The COVID-19 pandemic has forced numerous businesses such as department stores and supermarkets to limit the number of shoppers inside the store at any given time to minimize infection rates. We construct and analyze two models designed to optimize queue sizes and customer waiting times to ensure safety. In both models, customers arrive randomly at the store and, after receiving permission to enter, pass through two service phases: shopping and payment. Each customer spends a random period of time shopping (first phase) and then proceeds to the payment area of the store (second phase) where cashiers are assigned to serve customers. We propose a novel approach by which to calculate the risk of a customer being infected while queueing outside the store, while shopping, and while checking out with a cashier. The risk is proportional to the second factorial moment of the number of customers occupying the space in each phase of the shopping route. We derive equilibrium strategies for a Stackelberg game in which the authority acts as a leader who first chooses the maximum number of customers allowed inside the store to minimize the risk of infection. In the first model, store’ management chooses the number of cashiers to provide to minimize its operational costs and its customers’ implied waiting costs based on the number allowed in the store. In the second model, the store partitions its total space into two separate areas – one for shoppers and one for the cashiers and payers – to increase cashiers’ safety. Our findings and analysis are useful and applicable for authorities and businesses alike in their efforts to protect both customers and employees while reducing associated costs.
- Matrix geometric
- Stackelberg game