Find the mechanism before the formula
A Q-process describes the path law of a killed queue conditioned to survive indefinitely. Constructing that law is only the first step. The next question is how quickly it forgets its initial state and which states dominate a long surviving trajectory. Those are mixing and occupation questions, not merely another way to name the quasi-stationary distribution.
We will reuse a two-state conditioned queue, diagonalise its transition matrix, obtain an exact total-variation decay, change the starting state, derive the survival-biased occupation law, and mark where a single spectral gap stops being an adequate convergence certificate.
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Start with the question closest to the decision you need to make. Every route returns to the same worked evidence and transfer checks.
| Route | Chapters | Use it when |
|---|---|---|
| 1 | 1–3 | You need the Q-process, invariant law and meaning of mixing. |
| 2 | 4–5 | You want the exact eigenvalue and total-variation calculation. |
| 3 | 6–8 | You need changed starts, occupation measures and survival bias. |
| 4 | 9–10 | You must estimate rates and respect nonreversible or infinite-state limits. |
| 5 | 11–12 | You want FAQs and a reproducible mixing audit. |
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Build the model
Test the mechanism
1. Start with the conditioned path law, not the killed matrix
Let the Q-process transition matrix be PQ=[[0.75,0.25],[0.375,0.625]]. It came from a substochastic killed chain through a Doob h-transform. PQ is conservative: every row sums to one, because it already describes paths under the never-absorbed conditioning.
Mixing statements must refer to this transformed matrix. The dominant eigenvalue of the killed matrix controls survival decay, while the subdominant spectrum of PQ controls how conditioned paths forget their starting state. Using one in place of the other answers a different question.
- Name the killed event and the transformed law.
- State whether time is discrete or continuous.
- Keep the original survival decay separate from conditioned mixing.
- Report the initial state or initial distribution.
2. The invariant law is a survival-biased occupation law
For the transformed matrix, the invariant row vector π solves πPQ=π and π(1)+π(2)=1. Solving gives π=[0.60,0.40]. A long ergodic run of the Q-process therefore spends about sixty per cent of its time in state 1 and forty per cent in state 2.
In the standard finite-state setting, if α is the quasi-stationary left eigenvector of the killed chain and h is its positive right survival eigenvector, then π(i) is proportional to α(i)h(i). Here α=[0.50,0.50] and h=[1,2/3], so normalising [0.50,1/3] gives [0.60,0.40].
3. Mixing asks how fast an initial condition is forgotten
For an initial distribution μ, total-variation distance after n steps is one half of the absolute coordinate differences between μ(PQ)n and π. Convergence of this distance to zero is a mixing statement. It is stronger than saying that π is invariant.
A spectral gap gives a rate only after the relevant hypotheses and norm are specified. In a finite reversible chain, the gap between one and the largest nontrivial eigenvalue magnitude is especially transparent. Nonreversible chains can require singular values, pseudospectral bounds or coupling arguments instead.
| Object | Calculation | Meaning |
|---|---|---|
| Killed eigenvalue λ | Qh=λh | decay scale for unconditioned survival |
| Q-process invariant π | πPᑫ=π | long-run conditioned occupation |
| nontrivial eigenvalue | spectrum of Pᑫ | memory retained per step |
| total variation | ½Σ|μPⁿ−π| | observable distance from stationarity |
4. Worked example: the spectral gap is 0.625
The trace of PQ is 1.375 and its determinant is 0.375. Because a stochastic matrix has eigenvalue one, the other eigenvalue is 0.375. The absolute spectral gap for this two-state example is therefore 1−0.375=0.625.
The nonstationary component is multiplied by 0.375 at every step. A larger gap means faster contraction here, but the number 0.625 is not a universal mixing time. A tolerance, starting distribution and distance still have to be stated.
5. Worked example: calculate exact total-variation decay
Start in state 1, so μ0=[1,0]. Its difference from π is [0.40,−0.40]. In this two-state chain that direction is the nontrivial eigenvector direction, so after n steps the difference is 0.375n[0.40,−0.40].
The total-variation distance is therefore 0.40×0.375n. At n=1 it is 0.15; at n=2 it is 0.05625; at n=4 it is about 0.00791. This exact sequence is more informative than quoting the gap without a starting-state coefficient.
6. Transfer check: change the starting state
Starting in state 2 gives μ0=[0,1], whose total-variation distance from π is 0.60. The decay becomes 0.60×0.375n. The asymptotic factor is unchanged, but the initial coefficient is larger.
For a tolerance of 0.01, state 1 needs four steps because 0.40×0.375⁴≈0.00791. State 2 needs five steps because 0.60×0.375⁴≈0.01187 while the next step is about 0.00445. The same spectral gap can produce different finite-time answers.
7. Conditioned time averages converge to the quasi-ergodic law
Let AN(i)=N−1Σ from k=0 to N−1 of 1{Xk=i}. Under ergodicity of the Q-process, AN(i) converges to π(i). Equivalently, the time average of a bounded cost g converges to Σπ(i)g(i).
For a holding cost g=[2,7], the long-run conditioned average is 0.60×2+0.40×7=4.0 per step. This is a cost along unusually persistent paths. It is not the ordinary unconditional cost of the killed queue, whose episodes end and must be analysed with absorption or regenerative cycles.
8. Survival bias changes what an occupation measure represents
The quasi-stationary distribution α=[0.50,0.50] describes a late snapshot of the original chain given survival at that time. The Q-process invariant law π=[0.60,0.40] describes time spent along the indefinitely conditioned path law. The product αh tilts the snapshot toward states whose future survival potential is larger.
This distinction matters in persistent-congestion studies. A state can appear more often in long surviving episodes because it helps the episode continue, even if it is not overrepresented in a one-time surviving snapshot. State the sampling scheme before interpreting an empirical histogram.
9. Estimate mixing with diagnostics that expose rare-event weakness
For a known finite matrix, compute eigenvalues and direct powers, then compare the observed distance with the predicted rate. For simulation, run multiple starts, monitor transition counts and occupation averages, and report Monte Carlo uncertainty. A single long conditioned path can conceal slow modes.
If the Q-process is approximated by resampling particles from the killed chain, report effective sample size and genealogical collapse. Apparent rapid mixing can come from copying a few surviving ancestors rather than from the target dynamics. Increasing particle number and conditioning horizon should leave early-window estimates stable.
10. Limits and misconceptions
- The killed-chain dominant eigenvalue is not the Q-process spectral gap.
- An invariant distribution does not by itself prove a useful mixing rate.
- A gap controls asymptotic contraction only in the norm and setting covered by the theorem.
- Nonreversible matrices can mix differently from their eigenvalue magnitudes.
- Infinite queues need recurrence, integrability and domain conditions.
- Finite truncation can create an artificial gap or suppress a slow tail mode.
Periodicity can prevent ordinary convergence even when an invariant law exists. Nearly reducible phase structure can make finite-time mixing far slower than a coarse aggregate suggests. Check the full model rather than treating a single eigenvalue as a universal certificate.
11. Frequently asked questions
Is the quasi-ergodic distribution always π proportional to αh?
That identity is standard for a finite irreducible killed class with the required positive eigenvectors. General state spaces need the corresponding eigenmeasure, eigenfunction and integrability assumptions.
Does a larger spectral gap always mean fewer simulated steps?
It usually signals faster asymptotic contraction in the stated norm, but starting coefficients, nonnormality, periodicity and estimator variance still affect a practical run length.
Can I use π for ordinary service-level reporting?
Only if the decision genuinely concerns the survival-conditioned path law. Ordinary customers experience absorption, emptying and episode restarts, which require the unconditioned or regenerative model.
12. A reproducible conditioned-mixing checklist
| Stage | Evidence to retain |
|---|---|
| Reference | killed operator, absorption event and survival eigenpair |
| Transform | Pᑫ and row-sum checks |
| Stationarity | π and direct invariance check |
| Rate | nontrivial spectrum and stated norm |
| Transfer | at least two starting states and a tolerance |
| Occupation | sampling rule, cost average and survival-bias warning |
The gold standard is to show the exact distance sequence beside the spectral gap and occupation law. That makes it visible which part of the calculation describes survival, which describes memory loss, and which describes the states visited by an exceptionally persistent queue.
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Read next in this subject
- Doob h-Transforms, Q-Processes and Conditioned Queue Dynamics
- Quasi-Stationary Distributions, Killing and Yaglom Limits
Authoritative reading
- Champagnat and Villemonais, Q-processes and exponential convergence for moving boundaries
- He, Zhang and Zhu, quasi-ergodic distributions of absorbing Markov processes
- Champagnat and Villemonais, uniform convergence to the Q-process
- van Doorn and Pollett, quasi-stationary distributions for discrete-state models