## Numerical inversion of probability generating functions (1992)

Venue: | Oper. Res. Letters |

Citations: | 40 - 17 self |

### BibTeX

@ARTICLE{Abate92numericalinversion,

author = {Joseph Abate and Ward Whitt},

title = {Numerical inversion of probability generating functions},

journal = {Oper. Res. Letters},

year = {1992},

volume = {12},

pages = {245--251}

}

### Years of Citing Articles

### OpenURL

### Abstract

Random quantities of interest in operations research models can often be determined conveniently in the form of transforms. Hence, numerical transform inversion can be an effective way to obtain desired numerical values of cumulative distribution functions, probability density functions and probability mass functions. However, numerical transform inversion has not been widely used. This lack of use seems to be due, at least in part, to good simple numerical inversion algorithms not being well known. To help remedy this situation, in this paper we present a version of the Fourier-series method for numerically inverting probability generating functions. We obtain a simple algorithm with a convenient error bound from the discrete Poisson summation formula. The same general approach applies to other transforms. Key Words: numerical inversion of transforms, computational probability, generating functions, Fourier-series method, Poisson summation formula, discrete Fourier transform.

### Citations

1455 |
An Introduction to Probability Theory
- Feller
- 1971
(Show Context)
Citation Context ...nsider the Fourier transform f(u) = k = - S a k e iku , (5) which has an inverse a k = 2p 1 ___ 0 2p f(u) e - iku du , (6) as can easily be verified by substituting (5) into (6); see p. 511 of Feller =-=[5]-=-. With a k = q k r k for ks0 and a k = 0 for ks0, (6) reduces to (3). The error bound for the trapezoidal rule approximation to (6) now follows from the discrete Poisson summation formula. Theorem 2. ... |

172 |
Theory and Application
- RABINER, GOLD
- 1975
(Show Context)
Citation Context ...equence with terms a k p = j = - S a k + jm . (7) (The series in (7) converges absolutely since j = - Ssa js.) Next construct the discrete Fourier transform of {a k p }, see p. 51 of Rabiner and Gold =-=[12]-=-, to obtains- ask p = m 1 ___ j = 0 S m - 1 a j p e i2pkj/m = m 1 ___ j = 0 S m - 1 l = - S a j + lm e i2p jk/m = m 1 ___ j = - S a j e i2p jk/m = m 1 ___ f m 2pk ____ . Finally, from the inversion fo... |

149 | The Fourier-series method for inverting transforms of probability distributions. Queueing Systems Theory Appl
- Abate, Whitt
- 1992
(Show Context)
Citation Context ...algorithm for numerically inverting probability generating functions based on the Fourierseries method. Variants of the same method apply to other transforms, as can be seen from our longer review in =-=[1]-=-. We relate our algorithm to the literature in Remark 1 below; see [1] for further discussion. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by ... |

100 |
Asymptotic Methods in Enumeration
- Bender
- 1974
(Show Context)
Citation Context ... lines 30-34. As indicated in Remark 5 above, the transform can be used to determine the asymptotic behavior of p k and q k as ks. In particular, from (12) and p. 150 of Wilf [14] or p. 498 of Bender =-=[2], we find that -=-p k ~ a ks- "`b r 1 _____ k 3/2 G( - 1/2) b k _____________ = 2""`b rpk 3 b k __________ , (15) where p k ~ a k means that p k / a ks1 as ks, which agrees with what we get from (11) bys... |

34 |
Stochastic Service Systems
- Riordan
- 1962
(Show Context)
Citation Context ..._sk - 1 2k - 2 r k - 1 (1 + r) - 2k + 1 , ks1 , (11) and probability generating function P(z)sk = 0 S p k z k = "`b r 1 - """1 - bz ____________ , (12) where b = 4r / (1 + r) 2 ; s=-=ee p. 65 of Riordan [13]-=-. The tail probabilities q k = p k + 1 + p k + 2 + . . . (13) thus have generating function G(z)sk = 0 S q k z k = 1 - z 1 - P(z) _________ (14) for P(z) in (12). The displayed program in 1 computes q... |

26 |
An inversion technique for the Laplace transform
- Jagerman
- 2002
(Show Context)
Citation Context ...treat the case in which we need not havesq ks1 for all k. Essentially the same algorithm was proposed without error analysis by Cavers [3]. Nearly equivalent algorithms were also proposed by Jagerman =-=[7]-=-, [8] and Hosono [6]. Daigle [4] draws on the same ideas, but his algorithm is more complicated since he considers the special case with r = 1. (2) The algorithm LATTICE-POISSON is by no means the onl... |

9 |
Queue length distributions from probability generating functions via discrete Fourier transforms
- Daigle
- 1989
(Show Context)
Citation Context ...not havesq ks1 for all k. Essentially the same algorithm was proposed without error analysis by Cavers [3]. Nearly equivalent algorithms were also proposed by Jagerman [7], [8] and Hosono [6]. Daigle =-=[4]-=- draws on the same ideas, but his algorithm is more complicated since he considers the special case with r = 1. (2) The algorithm LATTICE-POISSON is by no means the only way to calculate q k from (1).... |

5 |
Differentiation formulas for analytic functions
- Lyness
- 1968
(Show Context)
Citation Context ...orem 1 as new, but it does not seem to be very well known. Indeed, the methods supporting Theorem 1 are classical, but we know of no explicit statement. The essential idea is expressed in 1 of Lyness =-=[10]-=-, but the focus there is on further analysis using the Mo .. bius function to treat the case in which we need not havesq ks1 for all k. Essentially the same algorithm was proposed without error analys... |

5 | UBASIC Version 8.12 (Faculty - Kida - 1990 |

4 |
On the fast Fourier transform inversion of probability generating functions
- Cavers
- 1978
(Show Context)
Citation Context ...us there is on further analysis using the Mo .. bius function to treat the case in which we need not havesq ks1 for all k. Essentially the same algorithm was proposed without error analysis by Cavers =-=[3]-=-. Nearly equivalent algorithms were also proposed by Jagerman [7], [8] and Hosono [6]. Daigle [4] draws on the same ideas, but his algorithm is more complicated since he considers the special case wit... |

4 |
UBASIC: A Public Domain BASIC for Mathematics
- Neumann
- 1989
(Show Context)
Citation Context ...tary cdf of the number of customers served in an M/M/1 busy period. UBASIC is a public-domain high-precision version of BASIC created by Kida [9] to do mathematics on a personal computer; see Neumann =-=[11]-=-. UBASIC permits complex numbers to be specified conveniently and it represents numbers and performs computations with up to 100-decimal-place accuracy. (Diskettes containing UBASIC and the algorithm ... |

3 |
Numerical algorithm for Taylor series expansion
- Hosono
- 1986
(Show Context)
Citation Context ...ich we need not havesq ks1 for all k. Essentially the same algorithm was proposed without error analysis by Cavers [3]. Nearly equivalent algorithms were also proposed by Jagerman [7], [8] and Hosono =-=[6]-=-. Daigle [4] draws on the same ideas, but his algorithm is more complicated since he considers the special case with r = 1. (2) The algorithm LATTICE-POISSON is by no means the only way to calculate q... |

2 |
UBASIC Version 8.12," Faculty
- Kida
- 1990
(Show Context)
Citation Context ...isplay below a UBASIC program to calculates- the complementary cdf of the number of customers served in an M/M/1 busy period. UBASIC is a public-domain high-precision version of BASIC created by Kida =-=[9]-=- to do mathematics on a personal computer; see Neumann [11]. UBASIC permits complex numbers to be specified conveniently and it represents numbers and performs computations with up to 100-decimal-plac... |

1 |
Generating functionology (Academic
- Wilf
- 1990
(Show Context)
Citation Context ... Theorem 1 is computed in lines 30-34. As indicated in Remark 5 above, the transform can be used to determine the asymptotic behavior of p k and q k as ks. In particular, from (12) and p. 150 of Wilf =-=[14] or p. 498 of B-=-ender [2], we find that p k ~ a ks- "`b r 1 _____ k 3/2 G( - 1/2) b k _____________ = 2""`b rpk 3 b k __________ , (15) where p k ~ a k means that p k / a ks1 as ks, which agrees with w... |