust" is load-bearing and cannot be assessed from the means alone: 12 of 20 wins could be consistent with substantial run-to-run overlap, and the Hospital row shows a large loss to Prophet. Please include per-log error bars (or per-run distributions) and at least pairwise significance tests (e.g., corrected paired tests) for the headline comparisons, and then qualify the robustness claim according to what those tests show.
[Section 4.5, Algorithm 3; Section 6 (Limitations)] The future segment assignment is a zero-order forecast: when the training label sequence L shows no recurring pattern, EstimateSegmentCluster reuses the most recent global segment cluster for the whole simulation horizon. This step determines which KDE submodels generate every simulated arrival, so it is load-bearing. The Hospital result and the paper's own Limitations paragraph confirm the risk: a sudden test-period decrease in arrivals is exactly the kind of trend this fallback cannot anticipate. To support the robustness claim in the regime the paper itself motivates (global dynamics that change after the training window), please report how often this fallback is active across the 20 logs, compare its behavior against simple alternatives (e.g., linear trend extrapolation or retaining multiple recent clusters) on those cases, and restrict the robustness claim to the regimes where it is actually demonstrated.
[Abstract; Section 5.2] The abstract's phrase "far more accurate and robust" is stronger than what Table 2 shows: AT-KDE is not the best method on 8 of 20 logs, and on Hospital the gap to Prophet is described as substantial. Please replace the unconditional wording with a quantitative, qualified statement (e.g., "best or near-best on most logs, with a known weakness on sudden test-period drops"), and make the conclusion's "superior accuracy and robustness" wording match the reported distribution of outcomes rather than the best-case interpretation.

minor comments (5)

[Section 4, Input definition] The input is described as "a sequence of N sequences," which appears to be a typo; it should say a sequence of N days (or N daily arrival sequences), since each t_i is the sequence of arrival timestamps on day i.
[Section 5.1, Hyperparameters] The statement that the chosen configuration "proved robust across diverse datasets" is not supported by any sensitivity analysis in the paper; please add an ablation or refer to a specific repository section that reports sensitivity results for omega, L, k_max, and the DBSCAN parameters.
[Section 5.1, Data Split] The temporal hold-out split is defined as the first 80% of cases versus the last 20% of cases; please clarify whether cases are sorted by arrival timestamp before splitting and whether this yields a calendar-time test interval, since an uneven arrival rate can make the test period length differ from 20% of the calendar time.
[Table 2, missing entries] Several benchmark entries are missing (e.g., LSTM and NPP on BPIC19, NPP on BPIC20I); please state whether these are whole-log convergence failures or run-level failures, and explain how missing values are treated in the per-log averaging and in the count of logs where each method is best.
[Section 3, Table 1] The motivating simulation-quality comparison is based on a single log and reports no confidence intervals; it would be clearer to present it as an illustrative case study rather than as evidence of a general relationship between arrival accuracy and simulation accuracy.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: AT-KDE is assessed out-of-sample against external benchmarks; the only self-citations are non-load-bearing.

full rationale

AT-KDE is evaluated on a temporal hold-out split (first 80% of cases for training, last 20% for testing) against external benchmarks (Mean, Best Distribution, Prophet, LSTM, Chronos, XGBoost, NPP) using the external CADD metric of Chapela-Campa et al. The test arrivals are never used for fitting: bandwidth is selected on the training portion only, and all segmentation, clustering, and KDE fitting are learned from training data. The central claim is therefore a measured out-of-sample comparison, not a derivation that reduces to its inputs. The only self-citations are [11] (AgentSimulator), used in Section 3 to illustrate the downstream effect of arrival models on simulation quality, and [12], the code repository; neither is used to justify AT-KDE's accuracy, and neither supplies a theorem or fitted value that the experimental comparison depends on. Algorithm 3's fallback (assuming the most recent global cluster when no recurring pattern exists) is a stated heuristic with an acknowledged limitation in the Limitations section; it is a modeling assumption, not a circular step, because it does not define the evaluation metric or the benchmark outcomes. No equation in the paper defines a predicted quantity in terms of the target, and no fitted parameter is renamed as a prediction. Hence there is no significant circularity; the minor self-citation is not load-bearing.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a set of chosen hyperparameters, most reported in Section 5.1, plus the domain assumption that partition subsets are homogeneous and that the most recent segment predicts the future. No new physical or conceptual entities are introduced.

free parameters (6)

bandwidth multiplier k = not reported (varies per subset)
Selected by validation on 20% of training data (Section 5.1); controls KDE smoothness
window size omega = 7 days
Chosen in Section 5.1; affects moving average and minimum segment length
intraday bins L = 3 (3-hour bins)
Chosen in Section 5.1; sets intraday resolution
max weekday clusters k_max = 6
Chosen in Section 5.1; upper bound on weekday clusters
sensitivity range Z = [0.1, 1]
Chosen in Section 5.1; controls change point detection strictness
DBSCAN eps and min_samples = unspecified
Required by Algorithm 1's ClusterSegments but not given in the paper

assumptions (4)

domain assumption Within each partition subset, inter-arrival times are sufficiently homogeneous to be modeled by a single KDE
Invoked in Section 4.4, where each subset gets one KDE from i.i.d. realizations
domain assumption Future arrivals follow the most recent global segment cluster when no recurring pattern is found
Used in Algorithm 3, Step 5 (Arrival Generation) and acknowledged as a limitation in Section 6
domain assumption Change points detected via IQR outlier rule reflect meaningful global dynamics
Section 4.1 uses 1.5*IQR whiskers to identify change points; assumes these correspond to real shifts
standard math Standard statistical results for KDE convergence, Silverman's rule, and clustering algorithms
Background for Sections 4.4 and 5.1

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Cite this review

Pith. "Pith review of A Divide-and-Conquer Approach for Modeling Arrival Times in Business Process Simulation." pith.science (2026). https://pith.science/paper/ILD6QIJS

@misc{pith2026250522381,
  author       = {Pith},
  title        = {Pith review of: A Divide-and-Conquer Approach for Modeling Arrival Times in Business Process Simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILD6QIJS}},
  note         = {Machine review of arXiv:2505.22381}
}

read the original abstract

Business Process Simulation (BPS) is a critical tool for analyzing and improving organizational processes by estimating the impact of process changes. A key component of BPS is the case-arrival model, which determines the pattern of new case entries into a process. Although accurate case-arrival modeling is essential for reliable simulations, as it influences waiting and overall cycle times, existing approaches often rely on oversimplified static distributions of inter-arrival times. These approaches fail to capture the dynamic and temporal complexities inherent in organizational environments, leading to less accurate and reliable outcomes. To address this limitation, we propose Auto Time Kernel Density Estimation (AT-KDE), a divide-and-conquer approach that models arrival times of processes by incorporating global dynamics, day-of-week variations, and intraday distributional changes, ensuring both precision and scalability. Experiments conducted across 20 diverse processes demonstrate that AT-KDE is far more accurate and robust than existing approaches while maintaining sensible execution time efficiency.

Figures

Figures reproduced from arXiv: 2505.22381 by the authors.

**Figure 2.** Workflow of the modeling methodology of our AT-KDE approach. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Distribution of arrivals of BPIC12 per hour of each day of the week. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Comparison of arrivals between AT-KDE, LSTM and Best Distribution. Outline of Weekday and Intraday Simulation Quality. To gain a better understanding of how the simulated arrival timestamps differ across approaches already used in BPS, we compare the hour-day distribut…

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Reference graph

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van der Aalst, W.M.P.: Business process simulation survival guide. In: vom Brocke, J., Rosemann, M. (eds.) Handbook on Business Process Management 1, Intro- duction, Methods, and Information Systems, 2nd Ed, pp. 337–370. International Handbooks on Information Systems, Springer (2015) A Divide-and-Conquer Approach for Modeling Arrival Times 17

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Camargo, M., Dumas, M., Rojas, O.G.: Learning accurate business process simu- lation models from event logs via automated process discovery and deep learning. In: CAiSE. Springer (2022)

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Chapela-Campa, D., Benchekroun, I., Baron, O., Dumas, M., Krass, D., Senderovich, A.: Can I trust my simulation model? measuring the quality of busi- ness process simulation models. In: BPM. Springer (2023)

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Springer (2013)

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