Capacity, Collision Avoidance and Shopping Rate under a Social Distancing Regime

Capacity restrictions in stores, maintained by mechanisms like spacing customer intake, became familiar features of retailing in the time of the pandemic. Shopping rates in a crowded store under a social distancing regime are prone to considerable slowdown. Inspired by the random particle collision concepts of statistical mechanics, we introduce a dynamical model of the evolution of the shopping rate as a function of a given customer intake rate. The slowdown of each individual customer is incorporated as an additive term to the baseline value of the shopping time, proportionally to the number of other customers in the store. We determine analytically and via simulation the trajectory of the model as it approaches a Little's law equilibrium and identify the point beyond which equilibrium cannot be achieved. By relating the customer shopping rate to the slowdown compared with the baseline, we can calculate the optimal intake rate leading to maximum equilibrium spending. This turns out to be the maximum rate compatible with equilibrium. The slowdown due to the largest possible number of shoppers is more than compensated for by the increased volume of shopping. This macroscopic model is validated by simulation experiments in which avoidance interactions between pairs of shoppers are responsible for shopping delays.