The impact of infection risk on customers' joining strategies

The risk of infection from the COVID-19 virus dictates businesses, such as supermarkets and department stores, to impose limits on the maximal number of customers allowed inside a store at any given time. These social distancing constraints generate long queues of waiting customers outside such businesses. This work investigates the impact of infection risk on arriving customers' strategic decisions regarding joining such queues. We consider a typical store where the floor is divided into two separate areas: (i) a shopping area with at most K shoppers allowed, and (ii) a payment area with c ≥ 1 parallel servers and an adjacent limited waiting space of size N ≥ 0. When the shopping area is full, a newly arriving customer observes only the outside queue and decides whether to join or balk. We investigate customers' individual joining strategies, as well as social optimization, with a utility function that takes into account not only the cost associated with waiting times (as in Naor's (1969) celebrated model), but also the cost related to the risk of infection. We propose an innovative risk measure that is a function of both the number of customers already in line, and those that a tagged customer 'meets' while waiting to enter the store. Consequently, expressions for mean waiting times and infection risk are derived and explicit formulas are obtained for limit values of the parameters. Our results can be used by authorities and customers alike to determine the maximal allowed queue sizes that ensure safety and reduce the risk of infection while minimizing associated costs.