GTH Algorithm, Censored Markov Chains, and $RG$-Factorization in Block-Form

Abstract: In 1985, Grassmann, Taksar, and Heyman published their celebrated paper, in which they introduced a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain, one of the key performance quantities in both theory and applications. This algorithm later became the well-known GTH algorithm (or the state-reduction method) in the literature, becoming one of the standard algorithms in applied probability. Later, this algorithm was extended to deal with the stationary distributions of block-structured Markov chains with repeating rows. In this paper, we focus on the block-form GTH algorithm and organize it into two parts. In the first part, we connect the block-form GTH algorithm to censored Markov chains and the block-form $RG$-factorization. We show that the forward block-elimination and back block-form substitution of the block-form GTH algorithm are equivalent to solving a system formulated using the $RG$-factorization in two steps. We also show that this connection remains valid when the block-form GTH algorithm is extended to infinite-state Markov chains. It is well known that censoring an infinite-state Markov chain to a finite state space yields a stationary distribution that provides a best approximation to the stationary distribution of the original infinite-state Markov chain. In the second part, we first derive an explicit expression for the censored Markov chain from the infinite state space to a finite space for Markov chains of $M/G/1$ type. Based on this expression, we propose a renormalized approximated censored transition matrix (RA-CM). The resulting stationary distribution is shown to be asymptotically optimal in terms of approximation error. We compare the approximation error of the RA-CM with the error arising from the last-block-column augmentation.

• The paper presents a unified framework combining the block-form GTH algorithm, censored Markov chains, and RG-factorization for accurate stationary distribution approximations.
• It introduces explicit censored matrix representations and a RA-CM method that achieves asymptotically optimal ℓ1 error bounds compared to traditional augmentation schemes.
• The approach offers practical insights for controlling errors in infinite-state Markov chains, benefiting applications like queueing networks and matrix-analytic methods.

GTH Algorithm, Censored Markov Chains, and $RG$-Factorization in Block-Form: Formal Analysis and Implications

Overview

The paper "GTH Algorithm, Censored Markov Chains, and $RG$-Factorization in Block-Form" (2604.14347) provides a technical synthesis and extension of core numerical and probabilistic techniques for analyzing stationary distributions in block-structured Markov chains, with extensive focus on Markov chains of $M/G/1$ type. It rigorously unifies the block-form GTH algorithm, censored chain methodology, and $RG$-factorization, and develops novel practical schemes for optimal approximation of infinite-state chains using finite-state truncations and augmentations.

Block-Form GTH Algorithm and Its Probabilistic Interpretation

The classical GTH algorithm offers a stable approach to computing stationary distributions for finite-state Markov chains, avoiding subtraction and instability present in Gaussian elimination. The block-form GTH algorithm generalizes this method for block-structured chains, enabling efficient treatment of models with level-phase decomposition (as in queueing networks and matrix-analytic methods). The algorithm consists of forward block elimination (reducing the system by censoring the highest level) and back block-form substitution, maintaining numerical stability in high-dimensional settings.

The paper rigorously demonstrates the equivalence between block elimination in the GTH algorithm and the construction of censored chains over shrinking subsets of the state space. Each elimination corresponds to “censoring” one level, resulting in a sequence of finite Markov chains whose stationary distributions converge to that of the original process under appropriate conditions.

Censored Markov Chains and Optimal Augmentation

Censored Markov chains—obtained by restricting trajectories to a subset and watching only times spent in that set—form the core theoretical foundation for the GTH algorithm’s probabilistic meaning. The paper recapitulates their formal structure, showing that the stationary distribution of the censored chain provides the best possible $\ell_1$ approximation error to the original infinite-state chain among all augmentation/truncation schemes.

It is proved that for any finite truncation size $N$, the censored process minimizes $\ell_1$ error compared to the original infinite-state chain, surpassing last-column and other augmentations. The construction enables practical assessment of approximation quality and guides the design of efficient finite-state kernels for large-scale Markov systems.

Connection to $RG$-Factorization and Linear Algebraic Structure

The block-form GTH algorithm is analytically recast through $RG$-factorization, a block-triangular decomposition of $I-P$ connecting $RG$0-measures (expected visits before hitting lower levels) and $RG$1-measures (first entrance probabilities into lower levels). This factorization is shown to be structurally equivalent to the two-step algorithmic process of forward elimination and back substitution, with precise mapping between probabilistic quantities and algebraic transformations.

For both finite and infinite block-structured chains, $RG$2-factorization provides a deterministic way to solve the stationary equations, offering insight into the relationship between numerical algorithms and probabilistic path expansions.

Explicit Representations for Censored Matrix in $RG$3-Type Chains

A significant technical contribution is the derivation of explicit formulas for the censored matrix of $RG$4-type chains. The construction relies on skip-free properties and path-based expansions: for each block entry in the censored matrix, the paper provides a detailed recursive formula summing over admissible nonnegative integer-valued paths subject to level constraints. The expansion covers all block entries and allows for practical implementation of censored kernels.

This enables computation and analysis of the fundamental matrix $RG$5 for the censoring operation, supporting both theoretical error bounds and practical approximation strategies.

Renormalized Approximated Censored Matrix (RA-CM): Asymptotic Optimality and Computable Error Bounds

Truncating the censored matrix to a finite sum of path kernels yields a sub-stochastic matrix, for which the paper introduces robust row-wise renormalization (RA-CM) to ensure stochasticity. It is shown that the RA-CM's stationary distribution converges, in $RG$6 norm, to that of the censored chain as the truncation depth increases.

Detailed computable error bounds are given, including uniform stopping criteria based on captured return masses, providing a practical mechanism for determining when the truncated censored kernel suffices for desired accuracy.

Key theoretical result: The stationary distribution of RA-CM is asymptotically optimal among all augmentations, minimizing approximation error. This is explicitly guaranteed under reasonable irreducibility and recurrence assumptions.

Comparative Analysis: Last-Block-Column Augmentation (LBCA) vs. RA-CM

The paper investigates the widely used last-block-column augmentation (LBCA) approach, showing that, while easy to implement and often accurate, it can be strictly suboptimal to the censored or RA-CM method in general. For $RG$7-type chains, detailed monotonicity and convergence results are provided, including block ratio identities and taboo probability structure, culminating in a proof of global $RG$8 convergence for LBCA stationary vectors to the original chain's stationary distribution.

Numerical experiments in an $RG$9 queueing system with batch arrivals and multiple vacation phases demonstrate that the RA-CM consistently attains smaller $M/G/1$0 errors than LBCA, with improvements quantified for a range of truncation levels. The paper notes that error improvement is parameter-sensitive and may increase under specific vacation schemes.

Practical and Theoretical Implications

The unified framework developed in this paper supports rigorous optimal truncations for infinite-state Markov chains, especially in applied stochastic modeling (queueing, reliability, telecommunication). The explicit censored matrix expansion and RA-CM renormalization scheme are broadly applicable to matrix-analytic methods, enabling precise and efficient finite-state approximations with controlled accuracy.

At the theoretical level, the tight correspondence between elimination, censoring, and $M/G/1$1-factorization informs the design of spectral analysis, perturbation bounds, and algorithmic stability, enhancing numerical methods in applied probability.

Extensions: The skip-free property is identified as crucial; further development for chains without this restriction (e.g., $M/G/1$2) is suggested as theoretically challenging but promising for general applicability.

Conclusion

The paper rigorously characterizes the relationships between the block-form GTH algorithm, censored Markov chains, and $M/G/1$3-factorization, extending classical techniques to block-structured settings and providing optimal and practical augmentation schemes for infinite-state chains. The renormalized approximated censored matrix (RA-CM) offers a provably optimal, computable, and flexible tool for stationary distribution approximation, surpassing standard augmentation methods in precision. The framework enriches both numerical algorithms and probabilistic understanding of structured Markov systems, and establishes directions for more general, non-skip-free chain analysis.