Abstract
For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahn, S., & Ramaswami, V. (2003). Fluid flow models & queues—a connection by stochastic coupling. Stochastic Models, 19(3), 325–348.
Ahn, S., & Ramaswami, V. (2004). Transient analysis of fluid flow models via stochastic coupling to a queue. Stochastic Models, 20(1), 71–101.
Ahn, S., & Ramaswami, V. (2005). Efficient algorithms for transient analysis of stochastic fluid flow models. Journal of Applied Probability, 42(2), 531–549.
Ahn, S., & Ramaswami, V. (2006). Transient analysis of fluid models via elementary level crossing arguments. Stochastic Models, 22(1), 129–147.
Ahn, S., Badescu, A. L., & Ramaswami, V. (2007). Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. Queueing Systems: Theory and Applications, 55(4), 207–222.
Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stochastic Models, 11, 1–20.
Asmussen, S., Avram, F., & Usábel, M. (2002). Erlangian approximations for finite time ruin probabilities. ASTIN Bulletin, 32, 267–281.
Badescu, A., Breuer, L., Drekic, S., Latouche, G., & Stanford, D. A. (2005). The surplus prior to ruin and the deficit at ruin for a correlated risk process. Scandinavian Actuarial Journal, 6, 433–445.
Bean, N. G., O’Reilly, M., & Taylor, P. G. (2005). Hitting probabilities and hitting times for stochastic fluid flows. Stochastic Processes and Their Applications, 115(9), 1530–1556.
da Silva Soares, A., & Latouche, G. (2002). Further results on the similarity between fluid queues and QBDs. In G. Latouche & P. Taylor (Eds.) Proceedings of the 4th international conference on matrix-analytic methods (pp. 89–106). River Edge: World Scientific.
Graham, A. (1981). Kronecker products and matrix calculus with applications. New York: Wiley.
Hirchman, I. I., & Widder, D. V. (1955). The convolution transform. Princeton: Princeton University Press.
Jagerman, D. L. (1978). An inversion technique for the Laplace transform with application to approximation. Bell System Technical Journal, 57(3), 669–710.
Jagerman, D. L. (1982). An inversion technique for the Laplace transform. Bell System Technical Journal, 61(8), 1995–2002.
Latouche, G., & Ramaswami, V. (1993). A logarithmic reduction algorithm for quasi-birth-death processes. Journal of Applied Probability, 30(3), 650–674.
Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. Philadelphia: SIAM and ASA.
Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models. Baltimore: John Hopkins University Press.
Ramaswami, V. (1999). Matrix analytic methods for stochastic fluid flows. In D. Smith & P. Hey (Eds.) Teletraffic engineering in a competitive world (Proceedings of the 16th international teletraffic congress) (pp. 1019–1030). Edinburgh: Elsevier Science B.V.
Ramaswami, V. (2006). Passage times in fluid models with application to risk processes. Methodology and Computing in Applied Probability, 8, 497–515.
Stanford, D. A., Avram, F., Badescu, A., Breuer, L., da Silva Soares, A., & Latouche, G. (2005a). Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models. ASTIN Bulletin, 35(1), 131–144.
Stanford, D. A., Latouche, G., Woolford, D. G., Boychuk, D., & Hunchak, A. (2005b). Erlangized fluid queues with application to uncontrolled fire perimeter. Stochastic Models, 21(23), 631–642.
Woolford, D. G. (2007). Stochastic models applied to lightning and wildfires in Ontario. Ph.D. thesis, University of Western Ontario, London, Ontario, Canada.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ramaswami, V., Woolford, D.G. & Stanford, D.A. The erlangization method for Markovian fluid flows. Ann Oper Res 160, 215–225 (2008). https://doi.org/10.1007/s10479-008-0309-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-008-0309-2