The Discrete Laplace Transform (DLT) Order: A Sensitive Approach to Comparing Discrete Residual Life Distributions with Applications to Queueing Systems

Abstract

In this paper, we propose a general methodology for implementing a stochastic ordering framework based on the Discrete Laplace Transform (DLT) to analyze and compare residual life distributions in discrete-time systems. We formally introduce the DLT-order, denoted X ≤DLT Y, and establish its key properties: reflexivity, transitivity, monotonicity, and closure under convolution and mixture. The DLT-order exhibits higher discriminatory power than classical stochastic orders—particularly when distributions are non-log-concave, heavy-tailed, or exhibit crossing survival functions. The framework is computationally efficient (with empirical estimation complexity O(n)) and operationally interpretable. Using real call center data, we demonstrate that DLT scores effectively differentiate between service classes even when mean waiting times are nearly identical. For instance, at discount rate s = 0.1, technical support calls yielded a DLT value of 3.545, compared to 3.503 for administrative inquiries, revealing latent disparities in residual waiting behavior. Sensitivity analyses confirm the robustness of the DLT-order under noise, outliers, and distributional shifts. Threshold-based operational rules (e.g., reallocating agents when DLT ≥ 1.5) led to an observed 18% reduction in average wait times in pilot deployments. Thus, the DLT-order fills a critical gap between discrete reliability theory and performance analytics, offering a repeatable, interpretable, and mathematically rigorous method to rank, compare, and optimize resources in discrete-time service operations.