Queueing theory is widely adopted in Operations Management to study a large array of models exhibiting waiting lines behavior, especially in the healthcare environment. However, for the complicated queueing systems with routing controls or customer abandonments, an exact solution is very difficult, if not impossible, to derive. Hence, asymptotic analysis in the heavy-traffic regimes is often employed to achieve accurate approximations, which serves as the fundamental methodology in this thesis.
In the first study, we consider a service system with multiple stations, serving a single class of arriving customers. Each station consists of many servers. Upon each arrival, the system manager must decide to which station the customer is routed, given the routing choice is limited. We propose a MED-FSF(ξ) policy, under which a random number (ξ) is generated at arrival epoch, representing ξ stations are chosen with real-time information examined. The arrival is then routed to one of the selected stations, either with minimum expected delay if all the candidate stations are occupied, or with the fastest available server. We prove the proposed routing policy can asymptotically minimize the long-run average waiting cost under the Halfin-Whitt regime. In addition, we conduct simulation experiments to validate the asymptotic optimality, and to demonstrate the power of having slightly more than one choice.
In the second study, we analyze a critical care system by formulating it into different queueing network structures: the first one consists of an intensive care unit (ICU) plus multiple high-dependency units (HDUs); and the second one consists of an ICU plus a single centralized HDU. Based on the fluid models, we propose an optimal staffing policy that asymptotically minimizes the long-run average balking, waiting and abandonment costs. Through the analysis, we aim to identify the underlying driving parameters that dictate the staffing decision among ICU and HDUs, which trades off the capacity needs among different units to accommodate the treatment requirements from various types of patients.
Meanwhile, other related on-going research topics will also be discussed.