Simple Markov Chain Steady State Demo

Your customer-success team tracks accounts across three states — active, at-risk, and churned — and every quarter some fraction moves between them. After enough quarters, the proportions stabilise: 55% active, 25% at-risk, 20% churned. That equilibrium is the steady state of a Markov chain, and it tells you the long-run composition of your portfolio given today’s transition matrix. The common mistake is planning headcount around this quarter’s snapshot rather than the steady state — if churn is accelerating, the current 70% active rate is temporary.

The steady state does not predict individual accounts. It predicts the aggregate distribution: if you run the system long enough with fixed transition probabilities, the fractions converge to these values regardless of where you started. That “regardless of start” property is what makes it useful for planning.

Each row of the transition matrix represents a current state; each column represents a next-period state. The entry P_ij is the probability of moving from state i to state j in one step. Rows must sum to 1.0 because the system has to go somewhere. To estimate the matrix from data, count transitions: if 200 customers were active last quarter and 30 moved to at-risk, P_{active→at-risk} = 30/200 = 0.15.

Two pitfalls. First, small state counts produce noisy probabilities — if only 8 customers were at-risk, one transition changes the rate by 12.5%. Use at least several quarters of data and pool the counts. Second, transition probabilities may not be stationary: a product improvement could cut the active-to-at-risk rate from 15% to 8%. If you suspect non-stationarity, compute the matrix for recent quarters separately and compare before using the steady-state result.

Convergence speed depends on the second-largest eigenvalue of the transition matrix. If that eigenvalue is close to 1 (say 0.95), convergence is slow — the chain takes dozens of steps to forget its starting state. If it is close to 0, convergence is fast and the steady state is reached in a few steps. For business models, slow convergence means your current customer mix will take many quarters to reach equilibrium, so short-term forecasts should not use the steady state directly.

A practical check: multiply the initial state vector by the matrix repeatedly and watch the fractions. If they stop changing (within 0.1%) by step 8, convergence is fast. If they are still drifting at step 30, the chain mixes slowly and the steady state is a long-run target rather than a near-term forecast. Plot the path — the curve from starting distribution to steady state shows exactly when each state stabilises.

An absorbing state is one you never leave: P_ii = 1. “Churned” is often absorbing in customer models — once gone, the account does not return. When the chain has an absorbing state, the steady-state distribution puts 100% of the population in that state eventually. The interesting question shifts from “what fraction ends up where?” to “how long until everyone is absorbed?”

The fundamental matrix N = (I − Q)⁻¹ (where Q is the sub-matrix of transient states) answers this. Each entry N_ij gives the expected number of periods spent in transient state j before absorption, starting from transient state i. Sum a row of N and you get the expected time to absorption from that state. For a customer model: starting from “active,” the expected number of quarters before churn might be 12 — that is the average customer lifespan implied by the transition rates.

Rows that do not sum to 1. This is the most frequent data-entry error. If row “active” sums to 0.95, the model silently leaks 5% of the population each step. Always normalise rows after entering raw counts.

Too many states. A 10-state matrix has 100 transition probabilities to estimate. Unless you have massive data, most entries will be noisy or zero. Collapse similar states (e.g., “low risk” and “medium risk” into “at-risk”) until every row has enough observations to produce stable rates.

Assuming stationarity over years. A transition matrix estimated from 2019–2022 data may not hold in 2024 if pricing, product, or market conditions changed. Test stationarity by comparing year-over-year matrices. If the rates drift more than 3–5 percentage points, use only recent data or build a time-varying model.

Core formulas for transition dynamics, steady-state computation, and absorbing-chain analysis:

Scenario: A SaaS company models accounts as Active (A), At-Risk (R), or Churned (C). Quarterly transition data gives: A→A = 0.80, A→R = 0.15, A→C = 0.05; R→A = 0.30, R→R = 0.40, R→C = 0.30; C→C = 1.00 (absorbing).

Absorption analysis: Since Churned is absorbing, every account eventually churns. The transient sub-matrix Q = [[0.80, 0.15], [0.30, 0.40]]. The fundamental matrix N = (I − Q)⁻¹ = [[7.14, 1.79], [2.14, 2.86]]. Starting from Active, expected quarters before churn = 7.14 + 1.79 = 8.93 quarters (≈ 2.2 years). Starting from At-Risk, expected quarters = 2.14 + 2.86 = 5.0 quarters.

Interpretation: An active account survives about 9 quarters on average; an at-risk account survives 5. If the success team can move the A→R rate from 15% to 10%, recalculating N shows the active lifespan jumps from 8.9 to 12.5 quarters — a 40% increase in customer lifetime from a 5-percentage-point retention improvement.