Summary
Preconditioned conjugate gradient methods are employed to find the steady-state probability distribution of Markovian queuing networks that have overflow capacity. Different singular preconditioners that can be handled by separation of variables are discussed. The resulting preconditioned systems are nonsingular. Numerical results show that the number of iterations required for convergence grows very slowly with the queue sizes.
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References
Astrakhantsev, G.: Method of Fictitious Domains for a Second-order Elliptic Equation with Natural Boundary Conditions. U.S.S.R. Comput. Maths. Math. Phys.18, 114–121 (1978)
Banegas, A.: Fast Poisson Solvers for Problems with Sparsity. Math. Comput.32, 441–446 (1978)
Bjørstad, P., Widlund, O.: Iterative Methods for the Solutions of Elliptic Problems on Regions Partitioned into Substructures. SIAM J. Numer. Anal.23, 1097–1120 (1986)
Buzbee, B., Dorr, F., George, J., Golub, G.: The Direct Solution of the Discrete Poisson Equations on Irregular Regions. SIAM J. Numer. Anal.8, 722–736 (1971)
Chan, R.: Iterative Methods for Overflow Queuing Models. NYU Comput. Sci. Dept. Tech. Report No. 171 (1985)
Chan, R.: Iterative Methods for Overflow Queuing Models I. Numer. Math.51, 143–180 (1987)
Concus, P., Golub, G., O'Leary, D.: A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations. Bunch, J.R., Rose, D.J., (eds.). Proc. Symposium on Sparse Matrix Computations. pp. 309–332. London: Academic Press 1976
Dryja, M.: A Capacitance Matrix Method for Dirichlet Problem on Polygon Region. Numer. Math.39, 51–64 (1982)
Elman, H.: Iterative Methods for Large, Sparse Nonsymmetric Systems of Linear Equations. Ph.D thesis. Department of Computer Science, Yale University 1982
Funderlic, R., Mankin, J.: Solution of Homogeneous Systems of Linear Equations arising from Compartmental Models. SIAM J. Sci. Statist. Comput.2, 375–383 (1981)
Kaufman, L., Seery, J., Morrison, J.: Overflow Models for Dimension PBX Feature Packages. Bell Syst. Tech. J.60, 661–676 (1981)
Kaufman, L.: Matrix Methods for Queuing Problems. SIAM J. Sci. Statist. Comput.4, 525–552 (1982)
O'Leary, D., Widlund, O.: Capacitance Matrix Methods for the Helmholtz Equation on General Three-dimensional Regions. Math. Comput.33, 849–879 (1979)
Parlett, B.: The Symmetric Eigenvalue Problem, 1st Ed. Prentice-Hall, Englewood Cliffs NJ 1980
Proskurowski, W., Widlund, O.: A Finite Element-Capacitance Matrix Method for the Neumann Problem for Laplace's Equation. SIAM J. Sci. Statist. Comput.1, 410–425 (1980)
Varga, R.: Matrix Iterative Analysis, 1st Ed. Prentice-Hall, Englewood Cliff NJ 1962
Young, D., Jea, K.: Generalized Conjugate Gradient Acceleration of Non-symmetric Iterative method. Linear Algebra Appl.34, 159–194 (1980)
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This research was supported in part by the National Science Foundation grant DCR-8405506 and DCR-8602563
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Chan, R.H. Iterative methods for overflow queuing models II. Numer. Math. 54, 57–78 (1988). https://doi.org/10.1007/BF01403891
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DOI: https://doi.org/10.1007/BF01403891