On Completion Times under Memoryless Catastrophe
Abstract
We study the completion time of a task subject to independent reset (catastrophe) at each step. The completion-time PGF depends on the base-process PGF through an affine relation, and we exploit this structure systematically. Our main result shows that, among age-based catastrophe mechanisms, geometric-tail catastrophe is exactly the class that yields uniform affine PGF structure; in continuous time, the characterization sharpens to Poisson resetting. We establish a sharp two-sided Kolmogorov bound of order p+|α| for the exponential approximation dK(T/E[T], Exp(1)), thereby closing a logarithmic gap. Applications to the coupon collector with reset coupons reveal a discontinuous Gumbel-to-Exponential transition under resetting, while a multi-phase model exhibits a Gaussian-to-exponential transition with exponential convergence rate.
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BibTeX
@misc{wang2026completion,
title = {{On Completion Times under Memoryless Catastrophe}},
author = {Sichen Wang and Zhipeng Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16566},
eprint = {2609.16566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16566},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Completion Times under Memoryless Catastrophe
AU - Sichen Wang
AU - Zhipeng Lu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16566
UR - https://arxiv.org/abs/2609.16566
AB - We study the completion time of a task subject to independent reset (catastrophe) at each step. The completion-time PGF depends on the base-process PGF through an affine relation, and we exploit this structure systematically. Our main result shows that, among age-based catastrophe mechanisms, geometric-tail catastrophe is exactly the class that yields uniform affine PGF structure; in continuous time, the characterization sharpens to Poisson resetting. We establish a sharp two-sided Kolmogorov bound of order p+|α| for the exponential approximation d_K(T/E[T], Exp(1)), thereby closing a logarithmic gap. Applications to the coupon collector with reset coupons reveal a discontinuous Gumbel-to-Exponential transition under resetting, while a multi-phase model exhibits a Gaussian-to-exponential transition with exponential convergence rate.
ER -