Markov chain problems, absorbing states and eigenvalues Problem set B

A bike-share scheme has three docking stations: X, Y and Z. Each bike is taken from one station and returned to one station the next day. The transition diagram shows the probabilities of where a bike is returned. Write down the transition matrix.

A robot vacuum cleaner is in one of three rooms, the kitchen (K), the lounge (L) or the hall (H), at the start of each hour. It moves according to the transition matrix \(T=\begin{pmatrix}0.24 & 0.32 & 0 \\ 0.76 & 0.28 & 0.52 \\ 0 & 0.4 & 0.48\end{pmatrix}\) (columns and rows in the order K, L, H). Draw the transition diagram, with the probability on each arrow.

A web server is Up (U) or Down (D) each hour. If it is up this hour, the probability that it is up next hour is 0.92. If it is down this hour, the probability that it is down next hour is 0.4. Write down the transition matrix, with states in the order U, D.

A delivery drone is based each night at one of three depots, P, Q or R. Where it is based tomorrow night depends only on where it is tonight, with transition matrix \(T=\begin{pmatrix}0.62 & 0.22 & 0.32 \\ 0.14 & 0.46 & 0.28 \\ 0.24 & 0.32 & 0.4\end{pmatrix}\) (columns and rows in the order P, Q, R).

1. (a) It is at depot Q on Monday night. Find the probability that it is at depot P on Wednesday night.
2. (b) It is at depot R on Monday night. Find the probability that it is at depot R on Wednesday night.

Each election a voter supports the Red party or the Blue party. The transition matrix is \(T=\begin{pmatrix}0.8 & 0.45 \\ 0.2 & 0.55\end{pmatrix}\) (columns and rows in the order Red, Blue). In the long term the proportions of voters for Red and for Blue are \(x\) and \(y\).

1. (a) Write down a system of equations for the long-term proportions.
2. (b) Solve the equations to find the exact long-term proportions.

A gym has 1200 members, each with one main class: Yoga (Y), Spin (S) or Weights (W). Each month, of the members in Yoga, 60% stay and 30% move to Spin (the rest move to Weights). Of the members in Spin, 20% move to Yoga and 50% stay. Of the members in Weights, 10% move to Yoga and 70% stay. Initially 480 members are in Yoga, 300 in Spin and 420 in Weights.

1. (a) Write down the transition matrix T, with the states in the order Y, S, W.
2. (b) Find the number of members in each class after 5 months. Give each answer to the nearest whole number.
3. (c) Find the number of members in each class in the long term, to the nearest whole number.

A factory machine is Working (W), Idle (I) or Faulty (F) each day, and tomorrow's state depends only on today's, with transition matrix \(T=\begin{pmatrix}0.78 & 0.42 & 0.32 \\ 0.12 & 0.48 & 0.22 \\ 0.1 & 0.1 & 0.46\end{pmatrix}\) (columns and rows in the order W, I, F). The machine is working today. Find the probability that it is faulty five days later.

Each year the water quality of a river is Good (G), Fair (F) or Poor (P). The transition matrix is \(T=\begin{pmatrix}0.62 & 0.24 & 0.08 \\ 0.28 & 0.52 & 0.28 \\ 0.1 & 0.24 & 0.64\end{pmatrix}\) (columns and rows in the order G, F, P). Use technology to find the long-term probability that the water quality is Fair.

A beetle is on a ladder with five rungs, numbered 1 (bottom) to 5 (top). Each minute it climbs up one rung with probability 0.55 or slips down one rung with probability 0.45. Rung 1 is a pool and rung 5 is its nest: once it reaches either one it stays there. The beetle starts on rung 2.

1. (a) Write down the transition matrix for the rung the beetle is on, with states 1 to 5.
2. (b) Find the probability that the beetle ends in its nest.

Learners on a language app are Daily or Occasional. Each week 0.2 of daily learners become occasional and 0.05 of occasional learners become daily. The transition matrix is \(T=\begin{pmatrix}0.8 & 0.05 \\ 0.2 & 0.95\end{pmatrix}\) (columns and rows in the order Daily, Occasional).

1. (a) Find the eigenvalues of T and a corresponding eigenvector for each.
2. (b) Write down matrices P and D such that \(T=PDP^{-1}\).

A shop is Busy or Quiet each hour. The transition matrix is \(T=\begin{pmatrix}0.75 & 0.15 \\ 0.25 & 0.85\end{pmatrix}\) (columns and rows in the order Busy, Quiet), and \(T=PDP^{-1}\) where \(P=\begin{pmatrix}3 & 1 \\ 5 & -1\end{pmatrix},\ D=\begin{pmatrix}1 & 0 \\ 0 & 0.6\end{pmatrix}\). At the start, the probability that the shop is busy is 0.9. Let \(f_n\) be the probability that the shop is busy after \(n\) hours.

1. (a) Find an expression for \(f_n\).
2. (b) Hence find the long-term probability that the shop is busy.

A pupil travels to school each day by Walking (W), Bus (B) or Cycling (C). The transition matrix is \(T = \begin{pmatrix} 0.55 & 0.2 & 0.2 \\ 0.1 & 0.65 & 0.15 \\ 0.35 & 0.15 & 0.65 \end{pmatrix}\), in that order.

1. (a) Use your calculator to find the three eigenvalues of \(T\).
2. (b) Write down an eigenvector for the eigenvalue 1.
3. (c) Hence find the steady-state vector, as exact fractions.