In this fictional monthly inspection model, a pump starting Healthy reaches the absorbing Failed state after an expected 6.875 steps, or 6.875 months. The same model gives a 15% probability of already being Failed after two inspections. Those are calculations from one stated transition matrix, not a prediction that a real pump will fail on a particular date.
Scope and terminology first
This note uses a finite, discrete-time, time-homogeneous Markov chain. Time advances in one-month inspection intervals, the pump occupies exactly one of three mutually exclusive states at each inspection, and the transition probabilities do not change from month to month. The next-state distribution depends only on the current recorded state under the model, not on the complete inspection history.
The NIST/SEMATECH e-Handbook of Statistical Methods defines a Markov chain in terms of discrete time, discrete states, a transition-probability matrix, and the Markov property: the current state summarizes the relevant past for the next transition. That assumption is substantial. If age, load, environment, repair history, or time spent Degraded changes the next-month probability, this three-state model is incomplete.
Three terms matter here:
- A transition probability
p(i,j)is the conditional probability of occupying statejnext month given stateinow. It is dimensionless; it is not a failure rate such asfailures/hour. - A row-stochastic matrix contains nonnegative probabilities and every row sums to
1. MathWorks' current Markov Chain Modeling documentation uses the same right-stochastic convention and advances a row-vector distribution asx(k+1) = x(k)P. - An absorbing state cannot be left once entered. Here Failed has a probability-
1self-loop. The model deliberately omits repair, replacement, and return to service.
Inputs and assumptions
Assume a fictional non-safety-critical water-transfer pump is inspected once per month. A reviewed inspection rubric assigns exactly one state:
| Current state | Next Healthy | Next Degraded | Next Failed | Row check |
|---|---|---|---|---|
| Healthy | 0.80 | 0.15 | 0.05 | 0.80 + 0.15 + 0.05 = 1.00 |
| Degraded | 0 | 0.60 | 0.40 | 0 + 0.60 + 0.40 = 1.00 |
| Failed | 0 | 0 | 1.00 | 0 + 0 + 1.00 = 1.00 |
The probabilities are invented to teach the calculation. They were not estimated from field data. Healthy means the pump meets the fictional inspection criteria; Degraded means it remains operable but fails at least one stated condition-monitoring criterion; Failed means it cannot perform the defined transfer function. These labels do not establish a maintenance standard, safety state, or fitness-for-service decision.
With the state order [Healthy, Degraded, Failed], the transition matrix is
P = [ 0.80 0.15 0.05
0.00 0.60 0.40
0.00 0.00 1.00 ]Every entry lies in [0,1], and the three row sums equal 1. A row sum of 0.95 would not mean “5% unknown”; it would mean the state space or data is incomplete and the matrix is invalid until the missing outcome is handled explicitly.
Calculate the two-month state distribution
Start Healthy, so the initial row vector is x(0) = [1, 0, 0]. One multiplication gives
x(1) = x(0)P = [0.80, 0.15, 0.05].After the second month, multiply again:
Healthy: 0.80×0.80 = 0.64
Degraded: 0.80×0.15 + 0.15×0.60 = 0.12 + 0.09 = 0.21
Failed: 0.80×0.05 + 0.15×0.40 + 0.05×1 = 0.04 + 0.06 + 0.05 = 0.15
x(2) = [0.64, 0.21, 0.15].The invariant is 0.64 + 0.21 + 0.15 = 1.00. Because Failed is absorbing, the 0.15 is the cumulative model probability of first failing in month 1 or month 2 and then remaining Failed, not a 15% per month rate.
Compute the expected time to absorption
Separate the transient-to-transient block Q and the transient-to-absorbing block R:
Q = [ 0.80 0.15 ] R = [ 0.05 ]
[ 0.00 0.60 ] [ 0.40 ]The Kemeny-Snell fundamental matrix is N = (I - Q)^-1:
I - Q = [ 0.20 -0.15 ]
[ 0.00 0.40 ]
N = [ 5.000 1.875 ]
[ 0.000 2.500 ]Each row sum of N is the expected number of steps before absorption from that transient state:
t(Healthy) = 5.000 + 1.875 = 6.875 months
t(Degraded) = 0.000 + 2.500 = 2.500 monthsThe MIT 6.436J / 15.085J Lecture 23 notes independently derive expected absorption time by conditioning on the first step. Applying that check here gives t_D = 1 + 0.60t_D, so t_D = 2.5; then t_H = 1 + 0.80t_H + 0.15t_D, so 0.20t_H = 1.375 and t_H = 6.875.
Finally, absorption probability is B = NR:
B = [ 5.000×0.05 + 1.875×0.40 ] = [ 1.00 ]
[ 0.000×0.05 + 2.500×0.40 ] [ 1.00 ]From either transient state, the model eventually reaches Failed with probability 1. That result follows from this topology and fixed matrix; it is not evidence that every real pump must fail before being repaired or retired.
Reproducible Schematex source
markov "Pump inspection states"
layout: layered
analysis: classify, absorbing
state Healthy "Healthy"
state Degraded "Degraded"
state Failed "Failed" absorbing
Healthy -> Healthy : 0.80
Healthy -> Degraded : 0.15
Healthy -> Failed : 0.05
Degraded -> Degraded : 0.60
Degraded -> Failed : 0.40
Failed -> Failed : 1.00Schematex 1.0.13 strictly parsed and rendered this source with no diagnostics on August 24, 2026. It assembled three rows, classified Healthy and Degraded as transient, cross-checked Failed as absorbing, formed N = (I-Q)^-1, and recorded t(Healthy)=6.875, t(Degraded)=2.5, and both absorption probabilities as 1 in the SVG description. The current Schematex Markov syntax reference documents the row-sum validation, state classification, stationary-distribution analysis, and absorbing-chain calculations. That owned page establishes renderer behavior; NIST, MIT, Kemeny and Snell, and MathWorks establish the probability model and calculations.
Checks and invariants
- State definition: the states are mutually exclusive, operationally observable, and collectively cover every modeled inspection result.
- Row sums: each current-state row sums to
1within the declared precision; no probability is negative or greater than1. - Conditioning direction: rows mean “current state” and columns mean “next state.” Transposing
Pchanges the model. - Absorbing assertion: Failed has only
Failed -> Failed : 1.00. Any path out would contradict the declared absorbing state. - Distribution conservation: every
x(k)remains nonnegative and sums to1; the two-step check above does. - Independent expected-time check: the fundamental-matrix row sums and first-step recurrences agree at
6.875and2.5. - Absorption check: each row of
Bsums to1because Failed is the only recurrent class reachable from the transient states. - Unit check: expected steps become months only because one modeled step is explicitly one month. The transition entries remain dimensionless probabilities.
Failure modes, limitations, and review boundary
Common errors are treating a probability as a rate, silently normalizing incomplete rows, mixing weekly and monthly observations, deriving transitions from too few events, allowing inspection rules to change over time while assuming a fixed matrix, and reading an expected value as a scheduled failure date. A three-state observed chain also cannot represent unobserved degradation, age dependence, competing failure modes, censored assets, repairs, maintenance decisions, or uncertainty in the estimated probabilities.
Paste the source into the Schematex playground and run two sensitivity checks. First change Degraded's row to 0.50 stay / 0.50 fail; t(Degraded) should fall to 2 months and t(Healthy) to 6.5. Then change it to 0.80 stay / 0.20 fail; the results should rise to 5 and 8.75 months. If the directions or row sums fail, correct the matrix before replacing any fictional value with reviewed inspection data.