Determine the long-run state of a transition model practice questions | General Mathematics (Year 12, VCE Units 3 & 4, Transition matrices, State recurrences and equilibrium)

The long-run state

In a steady state, the counts do not change from one step to the next, although transfers can continue. With a regular transition matrix, the state approaches its unique steady state for a fixed total. For a two-state model, balance the two off-diagonal flows and conserve the total

L_1:L_2=b:a

Balance the flows At a steady state, the total leaving each state equals the total entering it. In a two-state model, this balances the two flows \(\displaystyle aL_1=bL_2\)
Use the off-diagonal proportions For a regular two-state model, if a is the proportion leaving state 1 and b is the proportion leaving state 2, the long-run ratio is b to a. \(\displaystyle a=0.3,\quad b=0.6\Rightarrow L_1:L_2=0.6:0.3=2:1\)
Conserve the total The entries of every state matrix have the same total as the initial state \(\displaystyle L_1+L_2=S_{0,1}+S_{0,2}\)
Allow movement to continue A steady state means the totals stay fixed, not that no one moves between states \(\displaystyle 0.3 \times 40=0.6 \times 20=12\)

Worked example

The columns of T describe the current state and the rows describe the next state. The total is 120. Find the long-run state and the number moving each way

T=\begin{bmatrix}0.75&0.5\\0.25&0.5\end{bmatrix}

\(\displaystyle a=T_{21}=0.25,\quad b=T_{12}=0.5\)The off-diagonal entries give the proportions leaving state 1 and state 2.
\(\displaystyle L_1:L_2=b:a=0.5:0.25=2:1\)The long-run counts have the ratio of the second off-diagonal proportion to the first
\(\displaystyle L_1+L_2=120,\quad L_1=80,\quad L_2=40\)The ratio has three parts, so each part is 120 divided by 3, or 40.
\(\displaystyle 0.25 \times 80=20,\quad 0.5 \times 40=20\)The equal flows confirm the long-run state while movement continues

Common mistakes

Assuming an even split

✕\(\displaystyle L_1=L_2=45\text{ when the total is }90\)✓\(\displaystyle L_1:L_2=0.4:0.2=2:1\Rightarrow (L_1,L_2)=(60,30)\)

The long-run split is determined by the off-diagonal proportions, not automatically by equal parts of the total

Changing the conserved total

✕\(\displaystyle L_1+L_2=100\text{ when the initial total is }120\)✓\(\displaystyle L_1+L_2=120\)

Each transition-matrix column totals 1, so the total across all states is conserved

Reading an early state as the long-run state

✕\(\displaystyle S_0=\begin{bmatrix}20\\100\end{bmatrix}\Rightarrow S_{\text{long run}}=\begin{bmatrix}20\\100\end{bmatrix}\)✓\(\displaystyle S_{\text{long run}}=\begin{bmatrix}80\\40\end{bmatrix}\text{ because }L_1:L_2=2:1\)

The initial or an early state is not the long-run state unless its flows are already balanced

Difficulty levels

introFind a two-state long-run split and observe the settling across probe values

standardDescribe long-run behaviour for a two-state model approaching from a chosen side

challengeCompare two starting states in a three-state model and show that equal flows continue

examSelect one long-run count from even-split, initial-state and early-probe distractors

Questions about this topic

What does steady state mean in a transition model?+

The state counts remain unchanged from one application of the transition matrix to the next. People or items can still move between states when the flows balance

Why is the long-run ratio b to a?+

In a regular two-state model, if a leaves state 1 and b leaves state 2, balance requires a times the state 1 count to equal b times the state 2 count. Rearranging gives the ratio b to a. With more states, balance the total inflow and outflow for each state; individual pairs need not balance

Why does the total stay constant?+

Every column of the transition matrix totals 1, so all of the current population is distributed among the states at the next step. No count is created or lost

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 11: Transition matrices

In this chapter

11A Transition matrices and diagrams: Translate transition descriptions, diagrams and matrices 11A Transition matrices and diagrams: Check fixed-population and regular transition models 11B Interpreting transition matrices: Translate transition descriptions, diagrams and matrices Use a transition matrix to predict the next step 11C State recurrences and equilibrium: Check fixed-population and regular transition models Generate state matrices by recursion 11C State recurrences and equilibrium: Apply and reverse a state recurrence with external change Determine the long-run state of a transition model Construct a transition model with a specified equilibrium 11D Adjusted state recurrences: Apply and reverse a state recurrence with external change Construct and interpret Leslie matrices and life-cycle diagrams Generate and total Leslie population states Analyse long-run behaviour of a Leslie model