Capacity and Shopping Rate Under a Social Distancing Regime

Abstract

Capacity restrictions in stores, maintained by mechanisms like spacing customer intake at certain time intervals, have become familiar features in the time of the pandemic. The effect on total spending is not a linear function of reduced capacity, since shopping in a crowded store under a social distance regime is prone to considerable slowdown. In this thesis, We introduce a simple dynamical model of the evolution of shopping rate as a function of a given customer intake rate, starting with an empty store. The slowdown of each individual customer is incorporated as an additive term to a baseline value shopping time, proportional to the number of other customers in the store. We determine analytically and by simulation the trajectory of the model as it approaches a Little's Law equilibrium, and identify the point of phase change, beyond which equilibrium cannot be achieved. By relating customer shopping rate to the slowdown compared to 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 is not enough to justify a lower intake rate. Because the slowdown due to the largest possible number of shoppers is more than compensated for by the increased volume of shopping.