Abate, J. and Whitt, W. (1998). Computing Laplace transforms for numerical inversion via continued fractions. AT&T Labs, Florham Park.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... based analyses of specials processes are to be found in [18, 19, 35] for the distribution of transient characteristics in an M=M=1 system and in [20] for the tail behaviour of the cumulative waiting time in a busy period of an M=M=1 queue; see also the arguments developed by Abate and Whitt in [1, 2] in combination with their Laplace transform inversion algorithm specified in [3] The main objective of this paper as regards birth and death processes is to separate clearly the formal apparatus from the analytic probabilistic machinery, and neatly delineate parameters that are amenable to a ....

.... Delta generates symbolically all the sequences with components f . Three obvious combinatorial decompositions of paths then suffice to derive all the basic formulae. Arch decomposition: An excursion from and to level 0 consists of a sequence of arches , each made of either a c 0 or a a 0 H [1] 1;1 b 1 , so that H 0;0 = 1 Gamma c 0 Gamma a 0 H [1] 1;1 b 1 ) Gamma1 : COMBINATORICS OF BIRTH AND DEATH PROCESSES 7 which relativizes to height h: in general, one has the recursion 8 : H [j; h] j;j = i 1 Gamma c j Gamma a j H [j 1; h] j 1;j 1 b j 1 j Gamma1 ; H ....

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J. Abate and W. Whitt. Computing Laplace transforms for numerical inversion via continued fractions. INFORMS Journal on Computing, 7(1):36--43, 1995. To appear.

....1.4. For all p 0 and q 0, v(p, q; s) # 1 0 (1 sy) 1 b(p, q; y)dy = 2 F 1 (1, p; p q; s) 1.19) where 2 F 1 (a, b; c; z) is the Gauss hypergeometric function. It turns out that Theorem 1. 4 is very useful for computing BME Laplace transforms via continued fractions; see [8]. We can then apply Theorem 1.4 to obtain the following symmetry result for BME pdf s. Exploiting the Gauss hypergeometric function, we can also obtain this next result by an application of the Pfa# reflection law; see (5.101) on p. 217 of Graham, Knuth and Patashnik [22] or 15.3.4 of AS. ....

....transformation of the series in (2.6) yields v(p, q; s) 1 1 s # # n=0 # s 1 s # n (q) n (p q) n = 1 1 s v # q, p; s 1 s # . 2.15) Note that, for any s with Re(s) 0, s (1 s) 1, so that the series in (2.15) converges geometrically fast. See [7] and [8] for further discussion about how to compute the Laplace transforms. Theorem 2.2 provides an e#ective way to compute the pdf s and ccdf s for any p and q. As shown by Abate, Choudhury and Whitt [10] these Laguerre series can be di#cult to compute directly, but there are e#ective ways to enhance ....

Abate, J. and Whitt, W. (1998). Computing Laplace transforms for numerical inversion via continued fractions. AT&T Labs, Florham Park.

....For all p 0 and q 0, v(p; q; s) Z 1 0 (1 sy) Gamma1 b(p; q; y)dy = 2 F 1 (1; p; p q; Gammas) 1.19) where 2 F 1 (a; b; c; z) is the Gauss hypergeometric function. It turns out that Theorem 1. 4 is very useful for computing BME Laplace transforms via continued fractions; see [8]. We can then apply Theorem 1.4 to obtain the following symmetry result for BME pdf s. Exploiting the Gauss hypergeometric function, we can also obtain this next result by an application of the Pfaff reflection law; see (5.101) on p. 217 of Graham, Knuth and Patashnik [22] or 15.3.4 of AS. ....

....transformation of the series in (2.6) yields v(p; q; s) 1 1 s 1 X n=0 s 1 s n (q) n (p q) n = 1 1 s v q; p; Gammas 1 s : 2.15) Note that, for any s with Re(s) 0, js= 1 s)j 1, so that the series in (2.15) converges geometrically fast. See [7] and [8] for further discussion about how to compute the Laplace transforms. Theorem 2.2 provides an effective way to compute the pdf s and ccdf s for any p and q. As shown by Abate, Choudhury and Whitt [10] these Laguerre series can be difficult to compute directly, but there are effective ways to ....

Abate, J. and Whitt, W. (1998). Computing Laplace transforms for numerical inversion via continued fractions. AT&T Labs, Florham Park.

....process with the indicated rates in the limit as ffl 0. For example, we can apply Theorem 6.2 to calculate first passage time distributions from one level to another for the long jobs in the long time scale. We can exploit numerical transform inversion with Section 5 of Abate and Whitt [2] to obtain explicit numerical results. We can also describe the steady state behavior, for which we have insensitivity to the long job service requirement cdf. The insensitivity follows from Theorems 6.1 and 6.2 by the same argument used in Theorem 2.1. Theorem 6.3. Let Y ffl be a long job ....

J. Abate and W. Whitt, Computing Laplace transforms for numerical inversion via continued fractions. AT&T Labs, Florham Park, 1998. Submitted to INFORMS J. Computing.

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