Calculating Average Age in a Solera Process

[shinta]

Hi, I'm trying to figure out the average age in a solera process. The equations are:

An = (2/3)(An-1 + 1) + (1/3)(0)
Bn = (2/3)(Bn-1 + 1) + (1/3)(An-1 + 1)
Cn = (2/3)(Cn-1 + 1) + (1/3)(Bn-1 + 1)

With initial state:

A0 = 0
B0 = 0
C0 = 0

The question is, what is C as n goes to infinity?

 

[Fermat]

Find the limiting value of A_n as n -> infinity.
Substitute this value into the difference eqn for B_n.

Find the limiting value of B_n as n -> infinity.
Substitute this value into the difference eqn for C_n.

And finally, find the limiting value of C_n as n -> infinity.

I also wrote a small program to work out the values of the series, and got the limiting value for C_n as C_n = 8.00000...

 

[matt grime]

If you assume that they all converge to a, b and c respectively then they satisfy

a=(2/3)(a+1) + 1/3

b=(2/3)(b+1) +(1/3)(a+1)

c = (2/3)(c + 1) + (1/3)(b + 1)

which you can solve.

So, if there is a solution, that is what it is. You might need to prove that a limit exists, though.

 

[shinta]

Hey, thanks for the help guys.

Ya, I figured out later to just substitute a_n = a_n+1 and solve. I suppose I should prove that the series converges, but this is good enough for now.

Thanks!