## Modeling service-time distributions with non-exponential tails: beta mixtures of exponentials (1999)

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Venue: | STOCHASTIC MODELS |

Citations: | 8 - 3 self |

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@ARTICLE{Abate99modelingservice-time,

author = {Joseph Abate and Ward Whitt},

title = {Modeling service-time distributions with non-exponential tails: beta mixtures of exponentials},

journal = { STOCHASTIC MODELS },

year = {1999},

volume = {15},

pages = {517--546}

}

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### Abstract

Motivated by interest in probability density functions (pdf’s) with nonexponential tails in queueing and related areas, we introduce and investigate two classes of beta mixtures of exponential pdf’s. These classes include distributions introduced by Boxma and Cohen (1997) and Gaver and Jacobs (1998) to study queues with long-tail service-time distributions. When the standard beta pdf is used as the mixing pdf, we obtain pdf’s with an exponentially damped power tail, i.e., f(t) ∼ αt −q e −ηt as t → ∞. This pdf decays exponentially, but analysis is complicated by the power term. When the beta pdf of the second kind is used as the mixing pdf, we obtain pdf’s with a power tail, i.e., f(t) ∼ αt −q as t → ∞. We obtain explicit representations for the cumulative distributions functions, Laplace transforms, moments and asymptotics by exploiting connections to the Tricomi function. Properties of the power-tail class can be deduced directly from properties of the other class, because the power-tail pdf’s are undamped versions of the other pdf’s. The power-tail class can also be represented as gamma mixtures of Pareto pdf’s. Both classes of pdf’s have simple explicit Laguerre-series expansions.

### Citations

3016 |
Convergence of Probability Measures
- Billingsley
- 1968
(Show Context)
Citation Context ...akly to a cdf F if and only if either (i) F n (x) # F (x) for all x that are continuity points of F or (ii) # # 0 gdF n # # # 0 gdF for all continuous bounded real-valued functions g; see Billingsley =-=[14]-=-. Apply this with the integral representation of the cdf paralleling (1.5). Proof of Theorem 1.5. Observe that b(q, p; y) = b(p, q; 1 - y), 0 # y # 1, for all p > 0 and q > 0 and apply Theorem 1.4. He... |

1455 |
An Introduction to Probability Theory
- Feller
- 1971
(Show Context)
Citation Context ...y # 1, (1.1) and the beta pdf of the second kind b 2 (p, q; y) = #(p + q) #(p)#(q) y p-1 (1 + y) -(p+q) , y # 0, (1.2) for p > 0 and q > 0, where #(x) is the gamma function; e.g., see p. 50 of Feller =-=[19]-=- and p. 51 of Johnson and Kotz [26]. The beta pdf of the second kind is obtained by considering the random variable X/(1 -X), where X has a standard beta pdf. The (standard) beta mixture of exponentia... |

168 |
Divergent Series
- Hardy
- 1973
(Show Context)
Citation Context ... (2.6) does not always converge rapidly, we might make a transformation to obtain a more rapidly convergent series. If we use the Euler (E, 1) transformation for this purpose, e.g., see p. 7 of Hardy =-=[24]-=-, we obtain the same result. In fact, the Euler transformation in Theorem 2.3 provides a general way to construct Laguerre-series representations for pdf's and their Laplace transforms. We investigate... |

142 | Fitting mixtures of exponentials to longtail distributions to analyze network performance models
- Feldmann, Whitt
- 1997
(Show Context)
Citation Context ... waiting-time cdf when the service-time pdf is from one of these classes. The classes should also be su#ciently large that they cover an interesting range of cases. As discussed in Feldmann and Whitt =-=[18]-=- and references therein, there currently is great interest in power-tail pdf's in the study of communication network performance because measurements indicate that many distributions have this propert... |

55 | Waiting-time tail probabilities in queues with long-tail service-time distributions. Queueing Systems 16(3/4
- Abate, Choudhury, et al.
- 1994
(Show Context)
Citation Context ...ra#c complexity (e.g., long-range dependence and selfsimilarity) . The power-tail pdf's here are alternatives to the Pareto mixture of exponential (PME) pdf's introduced in Abate, Choudhury and Whitt =-=[9]-=-. This paper can be regarded as a continuation of the operational calculus for probability distributions via Laplace transforms in Abate and Whitt [4]. In particular, here we apply the stationary-exce... |

42 |
Tables of Laplace Transforms
- Oberhettinger, Badii
- 1973
(Show Context)
Citation Context ...4. The pdf v(1, 1/2; t) in Table 5 is determined from an integral representation of the Bessel function K 0 (t); see 9.6.23 and 13.6.21 of AS. For the transform, see p. 119 of Oberhettinger and Badii =-=[29]-=-. Note thatsv(1, 1/2; s) =s#(s) f(s), wheresf(s) = log( # s + # 1 + s)/ # s (6.7) and f(t) = E 1 (t)/2 # #t, t # 0 , (6.8) which has moments # n /(2n + 1). Hence, f(t) and #(t) are related as in Table... |

41 |
An Atlas of Functions
- Oldham, Myland, et al.
- 2009
(Show Context)
Citation Context ...ections to appropriate special functions. The key special function is the Tricomi function U(a, b, t), i.e., the second of the confluent hypergeometric functions; see Chapter 48 of Spanier and Oldham =-=[32]-=- or Chapter 13 of AS. In particular, the representation here follows form 13.2.6 of AS and (1.5) here after making the change of variables y = x - 1 in (1.5). Theorem 1.7. For all p > 0 and q > 0, v(p... |

40 |
Higher Transcendental Functions, Bateman Manuscript Project, Vol
- Erdélyi
- 1981
(Show Context)
Citation Context ...# - p; t/y)b(#, q; y)dy = # 1 0 y -1 b(#, q; t/y)#(1 + # - p; y)dy . (8.3) Proof. First express f(t) as f(t) = #(# + q)t #-p #(#)#(1 + # - p) e -t U(q, 2 - p; t) . Then apply (7) on p. 270 of Erdelyi =-=[17]-=- plus 15.3.5 and 13.1.29 of AS to obtain f(s) = 2 F 1 (1 + # - p, #, q + #; -s) = # # n=0 (1 + # - p) n (#) n (q + #) n (-s) n n! . We now give some other examples from Theorem 8.2. First, if # = 1/2,... |

39 | Asymptotics for M/G/1 low-priority waiting-time tail probabilities, Queueing Systems 25 - Abate, Whitt - 1997 |

34 | On the laguerre method for numerically inverting laplace transforms
- Abate, Choudhury, et al.
- 1995
(Show Context)
Citation Context ...f's and ccdf's admit explicit Laguerre-series expansions. These series representations can serve as an e#ective means of computation for any p and q, using the algorithm in Abate, Choudhury and Whitt =-=[10]-=-. In Section 3 we give asymptotic expansions as t ## and as t # 0 for the pdf's and cdf's, and asymptotic expansions for the moments as n # #. In Section 4 we observe that gamma pdf's with shape param... |

29 | The M/G/1 queue with heavy-tailed service time distribution
- Boxma, Cohen
- 1998
(Show Context)
Citation Context ...d Whitt [16]. We now apply Theorem 1.7 to obtain an interesting alternative characterization of B 2 ME ccdf's as gamma mixtures of Pareto distributions. (This is not a scale mixture.) Boxma and Cohen =-=[15]-=- introduce the subclass of B 2 ME distributions with p = 2-q in this form to study queues with long-tail servicetime distributions. (They also include an atom at the origin.) Theorem 1.8. The B 2 ME c... |

25 |
Transient behavior of regulated Brownian motion i: Starting at the origin
- Abate, Whitt
- 1987
(Show Context)
Citation Context ... (p + 1, q - 1, s) = q - 1 ps (1 -sv 2 (p, q; s)) . (1.17) It turns out that pdf's associated with the time-dependent behavior of reflected Brownian motion (RBM) previously studied in Abate and Whitt =-=[1, 2, 3, 4]-=- and elsewhere are BME pdf's (with the scale parameter chosen so that the beta pdf is on [0, 2]; e.g., see Theorem 4.2 of Abate and Whitt [3]. The previously exploited stationary-excess relations amon... |

21 | An operational calculus for probability distributions via Laplace transforms
- Abate, Whitt
- 1996
(Show Context)
Citation Context ...E) pdf's introduced in Abate, Choudhury and Whitt [9]. This paper can be regarded as a continuation of the operational calculus for probability distributions via Laplace transforms in Abate and Whitt =-=[4]-=-. In particular, here we apply the stationary-excess operator E , the stationarylifetime operator L, the unimodal operator U and the damping operator D; see (1.12), (5.7), (5.12) and (1.21) below. The... |

21 |
An Introduction to TRANSFORM THEORY
- Widder
- 1971
(Show Context)
Citation Context ...eltjes transform of x#(x), i.e., by changing the order of integration, f(s) = # # 0 x s + x #(x)dx . (4.4) We can then calculate x#(x) by inverting its Stieltjes transform, e.g., see p. 126 of Widder =-=[34]-=-. Starting with (4.2), we obtain #(x) = -1 #x Ims#(p; -x) = sin #p #x(x - 1) p = 1 #(p)#(1 - p)x(x - 1) p , x # 1 , (4.5) applying 6.1.17 and 4.3.4 of AS. We can also approach (and generalize) Theorem... |

17 |
An introduction of Probability theory
- Moran
- 1968
(Show Context)
Citation Context ... - 1) p = 1 #(p)#(1 - p)x(x - 1) p , x # 1 , (4.5) applying 6.1.17 and 4.3.4 of AS. We can also approach (and generalize) Theorem 4.1 another way, using the following lemma, e.g., see p. 329 of Moran =-=[28]-=-. Lemma 4.2. If X and Y are independent random variables with densities #(p; t) and #(q; t), then the ratio X/(X +Y ) has the beta b(p, q; y) pdf and this ratio is independent of the denominator X + Y... |

16 | Control and recovery from rare congestion events in a large multi-server system
- Duffield, Whitt
- 1997
(Show Context)
Citation Context ...heorems 1.4 and 1.7 is reminiscent of Srivastava and Kashyap [33], but BME and B2ME pdf’s are not discussed there. The function 2F1 was also used in a different way in Example 4 of Duffield and Whitt =-=[16]-=-. We now apply Theorem 1.7 to obtain an interesting alternative characterization of B2ME ccdf’s as gamma mixtures of Pareto distributions. (This is not a scale mixture.) Boxma and Cohen [15] introduce... |

15 |
Simple spectral representations for the M/M/1 queue, Queueing Syst
- Abate, Whitt
- 1988
(Show Context)
Citation Context ... (p + 1, q - 1, s) = q - 1 ps (1 -sv 2 (p, q; s)) . (1.17) It turns out that pdf's associated with the time-dependent behavior of reflected Brownian motion (RBM) previously studied in Abate and Whitt =-=[1, 2, 3, 4]-=- and elsewhere are BME pdf's (with the scale parameter chosen so that the beta pdf is on [0, 2]; e.g., see Theorem 4.2 of Abate and Whitt [3]. The previously exploited stationary-excess relations amon... |

14 | Computing Laplace transforms for numerical inversion via continued fractions
- Abate, Whitt
- 1999
(Show Context)
Citation Context ...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# r... |

13 | Asymptotic analysis of tail probabilities based on the computation of moments
- Abate, Choudhury, et al.
- 1995
(Show Context)
Citation Context ...n (p + q) n n!t n as t ## (3.2) and V c 2 (p, q; t) # #(p + q) #(p)t q # # n=0 (-1) n (q) n (p + q) n n!t n as t ## . (3.3) We can apply (3.2) and Theorem 5.3 of Abate, Choudhury, Lucantoni and Whitt =-=[11]-=- to obtain an asymptotic expansion for the moments. Alternatively, since the moments are available explicitly in (2.1), we can also apply 6.1.47 of AS for this purpose. Corollary. As n ##, m n (p, q) ... |

9 | Explicit M/G/1 waiting-time distributions for a class of long-taile service-time distributions
- Abate, Whitt
- 1999
(Show Context)
Citation Context ...service-time pdf is the B 2 ME pdf v 2 (1/2, 3/2; t). We extend this explicit representation to a larger class of service-time pdf's, all with the tail asymptotics f(t) # #t -3/2 , in Abate and Whitt =-=[6]-=-. In Proposition 8.2 of Abate and Whitt [4] we had previously obtained explicit solutions for the M/G/1 waiting-time distribution for a class of service-time pdf's including the BME pdf v(1/2, 3/2; t)... |

9 |
Control and recovery from rare congestion events in a large multi-server system
- Duffield, Whitt
- 1997
(Show Context)
Citation Context ...eorems 1.4 and 1.7 is reminiscent of Srivastava and Kashyap [33], but BME and B 2 ME pdf's are not discussed there. The function 2 F 1 was also used in a di#erent way in Example 4 of Du#eld and Whitt =-=[16]-=-. We now apply Theorem 1.7 to obtain an interesting alternative characterization of B 2 ME ccdf's as gamma mixtures of Pareto distributions. (This is not a scale mixture.) Boxma and Cohen [15] introdu... |

9 |
On queues in which customers are served in random order
- Kingman
- 1962
(Show Context)
Citation Context ...n ## = (2n)! 4 n . (7.6) The exponential mixture of exponentials arising when p = 1 is the heavytra #c limit for the waiting time in the M/G/1 queue with random order of service; see p. 89 of Kingman =-=[27]-=-. 8. Other Scale Mixtures As indicated in the introduction, scale mixtures are usefully viewed as products of independent random variables. The scale mixture can be represented as a random variable Z,... |

8 |
The correlation functions of
- Abate, Whitt
- 1988
(Show Context)
Citation Context ... (p + 1, q - 1, s) = q - 1 ps (1 -sv 2 (p, q; s)) . (1.17) It turns out that pdf's associated with the time-dependent behavior of reflected Brownian motion (RBM) previously studied in Abate and Whitt =-=[1, 2, 3, 4]-=- and elsewhere are BME pdf's (with the scale parameter chosen so that the beta pdf is on [0, 2]; e.g., see Theorem 4.2 of Abate and Whitt [3]. The previously exploited stationary-excess relations amon... |

6 | Infinite-series representations of Laplace transforms of probability density functions for numerical inversion
- Abate, Whitt
- 1999
(Show Context)
Citation Context ...he mixing representation shows that the B 2 ME and B 3 ME pdf's are similar, especially with regard to their tail behavior. We do not discuss the B 3 ME pdf further here, but we do in Abate and Whitt =-=[7]-=-. Let G c (t) # 1 - G(t) be the complementary cdf (ccdf) associated with a cdf G. By integrating in (1.3) and (1.4), we see that the associated ccdf's are simply related to the pdf's with di#erent par... |

6 |
Waiting times when service times are stable laws: tamed and wild
- Gaver, Jacobs
- 1998
(Show Context)
Citation Context ...Section 5 we given recurrence relations that facilitate determining new BME and B 2 ME pdf's given established ones. In Section 5 we also establish connections to pdf's introduced by Gaver and Jacobs =-=[20]-=- and Boxma and Cohen [15] to study the M/G/1 queue with a long-tail service-time pdf. In particular, we analytically invert a transform in Gaver and Jacobs [20], solving a problem they pose. In Sectio... |

4 |
Convergence of a sequence of transformations of distribution functions
- Harkness, Shantaram
- 1969
(Show Context)
Citation Context ...of b(p, q; y) approaches to a unit point mass at 1 as p ##. Given Theorem 1.3, the limit (6.4) can also be deduced from limits of iterates of the stationaryexcess operator; see Harkness and Shantaram =-=[25]-=-, Shantaram and Harkness [31], and van Beek and Braat [13]. We can obtain the transformsv(1, 1; s) as the limit ofsv(p, 2-p; s) as p # 1, using (5.17) and (5.18), i.e.,sv(p, 2 - p; s) = lim p#1 1 (1 -... |

3 |
On a certain class of limit distributions
- Shantaram, Harkness
- 1972
(Show Context)
Citation Context ... unit point mass at 1 as p ##. Given Theorem 1.3, the limit (6.4) can also be deduced from limits of iterates of the stationaryexcess operator; see Harkness and Shantaram [25], Shantaram and Harkness =-=[31]-=-, and van Beek and Braat [13]. We can obtain the transformsv(1, 1; s) as the limit ofsv(p, 2-p; s) as p # 1, using (5.17) and (5.18), i.e.,sv(p, 2 - p; s) = lim p#1 1 (1 - p)s ((1 + s) 1-p - 1) = log(... |

2 |
The limits of sequences of iterated overshoot distribution functions
- Beek, J
- 1973
(Show Context)
Citation Context .... Given Theorem 1.3, the limit (6.4) can also be deduced from limits of iterates of the stationaryexcess operator; see Harkness and Shantaram [25], Shantaram and Harkness [31], and van Beek and Braat =-=[13]-=-. We can obtain the transformsv(1, 1; s) as the limit ofsv(p, 2-p; s) as p # 1, using (5.17) and (5.18), i.e.,sv(p, 2 - p; s) = lim p#1 1 (1 - p)s ((1 + s) 1-p - 1) = log(1 + s) s . (6.5) We note that... |

2 |
Elementary presentation of the frequency distribution of certain statistical populations
- Sawkins
- 1940
(Show Context)
Citation Context ...) has the beta b(p, q; y) pdf and this ratio is independent of the denominator X + Y . We can apply Lemma 4.2 to establish the following generalizations of Theorem 4.1, evidently first due to Sawkins =-=[30]-=-. (We obtain Theorem 4.1 by letting p + q = 1.) Theorem 4.3. A beta b(p, q; y) scale mixture of gamma #(p + q; t) pdf's is gamma #(p; t); i.e., for all p > 0 and q > 0, #(p; t) = # 1 0 y -1 #(p + q; t... |

2 |
Special Functions in Queueing Theory
- Srivastava, Kashyap
- 1982
(Show Context)
Citation Context ..., t), t # 0 , (1.23) where U(a, b, t) is the Tricomi function. From the perspective of queueing theory, this link to special functions in Theorems 1.4 and 1.7 is reminiscent of Srivastava and Kashyap =-=[33]-=-, but BME and B 2 ME pdf's are not discussed there. The function 2 F 1 was also used in a di#erent way in Example 4 of Du#eld and Whitt [16]. We now apply Theorem 1.7 to obtain an interesting alternat... |