Calculator inputs
Use the form below to solve a two-state continuous-time forward Kolmogorov system. The page keeps a single-column overall flow, while the input fields use a responsive 3-column, 2-column, and 1-column grid.
Formula used
Generator matrix
Q = [ [ -q₁₂, q₁₂ ], [ q₂₁, -q₂₁ ] ]
Forward Kolmogorov equation
dp(t)/dt = p(t)Q
Component form
dp₁/dt = -q₁₂ p₁ + q₂₁ p₂
dp₂/dt = q₁₂ p₁ - q₂₁ p₂
Closed-form solution for two states
λ = q₁₂ + q₂₁
π₁ = q₂₁ / λ, π₂ = q₁₂ / λ
p₁(t) = π₁ + (p₁(0) - π₁)e-λt
p₂(t) = π₂ + (p₂(0) - π₂)e-λt
This calculator uses the exact analytical solution for a two-state continuous-time Markov chain. It returns transient probabilities, the derivative vector at time t, the transition matrix, equilibrium values, and a graph of the evolving distribution.
How to use this calculator
- Enter the initial probabilities for state 1 and state 2.
- Enter the transition rates q₁₂ and q₂₁.
- Provide the evaluation time t for the required solution point.
- Set the graph end time and number of graph points.
- Click Calculate to view probabilities above the form.
- Review the summary table, generator matrix, transition matrix, and chart.
- Use the CSV button to export graph data.
- Use the PDF button to save the result panel as a report.
Example data table
Sample case: start in state 1 with q₁₂ = 0.35, q₂₁ = 0.20, and evaluate at t = 5.
| p₁(0) | p₂(0) | q₁₂ | q₂₁ | t | p₁(t) | p₂(t) | dp₁/dt | dp₂/dt |
|---|---|---|---|---|---|---|---|---|
| 1.00 | 0.00 | 0.35 | 0.20 | 5.00 | 0.404318 | 0.595682 | -0.022375 | 0.022375 |
FAQs
1) What does this calculator compute?
It solves a two-state continuous-time forward Kolmogorov system. You get transient probabilities, the derivative vector, transition matrix values, stationary probabilities, decay constants, and a probability graph over time.
2) What is the forward Kolmogorov equation?
It describes how state probabilities evolve in time for a continuous-time Markov chain. Using the row-vector convention, the equation is dp(t)/dt = p(t)Q, where Q is the generator matrix.
3) Why does this page use a two-state model?
A two-state chain has an exact closed-form solution that is fast, stable, and easy to interpret. It is useful for reliability, switching, occupancy, binary status, and simple queueing approximations.
4) What do q₁₂ and q₂₁ represent?
They are instantaneous transition rates. q₁₂ moves probability from state 1 to state 2, while q₂₁ moves probability from state 2 back to state 1. Larger rates cause faster mixing.
5) Why are my initial probabilities normalized?
Probability vectors should sum to 1. If your entries do not, the calculator rescales them while preserving their relative proportions. That keeps the solution valid and prevents inconsistent outputs.
6) What happens when both rates are zero?
No transitions occur, so the initial distribution stays unchanged for all time. The transition matrix becomes the identity matrix, and there is no unique stationary distribution from rate balancing.
7) How is the stationary distribution found?
For positive total rate λ = q₁₂ + q₂₁, the stationary probabilities are π₁ = q₂₁/λ and π₂ = q₁₂/λ. The transient solution approaches these values as time grows.
8) How should I read the Plotly graph?
The solid curves show p₁(t) and p₂(t) over the chosen time interval. Dashed lines mark stationary values. When the solid lines flatten near the dashed lines, the chain is approaching equilibrium.