On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study non-steady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using well-known results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.
- H. Akaike, “Block toeplitz matrix inversion,” SIAM J. App. Math vol. 24 pp. 234-241, 1973.
- E. Anderson, Z. Bai, J. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du, Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide, SIAM: Pennsylvania, 1999.
- H. Ayhan, J. Limon-Robles, and M. A. Wortman, “An approach for computing tight numerical bounds on renewal functions,” IEEE Trans. Reliab vol. 48 pp. 182-188, 1999.
- R. P. Brent, “Parallel algorithms for Toeplitz systems,” In Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, edited by G. H. Golub and P. M. Van Dooren, vol. F70, pp. 7-92, Springer-Verlag: Berlin, 1991.
- R. P. Brent, “Stability of fast algorithms for structured linear systems,” In Fast Reliable Algorithms for Matrices with Structure, edited by T. Kailath and A. Sayed, SIAM: Pennsylvania, pp. 103-116, 1972.
- J. R. Bunch, “Stability of methods for solving toeplitz systems of equations,” SIAM J. Sci. and Stat. Computing vol. 6 pp. 349-364, 1985.
- S. Chandrasckaran and A. H. Sayed, “A fast stable solver for non-symmetric Toeplitz and Quasi-Toeplitz systems of linear equations,” SIAM J. Matrix Anal. and App. vol. 19, pp. 107-139, 1998.
- E. Çinlar, “Markov renewal theory,” Adv. App. Prob. vol. 1 pp. 123-187, 1969.
- E. Çinlar, “Markov renewal theory: a survey,” Mgmt. Sci. vol. 21 pp. 727-752, 1975a.
- E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall: NJ, 1975b.
- R. Disney and P. Kiessler, Traffic Processes in Queueing Networks, Johns Hopkins University Press: Baltimore MD, 1987.
- J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson, “An extended set of FORTRAN basic linear algebra subprograms,” ACM Trans. on Math. Soft. vol. 14 pp. 1-17, 1988.
- J. J. Du Croz and N. J. Higham, “Stability of methods for matrix inversion,” IMA J. of Numer. Anal. vol. 12 pp. 1-19, 1992.
- D. Elkins, A Fast Parallel Algorithm to Numerically Compute the Markov Renewal Kernel via Tight Lower and Upper Bounds. PhD thesis, Texas A&;M University, College Station, TX 77843, 2000.
- L. E. Garey and R. E. Shaw, “A parallel algorithm for solving toeplitz linear systems,” App. Math. and Computation vol. 100 pp. 241-247, 1999.
- G. H. Golub and C. F. Van Loan, Matrix Computations. Johns Hopkins University Press: Baltimore, MD, 3rd edition, 1996.
- D. J. Higham and N. J. Higham, “Backward error and condition of structured linear systems,” SIAM J. Matrix Anal. and App. vol. 13 pp. 162-175, 1992.
- N. J. Higham, “The accuracy of solutions to triangular systems,” SIAM J. Numer. Anal. vol. 26 pp. 1252-1265, 1989.
- N. J. Higham, “Stability of parallel triangular system solvers,” SIAM J. Sci. Computation vol. 16 pp. 400-413, 1995.
- J. Kohlas, Stochastic Methods of Operations Research, Cambridge University Press: Cambridge, 1982.
- R. K. Montoye and D. H. Lawrie, “A practical algorithm for the solution of triangular systems on a parallel processing system,” IEEE Trans. on Computers vol. C-31 pp. 1076-1082, 1982.
- M. Pinedo, Scheduling: Theory, Algorithms, and Systems, Prentice-Hall: Englewood Cliffs, NJ, 1995.
- R. Pyke, “Markov renewal processes: definitions and preliminary properties,” The Annals of Math. Stat. vol. 32 pp. 1231-1242, 1961a.
- R. Pyke, “Markov renewal processes with finitely many states,” The Annals of Math. Stat. vol. 32 pp. 1243-1259, 1961b.
- C. H. Romine and J. M. Ortega, “Parallel solution of triangular systems of equations,” Par. Comput. vol. 6 pp. 109-114, 1988.
- H. L. Royden, Real Analysis, Prentice-Hall: Englewood Cliffs, NJ, 3rd edition, 1963.
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, Inc.: New York, 3rd edition, 1976.
- A. H. Sameh and R. P. Brent, “Solving triangular systems on a parallel computer,” SIAM J. Numer. Anal. vol. 14 pp. 1101-1113, 1977.
- R. Schreiber, “Block algorithms for parallel machines,” In Numerical Algorithms for Modern Parallel Computer Architectures M. Schultz, editor, vol. 13 of IMA Vol. in Math. and its App., pp. 197-207, Springer-Verlag: New York, 1988.
- W. F. Trench, “An algorithm for the inversion of finite toeplitz matrices,” J. of SIAM vol. 12 pp. 515-522, 1964.
- E. E. Tyrtyshnikov, “Fast algorithms for block toeplitz matrices,” Soviet J. Num. Anal. and Math. Modeling vol. 1(2) pp. 121-139, 1985.
- J. M. Varah, “Backward error estimates for toeplitz systems,” SIAM J. Matrix Anal. and App. vol. 15(2) pp. 408-417, 1994.
- J. H. Wilkinson, “Error analysis of direct methods of matrix inversion,” J. ACM vol. 8 pp. 281-330, 1961.
- J. H. Wilkinson, Rounding Errors in Algebraic Processes. Notes on Applied Sci., No. 32, Her Majesty's Stationary Office: London, 1963.
- J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press: London, 1965.
- S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. ACM vol. 16 pp. 592-601, 1969.
- On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds
Methodology And Computing In Applied Probability
Volume 3, Issue 3 , pp 239-253
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Markov renewal process
- Markov renewal equation
- Markov renewal kernel
- Toeplitz matrix
- convolution integral equation