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Cubo (Temuco)

versão On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010 

CUBO A Mathematical Journal Vol.12, N° 02, (169–187). June 2010


A new solution algorithm for skip-free processes to the left

Claus Bauer

Dolby Laboratories, San Francisco, 94103, USA email:


This paper proposes a new solution algorithm for steady state models describing skip-free processes to the left where each level has one phase. The computational complexity of the algorithm is independent of the number of levels of the system. If the skip parameter of the skip-free process is significantly smaller than the number of levels of the system, our algorithm numerically outperforms existing algorithms for skip-free processes. The proposed algorithm is based on a novel method for applying generalized Fibonacci series to the solution of steady state models.

Key words and phrases: Skip-free processes, Markovian environment, stationary distribution


Este artículo propone un nuevo algoritmo solución para modelos estado-steady describiendo procesos libres-salto para la izquierda donde todo nivel tiene una fase. La complejidad computacional del algoritmo es independiente del número de niveles del sistema. Si el parámetro de salto de los procesos libre-salto es significativamente pequeño respecto del número de niveles del sistema, nuestro algoritmo numérico supera algoritmos existentes para procesos libre-salto. El algoritmo propuesto se basa en un método reciente para aplicar series de Fibonacci generalizados para la solución de modelos-steady.

AMS 2000 Subj. Class.: 60J10, 60J99


[1] A. Adhikari, Skip free processes. Ph.D thesis, Berkeley, 1986.         [ Links ]

[2] C. B. Boyer, Pascal's Formula for the Sums of Powers of the Integers. Scripta Math. 9, 237-244, 1943.         [ Links ]

[3] P.J. Brockwell, The extinction time of a birth, death and catastrophe process and of a related diffusion model. Advances Applied Probability, Vol. 17, 1985, 17 - 42.         [ Links ]

[4] M.F. Chen, Single birth processes, Chinese Ann. Math. Ser. A, 20:77-82, 1999.         [ Links ]

[5] D.P. Gaver, P.A. Jacobs and G. Latouche, Finite birth-and-death models in randomly changing environments. Advances in Applied Probability, 16:715-731, 1984.         [ Links ]

[6] W.K Grassmann and D.A. Stanford, Matrix analytic methods. In: Computational Probability, WK Grassmann (Ed.), Kluwer, 2000, pp 153-202.         [ Links ]

[7] L. Guen and A.M Makowski, Matrix-geometric solution for finite capacity queues with phasetype distributions. Performance'87, edited by Courtois, P.J.; Latouche, G.; Amsterdam, 1987, p. 269 - 282.         [ Links ]

[8] E. Hansen, M. Patrick and J. Rusnack, Some modifications of Laguere's method. BIT, 17(1977), 409 -417.         [ Links ]

[9] T. W. Hungerford, Algebra. Springer Publishing House, 1974.         [ Links ]

[10] W.G. Kelley and A.C. Peterson, Difference equations, An introduction with applications, Academic Press, Inc. 1991.         [ Links ]

[11] P.A. Jacobs, D.P. Gaver and G. Latouche, Finite markov chain models skip free in one direction. Naval Research Logistics Quarterly, 31, 1984, pp. 571-588.         [ Links ]

[12] E. Kuntz, Algebra., Vieweg Verlag, Braunschweig, 1991.         [ Links ]

[13] G. Latouche and V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling. Society for Industrial and Applied Mathematics, 1999.         [ Links ]

[14] M.F. Neuts, Matrix-geometric solutions in stochastic models. An Algorithmic Approach. The John Hopkins University Press, Baltimore, MD, 1981.         [ Links ]

[15] M.F. Neuts, Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, New York, 1989.         [ Links ]

[16] Y.V. Pan, Solving a polynomial equation : Some history and recent progress. Siam Rev., Vol. 39, No. 2, pp. 187 -220, June 1997.         [ Links ]

[17] K. Prachar, Primzahlverteilung., Berlin, Heidelberg, New York, Springer Verlag, 1978.         [ Links ]

[18] W.J. Stewart, On the use of numerical methods for ATM model. Modeling and performance evaluation of ATM technology, edited by Perros, H.; Pujolle, G.; Takahashi, p. 375 - 396.         [ Links ]

[19] D.A. Wolfram, Solving generalized Fibonacci recurrences. The Fibonacci Quarterly 36.2. May 1998, 129-45.         [ Links ]

[20] S.J. Yan and M.F. Chen, Multidimensional Q-processes. Chin. Ann. Math. Ser. A,7:90-110, 1986        [ Links ]

[21] J. Ye and S.Q. Li, Folding algorithm: A computational method for finite QBD processes with level dependent transitions. IEEE Transactions on Communications., 42:625-639, 1994.         [ Links ]

[22] J.K. Zhang, Generalized birth-death processes. Acta Mathematica Sinica, Vol. 46, 1984, 241 - 259.         [ Links ]

[23] Y.H. Zhang, Strong ergodicity for single-birth processes. Journal of Applied Probability, Vol. 38, 2001, 207 - 277         [ Links ]

Received: July 2008.

Revised: April 2009.

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