UMVUEs and Bayes estimators for various performance measures on a Poisson queue with discouraged arrivals

Abstract

To prevent overcrowding during the COVID-19 pandemic, numerous businesses, and public service systems have to limit the number of people entering the premises to reduce infection risks. Such a realistic situation prompts us to revisit an adaptive queueing model from a statistical perspective, which allows us to extensively analyze and explore the stochastic service system arising in the pandemic period. In order to avoid long waiting lines, we assume that the arrival rate of customers into the system depends on the system size instead of a constant rate. This article attempts to study the uniformly minimum variance unbiased estimators and closed-form Bayes estimators of various queueing characteristics, such as the probability that the server is busy, the proportion of lost customers, mean system length, and average queue length. The estimates and their behaviors are compared by Monte-Carlo simulation with different sample sizes. The simulation results show that we may choose different estimation techniques for different performance indicators to obtain a more precise estimate.