Continuous-time monitoring of queueing processes

In recent years, effective monitoring of categorical and count data has increasingly attracted attention of researchers in the area of statistical process control. However, most of the existing research model categorical and count data streams as independent and identically distributed data or serially correlated discrete time stochastic processes. Very limited research has been conducted for monitoring continuous-time stochastic processes (CTSPs). This paper develops a novel statistical monitoring method for CTSPs with a focus on queueing processes. The proposed method is based on detecting a change in the intensity function of such processes, using an approximate likelihood ratio test. The approximation method is both computationally easy for real-time implementation and well-suited for the introduction of penalization methods. Simulation results based on Markovian and non-Markovian queues show that the proposed methods effectively detect temporal changes in the queueing process. A case study focusing on monitoring the bed assignment process of patients visiting an emergency department demonstrates the efficacy of the proposed methods in a healthcare system. Note to practitioners The methods studied in this paper can be used by operations managers in service enterprises, such as healthcare and transportation industries, for monitoring the timeliness of service provided to customers. The proposed method requires arrival and departure timestamps of customers from a queueing system when the system is considered ideal. This data is then used to define a metric for evaluating the queueing system's performance in a real-time manner. The proposed method does not require the arrival rate of the customers to be time-homogeneous. The experimental results show that the method is agnostic to classical Markov process assumptions required in traditional performance modeling methods for queueing systems. The paper provides the details of applying the proposed method to a timeliness-of-care monitoring problem in the emergency department of a large academic medical center. This method is expected to be broadly applicable to other service systems as well.