A Congestion‐Discouraged Poisson Distribution for Operations Research: A New Count Model From State‐Dependent Order Inflow and Parallel Resolution

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Abstract
We introduce the congestion–discouraged Poisson (CDP) distribution, a two‐parameter count model derived from a state‐dependent birth–death process where arrivals are suppressed by system backlog. The distribution is specifically designed to model underdispersed count data common in operations research, such as service queues and inventory pipelines. We establish its fundamental properties, including ultra‐log‐concavity, a Stein‐type identity, and a simple probability recursion. Likelihood‐based inference is developed, with parameters providing direct operational insights into congestion costs and idle probabilities. In a simulation study and an empirical application to the WomenQueue dataset, the CDP model outperformed both Poisson and Conway–Maxwell–Poisson (CMP) benchmarks in AIC, BIC, and Pearson X 2 .The results confirm that the CDP distribution is a principled mechanism‐based tool that accurately captures congestion‐sensitive dynamics while remaining as tractable as classical benchmarks.
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  • Islamic University of Madinah

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