A Simple Model to Estimate Daily Visits of A Web Site

yeeyan.com, a translation 2.0 web site, was launched (actually, re-launched because we had a same-url-blog-site before) around the middle of December, 2006. A month later, on January 21st, 2007, I developed the following model to help our team to foresee the growth of daily visits of our web site.

List of Symbols:

V(t): number of daily visitors at time t
V_new(t): number of daily new visitors at time t
V_old(t): population size of old (sustained) visitors at time t
p: the probability that an old visitor visits the site in a day
r_new: drop rate of new visitors
r_old: drop rate of old visitors

Note that a new visitor is defined as a visitor who hasn't visited the web site for a certain period, for example, a month.

The model consists of two equations:

(1) V(t) = V_new(t) + p * V_old(t)

(2) V_old(t) = (1-r_old)*V_old(t-1) + (1-r_new)*V_new(t-1)

In stable status:

V_new(1-r_new) = V_old*r_old => V_old = V_new*(1-r_new)/V_old
Thus, V = [1+p*(1-r_new)/r_old]*V_new

Static Analysis:

V is proportional to V_new, which represents how effectively the web site is promoted to potential users.

V is also somewhat proportional to p, which is a reflection of how often the content of the web site is updated.

V is inverse proportional to r_old, which reflects how "sticky" the web site is (the smaller r_old, the stickier the web site). r_old is often seen as the most critical factor. Some people believe this should be the only factor of concern. Theoretically, if r_old was zero, V would be infinite.

Dynamic Analysis:

p has nothing to do with dynamic characteristic. It doesn't appear in the recursive equation (2); any change in p directly goes to V and there is no transit time.

r_old, again, is the major factor that affects the system dynamics. If you know something about control theory and Z-transform, you will learn that (1-r_old) is the only root of the system. When r_old gets closer to zero, the root gets closer to 1 and it takes longer time for the system to reach its stable status.

Discussion:

(1) All parameters can be obtained by using Google Analytics;

(2) Though all parameters are time varying, within a certain period, they change slowly and thus can be seen as constants;

(3) The estimation / forecast is valid only for short term or intermediate term. I tracked the records of daily visits of our web site, and they are very close to my estimation by now.