Calculator input
Enter the chain in canonical form: transient states first, absorbing states last.
Formula used
1) Canonical absorbing form
Arrange the transition matrix as
P = [[Q, R], [0, I]].
The matrix Q contains transient-to-transient moves.
The matrix R contains transient-to-absorbing moves.
2) Fundamental matrix
The core matrix is
N = (I − Q)−1.
Entry Nij gives the expected number of visits to transient state
j when starting from transient state
i.
3) Absorption probabilities
The matrix
B = NR
gives absorption probabilities.
Entry Bik is the probability of ending in absorbing state
k when starting from transient state
i.
4) Expected steps before absorption
Expected steps from a transient state come from
t = N1,
where 1 is a column of ones.
In this page, each value is the row sum of
N.
5) Steady state for an absorbing chain
If the starting distribution is
x₀ = [α, β],
then the limiting distribution becomes
x∞ = [0, β + αB].
All transient-state probabilities approach zero.
How to use this calculator
Example data table
This example matches the default 2 transient and 2 absorbing state matrix shown by the calculator.
| Item | Example value | Meaning |
|---|---|---|
| Transient states | 2 | T1 and T2 can leave and continue moving. |
| Absorbing states | 2 | A1 and A2 keep the chain fixed after entry. |
| Starting distribution | [1, 0, 0, 0] | The process begins fully in T1. |
| Absorption probability into A1 | 0.647059 | Starting from T1, A1 is the more likely final state. |
| Absorption probability into A2 | 0.352941 | Starting from T1, A2 is still possible. |
| Expected steps before absorption | 4.117647 | This is the expected transient duration from T1. |
Frequently asked questions
1) What does steady state mean for an absorbing chain?
It means the limiting distribution after many transitions. Every transient state receives probability zero, while absorbing states receive the final absorption probabilities.
2) Why must the matrix be in canonical order?
Canonical order places transient states first and absorbing states last. That makes it possible to isolate Q and R and then compute N, B, and the limiting distribution directly.
3) What happens if a row does not sum to 1?
The matrix would no longer represent valid transition probabilities. The calculator checks this and stops until each row adds to exactly 1 within a small tolerance.
4) Why do absorbing rows need identity values?
An absorbing state must remain in itself with probability 1. Every other outgoing probability from that row must be 0, which forms the identity block.
5) What does the fundamental matrix tell me?
The fundamental matrix shows expected visits among transient states before absorption. Row sums also give expected remaining steps before the process reaches an absorbing state.
6) Can the starting distribution include absorbing states?
Yes. Any starting probability already placed in absorbing states remains there. The final distribution adds those fixed amounts to the new absorption probabilities from transient states.
7) What does the absorption matrix B represent?
Each row of B lists the probabilities of ending in each absorbing state when starting from one transient state. Row sums should equal 1.
8) Why might the calculator reject my entries?
Common issues are invalid probabilities, rows not summing to 1, starting probabilities not summing to 1, or an invalid absorbing block that breaks the absorbing-chain structure.