Find the mechanism before the formula
Some queues are designed to end: a workload empties, a customer population disappears, a system fails or a routing network leaks into an absorbing state. Ordinary stationarity then collapses onto absorption. A quasi-stationary distribution asks a different question: among paths still alive, what shape can the queue approach?
We will separate absorption from conditioning, solve a two-state quasi-stationary example, calculate its survival law, transfer the calculation from a different starting state, and mark the existence, uniqueness and simulation limits behind the phrase “Yaglom limit”.
This guide belongs to the Gold Standard of Queueing Theory, the Gold Standard of Education and the How X Works hub.
Choose a reading route
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 killed-chain and conditional-distribution definitions. |
| 2 | 4–5 | You want the worked eigenvector and survival calculations. |
| 3 | 6–8 | You need Yaglom, quasi-ergodic and queueing interpretations. |
| 4 | 9–10 | You must compute and validate without hiding absorption. |
| 5 | 11–12 | You want FAQs and a reproducible quasi-stationary checklist. |
Chapter contents
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Build the model
Test the mechanism
1. Separate the transient queue from its absorbing outcome
Let state 0 be absorbing and let states 1,2,… describe an active queue. The process eventually enters 0, perhaps when the system empties, a tagged workload is completed or defective routing removes all jobs. Its ordinary limiting distribution is then a point mass at 0.
That answer is correct but unhelpful when the decision concerns long-lived active paths. Quasi-stationary analysis keeps the event τ>t visible, where τ is the absorption time, and studies the state distribution conditional on survival.
- Define the absorbing set physically.
- State whether time is discrete or continuous.
- Retain the initial distribution.
- Report survival probability separately from conditional shape.
2. A quasi-stationary distribution reproduces itself after conditioning
For a discrete-time chain with transient-state submatrix Q, a probability row vector α is quasi-stationary when Pα(X₁=j | τ>1)=αj. Algebraically, αQ=λ α for some 0<λ<1, with α1=1. The eigenvalue λ is the one-step survival probability under α.
The conditioning matters. Before normalisation, αQ has total mass λ; after dividing by λ, its shape returns to α. A stationary distribution instead satisfies πP=π on the full stochastic matrix and does not lose probability to absorption.
3. A Yaglom limit describes convergence from an initial state
A Yaglom limit is the limit of Pμ(Xn=j | τ>n) as n grows, starting from an initial law μ. When this conditional limit exists, it is a quasi-stationary distribution. The converse is not automatic: a chain can have several quasi-stationary distributions or conditional convergence can depend on μ.
Finite irreducible transient classes often have a clean Perron–Frobenius picture. Countably infinite queues need recurrence, spectral and entrance conditions. “The QSD” should therefore follow an existence and selection argument, not merely an eigenvector returned by software.
| Object | What converges or stays fixed | Do not confuse with |
|---|---|---|
| QSD α | conditional shape after one or all times | ordinary stationarity |
| λ | survival factor per step under α | traffic intensity |
| Yaglom limit | conditional law from a chosen start | every possible QSD |
| quasi-ergodic law | time average before absorption | snapshot conditional law |
4. Worked example: solve the dominant left eigenvector
Take Q=[[0.60,0.30],[0.20,0.50]]. Its row sums are 0.90 and 0.70; the missing mass is one-step absorption. The eigenvalues solve x²−1.10x+0.24=0, giving 0.80 and 0.30. The dominant survival eigenvalue is λ=0.80.
Write α=[a,b]. From αQ=0.80α, 0.60a+0.20b=0.80a, so b=a. With a+b=1, α=[0.50,0.50]. Indeed αQ=[0.40,0.40]=0.80α. Conditional on surviving one step, the distribution returns to [0.50,0.50].
5. Worked example: read the survival law and lifetime
Starting in α, survival for n steps is λn=0.80n. After five steps it is 0.32768. The conditional state remains half in each transient phase, even while the unconditioned active mass shrinks.
With absorption checked after each step, the lifetime is geometric with mean 1/(1−λ)=5 steps under this QSD start. This mean is not the expected queue length and λ is not an arrival-to-service ratio. It is the persistence rate of the killed process.
6. Transfer check: start away from quasi-stationarity
Start instead from μ=[1,0]. After one step μQ=[0.60,0.30], with survival 0.90 and conditional shape [0.667,0.333]. After two steps μQ²=[0.42,0.33], with survival 0.75 and conditional shape [0.56,0.44].
The conditional distribution moves toward [0.50,0.50], while survival from this start is not exactly 0.80n at short times. This calculation separates convergence of shape from decay of mass, the two signals that a single normalised plot can hide.
7. Killing can represent more than an empty queue
In a queueing network, absorption can mean all customers have departed, a tagged job has completed, a buffer overflow has killed the operating regime or defective routing has leaked work out of the observed network. The state and killing rule must match the question.
A QSD can reveal the typical congestion profile among systems that have not yet emptied or failed. It is a selection-biased population by design. It does not estimate what a randomly chosen system looks like without conditioning on survival.
8. Snapshot and occupation limits need not match
The quasi-stationary law describes the state at a late time given survival. A quasi-ergodic distribution describes the fraction of pre-absorption time spent in states, again under long survival. The latter usually combines left and right eigenvectors of Q.
This distinction matters operationally. A state can be common at the observation horizon but occupy less of the surviving path’s history, or vice versa. Name the estimator before comparing simulation output with an eigenvector.
9. Compute with mass conservation and spectral checks
For a finite chain, find the dominant eigenvalue of Q and a nonnegative normalised left eigenvector. Verify αQ−λα, α1=1 and λ=αQ1. A negative component or a residual that grows under higher precision signals a convention or numerical problem.
For large or infinite queues, truncation can change the selected QSD. Increase the boundary, monitor λ and low-state probabilities, and compare with Fleming–Viot particle simulation or direct conditional simulation. Retain confidence intervals because long survival is a rare event.
10. Limits and misconceptions
- Quasi-stationary does not mean the unconditioned process is stationary.
- A Yaglom limit may fail to exist or may depend on the starting law.
- Several QSDs can coexist on infinite state spaces.
- The dominant eigenvalue is a survival factor, not automatically a queue traffic intensity.
- Renormalising every simulation step can hide how rare survival has become.
- Finite truncation can select the wrong limiting distribution.
Conditioning on survival is useful precisely because it changes the population. Interpret results as a description of survivors, not as a neutral sample of all trajectories.
11. Frequently asked questions
Why not delete absorbed runs and average the rest?
That estimates a conditional law only if the deletion and time horizon are handled explicitly. It also needs the survival fraction and uncertainty, which can become poor when few runs remain.
Is every positive eigenvector a QSD?
It must be a probability vector, correspond to the killed transition operator and satisfy the correct left-versus-right convention. Infinite chains need additional domain and selection arguments.
Can a stable queue have a QSD?
Yes, if the model is killed at a target event such as emptying or completion. Ordinary stationary and quasi-stationary questions can coexist but describe different conditioning.
12. A reproducible quasi-stationary checklist
| Stage | Evidence to retain |
|---|---|
| Event | absorbing set and killing rule |
| Operator | transient submatrix or generator |
| Solution | λ and normalised left eigenvector |
| Transfer | conditional convergence from other starts |
| Mass | survival curve and lifetime scale |
| Robustness | truncation and simulation checks |
The gold standard is to display both disappearing mass and conditional shape, solve one example by hand and test a changed initial state. A QSD becomes informative when survival is never allowed to disappear from the explanation.
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