Linear Algebra Matrix Limits/Stochastic Process

lutheranian

1. The problem statement, all variables and given/known data

A diaper liner is placed in each diaper worn by a baby. If, after a diaper change, the liner is soiled, then it is discarded and replaced by a new liner. Otherwise, the liner is washed with the diapers and reused, except that each liner is discarded and replaced after its third use (even if it has never been soiled). The probability that the baby will soil any diaper liner is one third. If there are only new diaper liners at first, eventually what proportions of the diaper liners being used will be new, once used, and twice used? Hint: Assume that a diaper liner ready for use is in one of the three states: new, once used, or twice used. After its use, it then transforms into one of the three states described

2. Relevant equations


If A is a transition matrix and v is the initial state vector and

lim A^m = L as m -->[itex]\infty[/itex] then eventual state is Lv

3. The attempt at a solution

I set up the transition matrix with the first column/row corresponding to new liners, the second to once-used, and the third to twice-used, resulting in the following:

A= (1/3, 1/3, 1| 2/3, 0, 0 | 0, 2/3, 0)

The initial vector is v= (1, 0, 0)

I tried finding the limit of A^m as m --> [itex]\infty[/itex] using wolframalpha (which is allowed because the homework problems have messy numbers) but the computation times out every time.

Ray Vickson

You can get the limiting state-probabilities by setting up and solving a linear system of 3 equations in 3 unknowns. Using the standard convention (with the ROWS summing to 1---not the columns, as you have chosen), the limiting (row) vector, u, satisfies u = u.A and sum u(j) = 1. Omit one of the three equations u(j) = sum_{k} u(k)*A(k,j), j=1,2,3, and replace it by the normalizing condition sum u = 1.

RGV