Laplace Transforms of Stopping Times for Subordinator with Applications to Inventory Control

TL;DR

This paper develops an analytical framework using Laplace transforms of stopping times for subordinators, modeling demand in inventory systems with jumps, providing explicit formulas for reorder times and costs in complex demand scenarios.

Contribution

It introduces a unified mathematical approach for analyzing inventory control with demand modeled as a generalized Lévy process, including explicit formulas for key quantities.

Findings

Explicit Laplace transform for first-passage times in drifted Poisson processes

Closed-form expressions for mean and variance of reorder times with exponential and Gamma jumps

Analytical characterization of expected total costs in inventory systems

Abstract

Intermittent demand fluctuations pose significant challenges in disaster logistics and medical supply systems. In this study, we formulate cumulative demand as a generalized L\'evy process composed of a drift term, Poisson jumps, and compound Poisson jumps, and analyze a continuous-time inventory model. The proposed framework provides a unified formulation that encompasses both drifted Poisson processes and drifted compound Poisson processes. From a mathematical perspective, we treat the reorder time as a first-passage problem of a subordinator and derive its Laplace transform via the Laplace exponent. In particular, for the drifted Poisson case, we obtain an explicit representation of the inverse Laplace exponent using the Lambert W function, which yields an analytic expression for the Laplace transform of the first-passage time. Furthermore, when the jump sizes follow exponential and Gamma distributions, we derive explicit formulas for the mean and variance of the reorder times, thereby clarifying the moment structure of first-passage times for generalized L\'evy demand processes. From an operations research perspective, we explicitly characterize the expected total cost over a finite time horizon based on the distribution of cumulative demand. This study presents an analytical framework that integrates first-passage theory of L\'evy processes with continuous-time inventory control.