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An Introduction to Numerical Transform Inversion and Its Application to Probability Models

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Computational Probability

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 24))

Abstract

Numerical transform inversion has an odd place in computational probability. Historically, transforms were exploited extensively for solving queueing and related probability models, but only rarely was numerical inversion attempted. The model descriptions were usually left in the form of transforms. Vivid examples are the queueing books by Takács [Takács, 1962] and Cohen [Cohen, 1982]. When possible, probability distributions were calculated analytically by inverting transforms, e.g., by using tables of transform pairs. Also, moments of probability distributions were computed analytically by differentiating the transforms and, occasionally, approximations were developed by applying asymptotic methods to transforms, but only rarely did anyone try to compute probability distributions by numerically inverting the available transforms. However, there were exceptions, such as the early paper by Gaver [Gaver, 1966]. (For more on the history of numerical transform inversion, see our earlier survey [Abate and Whitt, 1992a].) Hence, in the application of probability models to engineering, transforms became regarded more as mathematical toys than practical tools. Indeed, the conventional wisdom was that numerical transform inversion was very difficult. Even numerical analysts were often doubtful of the numerical stability of inversion algorithms. In queueing, both theorists and practitioners lamented about the “Laplace curtain” obscuring our understanding of system behavior.

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References

  1. Abate, J., Choudhury, G. L., Lucantoni, D. M., and Whitt, W. (1995a). Asymptotic analysis of tail probabilities based on the computation of moments. Ann. Appl. Prob., 5: 983–1007.

    Article  Google Scholar 

  2. Abate, J., Choudhury, G. L., and Whitt, W. (1993). Calculation of the GI/G/1 waiting-time distribution and its cumulants from Pollaczek’s formulas. Archiv fir Elektronik and Übertragungstechnik, 47: 31 1321.

    Google Scholar 

  3. Abate, J., Choudhury, G. L., and Whitt, W. (1994). Waiting-time tail probabilities in queues with long-tail service-time distributions. Queueing Systems, 16: 311–338.

    Article  Google Scholar 

  4. Abate, J., Choudhury, G. L., and Whitt, W. (1995b). Exponential approximations for tail probabilities in queues, I: waiting times. Oper. Res., 43: 885–901.

    Article  Google Scholar 

  5. Abate, J., Choudhury, G. L., and Whitt, W. (1996). On the Laguerre method for numerically inverting Laplace transforms. INFORMS Journal on Computing, 8: 413–427.

    Article  Google Scholar 

  6. Abate, J., Choudhury, G. L., and Whitt, W. (1997). Numerical inversion of multidimensional Laplace transforms by the Laguerre method. Performance Evaluation, 31: 229–243.

    Article  Google Scholar 

  7. Abate, J. and Whitt, W. (1992a). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems, 10: 5–88.

    Article  Google Scholar 

  8. Abate, J. and Whitt, W. (1992b). Numerical inver- sion of probability generating functions. Oper. Res. Letters, 12: 245–251.

    Article  Google Scholar 

  9. Abate, J. and Whitt, W. (1992c). Solving probability transform functional equations for numerical inversion. Oper. Res. Letters, 12: 275–281.

    Article  Google Scholar 

  10. Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. on Computing, 7: 36–43.

    Article  Google Scholar 

  11. Abate, J. and Whitt, W. (1998). Calculating transient characteristics of the Erlang loss model by numerical transform inversion. Stochastic Models, 14.

    Google Scholar 

  12. Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.

    Google Scholar 

  13. Asmussen, S., Nerman, O., and Olsson, M. (1996). Fitting phase type distributions via the EM algorithm. Scand. J. Statist., 23: 419–441.

    Google Scholar 

  14. Bene, V. E. (1961). The covariance function of a simple trunk group with applications to traffic measurements. Bell System Tech. J., 40: 117–148.

    Google Scholar 

  15. Benes, V. E. (1965). Mathematical Theory of Connecting Networks and Telephone Traffic. Academic Press, New York.

    Google Scholar 

  16. Bertozzi, A. and McKenna, J. (1993). Multidimensional residues, generating functions, and their application to queueing networks. SIAM Review, 35: 239–268.

    Article  Google Scholar 

  17. Borovkov, A. A. (1984). Asymptotics Methods in Queueing Theory. Wiley, New York.

    Google Scholar 

  18. Chaudhry, M. L., Agarwal, M., and Templeton, J. G. C. (1992). Exact and approximate numerical solutions of steady-state distributions arising in the GI/G/1 queue. Queueing Systems, 10: 105–152.

    Article  Google Scholar 

  19. Choudhury, G. L., Leung, K. K., and Whitt, W. (1995a). An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models. Adv. Appl. Prob., 27: 1104–1143.

    Article  Google Scholar 

  20. Choudhury, G. L., Leung, K. K., and Whitt, W. (1995b). Calculating normalization constants of closed queueing networks by numerically inverting their generating functions. J. ACM, 42: 935–970.

    Article  Google Scholar 

  21. Choudhury, G. L., Leung, K. K., and Whitt, W. (1995c). Efficiently providing multiple grades of service with protection against overloads in shared resources. ATandT Tech. J., 74: 50–63.

    Google Scholar 

  22. Choudhury, G. L., Leung, K. K., and Whitt, W. (1995d). An inversion algorithm to compute blocking probabilities in loss networks with state-dependent rates. IEEE/ACM Trans. Networking, 3: 585601

    Google Scholar 

  23. Choudhury, G. L. and Lucantoni, D. M. (1996). Numerical computation of the moments of a probability distribution from its transform. Oper. Res., 44: 368–381.

    Article  Google Scholar 

  24. Choudhury, G. L., Lucantoni, D. M., and Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Prob., 4: 719–740.

    Article  Google Scholar 

  25. Choudhury, G. L., Lucantoni, D. M., and Whitt, W. (1996). Squeezing the most out of ATM. IEEE Trans. Commun., 44: 203–217.

    Article  Google Scholar 

  26. Choudhury, G. L., Lucantoni, D. M., and Whitt, W. (1997). Numerical solution of Mt/Gt/1 queues. Oper. Res., 45: 451–463.

    Article  Google Scholar 

  27. Choudhury and Whitt, 1995] Choudhury, G. L. and Whitt, W. (1995). Q2: A new performance analysis tool exploiting numerical transform inversion. In Proc. Third Int. Workshop on Modeling, Analysis and Simul. of Computer and Telecomm. Systems,pages 411–415, Durham, NC. (MASCOTS ’85).

    Google Scholar 

  28. Choudhury, G. L. and Whitt, W. (1996). Computing distributions and moments in polling models by numerical transform inversion. Performance Evaluation, 25: 267–292.

    Article  Google Scholar 

  29. Choudhury, G. L. and Whitt, W. (1997). Probabilistic scaling for the numerical inversion of non-probability transforms. INFORMS J. Computing, 9: 175–184.

    Article  Google Scholar 

  30. Cohen, J. W. (1982). The Single Server Queue. North-Holland, Amsterdam, second edition.

    Google Scholar 

  31. Conway, A. E. and Georganas, N. D. (1989). Queueing Networks — Exact Computational Algorithms: A Unified Theory Based on Decomposition and Aggregation. MIT Press, Cambridge, MA.

    Google Scholar 

  32. Darling, D. A. and Siegert, J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist., 24: 624–639.

    Article  Google Scholar 

  33. Davies, B. and Martin, B. L. (1979). Numerical inversion of the Laplace transform: A survey and comparison of methods. J. Comp. Phys., 33: 1–32.

    Article  Google Scholar 

  34. Davis, P. J. and Rabinowitz, P. (1984). Methods of Numerical Integration. Academic Press, New York, second edition.

    Google Scholar 

  35. Doetsch, G. (1961). Guide to Applications of Laplace Transforms. Van Nostrand, London.

    Google Scholar 

  36. Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer-Verlag, New York.

    Book  Google Scholar 

  37. Dubner, H. and Abate, J. (1968). Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J. ACM, 15: 115–123.

    Article  Google Scholar 

  38. Duffield, N. G. and Whitt, W. (1998). A source traffic model and its transient analysis for network control. Stochastic Models, 14.

    Google Scholar 

  39. Durbin, F. (1974). Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. Comput. J., 17: 371–376.

    Article  Google Scholar 

  40. Feldmann, A. and Whitt, W. (1997). Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation, 31: 245–279.

    Article  Google Scholar 

  41. Feller, W. (1971). An Introduction to Probability Theory and its Applications, volume II. Wiley, New York, second edition.

    Google Scholar 

  42. Gaver, D. P. (1966). Observing stochastic processes and approximate transform inversion. Operations Research, 14: 444–459.

    Article  Google Scholar 

  43. Gaver, D. P. and Jacobs, P. A. (1998). Waiting times when service times are stable laws: tamed and wild. In Shanthikumar, J. G. and (eds.), U. S., editors, Recent Contributions in Applied Probability and Stochastic Processes, Festschrift for Julian Keilson. Kluwer, Boston.

    Google Scholar 

  44. Giffin, W. C. (1975). Transform Techniques for Probability Modeling. Academic Press, New York.

    Google Scholar 

  45. Hosono, T. (1979). Numerical inversion of Laplace transform. J. Inst. Elec. Eng. Jpn., pages A54 - A64. 494 (In Japanese).

    Google Scholar 

  46. Hosono, T. (1981). Numerical inversion of Laplace transform and some applications to wave optics. Radio Sci., 16: 1015–1019.

    Article  Google Scholar 

  47. Hosono, 1984] Hosono, T. (1984). Fast Inversion of Laplace Transform by BASIC. Kyoritsu Publishers, Japan. (In Japanese).

    Google Scholar 

  48. Jagerman, D. L. (1974). Some properties of the Erlang loss function. Bell System Tech. J., 53: 525–551.

    Google Scholar 

  49. Jagerman, D. L. (1978). An inversion technique for the Laplace transform with applications. Bell System Tech. J., 57: 669–710.

    Google Scholar 

  50. Jagerman, D. L. (1982). An inversion technique for the Laplace transform. Bell Sys. Tech. J., 61: 1995–2002.

    Google Scholar 

  51. Johnsonbaugh, R. (1979). Summing an alternating series. Amer. Math. Monthly, 86: 637–648.

    Article  Google Scholar 

  52. Kao, E. P. C. (1997). An Introduction to Stochastic Processes. Duxbury Press, New York.

    Google Scholar 

  53. Keilson, J. (1979). Markov Chain Models — Rarity and Exponentiality. Springer-Verlag, New York.

    Book  Google Scholar 

  54. Keilson, J. and Ross, H. F. (1975). Passage time distributions for Gaussian Markov (Ornstein-Uhlenbech) statistical processes. Selected Tables in Mathematical Statistics, 3: 233–327.

    Google Scholar 

  55. Kingman, J. F. C. (1961). The single server queue in heavy traffic. Proc. Camb. Phil. Soc., 57: 902–904.

    Article  Google Scholar 

  56. Kleinrock, L. (1975). Queueing Systems, Volume I: Theory. Wiley, New York.

    Google Scholar 

  57. Knessl, C. (1990). On the transient behavior of the M/M/m/m loss model. Stochastic Models, 6: 749–776.

    Article  Google Scholar 

  58. Kobayashi, H. (1978). Modeling and Analysis: An Introduction to System Performance Evaluation Methodology. Addison-Wesley, Reading, MA.

    Google Scholar 

  59. Kwok, Y. K. and Barthez, D. (1989). An algorithm for the numerical inversion of the Laplace transform. Inverse Problems, 5: 1089–1095.

    Article  Google Scholar 

  60. Lam, S. S. and Lien, Y. L. (1983). A tree convolution algorithm for the solution of queueing networks. Commun. ACM, 26: 203215.

    Google Scholar 

  61. Lavenberg, S. S., editor (1983). Computer Performance Modeling Handbook. Academic Press, Orlando, FL.

    Google Scholar 

  62. Lucantoni, D. M., Choudhury, G. L., and Whitt, W. (1994). The transient BMAP/G/1 queue. Stochastic Models, 10: 145–182.

    Article  Google Scholar 

  63. Mitra, D. and Weiss, A. (1989). The transient behavior in Erlang’s model for large trunk groups and various traffic conditions. In Teletraffic Science for new cost-Effective Systems, Networks and Services, pages 1367–1374, Amsterdam. Elsevier-Science.

    Google Scholar 

  64. Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore.

    Google Scholar 

  65. Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.

    Google Scholar 

  66. O’Cinneide, C. A. (1997). Euler summation for Fourier series and Laplace transform inversion. Stochastic Models, to appear.

    Google Scholar 

  67. Pol, B. V. D. and Bremmer, H. (1987). Operational Calculus. Cambridge Press, reprinted Chelsea Press, New York.

    Google Scholar 

  68. Pollaczek, F. (1952). Fonctions caractéristiques de certaines répartitions définies au money de la notion d’ordre. Application à la théorie de attentes. C. R. Acad. Sci. Paris, 234: 2334–2336.

    Google Scholar 

  69. Pollaczek, F. (1965). Concerning an analytic method for the treatment of queueing problems. In Smith, W. L. and Wilkinson, W. E., editors, Proceedings of the Symposium on Congestion Theory, pages 1–25 and 34–42, Chapel Hill. The University of North Carolina Press.

    Google Scholar 

  70. Poularikas, A. D. (1996). The Transforms and Applications Handbook. CRC Press, Boca Raton, FL.

    Google Scholar 

  71. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1988). Numerical Recipes, FORTRAN Version. Cambridge University Press, Cambridge, England.

    Google Scholar 

  72. Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York.

    Google Scholar 

  73. Reiser, M. and Kobayashi, H. (1975). Queueing networks with multiple closed chains: theory and computational algorithms. IBM J. Res. Dev., 19: 283–294.

    Article  Google Scholar 

  74. Riordan, J. (1962). Stochastic Service Systems. Wiley, New York.

    Google Scholar 

  75. Simon, R. M., Stroot, M. T., and Weiss, G. H. (1972). Numerical inversion of Laplace transforms with applications to percentage labeled experiments. Comput. Biomed. Res., 6: 596–607.

    Article  Google Scholar 

  76. Smith, W. L. (1953). On the distribution of queueing times. Proc. Camb. Phil. Soc., 49: 449–461.

    Article  Google Scholar 

  77. Srikant, R. and Whitt, W. (1996). Simulation run lengths to estimate blocking probabilities. ACM J. TOMACS, 6: 7–52.

    Article  Google Scholar 

  78. Takâcs, L. (1962). Introduction to the Theory of Queues. Oxford University Press, New York.

    Google Scholar 

  79. Tolstov, G. P. (1976). Fourier Series. Dover, New York.

    Google Scholar 

  80. Weeks, W. T. (1966). Numerical inversion of Laplace transforms using Laguerre functions. J. ACM, 13: 419–426.

    Article  Google Scholar 

  81. Whitt, W. (1984). Heavy-traffic approximations for service systems with blocking. ATandT Bell Lab. Tech. J., 63: 689–708.

    Google Scholar 

  82. Wimp, J. (1981). Sequence Transformations and Their Applications. Academic Press, New York.

    Google Scholar 

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Authors and Affiliations

  1. 900 Hammond Road, Ridgewood, NJ, 07450-2908, USA

    Joseph Abate

  2. AT&T Labs, Room 1L-238 Holmdel, NJ, 07733-3030, USA

    Gagan L. Choudhury

  3. AT&T Labs, Room A117, 180 Park Avenue, Florham Park, NJ, 07932-0971, USA

    Ward Whitt

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  1. Joseph Abate
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  2. Gagan L. Choudhury
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  1. University of Saskatchewan, Canada

    Winfried K. Grassmann

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Abate, J., Choudhury, G.L., Whitt, W. (2000). An Introduction to Numerical Transform Inversion and Its Application to Probability Models. In: Grassmann, W.K. (eds) Computational Probability. International Series in Operations Research & Management Science, vol 24. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4828-4_8

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  • DOI: https://doi.org/10.1007/978-1-4757-4828-4_8

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