Calculator Inputs
Plotly Graph
Example Data Table
| State | To S1 | To S2 | To S3 | Row Sum | Departure Rate |
|---|---|---|---|---|---|
| S1 | -0.50 | 0.30 | 0.20 | 0.00 | 0.50 |
| S2 | 0.10 | -0.40 | 0.30 | 0.00 | 0.40 |
| S3 | 0.25 | 0.15 | -0.40 | 0.00 | 0.40 |
This example represents a continuous-time Markov chain. Diagonal entries are negative departure rates, off-diagonal values are jump intensities, and every row sums to zero.
Formula Used
1. Generator matrix rule
For a valid generator matrix Q, every off-diagonal entry qij with i ≠ j must satisfy qij ≥ 0, and each row must sum to zero.
2. Diagonal term
qii = -∑ qij for all j ≠ i
3. Expected holding time
Holding time in state i = 1 / (-qii) when -qii > 0
4. Transition matrix over time
P(t) = eQt
This calculator estimates P(t) through a finite matrix exponential series. That gives an interpretable approximation for state probabilities after the selected time horizon.
How to Use This Calculator
- Enter a square generator matrix with one row per line.
- Separate values using spaces, commas, or semicolons.
- Set the time horizon for transition probability estimation.
- Choose the starting state index for the probability output.
- Adjust precision and validation tolerance if needed.
- Optionally rename states using comma-separated labels.
- Submit the form to validate and analyze the matrix.
- Review row sums, departure rates, holding times, and probabilities.
- Use the chart and export buttons for reporting.
FAQs
1. What is a generator matrix?
A generator matrix describes transition intensities in a continuous-time Markov chain. Off-diagonal entries give jump rates between states, while diagonal values are negative and balance each row to zero.
2. Why must each row sum to zero?
Row sums must equal zero so total probability remains conserved over time. The negative diagonal exactly offsets all outbound transition rates from the same state.
3. Can off-diagonal entries be negative?
No. Off-diagonal entries represent transition intensities, so they must be zero or positive. Negative off-diagonal values violate the interpretation of a valid continuous-time jump process.
4. What does the diagonal entry mean?
The diagonal entry is the negative total departure rate from a state. Its magnitude determines how quickly the process tends to leave that state.
5. What is holding time?
Holding time is the expected time the chain stays in a state before jumping away. For state i, it equals 1 divided by the departure rate, when that rate is positive.
6. How is the transition matrix computed?
The transition matrix is computed from P(t) = e^(Qt). This page uses a finite series approximation, which is practical for many small and medium-sized matrices.
7. What does the starting state output show?
It shows the estimated probability of being in each target state after the chosen time horizon, assuming the process begins in the selected starting state.
8. When should I use this calculator?
Use it when modeling queues, reliability systems, biological switching, finance states, or any process that changes continuously in time with state-to-state jump rates.