The Fourier-series method for inverting transforms of probability distributions

Abate, Joseph; Whitt, Ward

doi:10.1007/bf01158520

Cited by 607 publications

(526 citation statements)

References 86 publications

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“…For this purpose, we applied the two-dimensional transform inversion algorithms in Choudhury, Lucantoni and Whitt [12]. These algorithms are based on the Fourier-series method [13], and are enhancements and generalizations of the Euler and LatticePoisson algorithms described there. We note that the same multidimensional transform inversion algorithms can be used to obtain numerical results from the expressions in this paper.…”

Section: Introductionmentioning

confidence: 99%

Further transient analysis of theBMAP/G/1 Queue

Lucantoni¹

1998

Communications in Statistics. Stochastic Models

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Previously, we derived the two-dimensional transforms of the emptiness function, the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes.We continue the transient analysis of this model in this paper by deriving explicit expressions for the transforms of the queue length at the -th departure (assuming a departure at time ), and the delay of the -th arrival (keeping track of the appropriate phase changes). Also, the departure process is characterized by the double transform of the probability that the -th departure occurs at time less than or equal to time .

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Section: Introductionmentioning

confidence: 99%

Further transient analysis of theBMAP/G/1 Queue

Lucantoni¹

1998

Communications in Statistics. Stochastic Models

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“…We recommend first computing the asymptotic parameters as discussed above to be used as a stopping criteria for an inversion algorithm such as the one presented in Abate and Whitt [68] for the inversion; see [8], [11], [12], [81] and [88] for applications and discussions of this.…”

Section: Numerical Transform Inversionmentioning

confidence: 99%

“…The waiting time distributions given by the transforms (14)- (16) can be computed by numerically solving the associated Volterra integral equations (see, e.g., Neuts [93] [2]) or by numerically inverting the transforms directly (see e.g., Abate and Whitt [68] and Choudhury, Lucantoni and Whitt [81]). …”

Section: Computing Waiting Time Distributionsmentioning

confidence: 99%

“…However, with the recent availability of improved transform inversion methods, it has become evident that numerical transform inversion can contribute significantly to the numerical solution of these models. Indeed, numerical transform inversion can provide extremely accurate results (see, e.g., Abate and Whitt [68] and Choudhury, Lucantoni and Whitt [12]). Hence, a good strategy for solving the BMAP/G/1 queue is to combine the standard matrix-analytic techniques with transform inversion routines.…”

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confidence: 99%

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The BMAP/G/1 queue: A tutorial

Lucantoni

Lecture Notes in Computer Science

149

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Abstract. We present an overview of recent results related to the single server queue with general independent and identically distributed service times and a batch Markovian arrival process (BMAP). The BMAP encompasses a wide range of arrival processes and yet, mathematically, the BMAP/G/1 model is a relatively simple matrix generalization of the M/G/1 queue. Stationary and transient distributions for the queue length and waiting time distributions are presented. We discuss numerical algorithms for computing these quantities, which exploit both matrix analytic results and numerical transform inversion. Two-dimensional transform inversion is used for the transient results. The idea of a BMAP is to keep the tractability of the Poisson arrival process but significantly generalize it in ways that allow the inclusion of dependent interarrival times, non-exponential interarrival-time distributions, and correlated batch sizes. The BMAP includes as special cases both phase type renewal processes (which include the Erlang, Ek, and hyperexponential, Hk, renewal processes) and non-renewal processes such as the Markov modulated Poisson process

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“…As W * (θ) is unlikely to have an analytical inversion, we invert it numerically using the Euler method [11] to obtain the response time pdf f W (t). The cumulative distribution function W (t) is also easily obtained by inverting W * (θ)/θ.…”

Section: Data Transfer Timementioning

confidence: 99%

A Response Time Distribution Model for Zoned RAID

Lebrecht

Dingle

Knottenbelt

Analytical and Stochastic Modeling Techniques and Applications

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Abstract. RAID systems are widely deployed, both as standalone storage solutions and as the building blocks of modern virtualised storage platforms. An accurate model of RAID system performance is therefore critical to understanding storage system performance. To this end, this paper presents a queueing network-based model of RAID systems comprised of zoned disks and operating at RAID level 0-1 or 5. The contribution over previous work is twofold. Firstly, our analysis approximates full I/O request response time distributions rather than just mean values. This provides the ability to reason about response time quantiles and higher moments of response time -both of which are useful in the context of modern quality of service requirements. Secondly, we validate our model against measurements from a real RAID system rather than a software simulation. The close agreement between predicted and observed response time distributions gives a high level of confidence in the validity of our model.

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The Fourier-series method for inverting transforms of probability distributions

Cited by 607 publications

References 86 publications

Further transient analysis of theBMAP/G/1 Queue

Further transient analysis of theBMAP/G/1 Queue

The BMAP/G/1 queue: A tutorial

A Response Time Distribution Model for Zoned RAID

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