This objective of this research project is to generate insights into near-optimal management of large-scale service systems such as customer call centers and hospitals. It will over analytical and numerical tools that help determine the amount of resource to meet certain performance constraints, under realistic assumptions on arrival times, service times, and customer behaviors. For a many-server queue with general service and patience time distributions that models a large-scale call center, (i) the PI will establish measure-valued diffusion limits under both the conventional and the hazard-rate scaling many-server heavy traffic. (ii) Using these limits, the PI will provide key insights into the sensitivity of service and patience time distributions for large-scale systems. (iii) The PI will develop a numerical algorithm to compute the stationary distribution of a diffusion limit; the key of the algorithm is to and a robust reference density. (iv) The PI will establish asymptotic relationships between abandonment processes and queue length processes; these relationships are essential to provide a modularized, continuous-mapping approach to proving diffusion limits; they also suggest statistical estimators for the patience time density at zero, a key system parameter. For hospital operations, the PI proposes a novel many-server queue model whose service times are critically linked to a patient discharge distribution; the discharge times are assumed to be independent, identically distributed (IID), and consequently the service times are not IID.
For a large-scale customer call center, customer expectation demands that a proper staffng level be maintained so that only a small to moderate fraction of customers abandon the system. For a hospital, it is important to maintain certain service levels; for example, for patients who are transferred from the emergency department to inpatient beds, only a small fraction of bed-requests have to wait six hours or longer. The call center model research will create a sharp understanding on the role of statistical distributions when the center is operating in a ealistic parameter regime; it will provide insights into estimating system parameters reliably and efficiently. The hospital model research promises key insights into bed-capacity management and near-optimal discharge distribution, overcoming many challenges including (a) the arrival process has a periodic arrival rate, (b) the service times are not iid, (c) the number of servers is large, (d) there is no finite-dimensional Markovian representation due to general distributional assumptions, and (c) extremely long service times, compared with the time-variations of the arrival rate.
The PI, together with his collaborators, developed analytical and numerical methods to analyze stochastic systems that model large scale service systems such as hospitals and customer call centers. The team developed diffusion models for a many-server queue that is operated in the quality- and efficiency-driven regime. The queue has many realistic features such as phase-type service time distribution and general customer patience time distribution. The diffusion processes are piecewise Ornstein-Uhlenbeck processes. Numerical algorithms have been developed to compute the stationary distribution for such a diffusion process. Computational results show that diffusion models are accurate in predicting system performance of a many-server queue with customer abandonment, at a small fraction of computational times by existing methods. The team conducted an empirical study of patient flow in a large urban hospital. The team proposed new queueing systems that include realistic features of a hospital such as two-time-scale service time that depends on the admission time and the discharge time of a patient. The team demonstrated that normal approximation is an accurate and practical tool to compute various time-dependent performance measures in a time-varying queueing system whose service times follow a two-time-scale model. A salient feature of the normal approximation is that its computational time is independent of the number of servers, yet the approximations are accurate for a range of systems having moderate or large number of servers. The normal approximation also provides a robust framework to model and analyze systems with overly-dispersed arrival processes that are not Poisson. The normal approximation provides an engineering tool for hospital managers to evaluate quickly the impact of various decisions on the time-of-day boarding time for emergency room patients awaiting inpatient beds. These decisions include the timing of discharge ward patients and the bed distribution among inpatient wards.
