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An operational calculus for probability distributions via Laplace transforms

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Joseph Abate  and
Ward Whitt
Joseph Abate*
Affiliation:
AT&T Bell Laboratories
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
* Postal address: 900 Hammond Rd., Ridgewood, NJ 07450-2908, USA.
** Postal address: AT&T Bell Laboratories, Room 2C-178, Murray Hill, NJ 07974-0636, USA.
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Abstract

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.

Keywords

UNIMODAL DISTRIBUTIONSINFINITELY DIVISIBLE DISTRIBUTIONSLEVY PROCESSESCOMPLETE MONOTONICITYCUMULANTSMOMENTSRANDOM SUMSINVERSE GAUSSIAN DISTRIBUTIONSRENEWAL PROCESSESSUBORDINATIONFIRST-PASSAGE TIMESBESSEL FUNCTIONSTHETA FUNCTIONSMIMI1 QUEUERANDOMIZED RANDOM WALKBROWNIAN MOTIONPOLLACZEK-KHINTCHINE FORMULA

MSC classification

Primary: 60E10: Characteristic functions; other transforms
Secondary: 60E07: Infinitely divisible distributions; stable distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 75 - 113
Copyright
Copyright © Applied Probability Trust 1996 

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References

Abate, J. and Whitt, A. (1987) Transient behavior of regulated Brownian motion I: starting at the origin. Adv. Appl. Prob. 19, 560–598.Google Scholar
Abate, J. and Whitt, W. (1988a) Transient behavior of the M/M/1 queue via Laplace transforms. Adv. Appl. Prob. 20, 145–178.Google Scholar
Abate, J. and Whitt, W. (1988b) Simple spectral representations for the M/M/1 queue. Queueing Systems 3, 321–346.CrossRefGoogle Scholar
Abate, J. and Whitt, W. (1988c) Approximations for the M/M/1 busy period. In Queueing Theory and its Applications, Liber Amicorum for J. W. Cohen, ed. Boxma, O. J. and Syski, R., pp. 149–191. North-Holland, Amsterdam.Google Scholar
Abate, J. and Whitt, W. (1988d) The correlation functions of RBM and M/M/1. Stoch. Models 4, 315–359.Google Scholar
Abate, J. and Whitt, W. (1992) The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–88.CrossRefGoogle Scholar
Abate, J. and Whitt, W. (1995) Numerical inversion of Laplace transforms of probability distributions. ORSA J. Computing 7, 36–43.Google Scholar
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. AEÜ 47, 311–321.Google Scholar
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.Google Scholar
Abate, J., Choudhury, G. L. and Whitt, W. (1995) Exponential approximations for tail probabilities in queues, I: waiting times. Operat. Res. 43, 885–901.Google Scholar
Abate, J., Choudhury, G. L., Lucantoni, D. M. and Whitt, W. (1995) Asymptotic analysis of tail probabilities based on the computation of moments. Ann. Appl. Prob. 5, to appear.Google Scholar
Abate, J., Kijima, M. and Whitt, W. (1991) Decompositions of the M/M/1 transition function. Queueing Systems 9, 323–336.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. (1972) Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. Google Scholar
Ackroyd, M. H. (1980) Computing the waiting time distribution for the G/G/1 queue by signal processing methods. IEEE Trans. Commun. 28, 52–58.Google Scholar
Asmussen, S. (1992) Phase-type representations in random walk and queueing problems. Ann. Prob. 20, 772–789.Google Scholar
Bellman, R. (1961) A Brief Introduction to Theta Functions. Holt Rinehart, New York.Google Scholar
Bernstein, S. N. (1928) Sur les fonctions absolument monotones. Acta Math. 51, 1–66.Google Scholar
Bingham, N. H. (1975) Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Bondesson, L. (1988) T1- and T2-classes of distributions. Encyclopedia of Statistical Sciences, ed. Johnson, N. L. and Kotz, S., 9, 157–159.Google Scholar
Bondesson, L. (1992) Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Springer, New York.Google Scholar
Choudhury, G. L. and Lucantoni, D. M. (1995) Numerical computation of the moments of a probability distribution from its transform. Operat. Res. To appear.Google Scholar
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. 7, 719–740.Google Scholar
Choudhury, G. L., Lucantoni, D. M. and Whitt, W. (1996) Numerical solution of Mt/Gt/1 queues. Operat. Res. To appear.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes, Wiley, New York.Google Scholar
Dharmadhikari, S. and Joag-Dev, K. (1988) Unimodality, Convexity and Applications. Academic Press, Boston.Google Scholar
Embrechts, P. (1983) A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Prob. 20, 537–544.Google Scholar
Embrechts, P. and Klüppelberg, C. (1993) Some aspects of insurance mathematics. Dept. of Mathematics, ETH-Zentrum, Zürich.Google Scholar
Embrechts, P. and Villasenor, J. A. (1988) Ruin estimates for large claims. Insurance: Math. and Econ. 7, 269–274.Google Scholar
Feller, W. (1966) Infinitely divisible distributions and Bessel functions associated with random walks. SIAM J. Appl. Math. 14, 864–875.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
Gaver, D. P. Jr. (1954) The influence of service times in queueing processes. Operat. Res. 2, 139–149.Google Scholar
Gaver, D. P. Jr. and Jacobs, P. A. (1988) Nonparametric estimation of the probability of a long delay in the M/G/1 queue. J. R. Statist. Soc. B 50, 392–402.Google Scholar
Giffin, W. C. (1975) Transform Techniques for Probability Modeling. Academic Press, New York.Google Scholar
Graham, R. L., Knuth, D. E. and Patashnik, O. (1989) Concrete Mathematics. Addison-Wesley, Reading, MA.Google Scholar
Grübel, R. (1991) Algorithm AS265: G/G/1 via fast Fourier transform. J. R. Statist. Soc. C 40, 355–365.Google Scholar
Grübel, R. and Pitts, S. M. (1992) A functional approach to the stationary waiting-time and idle period distributions of the GI/G/1 queue. Ann. Prob. 20, 1754–1778.Google Scholar
Heyman, D. P. (1974) An approximation for the busy period of the M/G/1 queue using a diffusion model. J. Appl. Prob. 11, 159–169.Google Scholar
Johnson, N. L. and Kotz, S. (1970a) Continuous Univariate Distributions-1. Wiley, New York.Google Scholar
Johnson, N. L. and Kotz, S. (1970b) Continuous Univariate Distributions-2. Houghton Miffin, Boston.Google Scholar
Jorgensen, B. (1982) Statistical Properties of the Generalized Inverse Gaussian Distributions. Lecture Notes in Statistics 9. Springer, New York.CrossRefGoogle Scholar
Keener, R. W. (1994) Quadrature routines for ladder variables. Ann. Appl. Prob. 4, 570–590.Google Scholar
Kella, O. and Whitt, W. (1991) Queues with server vacations and Lévy processes with secondary jump input. Ann. Appl. Prob. 1, 104–117.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992) Useful martingales for stochastic processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
Kendall, M. G. and Stuart, A. (1987) The Advanced Theory of Statistics, Vol. 1, 5th edn. Oxford University Press, New York.Google Scholar
Keilson, J. (1979) Markov Chain Models–Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Khintchine, A. Y. (1938) On unimodal distributions. Izv. Nauchno-Isled. Inst. Mat. Mech. Tomsk. Gos. Univ. 2, 1–7.Google Scholar
Konheim, A. G. (1975) An elementary solution of the queueing system G/G/1. SIAM J. Comput. 4, 540–545.Google Scholar
Lucantoni, D. M., Choudhury, G. L. and Whitt, W. (1994) The transient BMAP/G/1 queue. Stoch. Models 10, 145–182.Google Scholar
Lukacs, E. (1970) Characteristic Functions, 2nd edn. Hafner, New York.Google Scholar
Moran, P. A. P. (1959) The Theory of Storage. Methuen, London.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore.Google Scholar
Oberhettinger, F. and Badii, L. (1973) Tables of Laplace Transforms. Springer-Verlag, New York.Google Scholar
Odlyzko, A. M. (1993) Asymptotic enumeration methods. In Handbook of Combinatorics, ed. Graham, R. L., Gröschel, M. and Lovasz, L. Elsevier, Amsterdam.Google Scholar
Prabhu, N. U. (1980) Stochastic Storage Processes. Springer, New York.Google Scholar
Riordan, J. (1958) Introduction to Combinatorial Analysis. Wiley, New York.Google Scholar
Riordan, J. (1968) Combinatorial Identities. Wiley, New York.Google Scholar
Shepp, L. A. (1962) Symmetric random walk. Trans. Amer. Math. Soc. 104, 144–153.Google Scholar
Steutel, F. W. (1973) Some recent results in infinite divisibility. Stoch. Proc. Appl. 1, 125–143.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Thorin, O. (1977a) On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. 60, 31–40.Google Scholar
Thorin, O. (1977b) On the infinite divisibility of the lognormal distribution. Scand. Actuarial J. 60, 121–148.Google Scholar
Whitt, W. (1985) The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar
Wilf, H. S. (1994) Generatingfunctionology. 2nd edn. Academic Press, New York.Google Scholar