Analyse long-run behaviour of a Leslie model
A treatment model records expected numbers of filters in successive service stages
All modelled items are counted. A first-row entry rate means the expected number entering the first stage per current item in one month. Survival entries are proportions for moving to the next stage
The stages, in order, are unused filters, working filters, and checked filters. One application of the Leslie matrix represents one month
\[L = \begin{bmatrix} \frac{1}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \end{bmatrix}\]
\[S_0 = \begin{bmatrix} 360 \\ 180 \\ 90 \end{bmatrix}\]
Use \(S_{n+1} = L S_n\). The displayed probe periods are 0, 1, 2. Give finite ratios or percentage changes to 1 decimal place where requested