Model absorbing dynamics in Markov or random walks. Use matrices to estimate mean absorption time. Compare scenarios, export outputs, and validate with examples quickly.
This calculator models a discrete-time Markov process with transition matrix P. Absorbing states are states that, once entered, cannot be left.
Example 4-state chain where state 4 is absorbing.
| From \ To | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 0.5 | 0.5 | 0 | 0 |
| 2 | 0.2 | 0.3 | 0.5 | 0 |
| 3 | 0 | 0.2 | 0.3 | 0.5 |
| 4 | 0 | 0 | 0 | 1 |
In a discrete-time Markov process, an absorbing state is terminal: once entered, the probability of leaving is zero. Absorbing time is the number of steps required to reach any absorbing state from a chosen start state or start distribution. If each step represents a physical interval Δt, convert to time with T = steps · Δt.
The transition matrix must be square, non‑negative, and row‑stochastic (each row sums to 1 within tolerance). For measured data, small rounding drift is handled by row‑sum tolerance. Absorbing rows often resemble [0, …, 0, 1, 0, …]. You can list absorbing indices or let the calculator auto‑detect rows with near‑unit self‑probability.
After reordering states into transient and absorbing sets, the transient block Q contains transitions that do not terminate. The fundamental matrix N = (I − Q)^{-1} accumulates expected visits to transient states. Expected steps to absorption are t = N · 1, where 1 is an all‑ones vector.
The mean can hide dispersion, so the calculator reports variance using v = (2N − I)·t − t ⊙ t. Large variance is common when transient loops are “sticky” (for example, a self‑transition near 0.99), producing long absorption-time tails.
With multiple absorbing states (distinct endpoints or outcomes), absorption probabilities explain where the system is likely to finish. Using B = N · R, the calculator returns the probability of ending in each absorbing state from each transient start.
For very large state spaces, matrix inversion can be costly. Monte Carlo simulation estimates absorbing time by sampling many trajectories until absorption or a step cap. Uncertainty typically shrinks like 1/√trials, so more trials improve stability but increase runtime.
Near‑singular I − Q indicates slow absorption and can amplify rounding error. Useful checks include verifying row sums, ensuring each transient state can reach absorption, and comparing exact and Monte Carlo results on a smaller representative model.
Absorbing time models appear in random walks with boundaries, diffusion with traps, chemical reaction pathways, reliability chains, and queueing systems with terminating conditions. Pairing expected time, variance, and absorption probabilities helps evaluate both speed and risk of termination.
Steps are discrete transitions of the Markov process. If each step represents a physical interval Δt, the calculator multiplies expected steps by Δt to report expected time in your chosen units.
Use auto-detect first, then verify. A truly absorbing state has probability 1 of staying in the same state and 0 to others. If your model has approximate absorption, list the intended terminal states manually.
Absorbing-time results assume each row is a probability distribution. If a row sum differs from 1 beyond tolerance, the matrix is not stochastic. Fix data entry, normalize the row, or check for missing transitions.
Use Exact for small to medium matrices when inversion is stable. Use Monte Carlo when the state space is large, when you want a quick estimate, or when exact inversion becomes slow or numerically fragile.
It indicates absorption times are highly spread out. This often happens when the chain can linger in transient loops with high self-transition probability. In such cases, the mean may be less representative than percentiles from simulation.
Yes. Provide a start distribution that sums to 1. The calculator forms a weighted average of expected steps, variances, and absorption probabilities across transient start states.
Then absorption may be impossible from those starts, and the expected time can diverge. Modify your model so every transient state has a path to absorption, or reclassify unreachable components appropriately.
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