pith. sign in

arxiv: 2604.14347 · v1 · submitted 2026-04-15 · 🧮 math.NA · cs.NA

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

Qihui Bu , Yiqiang Q. Zhao This is my paper

Pith reviewed 2026-05-10 12:05 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords block-form GTH algorithmcensored Markov chainsRG-factorizationM/G/1 typestationary distributionapproximation errorblock-structured chainsrenormalized censored matrix
0
0 comments X

The pith

The block-form GTH algorithm for Markov chains is equivalent to a two-step RG-factorization solution on censored chains and yields an asymptotically optimal approximation for infinite M/G/1-type cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the forward block-elimination phase and the backward block-form substitution phase of the block-form GTH algorithm match exactly the two steps needed to solve a linear system expressed through the block RG-factorization of a censored Markov chain. This equivalence is proved for finite chains and then extended without change to infinite-state chains that admit a repeating block structure. For chains of M/G/1 type the paper gives an explicit formula for the censored transition matrix obtained by truncating the infinite space, and uses that formula to construct a renormalized approximated censored matrix whose resulting stationary distribution converges to the true one with asymptotically smaller error than the common last-block-column augmentation method.

Core claim

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. For Markov chains of M/G/1 type an explicit expression for the censored transition matrix from the infinite state space to a finite space is derived; the renormalized approximated censored transition matrix constructed from this expression produces a stationary distribution that is asymptotically optimal in approximation error compared with the error obtained from last-block-column augmentation.

What carries the argument

The block RG-factorization of the censored transition matrix, which converts the two phases of the block GTH algorithm into the forward and backward solution steps of a linear system and supplies the explicit censored matrix needed to build the RA-CM.

If this is right

  • The equivalence extends the GTH algorithm to infinite-state chains while retaining its numerical stability properties.
  • The RA-CM stationary distribution converges to the true stationary distribution with smaller asymptotic error than the last-block-column augmentation for any M/G/1-type chain.
  • Censoring an infinite repeating-block chain to a finite level produces the stationary distribution that is optimal among all finite truncations in the sense of approximation error.
  • The same two-step RG-factorization equivalence remains valid once the standard conditions of irreducibility and positive recurrence are assumed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Software implementations could replace the elimination and substitution loops with calls to existing RG-factorization routines to simplify code for block-structured problems.
  • The explicit censored-matrix formula derived for M/G/1 type suggests that analogous closed-form expressions might be obtainable for GI/M/1 type or other repeating structures, yielding similar optimal approximations.
  • Numerical tests on queueing models with known closed-form stationary distributions could directly measure the rate at which the RA-CM error vanishes compared with augmentation.

Load-bearing premise

The Markov chain possesses a repeating block structure of M/G/1 type so that an explicit closed-form expression for the censored transition matrix exists.

What would settle it

For a concrete infinite M/G/1-type chain such as a GI/M/1 queue, compute the total-variation distance of the stationary distribution from the RA-CM and from last-block-column augmentation at successively larger truncation levels and check whether the RA-CM distance decays faster as the level tends to infinity.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper connects the block-form GTH algorithm to censored Markov chains and block-form RG-factorization, showing that its forward block-elimination and back block-form substitution steps are equivalent to solving a system via RG-factorization in two steps; this equivalence extends to infinite-state chains. For M/G/1-type Markov chains, an explicit expression for the censored transition matrix (from infinite to finite space) is derived, from which a renormalized approximated censored transition matrix (RA-CM) is proposed whose stationary distribution is asymptotically optimal in approximation error relative to last-block-column augmentation.

Significance. If the derivations and optimality result hold, the work supplies a rigorous probabilistic interpretation and extension of the GTH algorithm via censoring and RG-factorization, which are standard tools in applied probability. The explicit censored-matrix expression and the asymptotic-optimality claim for RA-CM represent concrete advances for approximating stationary distributions of infinite-state block-structured chains arising in queueing models; the comparison to augmentation methods supplies a falsifiable performance benchmark.

major comments (2)
  1. [Section on explicit censored matrix (second part of the paper)] The section deriving the explicit censored transition matrix for infinite M/G/1-type chains: the derivation invokes standard irreducibility and positive recurrence but does not verify these properties for the censored chain or detail level-dependent boundary adjustments at the truncation point; without this, the renormalization step used to form the RA-CM may not preserve the minimal nonnegative solution properties required by the RG-factorization, undermining the subsequent asymptotic-optimality argument.
  2. [Section introducing and analyzing the RA-CM] The proof that the RA-CM stationary distribution is asymptotically optimal in approximation error (compared with last-block-column augmentation): the argument relies on renormalization preserving the censored chain's error bounds and tail behavior, yet the manuscript provides no explicit error-bound analysis or counter-example check for cases where boundary effects at truncation are incompletely captured by the repeating-block expression; this is load-bearing for the central claim of superiority.
minor comments (2)
  1. [Abstract] The abstract states that the RA-CM result is 'asymptotically optimal in terms of approximation error' but does not specify the norm (e.g., total variation, l1) or the precise sense of 'asymptotic' (e.g., as the truncation level tends to infinity); adding this would improve clarity.
  2. [Section on RG-factorization connection] Notation for the block-form RG-factorization (R and G matrices) is introduced without an early equation reference; cross-referencing the definitions when they are first used in the equivalence proof would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript on the connections between the block-form GTH algorithm, censored Markov chains, and RG-factorization, along with the RA-CM approximation for M/G/1-type chains. The major comments identify areas where additional rigor on chain properties and error analysis would strengthen the presentation. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section on explicit censored matrix (second part of the paper)] The section deriving the explicit censored transition matrix for infinite M/G/1-type chains: the derivation invokes standard irreducibility and positive recurrence but does not verify these properties for the censored chain or detail level-dependent boundary adjustments at the truncation point; without this, the renormalization step used to form the RA-CM may not preserve the minimal nonnegative solution properties required by the RG-factorization, undermining the subsequent asymptotic-optimality argument.

    Authors: We agree that the manuscript would benefit from explicit verification of these properties. For M/G/1-type chains satisfying the standard stability condition (negative drift), the censored chain on the finite level set remains irreducible on its communicating class and positive recurrent, as the return probability to the censored set is bounded away from zero. The explicit expression for the censored matrix already encodes the level-dependent adjustment in the final block row via the tail probabilities. In the revised version we add a lemma that formally verifies irreducibility and positive recurrence for the censored chain and proves that row renormalization preserves the minimal nonnegative solution property of the RG-factorization, because the G and R factors in the repeating blocks are unchanged and the boundary adjustment remains a valid stochastic completion. revision: yes

  2. Referee: [Section introducing and analyzing the RA-CM] The proof that the RA-CM stationary distribution is asymptotically optimal in approximation error (compared with last-block-column augmentation): the argument relies on renormalization preserving the censored chain's error bounds and tail behavior, yet the manuscript provides no explicit error-bound analysis or counter-example check for cases where boundary effects at truncation are incompletely captured by the repeating-block expression; this is load-bearing for the central claim of superiority.

    Authors: The current argument establishes asymptotic optimality by showing that the RA-CM converges to the exact censored matrix as the truncation level tends to infinity, inheriting the same tail decay. We acknowledge that an explicit finite-N error bound and targeted checks for boundary effects are absent. In the revision we will insert a theorem giving an explicit total-variation bound on the stationary-distribution error in terms of the truncation level and the geometric decay rate, together with additional numerical experiments that examine high-load regimes where boundary effects are pronounced. These additions will make the superiority claim over last-block-column augmentation fully rigorous while preserving the probabilistic interpretation via censoring. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are explicit and self-contained.

full rationale

The paper establishes the equivalence between block-form GTH forward/back steps and RG-factorization solution by direct construction within the manuscript, then derives an explicit censored transition matrix expression for M/G/1-type chains before defining RA-CM via renormalization and proving its asymptotic optimality through error comparison to augmentation. These steps use standard irreducibility/positive recurrence assumptions and explicit block-structure algebra rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation that reduces the central claims to unverified inputs. Background references to prior RG-factorization results supply context but are not invoked to force uniqueness or smuggle ansatzes; the new explicit expressions and optimality arguments stand independently.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard assumptions of Markov chain theory (irreducibility, positive recurrence) and the existence of the RG-factorization for block-structured chains; no new free parameters are introduced beyond the cutoff level for the finite approximation.

axioms (2)
  • domain assumption The Markov chain is irreducible and positive recurrent so that a unique stationary distribution exists.
    Invoked implicitly when discussing stationary probabilities and censoring approximations.
  • domain assumption The chain admits a repeating block structure of M/G/1 type allowing an explicit censored transition matrix.
    Required for the second part's explicit expression and RA-CM construction.
invented entities (1)
  • RA-CM (renormalized approximated censored transition matrix) no independent evidence
    purpose: Adjusted finite censored matrix whose stationary distribution approximates the infinite-chain distribution with asymptotic optimality.
    Introduced in the second part as a new construction; independent evidence would be numerical verification on concrete M/G/1 examples.

pith-pipeline@v0.9.0 · 5623 in / 1628 out tokens · 51178 ms · 2026-05-10T12:05:10.409681+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    (1977) Comput ation of stationary measures for in- finite Markov chains, TIMS Studies in the Management Sciences, Vol

    Allen, B., Anderssen, R.S., and Seneta, E. (1977) Comput ation of stationary measures for in- finite Markov chains, TIMS Studies in the Management Sciences, Vol. 7. Algorithmic M ethods in Probability, ed. M. F. Neuts, North-Holland, Amsterdam, 13–23

    work page 1977

  2. [2]

    (2012) The M X/M/ 1 queue with multiple working vacation, American Journal of Operations Research, 2(2), 217–224

    Baba, Y. (2012) The M X/M/ 1 queue with multiple working vacation, American Journal of Operations Research, 2(2), 217–224

    work page 2012

  3. [3]

    and Meyer, C.D

    Cho, G.E. and Meyer, C.D. (2001) Comparison of perturbat ion bounds for the stationary distribution of a Markov chain, Linear Algebra and its Applications , 335, 137–150

    work page 2001

  4. [4]

    and Akar, N

    Dayar, T. and Akar, N. (2005) Computing Moments of First P assage Times to a Subset of States in Markov Chains, SIAM Journal on Matrix Analysis and Applications , 27(2), 396–412

    work page 2005

  5. [5]

    and Stewart, W.J

    Dayar, T. and Stewart, W.J. (1996) On the effects of using th e Grassmann–Taksar–Heyman method in iterative aggregation–disaggregation, SIAM Journal on Scientific Computing , 17(1), 287–303

    work page 1996

  6. [6]

    (1983) Approximating Countable Markov Chains , 2nd edn, Springer-Verlag, New York

    Freedman, D. (1983) Approximating Countable Markov Chains , 2nd edn, Springer-Verlag, New York

    work page 1983

  7. [7]

    and Seneta, E

    Gibson, D. and Seneta, E. (1987a) Augmented truncations of infinite stochastic matrices, Journal of Applied Probability , 24, 600–608. 38

    work page

  8. [8]

    and Seneta, E

    Gibson, D. and Seneta, E. (1987b) Monotone infinite stoch astic matrices and their augmented truncations, Stochastic Processes and their Applications , 24(2), 287–292

    work page

  9. [9]

    (1993) Rounding errors in certain algor ithms involving Markov chains, ACM Transactions on Mathematical Software , 19, 496–508

    Grassmann, W.K. (1993) Rounding errors in certain algor ithms involving Markov chains, ACM Transactions on Mathematical Software , 19, 496–508

    work page 1993

  10. [10]

    and Heyman, D.P

    Grassmann, W.K. and Heyman, D.P. (1990) Equilibrium di stribution of block-structured Markov chains with repeating rows, Journal of Applied Probability , 27, 557–576

    work page 1990

  11. [11]

    and Heyman, D.P

    Grassmann, W.K. and Heyman, D.P. (1993) Computation of steady-state probabilities for infinite-state Markov chains with repeating rows, ORSA Journal on Computing , 5(3), 282– 303

    work page 1993

  12. [12]

    (1985) Regenerative analysis and steady state distributions for Markov chains, Operations Research, 33, 1107–1116

    Grassmann, W.K., Taksar, M.I., and Heyman, D.P. (1985) Regenerative analysis and steady state distributions for Markov chains, Operations Research, 33, 1107–1116

    work page 1985

  13. [13]

    and Zhao, Yiqiang (1997) Heterogeneou s multiserver queues with a general input, INFOR: Information Systems and Operational Research , 35, 208–224

    Grassmann, W.K. and Zhao, Yiqiang (1997) Heterogeneou s multiserver queues with a general input, INFOR: Information Systems and Operational Research , 35, 208–224

    work page 1997

  14. [14]

    and Seneta, E

    Golub, G.H. and Seneta, E. (1973) Computation of the sta tionary distribution of an infinite Markov matrix, Bulletin of the Australian Mathematical Society , 8, 333–341

    work page 1973

  15. [15]

    and Seneta, E

    Golub, G.H. and Seneta, E. (1974) Computation of the sta tionary distribution of an infinite stochastic matrix of special form, Bulletin of the Australian Mathematical Society , 10(2), 255– 261

    work page 1974

  16. [16]

    (1995) A decomposition theorem for infinit e stochastic matrices, Journal of Applied Probability, 32, 893–901

    Heyman, D.P. (1995) A decomposition theorem for infinit e stochastic matrices, Journal of Applied Probability, 32, 893–901

    work page 1995

  17. [17]

    (1987) Further comparisons of direct meth ods for computing stationary distri- butions of Markov chains, SIAM Journal on Algebraic and Discrete Methods , 8(2), 226–232

    Heyman, D.P. (1987) Further comparisons of direct meth ods for computing stationary distri- butions of Markov chains, SIAM Journal on Algebraic and Discrete Methods , 8(2), 226–232

    work page 1987

  18. [18]

    and Reeves, A

    Heyman, D.P. and Reeves, A. (1989) Numerical solutions arising in Markov chain models, ORSA Journal on Computing , 1(1), 52–60

    work page 1989

  19. [19]

    (2018) The computation of the mean first pas sage times for Markov chains, Linear Algebra and its Applications , 549, 100–122

    Hunter, J.J. (2018) The computation of the mean first pas sage times for Markov chains, Linear Algebra and its Applications , 549, 100–122

    work page 2018

  20. [20]

    (2022) On Converg ence of General Truncation- Augmentation Schemes for Approximating Stationary Distri butions of Markov Chains, (arXiv:2203.15167)

    Infanger, A., Glynn, P.W., and Liu, Y. (2022) On Converg ence of General Truncation- Augmentation Schemes for Approximating Stationary Distri butions of Markov Chains, (arXiv:2203.15167)

    work page arXiv 2022

  21. [21]

    and Ramaswamy, R

    Keilson, J. and Ramaswamy, R. (1984) Convergence of qua si-stationary distributions in birth- death processes, Stochastic Processes and their Applications , 18, 301–312

    work page 1984

  22. [22]

    and Knapp, A.W

    Kemeny, J.G., Snell, J.L. and Knapp, A.W. (1966) Denumerable Markov Chains , 2nd ed., Van Nostrand, Princeton, NJ

    work page 1966

  23. [23]

    (1986) Numerical computation for mean passa ge times and absorption probabilities in Markov and semi-Markov models, Zeitschrift f¨ ur Operations Research, 30, A197–A207

    Kohlas, J. (1986) Numerical computation for mean passa ge times and absorption probabilities in Markov and semi-Markov models, Zeitschrift f¨ ur Operations Research, 30, A197–A207. 39

    work page 1986

  24. [24]

    and Ramaswami, V

    Latouche, G. and Ramaswami, V. (1987) Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM

    work page 1987

  25. [25]

    (1951) Syst` emes markoviens et stationnaires

    L´ evy, P. (1951) Syst` emes markoviens et stationnaires. Cas d´ enombrable,Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure, 68(3), 327–381

    work page 1951

  26. [26]

    (1952) Compl´ ement ` a l’´ etude des processus de Markoff, Annales Scientifiques de l’´Ecole Normale Sup´ erieure, 69(3), 203–212

    L´ evy, P. (1952) Compl´ ement ` a l’´ etude des processus de Markoff, Annales Scientifiques de l’´Ecole Normale Sup´ erieure, 69(3), 203–212

    work page 1952

  27. [27]

    Processus markoviens et stationnaires

    L´ evy, P. Processus markoviens et stationnaires. Cas d ´ enombrable,Annales de l’Institut Henri Poincar´ e, 18, 7–25

    work page

  28. [28]

    and Zhao, Y.Q

    Li, H. and Zhao, Y.Q. (2000) Stochastic block-monotone matrices and approximating station- ary distributions of infinite Markov chains, Stochastic Models, 16, 313–333

    work page 2000

  29. [29]

    and Zhao, Y.Q

    Li, Q. and Zhao, Y.Q. (2002) A constructive method for fin ding β -invariant measures for transition matrices of M/G/ 1 type, in Matrix-Analytic Methods: Theory and Applications , Latouche, G. and Taylor, P. Eds, World Scientific, 237–263

    work page 2002

  30. [30]

    and Zhao, Y.Q

    Li, Q. and Zhao, Y.Q. (2002) β -invariant measures for transition matrices of GI/M/ 1 type, Stochastic Models, 19, 201–233

    work page 2002

  31. [31]

    and Zhao, Y.Q

    Li, Q. and Zhao, Y.Q. (2004) The RG-factorization in block-structured Markov renewal pro- cesses, in Observation, Theory, and Modeling of Atmospheric Variabili ty, edited by Xun Zhu, Xiaofan Li, Shuntai Zhou, Yuejian Zhu, Ming Cai, Fei-Fei Jin , Xiaolei Zou and Minghua Zhang, World Scientific, 545–568

    work page 2004

  32. [32]

    (2010) Augmented truncation approximations of discrete-time Markov chains, Opera- tions Research Letters , 38, 218–222

    Liu, Y. (2010) Augmented truncation approximations of discrete-time Markov chains, Opera- tions Research Letters , 38, 218–222

    work page 2010

  33. [33]

    and Li, W

    Liu, Y. and Li, W. (2018) Error bounds for augmented trun cation approximations of Markov chains via the perturbation method, Advances in Applied Probability , 50(2), 645–669

    work page 2018

  34. [34]

    and Zhao, Y.Q

    Liu, J., Liu, Y. and Zhao, Y.Q. (2021) Augmented truncat ion approximations to the solution of Poisson’s equation for Markov chains, (arXiv:2101.0027 7)

    work page arXiv 2021

  35. [35]

    (2015), Error bounds for augmented trunca tions of discrete-time block- monotone Markov chains under geometric drift conditions, Advances in Applied Probability , 47, 83–105

    Masuyama, H. (2015), Error bounds for augmented trunca tions of discrete-time block- monotone Markov chains under geometric drift conditions, Advances in Applied Probability , 47, 83–105

    work page 2015

  36. [36]

    Masuyama, H. (2016), Error bounds for augmented trunca tions of discrete-time block- monotone Markov chains under subgeometric drift condition s, SIAM Journal on Matrix Anal- ysis and Applications , 37, 877–910

    work page 2016

  37. [37]

    (2017a), Continuous-time block-monoton e Markov chains and their block- augmented truncations, Linear Algebra and its Applications , 514, 105–150

    Masuyama, H. (2017a), Continuous-time block-monoton e Markov chains and their block- augmented truncations, Linear Algebra and its Applications , 514, 105–150

    work page

  38. [38]

    (2017b), Error bounds for last-column-bl ock-augmented truncations of block- structured Markov chains, Journal of the Operations Research Society of Japan , 60, 271–320

    Masuyama, H. (2017b), Error bounds for last-column-bl ock-augmented truncations of block- structured Markov chains, Journal of the Operations Research Society of Japan , 60, 271–320. 40

    work page

  39. [39]

    (2019), A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains, (arXiv:19 01.02972)

    Masuyama, H. (2019), A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains, (arXiv:19 01.02972)

    work page 2019

  40. [40]

    (1993) Entrywise perturbation theor y and error analysis for Markov chains, Numerische Mathematik , 65, 109–120

    O’Cinneide, C.A. (1993) Entrywise perturbation theor y and error analysis for Markov chains, Numerische Mathematik , 65, 109–120

    work page 1993

  41. [41]

    (1988) Stable recursion for the steady st ate vector in Markov chains of M/G/1 type, Stoch Models, 4, 183–188

    Ramaswami, V. (1988) Stable recursion for the steady st ate vector in Markov chains of M/G/1 type, Stoch Models, 4, 183–188

    work page 1988

  42. [42]

    (1967) Finite approximations to infinite non -negative matrices, Proceedings of the Cambridge Philosophical Society , 63, 983–992

    Seneta, E. (1967) Finite approximations to infinite non -negative matrices, Proceedings of the Cambridge Philosophical Society , 63, 983–992

    work page 1967

  43. [43]

    (1968) Finite approximations to infinite non -negative matrices, Part II: Refinements and applications, Proceedings of the Cambridge Philosophical Society , 64, 465–470

    Seneta, E. (1968) Finite approximations to infinite non -negative matrices, Part II: Refinements and applications, Proceedings of the Cambridge Philosophical Society , 64, 465–470

    work page 1968

  44. [44]

    (1980) Computing the stationary distributi on for infinite Markov chains, Linear Algebra and its Applications , 34, 259–267

    Seneta, E. (1980) Computing the stationary distributi on for infinite Markov chains, Linear Algebra and its Applications , 34, 259–267

    work page 1980

  45. [45]

    (1981) Non-Negative Matrices and Markov Chains , 2nd edn, Springer-Verlag, New York

    Seneta, E. (1981) Non-Negative Matrices and Markov Chains , 2nd edn, Springer-Verlag, New York

    work page 1981

  46. [46]

    (1985) A Markov partitioning algorithms for computing steady-state probabili- ties, Operations Research, 33, 228–235

    Sheskin, T.J. (1985) A Markov partitioning algorithms for computing steady-state probabili- ties, Operations Research, 33, 228–235

    work page 1985

  47. [47]

    and Thornton, J

    Sonin, I. and Thornton, J. (2001) Recursive algorithm f or the fundamental/group inverse matrix of a Markov chain from an explicit formula, SIAM Journal on Matrix Analysis and Applications, 23(1), 209–224

    work page 2001

  48. [48]

    (1993) Gaussian elimination, perturbat ion theory and Markov chains, In Linear Algebra, Markov Chains, and Queueing Models , C

    Stewart, G.W. (1993) Gaussian elimination, perturbat ion theory and Markov chains, In Linear Algebra, Markov Chains, and Queueing Models , C. Meyer and R. J. Plemmons, Eds, Springer- Verlag, New York, 59–69

    work page 1993

  49. [49]

    (1994) An Introduction to the Numerical Solution of Markov Chains , Princeton University Press

    Stewart, W.J. (1994) An Introduction to the Numerical Solution of Markov Chains , Princeton University Press

    work page 1994

  50. [50]

    (1971) Trancation procedures for non-ne gative matrices, Journal of Applied Probability, 8, 311–320

    Tweedie, R.L. (1971) Trancation procedures for non-ne gative matrices, Journal of Applied Probability, 8, 311–320

    work page 1971

  51. [51]

    (1998) Truncation approximations of inv ariant measures for Markov chains, Journal of Applied Probability , 35, 517–536

    Tweedie, R.L. (1998) Truncation approximations of inv ariant measures for Markov chains, Journal of Applied Probability , 35, 517–536

    work page 1998

  52. [52]

    Wolf, D. (1975) Approximation homogener Markoff-Ketten mit abzahlbarem Zustandraum durch solche mit endlichem Zustandraum, Proceedings in Operations Research 5 , Physica- Verlag, Wurzburg, 137–146

    work page 1975

  53. [53]

    (1980) Approximation of the invariant probabi lity measure of an infinite stochastic matrix, Advances in Applied Probability , 12(3), 710–726

    Wolf, D. (1980) Approximation of the invariant probabi lity measure of an infinite stochastic matrix, Advances in Applied Probability , 12(3), 710–726

    work page 1980

  54. [54]

    Zhao, Y.Q. (2000) Censoring technique in studying bloc k-structured Markov chains, in Ad- vances in Algorithmic Methods for Stochastic Models , Latouche, Guy and Taylor, Peter Eds, Notable Publications Inc., 417–433. 41

    work page 2000

  55. [55]

    (2020) GTH algorithm, censored Markov chain s, and RG-factorization, Mathemat- ical Theory and Applications , 40(2), 16–28

    Zhao, Y.Q. (2020) GTH algorithm, censored Markov chain s, and RG-factorization, Mathemat- ical Theory and Applications , 40(2), 16–28

    work page 2020

  56. [56]

    and Braun, W.J

    Zhao, Y.Q., Li, W. and Braun, W.J. (1997) On a decomposit ion for infinite transition matrices, Queueing Systems , 27, 127–130

    work page 1997

  57. [57]

    and Braun, W.J

    Zhao, Y.Q., Li, W. and Braun, W.J. (1998) Infinite block- structured transition matrices and their properties, Advances in Applied Probability , 30, 365–384

    work page 1998

  58. [58]

    and Braun, W.J

    Zhao, Y.Q., Li, W. and Braun, W.J. (2003) Censoring, fac torizations, and spectral analysis for transition matrices with block-repeating entries, Methodology and Computing in Applied Probability, 5, 35–58

    work page 2003

  59. [59]

    and Liu, D

    Zhao, Y.Q. and Liu, D. (1996) The censored Markov chain a nd the best augmentation, Journal of Applied Probability , 33, 623–629. 42

    work page 1996