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11
Asymptotics for M/G/1 lowpriority waitingtime tail probabilities
, 1997
"... We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptiveresume disciplines. We show that the lowpriority steadystate waitingtime can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waitingtime distribution. ..."
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Cited by 39 (6 self)
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We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptiveresume disciplines. We show that the lowpriority steadystate waitingtime can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waitingtime distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have nonexponential asymptotics. This phenomenon even occurs when both servicetime distributions are exponential. When nonexponential asymptotics holds, the asymptotic form tends to be determined by the nonexponential asymptotics for the highpriority busyperiod distribution. We obtain asymptotic expansions for the lowpriority waitingtime distribution by obtaining an asymptotic expansion for the busyperiod transform from Kendall’s functional equation. We identify the boundary between the exponential and nonexponential asymptotic regions. For the special cases of an exponential highpriority servicetime distribution and of common general servicetime distributions, we obtain convenient explicit forms for the lowpriority waitingtime transform. We also establish asymptotic results for cases with longtail servicetime distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the nonexponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waitingtime transform.
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
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On the Laguerre method for numerically inverting Laplace transforms
 INFORMS Journal on Computing
, 1996
"... The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 TricomiWidder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coeff ..."
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Cited by 34 (7 self)
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The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 TricomiWidder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (1) using our previously developed variant of the Fourierseries method to calculate the coefficients of the Laguerre generating function, (2) developing systematic methods for scaling, and (3) using Wynn’s ɛalgorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux’s theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples. Subject classifications: Mathematics, functions: Laplace transforms. Probability, distributions: calculation by transform inversion. Queues, algorithms: Laplace transform inversion.
Numerical inversion of multidimensional Laplace transforms by the Laguerre method
 Eval
, 1998
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InfiniteSeries Representations Of Laplace Transforms Of Probability Density Functions For Numerical Inversion
, 1998
"... In order to numerically invert Laplace transforms to calculate probability density functions (pdf's) and cumulative distribution functions (cdf's) in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., o ..."
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Cited by 6 (1 self)
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In order to numerically invert Laplace transforms to calculate probability density functions (pdf's) and cumulative distribution functions (cdf's) in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., of a waitingtime cdf) can be computed when the Laplace transform values of component pdf's (e.g., of a servicetime pdf) can be computed. However, there are few explicit expressions for Laplace transforms of component pdf's available when the pdf does not have a pure exponential tail. In order to remedy this problem, we propose the construction of infiniteseries representations for Laplace transforms of pdf's and show how they can be used to calculate transform values. We use the Laplace transforms of exponential pdf's, Laguerre functions and Erlang pdf's as basis elements in the series representations. We develop several specific parametric families of pdf's in this infiniteseries framework. We show how to determine the asymptotic form of the pdf from the series representation and how to truncate so as to preserve the asymptotic form for a time of interest. 1.
Queueing models with multiple waiting lines
 Queueing Systems
, 2001
"... This paper discusses analytic solution methods for queueing models with multiple waiting lines. The methods are briefly illustrated, using key models like the 2 × 2 switch, the shortest queue and the cyclic polling system. AMS subject classification: 60K25, 90B22. ..."
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Cited by 2 (1 self)
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This paper discusses analytic solution methods for queueing models with multiple waiting lines. The methods are briefly illustrated, using key models like the 2 × 2 switch, the shortest queue and the cyclic polling system. AMS subject classification: 60K25, 90B22.
Laplace Transforms Of Probability Density Functions With Series Representations
 AT&T Labs
, 1998
"... In order to numerically invert Laplace transforms to calculate probability distributions in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., of a waitingtime distribution) can be computed when the La ..."
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Cited by 1 (1 self)
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In order to numerically invert Laplace transforms to calculate probability distributions in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., of a waitingtime distribution) can be computed when the Laplace transform values of component probability density functions (pdf's) (e.g., of a servicetime pdf) can be computed. However, in some cases explicit expressions for Laplace transforms of component pdf's are not available. Hence, we propose the construction of infiniteseries representations for Laplace transforms of pdf's and show how they can be used to calculate transform values. We use the Laplace transforms of Laguerre functions, Erlang pdf's and exponential pdf's as basis elements in the series representation. We develop several specific parametric families of pdf's in this infinite series framework. We show how to determine the asymptotic form of the pdf from the series representat...
The twoqueue E/1L polling model with regularly varying service and/or switchover
, 2001
"... times ..."
Asymptotics for Polling Models with Limited Service Policies
"... In this paper we find exact asymptotic expressions for the event that the total queue length is large for a k i limited exponential polling model with equal service rates and two classes of customers. It is found that this behaviour divides into two very di#erent regimes, depending on the arrival r ..."
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In this paper we find exact asymptotic expressions for the event that the total queue length is large for a k i limited exponential polling model with equal service rates and two classes of customers. It is found that this behaviour divides into two very di#erent regimes, depending on the arrival rates to the system. 1 Introduction Polling models have seen wide application in modelling systems in the diverse areas of telecommunications, transportation, computer performance, inventory, etc. (for a couple of general references, see Boxma and Takagi [1] and Levy and Sidi [5]). A (single server) polling model is simple to describe: there are several classes of customers arriving to a single server which operates under a particular service policy. Some typical examples are the socalled exhaustive policy, in which the server empties the system of a class of customers before moving on to the next class and limited policies, in which the server has a limit on the number of customers of a c...
INFINITESERIES REPRESENTATIONS OF LAPLACE TRANSFORMS OF PROBABILITY DENSITY FUNCTIONS FOR NUMERICAL INVERSION
, 1999
"... In order to numerically invert Laplace transforms to calculate probability density functions (pdf’s) and cumulative distribution functions (cdf’s) in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., ..."
Abstract
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In order to numerically invert Laplace transforms to calculate probability density functions (pdf’s) and cumulative distribution functions (cdf’s) in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., of a waitingtime cdf) can be computed when the Laplace transform values of component pdf’s (e.g., of a servicetime pdf) can be computed. However, there are few explicit expressions for Laplace transforms of component pdf’s available when the pdf does not have a pure exponential tail. In order to remedy this problem, we propose the construction of infiniteseries representations for Laplace transforms of pdf’s and show how they can be used to calculate transform values. We use the Laplace transforms of exponential pdf’s, Laguerre functions and Erlang pdf’s as basis elements in the series representations. We develop several specific parametric families of pdf’s in this infiniteseries framework. We show how to determine the asymptotic form of the pdf from the series representation and how to truncate so as to preserve the asymptotic form for a time of interest.