Queues with Censored Demand and Autoregressive Net Input

We develop methods for simulating a queue with an autoregressive net-input process and for recovering characteristics of such a net-input process from samples of queue lengths. We apply these methods to the problem of estimating the censored (unsatisfied) demand for the queue’s content and show how to model a queue for which the censoring of demand is graduated in a neighborhood of the queue’s zero lower bound. As an example, we estimate the monthly unsatisfied demand for U.S. nonfarm jobs based on samples of job openings through a period including the last two recessions.

💡 Research Summary

The paper develops a comprehensive framework for modeling, simulating, and estimating queues whose net‑input follows an autoregressive (AR) process and whose demand may be censored when the queue reaches its lower bound. Traditional queueing theory typically assumes independent arrivals (often Poisson) and a fixed service rate, but many real‑world systems—especially in economics and operations—exhibit temporally correlated demand and a mechanism that blocks or “censors” excess demand once capacity is exhausted. The authors therefore construct a state‑space representation that integrates an AR(1) net‑input, a deterministic service capacity, and a censoring function that can be either hard (complete blocking) or graduated (partial, smoothly diminishing blocking as the queue approaches zero).

The net‑input (X_t) is modeled as (X_t = \phi X_{t-1} + \epsilon_t) with (\epsilon_t \sim N(0,\sigma^2)). The queue length evolves according to (Q_{t+1}= \max{Q_t + X_t - C, 0}), where (C) is a constant outflow. When (Q_t) is zero, any surplus (X_t - C) is not observed; this unobserved portion is denoted (U_t) (censored demand). In the hard‑censoring case, (U_t = \max{- (Q_t + X_t - C), 0}). The graduated‑censoring extension introduces a scaling function (g(Q_t)\in