# Connecting irreducible and absorbing Markov chains with applications

Gerardo Rubino

► To cite this version:

Gerardo Rubino. Connecting irreducible and absorbing Markov chains with applications. JMM 2025 - Joint Mathematical Meeting of the American Mathematical Society, American Mathematical Society, Jan 2025, Seattle, WA, United States. [https://arxiv.org/abs/2001.04477](https://arxiv.org/abs/2001.04477)

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https://inria.hal.science/hal-05451109v1

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# Connecting irreducible and absorbing Markov chains with applications v1

G. Rubino, Inria, Rennes, France

JMM 2025

Seattle, January 9, 2025

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Talk's content

1 Computing in an absorbing chain

2 Two facts on irreducible CTMCs

3 Connecting absorbing and irreducible chains

4 Cuts in absorbing chains

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Abstract

Sigmund duality makes a connection between arbitrary irreducible Markov chains and specific absorbing ones. This talk is devoted to a different way of connecting both classes of chains, that proves useful in the analysis of their steady-state regime (for absorbing chains, instead of the useless steady-state probabilities we work with absorption probabilities and with mean absorption times). The connection we discuss also considers arbitrary irreducible models, which are associated with specific absorbing chains, but with a totally different structure than when using duality.

In the talk, we will show examples of uses of this approach. For instance, it leads to new ideas for the numerical analysis of irreducible models in equilibrium. Then, we will develop another set of results by showing how the connection of the two types of chains allows obtaining an equivalent to Kelly's Lemma (that concerns the equilibrium through cuts in the irreducible case), but now on absorbing models and using mean hitting times. We will also explore the case of gambler ruin problems through the previously mentioned connection.

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# Introduction

* In the quantitative analysis of systems, two broad territories appear: performance analysis ("how much" the system does, *assuming it is perfect*) and dependability studies (how it behaves face to failures and in some cases repairs, *ignoring its work, its performance*).

* In performance, most analysis are done with irreducible models, stable and in equilibrium. In dependability, studies are done both with reducible models (typically when there is no repair facility, or when repairs are *partial*) and irreducible ones, both in their transient phase, and in equilibrium, in the stable case.

* We are going to discuss connections between irreducible and absorbing models, in equilibrium for the former, and looking at what happens during the models' "life-times" (time before absorption) for the latter.

* This will be done around the main tool for these tasks, Markovian models, and in continuous time because this is the usual setting in applications (but everything holds also in the discrete time case).

* To simplify the presentation, we will work with the finite case only.

* The corresponding paper will be uploaded soon to arXiv.

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Talk's content

1 Computing in an absorbing chain

2 Two facts on irreducible CTMCs

3 Connecting absorbing and irreducible chains

4 Cuts in absorbing chains

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Initial model X

* Let X be a Continuous Time Markov Chain (CTMC) with finite state space $\mathcal{X}$, transition rates $Q_{x,y}$, a single absorbing state $a \in \mathcal{X}$ and the rest of states transient. We denote by $\mathcal{T}$ the set of transient states. We denote $\alpha_x = \mathbb{P}(X(0) = x)$, with $\alpha_a = 0$.

* We denote by $P_{x,y}$ the corresponding probability transition rates from $x$ to $y$, that is, $P_{x,y} = Q_{x,y}/d_x$ for any state $x \in \mathcal{T}$ and all $y$, where $d_x$ = departure rate from $x$, $= \sum_y Q_{x,y}$. See that $d_x > 0$. For state $a$, $d_a = 0$ and we set $P_{a,a} = 1$ ($P_{a,x} = 0$ for all $x \neq a$).

$\mathcal{T}$: transient states

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* The absorption time $T$ of $X$ is

$$T = \inf\{t \ge 0: X(t) \notin T\} = \int_0^\infty 1(X(t) \in T) \, dt.$$

In words, $T$ is the time taken by $X$ to reach state $a$.

* We will be mainly interested in the mean, $\mathbb{E}(T)$, and in the conditional means $\mathbb{E}_i(T) = \mathbb{E}(T | X(0) = i)$.
Since $\mathbb{E}_a(T) = 0$, we have $\mathbb{E}(T) = \sum_{i \in T} \alpha_i \mathbb{E}_i(T)$.

* Let $S_x$ be the total time spent by $X$ in state $x$ until absorption,

$$S_x = \int_0^T 1(X(t) = x) \, dt.$$

Observe that $S_a = 0$ and that $T = \sum_{x \in T} S_x$.

* We are interested in the mean $\mathbb{E}(S_x)$, and in the conditional mean $\mathbb{E}_i(S_x)$. Observe that $\mathbb{E}(S_a) = \mathbb{E}_a(S_x) = 0$ for all state $x$.

* We have $\mathbb{E}(T) = \sum_{x \in T} \mathbb{E}(S_x)$, and $\mathbb{E}(S_x) = \sum_{x \in T} \alpha_i \mathbb{E}_i(S_x)$.

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Computing the conditional $E_i(T)$s

Denote $E_i(T) =: \tau_i$. To obtain the $\tau_i$s, we must solve the following linear system: for all $i \in T$,

$$ \tau_i = h_i + \sum_{j \in T} P_{i,j} \tau_j, $$

where $h_i$ is the mean holding time in state $i$, that is, $h_i = 1/d_i$. Observe that for any $i \in T$, we have $P_{i,i} = 0$.

In matrix terms, define vector $\tau$ as the **row vector** of the $\tau_i$s, matrix $H = \text{diag}(h_i)$, vector $\mathbf{1} = (1, 1, ..., 1)$, and $P = (P_{x,y})$, **everything limited to (indexed on) the subset of transient states T**.

We can then write

$$ \tau^{\text{T}} = (..., \tau_i, ...)^{\text{T}} = H \mathbf{1}^{\text{T}} + P \tau^{\text{T}}. $$

From this,

$$ \tau^{\text{T}} = (I - P)^{-1} H \mathbf{1}^{\text{T}}. $$

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Computing the conditional $E_i(S_\chi)$s

Denote $E_i(S_\chi) := \sigma_{i,x}$. To compute the $\sigma_{i,x}$s, we have to solve the following linear system:

$$ \text{for } i, x \in \mathcal{T}, i \neq x, \quad \sigma_{i,x} = \sum_{j \in \mathcal{T}} P_{i,j} \sigma_{j,x} $$

$$ \text{for } x \in \mathcal{T}, \quad \sigma_{x,x} = h_x + \sum_{j \in \mathcal{T}} P_{x,j} \sigma_{j,x} $$

In matrix terms, if $\sigma = (\sigma_{i,x})_{i,j \in \mathcal{T}}$, then $\sigma = H + P\sigma$, which leads to

$$ \sigma = (I - P)^{-1} H. $$

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# Comments

* Once the row vector $\tau$ known, $\mathbb{E}(T) = \alpha\tau^T$.

* Once matrix $\sigma$ known,

* we have

$$(..., \sigma_x, ...) = \alpha\sigma,$$

where the row vector $\alpha$ is also indexed on $\mathcal{T}$.

* Also, $\tau^T = \sigma\mathbf{1}^T$ and $\mathbb{E}(T) = \alpha\tau^T = \alpha\sigma\mathbf{1}^T = \alpha(I - P)^{-1}H\mathbf{1}^T$.

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Talk's content

1 Computing in an absorbing chain

2 Two facts on irreducible CTMCs

3 Connecting absorbing and irreducible chains

4 Cuts in absorbing chains

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# Cuts in irreducible chains

* Let $U$ be an irreducible CTMC with state space $U$ and transition rates $Q$, ergodic. Let $\pi$ be the stationary distribution of $U$ and $(A, B)$ a partition of $U$. We then have the balance equation

$$ \sum_{x \in A} \pi_x Q_{x,B} = \sum_{y \in B} \pi_y Q_{y,A}, $$

which can be read “in equilibrium, the probabilistic flow from $A$ to $B$ equals the probabilistic flow from $B$ to $A$”.

* This is Lemma 1.4 in Kelly's book "Reversibility and Stochastic Networks".

* We will explore here "associated" relations, in some sense, defined on absorbing chains.

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# Pseudo-aggregation

* Let $U$ be an irreducible CTMC with state space $U$ and transition rates $Q$, ergodic, and a partition $\mathcal{P} = (B_0, B_1, B_2, \ldots)$ of $U$; the stationary distribution of $U$ is $\pi$.

* Define $V$, the pseudo-aggregation of $U$ with respect to $\mathcal{P}$, as the CTMC defined on $\mathcal{P}$ with rates

$$r_{B,B'} = \frac{\sum_{x \in B} \pi_x Q_{x,B'}}{\pi_B},$$

plus, if needed, an arbitrary distribution of $V(0)$ (e.g., $\mathbb{P}(V(0) = B) = \mathbb{P}(U(0) \in B)$, or $\mathbb{P}(V(0) = B) = \pi_B$).

* Then, $V$ is ergodic, with stationary distribution $\hat{\pi}$, where for any $B \in \mathcal{P}$, $\hat{\pi}_B = \sum_{x \in B} \pi_x = \pi_B$.

* See that $V$ is Markov by **construction**. The aggregation of $U$ w.r.t. $\mathcal{P}$ is, in general, not Markovian (in general, it is “nothing”).

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# Some references

Concerning pseudo-aggregation and their context, see

* "*Sojourn times in Finite Markov processes*", G. Rubino and B. Sericola, Journal of Applied Probability, 26, pages 744-756, 1989.

and then

* "*A finite characterization of weak lumpable Markov processes. Part I: the discrete time case*", G. Rubino and B. Sericola, Stochastic Processes and their Applications, 38, pages 195-204, 1991.

* "*A finite characterization of weak lumpable Markov processes. Part II: the continuous time case*", G. Rubino and B. Sericola, Stochastic Processes and their Applications, 45, pages 115-125, 1993.

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Talk's content

1 Computing in an absorbing chain

2 Two facts on irreducible CTMCs

3 Connecting absorbing and irreducible chains

4 Cuts in absorbing chains

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## From X to Y

From X let us build Y by adding transitions from a to any $x \in T$ with rate $r\alpha_x$ ($r > 0$ is arbitrary).

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* Fact 1: Y has a single recurrent class (but it may have transient states). Its stationary distribution is denoted by $\pi$.

* Fact 2: If $\pi$ is the stationary distribution of Y, we have $\pi_a > 0$.

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## From X to $Y^{(i)}$

We also denote by $Y^{(i)}$ the same type of construction but simply adding a single transition from $a$ to $i \in T$, with rate $r$.

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Facts 1 and 2 hold obviously for $Y^{(i)}$ too, whose stationary distribution is denoted by $\pi^{(i)}$; in particular, $\pi_a^{(i)} > 0$.

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# Relations between X and Y, Y<sup>(i)</sup>

* Consider the pseudo-aggregation of Y into two macro-states a and b, where the latter is the result of collapsing all the transient states. We have

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* Since the mean holding time of this new 2-state process in state b is $E(T)$ we have the balance relation

$$ \hat{\pi}_b \frac{1}{E(T)} = \hat{\pi}_a r, $$

where $\hat{\pi}_b = \sum_{i \in T} \pi_i = \pi_B$ and $\hat{\pi}_a = \pi_a$.

* Replacing $\hat{\pi}_b$ by $1 - \pi_a$, we get $E(T) = \frac{1 - \pi_a}{r \pi_a}$, or equivalently,

$$ \pi_a = \frac{1}{1 + r E(T)}. $$

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* If we use $Y^{(i)}$ instead of $Y$, we obtain

$$ \pi_a^{(i)} = \frac{1}{1 + r \tau_i}, $$

or equivalently,

$$ \tau_i = \frac{1 - \pi_a^{(i)}}{r \pi_a^{(i)}}. $$

* If we fix a state $i$ and use the regenerative properties of $Y^{(i)}$ w.r.t. state $a$, we obtain

$$ \frac{\sigma_{i,x}}{\frac{1}{r} + \tau_i} = \pi_x^{(i)}. $$

Reasoning similarly in $Y$ now, we have

$$ \frac{\mathbb{E}(S_x)}{\frac{1}{r} + \mathbb{E}(T)} = \pi_x. $$

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Conclusion

* Given an absorbing Markov chain X as before, we can compute the quantities $E(T)$ and $E(S_x)$ for every state x by working with the irreducible model Y in steady-state.

* We can also go the other way, computing in an irreducible model by working with an absorbing one. For that purpose, we rename Y the given irreducible chain, we choose any state, we rename it a, and we remove the transitions from a, building an absorbing chain that we call X. Then, we use the previously given relations.

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Applications

(i) New fast numerical techniques to compute the stationary distribution of irreducible chains.

(ii) New tools to compute bounds of quantities calculated using the stationary distribution of irreducible chains (for instance, the stationary probability of being in a subset of states).

(iii) New results for absorbing models similar to the balance relations through cuts in irreducible ones.

In the rest of the talk, we will develop only point (iii).

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Talk's content

1 Computing in an absorbing chain

2 Two facts on irreducible CTMCs

3 Connecting absorbing and irreducible chains

4 Cuts in absorbing chains

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Preliminary relations

* We are in the same setting as at the beginning, with chains X (absorbing), Y and $Y^{(i)}$ (both recurrent, perhaps with some transient states).

* Working with $Y^{(i)}$, we write the equilibrium relation for transient state x:

$$ \pi_x^{(i)} d_x = \sum_{y \in \mathcal{T}} \pi_y^{(i)} Q_{y,x} + \pi_a^{(i)} r1(x = i). $$

* We know that $\pi_z^{(i)} = r\sigma_{i,z}\pi_a^{(i)}$, for $z \neq a$ (see slide 19). Putting this in previous relation, we get, for every transient state x,

$$ \sigma_{i,x} d_x = \sum_{y \in \mathcal{T}} \sigma_{i,y} Q_{y,x} + 1(x = i). $$

* Multiplying by $\alpha_i$ and summing on $i \in \mathcal{T}$, we obtain

$$ \text{for all } x \neq a, \quad \mathbb{E}(S_x)d_x = \sum_{y \in \mathcal{T}} \mathbb{E}(S_y)Q_{y,x} + \alpha_x. $$

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As a side effect, with a little algebra, we have

$$ \sum_{x \in \mathcal{T}} \sigma_{i,x} Q_{x,a} = 1. $$

Multiplying by $\alpha_i$ and summing on $i \in \mathcal{T}$,

$$ \sum_{y \in \mathcal{T}} \sigma_y Q_{y,a} = 1. $$

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First result

* Let us now decompose the transient states of X into two non-empty classes B and C. Choose some state $i \in B$ and consider, in $Y^{(i)}$, the equilibrium relation w.r.t. the partition $\{B, C \cup \{a\}\}$. Use then the bridges between X, Y, $Y^{(i)}$, and obtain

$$ \sum_{j \in B} \sigma_{i,j} (Q_{j,C} + Q_{j,a}) = \sum_{k \in C} \sigma_{i,k} Q_{k,B} + 1. $$

* Different formulation: let $N_{B,C} = $ the r.v. “# of transitions from B to C in process X before absorption”, and analogously for $N_{C,B}$, $N_{B,a}$ and $N_{C,a}$. Observe that $N_{B,a} + N_{C,a} = 1$, so, $N_{B,a}$ and $N_{C,a}$ are two (dependent) Binary random variables.

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* Choosing $i \in B$, we have

$$ \sum_{j \in B} \sigma_{i,j} Q_{j,C} = \mathbb{E}_i(N_{B,C}), $$

$$ \sum_{k \in C} \sigma_{i,k} Q_{k,B} = \mathbb{E}_i(N_{C,B}) $$

and

$$ \sum_{j \in B} \sigma_{i,j} Q_{j,a} = \mathbb{E}_i(N_{B,a}). $$

* We also have $\mathbb{E}_i(N_{B,a}) = \mathbb{P}_i(X \text{ gets absorbed from } B)$.

* From these, we obtain

$$ \mathbb{E}_i(N_{B,C}) = \mathbb{E}_i(N_{C,B}) + \mathbb{P}_i(X \text{ gets absorbed from } C) \quad (i \in B). $$

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Second result

• Similarly as before, and deconditioning,

$$
\mathbb{E}(N_{B,C}) + \mathbb{P}(X \text{ gets absorbed from } B) = \mathbb{E}(N_{C,B}) + \mathbb{P}(X(0) \in B).
$$

• By symmetry, we also have

$$
\mathbb{E}(N_{B,C}) + \mathbb{P}(X(0) \in C) = \mathbb{E}(N_{C,B}) + \mathbb{P}(X \text{ gets absorbed from } C).
$$

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Conclusions

* Basically, we first make a connection between absorbing and chains with a recurrent class (in particular, with irreducible chains), which allows to move from one type of process to the other one.

* This allows to solve problems in one type of models with techniques coming from the other one.

* It also allows to obtain new results in the area.

* In the paper, all the details and also examples illustrating the different sets of results.

* There will also be some material about applications of the connection between absorbing and irreducible (or at least, with a recurrent class) chains.

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Some used examples

<figure><img src="image_206.png" /><figcaption>Figure: First basic example</figcaption></figure>

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Examples

<figure><img src="image_214.png" /><figcaption>Figure: Second basic example: a star-shaped model</figcaption></figure>

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Examples

<figure><img src="image_220.png" /><figcaption>Figure: Third basic example</figcaption></figure>

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* Work in progress

* Currently, working on

* two main applications: numerical analysis and bounding techniques,

* and on some extensions (infinite state space, multiple absorbing states).

* The application to bounding techniques extends and improves, in particular,

* "*Bound computation of dependability and performability measures*", S. Mahévas and G. Rubino, IEEE Transactions on Computers, Vol. 50, No. 5, p. 399-413, June 2001,

* "*Bounding the mean cumulated reward up to absorption*", A. P. Couto da Silva and G. Rubino, in A. Langville and W. J. Stewart, Eds, Markov Anniversary Meeting, Chap. 22, pp. 169-188, Boson Books, 2006, and

* Chapter 10 of "*Markov Chains and Dependability Theory*", by G. Rubino and B. Sericola, Cambridge University Press, 2014

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