Abstract
Classical iterative methods for the solution of a linear system of equations as in (1.3) start with an initial approximation. At each iteration, they multiply the current approximation with a particular matrix to obtain a new approximation with the objective that the approximations eventually converge to the true solution [83, 130]. These methods are the building blocks of all advanced iterative methods. The matrix used in the iterative multiplication process is obtained at the outset by splitting the coefficient matrix of the linear system, which is Q in our setting. Therefore, we begin by splitting the smaller matrices that form the Kronecker products as in [146] and show how classical iterative methods can be formulated in terms of these smaller matrices.
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References
Akyildiz, I.F.: Mean value analysis for blocking queueing networks. IEEE Trans. Softw. Eng. 14, 418–428 (1988)
Aldous, D., Shepp, L.: The least variable phase type distribution is Erlang. Stoch. Model. 3, 467–473 (1987)
APNN–Toolbox. http://www4.cs.uni-dortmund.de/APNN-TOOLBOX/ (2004). Accessed 4 Apr 2012
Bao, Y., Bozkurt, I.N., Dayar, T., Sun, X., Trivedi, K.S.: Decompositional analysis of Kronecker structured Markov chains. Electron. Trans. Numer. Anal. 31, 271–294 (2008)
Barker, V.A.: Numerical solution of sparse singular linear systems of equations arising from ergodic Markov chains. Comm. Stat. Stoch. Model. 5, 355–381 (1989)
Baumann, H., Sandmann, W.: Numerical solution of level dependent quasi-birth-and-death processes. In: International Conference on Computational Science, Procedia Computer Science, vol. 1, pp. 1555–1563. Elsevier, Amsterdam (2010)
Bause, F., Buchholz, P., Kemper, P.: A toolbox for functional and quantitative analysis of DEDS. In: Puigjaner, R., Savino, N.N., Serra, B. (eds.) Quantitative Evaluation of Computing and Communication Systems, Lecture Notes in Computer Science, vol. 1469, pp. 356–359. Springer, Berlin Heidelberg New York (1998)
Benoit, A., Brenner, L., Fernandes, P., Plateau, B., Stewart, W.J.: The Peps software tool. In: Kemper, P., Sanders, W.H. (eds.) Computer Performance Evaluation: Modelling Techniques and Tools, Lecture Notes in Computer Science, vol. 2794, pp. 98–115. Springer, Heidelberg (2003)
Benoit, A., Brenner, L., Fernandes, P., Plateau, B.: Aggregation of stochastic automata networks with replicas. Linear Algebr. Appl. 386, 111–136 (2004)
Benoit, A., Fernandes, P., Plateau, B., Stewart, W.J.: On the benefits of using functional transitions and Kronecker algebra. Perform. Eval. 58, 367–390 (2004)
Benoit, A., Plateau, B., Stewart, W.J.: Memory-efficient Kronecker algorithms with applications to the modelling of parallel systems. Futur. Gener. Comput. Syst. 22, 838–847 (2006)
Benzi, M: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, Pennslyvania (1994)
Bini, D.A., Latouche, G., Meini, B.: Numerical Methods for Structured Markov Chains. Oxford University, Oxford (2005)
Brenner, L., Fernandes, P., Plateau, B., Sbeity, I.: PEPS 2007 – Stochastic automata networks software tool. In: Proceedings of the Fourth International Conference on Quantitative Evaluation of Computer Systems and Technologies, pp. 163–164. IEEE Computer Society, Edinburgh (2007)
Brenner, L., Fernandes, P., Fourneau, J.-M., Plateau, B.: Modelling Grid5000 point availability with SAN. Electron. Notes Theor. Comput. Sci. 232, 165–178 (2009)
Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia, Pennslyvania (2000)
Bright, L., Taylor, P.G.: Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stoch. Model. 11, 497–525 (1995)
Buchholz, P.: A class of hierarchical queueing networks and their analysis. Queue. Syst. 15, 59–80 (1994)
Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. J. Appl. Probab. 31, 59–75 (1994)
Buchholz, P.: Hierarchical Markovian models: symmetries and reduction. Perform. Eval. 22, 93–110 (1995)
Buchholz, P.: An aggregation ∖ disaggregation algorithm for stochastic automata networks. Probab. Eng. Inf. Sci. 11, 229–253 (1997)
Buchholz, P.: Exact performance equivalence: An equivalence relation for stochastic automata. Theor. Comput. Sci. 215, 263–287 (1999)
Buchholz, P.: Hierarchical structuring of superposed GSPNs. IEEE Trans. Softw. Eng. 25, 166–181 (1999)
Buchholz, P.: Structured analysis approaches for large Markov chains. Appl. Numer. Math. 31, 375–404 (1999)
Buchholz, P.: Projection methods for the analysis of stochastic automata networks. In: Plateau, B., Stewart, W.J., Silva, M. (eds.) Numerical Solution of Markov Chains, pp. 149–168. Prensas Universitarias de Zaragoza, Zaragoza (1999)
Buchholz, P.: An adaptive aggregation/disaggregation algorithm for hierarchical Markovian models. Eur. J. Oper. Res. 116, 545–564 (1999)
Buchholz, P.: Multilevel solutions for structured Markov chains. SIAM J. Matrix Anal. Appl. 22, 342–357 (2000)
Buchholz, P.: Efficient computation of equivalent and reduced representations for stochastic automata. Comput. Syst. Sci. Eng. 15, 93–103 (2000)
Buchholz, P.: An iterative bounding method for stochastic automata networks. Perform. Eval. 49, 211–226 (2002)
Buchholz, P.: Adaptive decomposition and approximation for the analysis of stochastic Petri nets. Perform. Eval. 56, 23–52 (2004)
Buchholz, P., Dayar, T.: Block SOR for Kronecker structured Markovian representations. Linear Algebr. Appl. 386, 83–109 (2004)
Buchholz, P., Dayar, T.: Comparison of multilevel methods for Kronecker structured Markovian representations. Computing 73, 349–371 (2004)
Buchholz, P., Dayar, T.: Block SOR preconditioned projection methods for Kronecker structured Markovian representations. SIAM J. Sci. Comput. 26, 1289–1313 (2005)
Buchholz, P., Dayar, T.: On the convergence of a class of multilevel methods for large, sparse Markov chains. SIAM J. Matrix Anal. Appl. 29, 1025–1049 (2007)
Buchholz, P., Kemper, P.: On generating a hierarchy for GSPN analysis. Perform. Eval. Rev. 26, 5–14 (1998)
Buchholz, P., Kemper, P.: Kronecker based representations of large Markov chains. In: Haverkort, B., Hermanns, H., Siegle, M. (eds.) Validation of Stochastic Systems, Lecture Notes in Computer Science, vol. 2925, pp. 256–295. Springer, Berlin Heidelberg New York (2004)
Buchholz, P., Ciardo, G., Donatelli, S., Kemper, P.: Complexity of memory-efficient Kronecker operations with applications to the solution of Markov models. INFORMS J. Comput. 12, 203–222 (2000)
Campos, J., Donatelli, S., Silva, M.: Structured solution of asynchronously communicating stochastic models. IEEE Trans. Softw. Eng. 25, 147–165 (1999)
Cao, W.-L., Stewart, W.J.: Aggregation/disaggregation methods for nearly uncoupled Markov chains. J. ACM 32, 702–719 (1985)
Chan, R.H., Ching, W.K.: Circulant preconditioners for stochastic automata networks. Numer. Math. 87, 35–57 (2000)
Chung, M.-Y., Ciardo, G., Donatelli, S., He, N., Plateau, B., Stewart, W., Sulaiman, E., Yu, J.: A comparison of structural formalisms for modeling large Markov models. In: Proceedings of the 18th International Parallel and Distributed Processing Symposium, pp. 196b. IEEE Computer Society, Edinburgh (2004)
Ciardo, G., Miner, A.S.: A data structure for the efficient Kronecker solution of GSPNs. In: Buchholz, P., Silva, M. (eds.) Proceedings of the 8th International Workshop on Petri Nets and Performance Models, pp. 22–31. IEEE Computer Society, Edinburgh (1999)
Ciardo, G., Jones, R.L., Miner, A.S., Siminiceanu, R.: Logical and stochastic modeling with SMART. In: Kemper, P., Sanders, W.H. (eds.) Computer Performance Evaluation: Modelling Techniques and Tools, Lecture Notes in Computer Science, vol. 2794, pp. 78–97. Springer, Heidelberg (2003)
Clark, G., Gilmore, S., Hillston, J., Thomas, N.: Experiences with the PEPA performance modelling tools. IEE Softw. 146, 11–19 (1999)
Courtois, P.-J., Semal, P.: Bounds for the positive eigenvectors of nonnegative matrices and for their approximations by decomposition. J. ACM 31, 804-825 (1984)
Czekster, R.M., Fernandes, P., Vincent, J.-M., Webber, T.: Split: a flexible and efficient algorithm to vector–descriptor product. In: Glynn, P.W. (ed.) Proceedings of the 2nd International Conference on Performance Evaluation Methodologies and Tools, 83. Nantes, ACM International Conference Proceeding Series (2007)
Czekster, R.M., Fernandes, P., Webber, T.: GTAexpress: A software package to handle Kronecker descriptors. In: Proceedings of the Sixth International Conference on Quantitative Evaluation of Computer Systems and Technologies, pp. 281–282. IEEE Computer Society, Budapest (2009)
Dao-Thi, T.-H., Fourneau, J.-F.: Stochastic automata networks with master/slave synchronization: Product form and tensor. In: Al-Begain, K., Fiems, D., Horvaáthe, G. (eds.) Proceedings of the 16th International Conference on Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science, vol. 5513, pp. 279–293. Springer, Heidelberg (2009)
Davio, M.: Kronecker products and shuffle algebra. IEEE Trans. Comput. C-30, 116–125 (1981)
Davis, T.A., Gilbert, J.R., Larimore, S., Ng, E.: Algorithm 836: COLAMD, a column approximate minimum degree ordering algorithm. ACM Trans. Math. Softw. 30, 377–380 (2004)
Dayar, T.: State space orderings for Gauss–Seidel in Markov chains revisited. SIAM J. Sci. Comput. 19, 148–154 (1998)
Dayar, T.: Permuting Markov chains to nearly completely decomposable form. Technical Report BU–CEIS–9808, Department of Computer Engineering and Information Science, Bilkent University, Ankara (1998)
Dayar, T.: Effects of reordering and lumping in the analysis of discrete–time SANs. In: Gardy, D., Mokkadem, A. (eds.) Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, pp. 209–220. Birkhauser, Switzerland (2000)
Dayar, T.: Analyzing Markov chains based on Kronecker products. In: Langville, A.N., Stewart, W.J. (eds.) MAM 2006: Markov Anniversary Meeting, pp. 279–300. Boson Books, Raleigh, North Carolina (2006)
Dayar, T., Meriç, A.: Kronecker representation and decompositional analysis of closed queueing networks with phase-type service distributions and arbitrary buffer sizes. Ann. Oper. Res. 164, 193–210 (2008)
Dayar, T., Orhan, M.C.: LDQBD solver version 2. http://www.cs.bilkent.edu.tr/~tugrul/software.html (2011). Accessed 4 Apr 2012
Dayar, T., Stewart, W.J.: Quasi lumpability, lower-bounding coupling matrices, and nearly completely decomposable Markov chains. SIAM J. Matrix Anal. Appl. 18, 482–498 (1997)
Dayar, T., Stewart, W.J.: Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains. SIAM J. Sci. Comput. 21, 1691–1705 (2000)
Dayar, T., Pentakalos, O.I., Stephens, A.B.: Analytical modeling of robotic tape libraries using stochastic automata. Technical Report TR–97–189, Center of Excellence in Space Data & Information Systems, NASA/Goddard Space Flight Center, Greenbelt, Maryland (1997)
Dayar, T., Hermanns, H., Spieler, D., Wolf, V.: Bounding the equilibrium distribution of Markov population models. Numer. Linear Algebr. Appl. 18, 931–946 (2011)
Dayar, T., Sandmann, W., Spieler, D., Wolf, V.: Infinite level–dependent QBDs and matrix analytic solutions for stochastic chemical kinetics. Adv. Appl. Probab. 43, 1005–1026 (2011)
Donatelli, S.: Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space. Perform. Eval. 18, 21–26 (1993)
Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Clarendon, Oxford (1986)
Fernandes, P., Plateau, B.: Triangular solution of linear systems in tensor product format. Perform. Eval. Rev. 28(4), 30–32 (2001)
Fernandes, P., Plateau, B., Stewart, W.J.: Efficient descriptor–vector multiplications in stochastic automata networks. J. ACM 45, 381–414 (1998)
Fernandes, F., Plateau, B., Stewart, W.J.: Optimizing tensor product computations in stochastic automata networks. RAIRO Oper. Res. 32, 325–351 (1998)
Fletcher, R.: Conjugate gradient methods for indefinite systems. In: Watson, G.A. (ed.) Proceedings of the Dundee Conference on Numerical Analysis, Lecture Notes in Mathematics, vol. 506, pp. 73–89. Springer, Heidelberg (1976)
Fourneau, J.-M.: Discrete time stochastic automata networks: using structural properties and stochastic bounds to simplify the SAN. In: Glynn, P.W. (ed.) Proceedings of the 2nd International Conference on Performance Evaluation Methodologies and Tools, 84. Nantes, ACM International Conference Proceeding Series (2007)
Fourneau, J.-M.: Product form steady-state distribution for stochastic automata networks with domino synchronizations. In: Thomas, N., Juiz, C. (eds.) Proceedings of the 5th European Performance Engineering Workshop, Lecture Notes in Computer Science, vol. 5261, pp. 110–124. Springer, Berlin Heidelberg New York (2008)
Fourneau, J.-M.: Collaboration of discrete-time Markov chains: tensor and product form. Perform Eval. 67, 779–796 (2010)
Fourneau, J.-M., Quessette, F.: Graphs and stochastic automata networks. In: Stewart, W.J. (ed.) Computations with Markov Chains. In: Proceedings of the 2nd International Workshop on the Numerical Solution of Markov Chains, pp. 217–235. Kluwer, Boston (1995)
Fourneau, J.-M., Maisonniaux, H., Pekergin, N., Véque, V.: Performance evaluation of a buffer policy with stochastic automata networks. In: IFIP Workshop on Modelling and Performance Evaluation of ATM Technology, vol. C–15, pp. 433–451. La Martinique, IFIP Transactions North-Holland, Amsterdam (1993)
Fourneau, J.-M., Kloul, L., Pekergin, N., Quessette, F., Véque, V.: Modelling buffer admission mechanisms using stochastic automata networks. Rev. Ann. Télécommun. 49, 337–349 (1994)
Fourneau, J.-M., Plateau, B., Stewart, W.J.: An algebraic condition for product form in stochastic automata networks without synchronizations. Perform. Eval. 65, 854–868 (2008)
Freund, R.W., Nachtigal, N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991)
Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University, Baltimore (1996)
Gordon, J.W., Newell, G.F.: Closed queueing systems with exponential servers. Oper. Res. 15, 252–267 (1967)
Grassmann, W.K.: Transient solutions in Markovian queueing systems. Comput. Oper. Res. 4, 47–56 (1977)
Grassmann, W.K. (ed.): Computational Probability. Kluwer, Norwell, MA (2000)
Grassmann, W.K., Stanford, D.A.: Matrix analytic methods. In: Grassmann, W.K. (ed.) Computational Probability, pp. 153–204. Kluwer, Norwell, MA (2000)
Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, Pennslyvania (1997)
Gross, D., Miller, D.R.: The randomization technique as a modeling tool and solution procedure for transient Markov processes. Oper. Res. 32, 343–361 (1984)
Gusak, O., Dayar, T.: Iterative aggregation–disaggregation versus block Gauss–Seidel on continuous-time stochastic automata networks with unfavorable partitionings. In: Obaidat, M.S., Davoli, F. (eds.) Proceedings of the 2001 International Symposium on Performance Evaluation of Computer and Telecommunication Systems, pp. 617–623. Orlando, Florida (2001)
Gusak, O., Dayar, T., Fourneau, J.-M.: Stochastic automata networks and near complete decomposability. SIAM J. Matrix Anal. Appl. 23, 581–599 (2001)
Gusak, O., Dayar, T., Fourneau, J.-M.: Lumpable continuous-time stochastic automata networks. Eur. J. Oper. Res. 148, 436–451 (2003)
Gusak, O., Dayar, T., Fourneau, J.-M.: Iterative disaggregation for a class of lumpable discrete-time stochastic automata networks. Perform. Eval. 53, 43–69 (2003)
Haddad, S., Moreaux, P.: Asynchronous composition of high–level Petri nets: a quantitative approach. In: Billington, J., Reisig, W. (eds.) Proceedings of the 17th International Conference on Application and Theory of Petri Nets, Lecture Notes in Computer Science, vol. 1091, pp. 192–211. Springer, Heidelberg (1996)
Haverkort, B.R.: Performance of Computer Communication Systems: A Model-Based Approach. Wiley, New York (1998)
Hillston, J., Kloul, L.: An efficient Kronecker representation for PEPA models. In: de Alfaro, L., Gilmore, S. (eds.) Proceedings of the 1st Process Algebras and Performance Modeling, Probabilistic Methods in Verification Workshop, Lecture Notes in Computer Science, vol. 2165, pp. 120–135. Springer, Berlin Heidelberg New York (2001)
Horton, G., Leutenegger, S.: A multi-level solution algorithm for steady state Markov chains. Perform. Eval. Rev. 22(1), 191–200 (1994)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, Berlin Heidelberg New York (1983)
Kemper, P.: Numerical analysis of superposed GSPNs. IEEE Trans. Softw. Eng. 22, 615–628 (1996)
Koury, J.R., McAllister, D.F., Stewart, W.J.: Iterative methods for computing stationary distributions of nearly completely decomposable Markov chains. SIAM J. Algebr. Discrete Math. 5, 164–186 (1984)
Krieger, U.: Numerical solution of large finite Markov chains by algebraic multigrid techniques. In: Stewart, W.J. (ed.) Computations with Markov Chains, pp. 403–424. Kluwer, Boston (1995)
Kurtz, T.G.: The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57, 2976–2978 (1972)
Langville, A.N., Stewart, W.J.: The Kronecker product and stochastic automata networks. J. Comput. Appl. Math. 167, 429–447 (2004)
Langville, A.N., Stewart, W.J.: Testing the nearest Kronecker product preconditioner on Markov chains and stochastic automata networks. INFORMS J. Comput. 16, 300–315 (2004)
Langville, A.N., Stewart, W.J.: A Kronecker product approximate preconditioner for SANs. Numer. Linear Algebr. Appl. 11, 723–752 (2004)
Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia, Pennslyvania (1999)
Lee, T.L., Li, T.Y., Tsai, C.H.: HOM4PS-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83, 109–133 (2008)
Li, H., Cao, Y., Petzold, L.R., Gillespie, D.: Algorithms and software for stochastic simulation of biochemical reacting systems. Biotechnol. Prog. 24, 56–62 (2008)
Loinger, A., Biham, O.: Stochastic simulations of the repressilator circuit. Phys. Rev. E 76, 051917 (2007)
Marek, I., Mayer, P.: Convergence analysis of an iterative aggregation/disaggregation method for computing stationary probability vectors of stochastic matrices. Numer. Linear Algebr. Appl. 5, 253–274 (1998)
Marek, I., Pultarová, I.: A note on local and global convergence analysis of iterative aggregation–disaggregation methods. Linear Algebra Appl. 413, 327-341 (2006)
Marie, A.R.: An approximate analytical method for general queueing networks. IEEE Trans. Softw. Eng. 5, 530–538 (1979)
Meriç, A.: Kronecker Representation and Decompositional Analysis of Closed Queueing Networks with Phase–Type Service Distributions and Arbitrary Buffer Sizes. M.S. Thesis, Department of Computer Engineering, Bilkent University, Ankara, Turkey (2007)
Meriç, A.: Software for Kronecker Representation and Decompositional Analysis of Closed Queueing Networks with Phase-Type Service Distributions and Arbitrary Buffer Sizes. http://www.cs.bilkent.edu.tr/~tugrul/software.html (2007). Accessed 4 Apr 2012
Meyer, C.D.: Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev. 31, 240–272 (1989)
Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)
Migallón, V., Penadés, J., Syzld, D.B.: Block two-stage methods for singular systems and Markov chains. Numer. Linear Algebr. Appl. 3, 413–426 (1996)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithhmic Approach. Johns Hopkins University Press, Baltimore (1981)
Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York (1989)
Orhan, M.C.: Kronecker-based Infinite Level-Dependent QBDs: Matrix Analytic Solution versus Simulation. M.S. Thesis, Department of Computer Engineering, Bilkent University, Ankara, Turkey (2011)
Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16, 973–989 (1987)
PEPA Home Page. http://www.dcs.ed.ac.uk/pepa/tools/ (2005). Accessed 4 Apr 2012
PEPS Home Page. http://www-id.imag.fr/Logiciels/peps (2007). Accessed 4 Apr 2012
Plateau, B.: On the stochastic structure of parallelism and synchronization models for distributed algorithms. Perform. Eval. Rev. 13(2), 147–154 (1985)
Plateau, B., Atif, K.: Stochastic automata network for modeling parallel systems. IEEE Trans. Softw. Eng. 17, 1093–1108 (1991)
Plateau, B., Fourneau, J.-M.: A methodology for solving Markov models of parallel systems. J. Parallel Distrib. Comput. 12, 370–387 (1991)
Plateau, B., Stewart, W.J.: Stochastic automata networks. In: W.K. Grassmann, W.K. (ed.) Computational Probability, pp. 113–152. Kluwer, Norwell, MA (2000)
Plateau, B.D., Tripathi, S.K.: Performance analysis of synchronization for two communicating processes. Perform. Eval. 8, 305–320 (1988)
Plateau, B., Fourneau, J.-M., Lee, K.-H.: PEPS: A package for solving complex Markov models of parallel systems. In: Puigjaner, R., Ptier, D. (eds.) Modeling Techniques and Tools for Computer Performance Evaluation, pp. 291–305. Palma de Mallorca (1988)
Pultarová, I., Marek, I.: Convergence of multi-level iterative aggregation–disaggregation methods. J. Comp. Appl. Math 236, 354–363 (2011)
Ramaswami, V., Taylor, P.G.: Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stoch. Model. 12, 143–164 (1996)
Ruge, J.W., Stüben, K.: Algebraic multigrid. In: McCormick, S.F. (ed.) Multigrid Methods, Frontiers in Applied Mathematics 3, pp. 73–130. SIAM, Philadelphia (1987)
Saad, Y.: Projection methods for the numerical solution of Markov chain models. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains, pp. 455–471. Marcel Dekker, New York (1991)
Saad, Y.: Preconditioned Krylov subspace methods for the numerical solution of Markov chains. In: Stewart, W.J. (ed.) Computations with Markov Chains. In: Proceedings of the 2nd International Workshop on the Numerical Solution of Markov Chains, pp. 49–64. Kluwer, Boston (1995)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Saad, Y., Schultz, M.H.: GMRES: a generalized minimum residual algorithm for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
Sbeity, I., Plateau, B.: Structured stochastic modeling and performance analysis of a multiprocessor system. In: Langville, A.N., Stewart, W.J. (eds.) MAM 2006: Markov Anniversary Meeting, pp. 301–314. Boson Books, Raleigh, NC (2006)
Sbeity, I., Brenner, L., Plateau, B., Stewart, W.J.: Phase-type distributions in stochastic automata networks. Eur. J. Oper. Res. 186, 1008–1028 (2008)
Scarpa, M., Bobbio, A.: Kronecker representation of stochastic Petri nets with discrete PH distributions. In: Proceedings of the IEEE International Computer Performance and Dependability Symposium, pp. 52–61. IEEE Computer Society, Budapest (1998)
Seneta E.: Non-negative Matrices: An Introduction to Theory and Applications. Allen & Unwin, London (1973)
SMART Project Home page. http://www.cs.ucr.edu/~ciardo/SMART (2004). Accessed 4 April 2012
Sonneveld, P.: CGS: A fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 36–52 (1989)
Stewart, G.W., Stewart, W.J., McAllister, D.F.: A two-stage iteration for solving nearly completely decomposable Markov chains. In: Golub, G.H., Greenbaum, A., Luskin, M. (eds.) The IMA Volumes in Mathematics and its Applications 60: Recent Advances in Iterative Methods, pp. 201–216. Springer, Berlin Heidelberg New York (1994)
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, NJ (1994)
Stewart, W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press, Princeton, NJ (2009)
Stewart, W.J., Atif, K., Plateau, B.: The numerical solution of stochastic automata networks. Eur. J. Oper. Res. 86, 503–525 (1995)
StochKit. http://engineering.ucsb.edu/~cse/StochKit/ (2012). Accessed 4 Apr 2012
Tewarson, R.P.: Sparse Matrices. Academic, New York (1973)
Touzene, A.: A tensor sum preconditioner for stochastic automata networks. INFORMS J. Comput. 20, 234–242 (2008)
Tweedie, R.L.: Sufficient conditions for regularity, recurrence and ergodicity of Markov processes. Math. Proc. Camb. Philos. Soc. 78, 125–136 (1975)
Uysal, E., Dayar, T.: Iterative methods based on splittings for stochastic automata networks. Eur. J. Oper. Res. 110, 166–186 (1998)
van der Vorst, H.A.: BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)
Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)
Vèque, V., Ben–Othman, J.: MRAP: A multiservices resource allocation policy for wireless ATM network. Comput. Netw. ISDN Syst. 29, 2187–2200 (1998)
Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)
Wolf, V.: Modelling of biochemical reactions by stochastic automata networks. Electron. Notes Theor. Comput. Sci. 171, 197–208 (2007)
Yao, D.D., Buzacott, J.A.: The exponentialization approach to flexible manufacturing systems models with general processing times. Eur. J. Oper. Res. 24, 410–416 (1986)
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Dayar, T. (2012). Iterative Methods. In: Analyzing Markov Chains using Kronecker Products. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4190-8_3
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