A matrix continued fraction algorithm for the multiserver repeated order queue

Hanschke, Thomas

doi:10.1016/s0895-7177(99)00139-9

Cited by 36 publications

(59 citation statements)

References 12 publications

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“…Results of numerical experiments we provide on a set of problems from the literature suggest, among a number of alternative approaches, that the zero matrix as an approximation to the matrix LDQBD processes for stochastic chemical kinetics 1007 of conditional expected sojourn times at level high + 1 [1], [14] fares very well in terms of accuracy. The solution is obtained rapidly with a residual norm in the order of 10 −16 when the stationary probability mass is located in lower level numbers.…”

Section: T Dayar Et Almentioning

confidence: 99%

“…(A3) An alternative discussed in [1] and [14] is to letR high be the zero matrix. The existence ofR high−1 in (12) is guaranteed by the fact that −Q high,high is a nonsingular M-matrix.…”

Section: ≥0mentioning

confidence: 99%

“…In the case of level-dependent QBD (LDQBD) processes, the transition rates may depend on the level. Applying suitable state space truncation and augmentation procedures [10], [17], [30], [32], the stationary distribution of infinite LDQBD processes can be approximated by matrix-analytic methods [1], [4], [9], [14].…”

Section: Introductionmentioning

confidence: 99%

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Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

et al. 2011

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Systems of stochastic chemical kinetics are modeled as infinite level-dependent quasibirth-and-death (LDQBD) processes. For these systems, in contrast to many other applications, levels have an increasing number of states as the level number increases and the probability mass may reside arbitrarily far away from lower levels. Ideas from Lyapunov theory are combined with existing matrix-analytic formulations to obtain accurate approximations to the stationary probability distribution when the infinite LDQBD process is ergodic. Results of numerical experiments on a set of problems are provided.

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Section: T Dayar Et Almentioning

confidence: 99%

Section: ≥0mentioning

confidence: 99%

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Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

et al. 2011

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“…because we have to solve an infinite dimensional system of equations. As for the BSMCs with the special structures mentioned above, we can establish the stochastically interpretable expression of the stationary distribution vector by matrix analytic methods (Grassmann and Heyman [21], Latouche and Ramaswami [34], Neuts [53], Zhao et al [65]) and can also obtain the analytical expression of the stationary distribution vector by continued fraction approaches (Hanschke [23], Pearce [54]). However, the construction of such expressions requires an infinite number of computational steps involving an infinite number of block matrices that characterize those BSMCs.…”

Section: Where (K I; ℓ J) Denotes Ordered Pair ((K I) (ℓ J)) Simentioning

confidence: 99%

Error Bounds for Last-Column-Block-Augmented Truncations of Block-Structured Markov Chains

Masuyama

2017

JORSJ

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This paper discusses the error estimation of the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of block-structured Markov chains (BSMCs) in continuous time. We first derive upper bounds for the absolute difference between the time-averaged functionals of a BSMC and its LC-block-augmented truncation, under the assumption that the BSMC satisfies the general f -modulated drift condition. We then establish computable bounds for a special case where the BSMC is exponentially ergodic. To derive such computable bounds for the general case, we propose a method that reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the properties of the bounds through the numerical results on an M/M/s retrial queue, which is a representative example of LD-QBDs. Finally, we present computable perturbation bounds for the stationary distribution vectors of BSMCs.

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“…irreducibility, see [30] for more details). In the context of (that is, the corresponding Markov chain is a quasi-birth-death process) by means of matrix-valued continued fractions has been studied in [31]. The resulting algorithm is similar to the matrix-analytic methods studied in [32] [33].…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

Inherent Numerical Instability in Computing Invariant Measures of Markov Chains

Baumann¹,

Hanschke²

2017

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Invariant measures of Markov chains in discrete or continuous time with a countable set of states are characterized by its steady state recurrence relations. Exemplarily, we consider transition matrices and Q-matrices with upper bandwidth n and lower bandwidth 1 where the invariant measures satisfy an (n + 1)-order linear difference equation. Markov chains of this type arise from applications to queueing problems and population dynamics. It is the purpose of this paper to point out that the forward use of this difference equation is subject to some hitherto unobserved aspects. By means of the concept of generalized continued fractions (GCFs), we prove that each invariant measure is a dominated solution of the difference equation such that forward computation becomes numerically unstable. Furthermore, the GCF-based approach provides a decoupled recursion in which the phenomenon of numerical instability does not appear. The procedure results in an iteration scheme for successively computing approximants of the desired invariant measure depending on some truncation level N. Increasing N leads to the desired solution. A comparison study of forward computation and the GCF-based approach is given for Q-matrices with upper bandwidth 1 and 2.

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A matrix continued fraction algorithm for the multiserver repeated order queue

Cited by 36 publications

References 12 publications

Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

Error Bounds for Last-Column-Block-Augmented Truncations of Block-Structured Markov Chains

Inherent Numerical Instability in Computing Invariant Measures of Markov Chains

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