Structured Markov chains solver: tool extension

Бини, Дарио Андреа; Meini, Beatrice; Steffé, S.; Houdt, Benny Van

doi:10.4108/icst.valuetools2009.7443

Cited by 6 publications

(3 citation statements)

References 14 publications

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“…For the standard QBD, the matrix analytic method is a mature method to obtain the equilibrium probabilities. Implementations of the matrix analytical method are available in Bini and coworkers [8][9][10]. The computational complexity for QBDs is in general O(N 3 ), where N is the number of phases in the QBD, see Latouche and Ramaswami [27] [6,7,23,33,35].…”

Section: Two-node Queue With Finite Buffers At One Queuementioning

confidence: 99%

Performance Measures for the Two-Node Queue With Finite Buffers

Chen¹,

Bai

Boucherie

2019

Prob. Eng. Inf. Sci.

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We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. We also apply our approximation scheme to a coupled-queue in which only one of the buffers has finite capacity.The two-node queue is one of the most extensively studied topics in queueing theory. It can be often modeled as a two-dimensional random walk on the quarter-plane. Hence, it is sufficient to find performance measures of the corresponding two-dimensional random walk if we are interested in the performance of the two-node queue. In this work we analyze the steadystate performance of a two node queue for the particular case that one or

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Section: Two-node Queue With Finite Buffers At One Queuementioning

confidence: 99%

Performance Measures for the Two-Node Queue With Finite Buffers

Chen¹,

Bai

Boucherie

2019

Prob. Eng. Inf. Sci.

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“…The matrix polynomial A(z) has 5 eigenvalues of modulus less than or equal to 1, that is 0.15140 -0.01978i 0.15140 + 0.01978i 0.20936 -0.09002i 0.20936 + 0.09002i 1.00000 + 0.00000i moreover, the eigenvalue of smallest modulus among those which lie outside the unit disk is 1.01258. Applying cyclic reduction to approximate the minimal solution of the matrix equation 4 i=0 A i X i = 0, relying on the software of [6], requires 12 iterations. Applying the same method with the same software to the equation 4 i=0 A i X i = 0 obtained by shifting the eigenvalue 1 to zero provides the solution in just 6 iterations.…”

Section: Quadratic Matrix Polynomials and Matrix Equationsmentioning

confidence: 99%

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Бини

Meini

2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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Given a square matrix A, Brauer's theorem [Duke Math. J. 19 (1952), [75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91] shows how to modify one single eigenvalue of A via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer's theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series A(z) together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function A(z) has a canonical factorization A(z) = U (z) L(z −1 ) and we provide explicit expressions of the factors U (z) and L(z). Similar conditions and expressions are given for the factorization of A(z −1 ). Some applications are discussed. * Work supported by Gruppo Nazionale di Calcolo Scientifico (GNCS) of INdAM, and by the PRA project "Mathematical models and computational methods for complex networks" funded by the University of Pisa. inverse eigenvalue problem [10], [21], to accelerating the convergence of iterative methods for solving matrix equations in particular quadratic equations [20], equations related to QBD Markov chains [17], [14], to M/G/1-type Markov chains [5], [3], or for algebraic Riccati equations in [15], [16], [19], and [2].In this paper we revisit Brauer's theorem in functional form and generalize it to a non-degenerate n × n matrix function A(z) analytic in the open annulusHere, non-degenerate means that det A(z) is not identically zero. Throughout the paper we assume that all the matrix functions are nondegenerate.Let λ ∈ A r1,r2 be an eigenvalue of A(z), i.e., such that det A(λ) = 0 and u ∈ C n , u = 0, be a corresponding eigenvector, i.e., such that A(λ)u = 0. We prove that for any v ∈ C n such that v * u = 1 and for any µ ∈ C, the functionwhere v * denotes the transpose conjugate of v. Moreover, A(z) has all the eigenvalues of A(z), except for λ which, if µ ∈ A r1,r2 , is replaced by µ, otherwise is removed. We provide explicit expressions which relate the coefficients A i of A(z) = i∈Z z i A i to those of A(z).This transformation, called right shift, is obtained by using a right eigenvector u of A(z). A similar version, called left shift, is given by using a left eigenvector v of A(z). Similarly, a combination of the left and the right shift, leads to a transformation called double shift, which enables one to shift a pair of eigenvalues by operating to the right and to the left of A(z).This result, restricted to a matrix Laurent polynomial A(z) = k −h z i A i , provides a shifted function which is still a matrix Laurent polynomial A(z) = k i=−h z i A i . In the particular case where A(z) = zI − A, we find that A(z) = zI − A is such that A is the matrix given in the classical theorem of Brauer [9]. Therefore, these results provide an extension of Brauer's Theorem 1.1 to matrix Laurent series and to matrix polynomials.A further generalization is obtained by performing the right, left, or double shift simultaneously to a packet of eigenvalues once we are given an invariant (right or le...

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“…To this purpose, many numerical methods, with different properties, have been designed in the last years (see for instance [1,2,3,4]). However, many of these numerical methods are defined for general block coefficients A−1, A0 and A1, and do not exploit the possible structure of these blocks.…”

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confidence: 99%

A compressed cyclic reduction for QBDs with low rank upper and lower transitions

Бини

Favati

Meini

2012

SIGMETRICS Perform. Eval. Rev.

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Consider a quasi-birth-and-death (QBD) Markov chain [6], having probability transition matrix P = 2 6 6 6 6 6 6 4 B0 B1 0 B−1 A0 A1 A−1 A0 A1 A−1 A0 . . . 0 . . . . . . 3 7 7 7 7 7 7 5, where B i, Ai, i = −1, 0, 1, are m × m matrices. In the numerical solution of QBD Markov chains a crucial step is the efficient computation of the minimal nonnegative solution R of the quadratic matrix equationTo this purpose, many numerical methods, with different properties, have been designed in the last years (see for instance [1, 2, 3, 4]). However, many of these numerical methods are defined for general block coefficients A−1, A0 and A1, and do not exploit the possible structure of these blocks.Recently, some attention has been addressed to the case where A−1 has only few non-null columns, or A1 has only few non-null rows. These properties are satisfied when the QBD has restricted transitions to higher (or lower) levels.In particular, in [7] the authors exploit these properties of the matrix A−1, or A1, to formulate the QBD in terms of an M/G/1 type Markov chain, where the block matrices have size smaller than m; in particular, when both A−1 and A1 have the desired property, the latter M/G/1 type Markov chain reduces to a QBD. In [5] the structure of A−1 is used in order to reduce the computational cost of some algorithms for computing R.Here we assume that both the matrices A−1 and A1 have small rank with respect to their size m. In particular, if A−1 and A1 have only few non-null columns and rows, respectively, they have small rank. We show that, under this assumption, the matrix R can be computed by using the cyclic reduction algorithm, where the matrices A (k) i , i = −1, 0, 1, generated at the kth step of the algorithm, can be represented by small rank matrices. In particular, if r−1 is the rank of A−1, and if r1 is the rank of A1, then each step of cyclic reduction can be performed by means of O((r−1+r1) 3 ) arithmetic operations. This cost estimate must be compared with the cost of O(m 3 ) arithmetic operations, needed without exploiting the structure of A−1 and A1. Therefore, if r−1 and r1 are much smaller than m, the advantage is evident.It remains an open issue to understand how the structure can be exploited in the case where only one between A−1 and A1 has low rank.

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Structured Markov chains solver: tool extension

Cited by 6 publications

References 14 publications

Performance Measures for the Two-Node Queue With Finite Buffers

Performance Measures for the Two-Node Queue With Finite Buffers

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

A compressed cyclic reduction for QBDs with low rank upper and lower transitions

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