A mathematical model on public library loan incorporating an inverse size effect

The purpose of this paper is to propose a new mathematical model for predicting the amount of loan at public libraries, and to investigate the validity of the model by using statistical data on public libraries at Saitama Prefecture in Japan. The model is L=aCPexp (-bC), where L is the number of times books are circulated in a library, C is the number of books held at the library, P is the population within the service area, and a and b are parameters. The model explains well statistical variation of the amount of loan at Saitama Prefecture. It can be said that the result shows a sort of "inverse effect" of size of collection or population, which is represented by exp (-bC) in the model, on the increase in loan. Also, a simple simulation based on the model is tried to assess the effectiveness of new sites for access.