Abstract
Birth death processes with rational birth and death rates have been studied by Maki (1976). In this paper a fluid queue driven by a birth death process with infinite state space and absorption is studied where the birth and death rates are rational functions of linear polynomials. We obtain an explicit transient solution for the fluid queue model using continued fraction approach to solve the underlying system of partial differential equations. For specific value of the parameter the considered model reduces to the model discussed by Parthasarathy, Sericola and Vijayashree (2005) and the results coincide. We also analyze the behavior of the fluid queue in a long run.
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References
Karlin, S.: A First Course in Stochastic Processes. Academic Press, New York (1969)
Maki, D.P.: On birth death processes with rational rates. SIAM J. Math. Anal. 7, 29–36 (1976)
Parthasarathy, P.R., Sericola, B., Vijayashree, K.V.: Transient analysis of a fluid queue driven by a birth and death process suggested by a chain sequence. J. Appl. Math. Stoch. Anal. 2005(2), 143–158 (2005)
Parthasarathy, P.R., Sericola, B., Vijayashree, K.V.: Exact transient solution of an M/M/1 driven fluid queue. Int. J. Comput. Math. 82 (6), 1–13 (2005)
Sericola, B.: Transient analysis of stochastic fluid models. Perf. Eval. 32, 245–263 (1998)
Sericola, B.: A finite buffer fluid queue driven by a Markovian queue. Queueing Syst. 38, 213–220 (2001)
Sericola, B., Guillemin, F.: Transient analysis of a fluid buffer driven by a birth and death process. In: 19th International Teletraffic Congress (ITC’19), Beijing (2005)
Sericola, B., Tuffin, B.: A fluid queue driven by a Markovian queue. Queueing Syst. 31, 253–264 (1999)
Anick, D., Mitra, D., Sondhi, M.: Stochastic theory of data-handling system with multiple sources. Bell Syst. Tech. J. 61, 1871–1894 (1977)
Ren, Q., Kobayashi, H.: Transient solutions for the buffer behavior in statistical multiplexing. Perform. Eval. 21, 65–87 (1995)
Mao, G., Habibi, D.: Heterogeneous on-off sources in a bufferless fluid flow model. In: IFFF International Conference on Networks, pp. 307–317 (2000)
Tanaka, T., Yashida, O., Takahashi, Y.: Transient analysis of fluid models for ATM statistical multiplexer. Perform. Eval. 23, 145–162 (1995)
Parthasarathy, P.R., Vijayashree, K.V., Lenin, R.B.: An M/M/1 driven fluid queue-continued fraction approach. Queuing Syst. Theory Appl. 42, 189–199 (2002)
van Doorn, E.A., Scheinhardt, W.R.W.: A fluid queue driven by an infinite state birth- death process. In: Proceedings of the ITC 15, Teletraffic Contributions for the Information Age. Elsevier, pp. 465–475 (1997)
Lenin, R.B., Parthasarathy, P.R.: A birth-death process suggested by a chain sequence. J. Comput. Math. Appl. 40 (2–3), 239–247 (2000)
Arunachalam, V., Vandana Gupta, S., Dharmaraja, A.: Fluid queue modulated by two independent birth death processes. J. Comput. Math. Appl. 60, 2433–2444 (2010)
Parthasarathy, P.R., Lenin, R.B.: Fluid queues driven by an M/M/1/N queue. Math. Probl. Eng. 6, 439–460 (2000)
Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. Elsevier, Netherlands (1992)
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One of the authors (SD) would like to thank the National Board for Higher Mathematics, India, for financial support given to them during the preparation of the paper.
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Kapoor, S., Selvamuthu, D. On the exact transient solution of fluid queue driven by a birth death process with specific rational rates and absorption. OPSEARCH 52, 746–755 (2015). https://doi.org/10.1007/s12597-015-0199-4
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DOI: https://doi.org/10.1007/s12597-015-0199-4