Probabilistic detection model for unqualified units under encouraged arrivals in a feedback Markovian queue: a search-theoretic approach with discounted effort and waiting time reduction

This study develops a probabilistic framework for detecting non-servable units in a feedback Markovian queue with finite capacity. Such units arise from system-level constraints (e.g. technical, temporal, or quality-related factors) and contribute to congestion, increased waiting times, and potential customer abandonment. Arrivals follow a Poisson process, while balking and retention of reneged units are incorporated into the model. The system dynamics are formulated using differential equations and solved via Laplace transforms. To mitigate congestion effects, an encouraged arrival mechanism based on incentive strategies (e.g. discounts) is introduced. An optimization problem is formulated to minimize the total detection cost under constraints related to the incentive parameter, employing an exponential detection function. An algorithm is proposed to compute the transient state probabilities and evaluate the likelihood of detecting non-servable units over time. A numerical example, supported by sensitivity analysis, illustrates the model’s effectiveness and robustness in improving detection performance.