Problem definition: Physician productivity (measured by new patients seen per hour) in emergency departments (EDs) exhibits a distinct time-varying pattern. We examine the factors contributing to this phenomenon empirically and analytically, and investigate the implications of incorporating time-varying service rates into ED modeling and physician staffing. Methodology/results: Using data from a Canadian ED, we provide empirical evidence that the "shift hour" (time elapsed since the shift started) is the most significant predictor of physician productivity. We then model the new patient pickup decisions using an optimal control framework. Assuming physicians' objective is to maximize throughput while avoiding handoffs or overtime, the optimal policy implies adjusting pickup rates throughout a shift is a rational decision. We then investigate the impact of incorporating the time-varying physician productivity in ED modeling and staffing. Validated using data from two Canadian EDs, our simulation results demonstrate that a multi-server queueing model with shift-hour-dependent service rates, despite its lower dimensionality, can accurately capture time-of-day-dependent patient waiting times, whereas models assuming constant service rates deviate significantly from observed data. Furthermore, a case study reveals that staffing plans optimized for time-varying rates outperform current practices, leading to substantial cost savings. Managerial implications: Our findings underscore the necessity of accounting for the time-varying nature of physician productivity. Hospital administrators and schedulers should incorporate this shift-hour dependency into staffing decisions to enhance resource allocation and ED operational efficiency.

For a queueing system with multiple customer types differing in service-time distributions and waiting costs, it is well known that the cµ-rule is optimal if costs for waiting are incurred linearly with time. In this paper, we seek to identify policies that minimize the long-run average cost under nonlinear waiting cost functions within the set of fixed priority policies that only use the type identities of customers independently of the system state. For a single-server queueing system with Poisson arrivals and two or more customer types, we first show that some form of the cµ-rule holds with the caveat that the indices are complex, depending on the arrival rate, higher moments of service time, and proportions of customer types. Under quadratic cost functions, we provide a set of conditions that determine whether to give priority to one type over the other or not to give priority but serve them according to first-come, first-served (FCFS). These conditions lead to useful insights into when strict (and fixed) priority policies should be preferred over FCFS and when they should be avoided. For example, we find that, when traffic is heavy, service times are highly variable, and the customer types are not heterogenous, so then prioritizing one type over the other (especially a proportionally dominant type) would be worse than not assigning any priority. By means of a numerical study, we generate further insights into more specific conditions under which fixed priority policies can be considered as an alternative to FCFS. This paper was accepted by Baris Ata, stochastic models and simulation.