Kardjian Brothers, Inc. has invented a fundamentally new way to control the photolithography process used in manufacturing computer chips. Before they sell or license this technological breakthrough, they would like to have some idea of how rapidly it might gain market share. Their analysis is based on the Bass diffusion model, which rests on three assumptions:

While there are many interpretations of this general model, it is common to attribute the innovation effect to marketing efforts such as advertising, which convert some percentage of potential adopters every time period. The imitation effect is thought to operate through word of mouth, in which every contact between an adopter and a potential adopter results in a new adoption with some probability.

The following model is consistent with these assumptions:

where

Assume an initial population of 1,000 and parameter values of 0.03 for p and 0.38 for q. (These are the average values found over dozens of new products and technologies ranging from toasters to CT scanners.)

a. Build a deterministic model for the number of adopters over time. In what year will the cumulative numbers of adopters first exceed half the original population?
b. Build a simulation model for the number of adopters over time, assuming the following uniform distributions for the parameters: p lies between 0 and 0.5; q lies between 0.28 and 0.48. What is the average number of years it takes for the cumulative number of adopters to reach 50 percent of the initial population?
c. What is the probability that it will take more than 10 years for the cumulative number of adopters to reach 50 percent of the initial population?