A deductive derivation and a generalization of a negative binomial distribution model describing the frequency of book circulation

Recently, some models describing the frequency distribution of the book circulation are proposed. In a previous paper, authors also propose a new model of book circulation distribution which is based on a negative binomial distribution. This model is able to describe varians of book circulation distribution with time. This paper has two purposes; 1) as a model of the book circulation, mathematically deducting a negative binomial distribution, 2) discussing the validity of the book circulation model which authors proposed and generalizing this model for the change with time. Consequently, two components of a compound Poisson distribution, a Poisson distribution and a gamma distribution, are deductively derived by considering the nature of book loans, and a negative binomial distribution are deductively derived perfectly. And the two parameters of this gamma distribution are assumed as functions of time variation for describing varians of book circulation with time. The validity of this assumption and the function of these parameters are discussed. As this function, two expressions are used; 1) the usual expression of obsolescence; a exponential function, 2) an alternative for the expression of obsolescence which is derived from a extended obsolescence model. As a result, a generalized model of book circulation distribution with a extended expression of obsolescence is constructed.