Servers in many real queueing systems do not work at a constant speed. They adapt to the system state by speeding up when the system is highly loaded or slowing down when load has been high for an extended time period. Their speed can also be constrained by other factors, such as geography or a downstream blockage. We develop a state-dependent queueing model in which the service rate depends on the system "load" and "overwork." Overwork refers to a situation where the system has been under a heavy load for an extended time period. We quantify load as the number of users in the system and we operationalize overwork with a state variable that is incremented with each service completion in a high-load period and decremented at a rate that is proportional to the number of idle servers during low-load periods. Our model is a quasi-birth-and-death process with a special structure that we exploit to develop efficient and easy-toimplement algorithms to compute system performance measures. We use the analytical model and simulation to demonstrate how using models that ignore adaptive server behavior can result in inconsistencies between planned and realized performance and can lead to suboptimal, unstable, or oscillatory staffing decisions.
The COVID‐19 pandemic has had profound effects on grocery retailers, forcing them to make many operational changes in response to public health concerns and the shift in customers' shopping behavior. Grocery retailers need to understand the impact of pandemic conditions on their operations, but the literature has not modeled and analyzed this issue. We bridge this gap through economic models that consider the documented changes in the customers' shopping behavior during the COVID‐19 pandemic, including less frequent in‐store shopping and bulk‐shopping tendency. We capture the impact of occupancy limitation guidelines on grocery retailers' service capacity, customers' shopping behavior, and, consequently, on the retailers' store traffic and profit. We find that though store occupancy limitations reduce the in‐store foot traffic (which helps with curbing the disease spread), interestingly, they do not necessarily result in a profit decline. Under occupancy limitations and when the retailer offers the delivery or curbside pickup service, our analyses highlight the externality impact of online customers on the shopping behavior of in‐store customers. When the retailer adds the delivery service, such externalities may increase the store traffic (higher infection risk inside the grocery store) and reduce the retailer's profit. When the retailer adds the curbside pickup instead, it has more control over the impact of externalities, which helps in lowering the store traffic and increasing the profit. Our results offer valuable insights into how retailers should regard occupancy limitations and health safety measures. Our results also highlight conditions under which various operating modes may help retailers reduce infection risk and achieve higher profit.
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