Modelling organic customer growth

Question

I'm currently making a model of a start-up company with the goal of describing the number of customers the company has at a given year (variable t).

These are the assumptions:

• The company has 15 customers year 1. (variable a)
• On average, each existing customer invites 2 friends their first year who then also become customers. (variable r)

I've used $c(t)=ar^t$ to describe the number of customers c at time t.

The problem

The problem with the function is that eventually there will be more customers than there are people on this planet. So I want to add customer population as a variable, putting a limit on how much it can grow.

I think the more accurate representation would be an S-shaped curve but I don't have the mathematical knowledge to represent this as a function.

So, how can I account for the fact that there is a limited number in the customer population?

Answer 1

You can use a logistic function to model customer growth with the limitation of a maximum population. The general formula for a logistic function is:

$\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}$

Where:

• x_0 = value of the sigmoid's midpoint;
• L = the curve's maximum value;
• k = the logistic growth rate or steepness of the curve.

Hope this helps.