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132
Fitting Mixtures Of Exponentials To LongTail Distributions To Analyze Network Performance Models
, 1997
"... Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and interva ..."
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Cited by 206 (14 self)
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Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have longtail distributions, being well described by distributions such as the Pareto and Weibull. It is known that longtail distributions can have a dramatic effect upon performance, e.g., longtail servicetime distributions cause longtail waitingtime distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component longtail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a longtail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...
A diffusion approximation for the G/GI/n/m queue
 Operations Research
"... informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queuelength stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra ..."
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Cited by 40 (9 self)
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informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queuelength stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra waiting spaces). We use the steadystate distribution of that diffusion process to obtain approximations for steadystate performance measures of the queueing model, focusing especially upon the steadystate delay probability. The approximations are based on heavytraffic limits in which n tends to infinity as the traffic intensity increases. Thus, the approximations are intended for large n. For the GI/M/n/ � special case, Halfin and Whitt (1981) showed that scaled versions of the queuelength process converge to a diffusion process when the traffic intensity �n approaches 1 with �1 − �n � √ n → � for 0 <�<�. A companion paper, Whitt (2005), extends that limit to a special class of G/GI/n/mn models in which the number of waiting places depends on n and the servicetime distribution is a mixture of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. Finite waiting rooms are treated by incorporating the additional limit mn / √ n → � for 0 <� � �. The approximation for the more general G/GI/n/m model developed here is consistent
A ClosedForm Solution for Mapping General Distributions to Minimal PH Distributions
 In Performance TOOLS
, 2003
"... Approximating general distributions by phasetype (PH) distributions is a popular technique in queueing analysis, since the Markovian property of PH distributions often allows analytical tractability. ..."
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Cited by 33 (18 self)
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Approximating general distributions by phasetype (PH) distributions is a popular technique in queueing analysis, since the Markovian property of PH distributions often allows analytical tractability.
Matching Moments For Acyclic Discrete And Continuous PhaseType Distributions Of Second Order
, 2002
"... The problem of matching moments to phasetype (PH) distributions occurs in many applications. Often, low dimensions of the selected distributions are desired in order to meet state space constraints. It is obvious that the three parameters of acyclic PH distributions of second order  be they contin ..."
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Cited by 31 (5 self)
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The problem of matching moments to phasetype (PH) distributions occurs in many applications. Often, low dimensions of the selected distributions are desired in order to meet state space constraints. It is obvious that the three parameters of acyclic PH distributions of second order  be they continuous (ACPH(2)) or discrete (ADPH(2))  can be fitted to three given moments provided that these are feasible. For both types of PH distributions, this paper provides the permissible ranges by giving the immanent lower and upper (if existing) bounds for the first three moments. For moments which obey these bounds an exact and minimal (with respect to the dimension of the representation) analytic mapping of three moments into ACPH(2) or ADPH(2) distributions is presented. Besides the unified treatment of the discrete and continuous cases, the contribution of this paper mainly consists in the presentation of exhaustive analytic thirdmoment bounds, which allow to go beyond existing loworder momentfitting techniques  with respect to either the range of applicability or the precision or the order of the resulting PH distributions.
A closed form solution for mapping general distributions to minimal ph distributions
 In International Conference on Performance Tools – TOOLS 2003
, 2003
"... Approximating general distributions by phasetype (PH) distributions is a popular technique in stochastic analysis, since the Markovian property of PH distributions often allows analytical tractability. This paper proposes an algorithm for mapping a general distribution, G, to a PH distribution, whi ..."
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Cited by 26 (1 self)
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Approximating general distributions by phasetype (PH) distributions is a popular technique in stochastic analysis, since the Markovian property of PH distributions often allows analytical tractability. This paper proposes an algorithm for mapping a general distribution, G, to a PH distribution, which matches the first three moments of G. Efficiency of our algorithm hinges on narrowing the search space to a particular subset of the PH distributions, which we refer to as EC distributions. The class of EC distributions has a small number of parameters, and we provide closedform solutions for these. Our solution applies to any distribution whose first three moments can be matched by a PH distribution. Also, our resulting EC distribution requires a nearly minimal number of phases, within one of the minimal number of phases required by any acyclic PH distribution. Key words: PH distribution, moment matching, closed form, normalized moment PACS: 1
Analysis of multiserver systems via dimensionality reduction of Markov chains
 School of Computer Science, Carnegie Mellon University
, 2005
"... The performance analysis of multiserver systems is notoriously hard, especially when the system involves resource sharing or prioritization. We provide two new analytical tools for the performance analysis of multiserver systems: moment matching algorithms and dimensionality reduction of Markov chai ..."
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Cited by 21 (5 self)
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The performance analysis of multiserver systems is notoriously hard, especially when the system involves resource sharing or prioritization. We provide two new analytical tools for the performance analysis of multiserver systems: moment matching algorithms and dimensionality reduction of Markov chains (DR). Moment matching algorithms allow us to approximate a general distribution with a phase type (PH) distribution. Our moment matching algorithms improve upon existing ones with respect to the computational efficiency (we provide closed form solutions) as well as the quality and generality of the solution (the first three moments of almost any nonnegative distribution are matched). Approximating job size and interarrival time distributions by PH distributions enables modeling a multiserver system by a Markov chain, so that the performance of the system is given by analyzing the Markov chain. However, when the multiserver system involves resource sharing or prioritization, the Markov chain often has a multidimensionally infinite state space, which makes the analysis computationally hard. DR allows us to closely approximate a multidimensionally infinite Markov chain with a Markov
A Review of Open Queueing Network Models of Manufacturing Systems
 QUEUEING SYSTEMS
, 1992
"... In this paper we review open queueing network models of manufacturing systems. The paper consists of two parts. In the first part we discuss design and planning problems arising in manufacturing. In doing so we focus on those problems that are best addressed by queueing network models. In the seco ..."
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Cited by 21 (2 self)
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In this paper we review open queueing network models of manufacturing systems. The paper consists of two parts. In the first part we discuss design and planning problems arising in manufacturing. In doing so we focus on those problems that are best addressed by queueing network models. In the second part of the paper we describe the developments in queueing network methodology. We are primarily concerned with features such as general service times, deterministic product routings, and machine failures features that are prevalent in manufacturing settings. Since these features have eluded exact analysis, approximation procedures have been proposed. In the second part of this paper we review the developments in approximation procedures and highlight the assumptions that underlie these approaches. A significant development in the study of queueing network models is the discovery (empirical) that under conditions that are not very restrictive in practice: (i) equilibrium expected queue lengths behave as if they are convex functions of the processing rate of the server, and (ii) altering the processing rate at one station has minimal effect on the equilibrium expected queue lengths at other stations in the network. As a result researchers have been able to approximate some of the optimal design problems by convex programs. In the second part of this paper we describe these developments. Inspite of the advances made in the analysis of open queueing networks, several of the problems described in the first part of the paper cannot be analyzed without further progress in 2 methodology. One of the objectives of this paper is to expose the gap between the problems arising in manufacturing and the analytical tools that are currently available. We hope that by first describing the problems and then discussing the methodological developments the gap becomes apparent to the reader.
Towards better multiclass parametricdecomposition approximations for open queueing networks
 Annals of Operations Research
, 1994
"... Methods are developed for approximately characterizing the departure process of each customer class from a multiclass singleserver queue with unlimited waiting space and the firstin firstout service discipline. The model is Σ(GI i / GI i)/1 with a nonPoisson renewal arrival process and a nonex ..."
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Cited by 19 (0 self)
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Methods are developed for approximately characterizing the departure process of each customer class from a multiclass singleserver queue with unlimited waiting space and the firstin firstout service discipline. The model is Σ(GI i / GI i)/1 with a nonPoisson renewal arrival process and a nonexponential servicetime distribution for each class. The methods provide a basis for improving parametricdecomposition approximations for analyzing nonMarkov open queueing networks with multiple classes. For example, parametricdecomposition approximations are the Queueing Network Analyzer (QNA). The specific approximations here extend ones developed by G. Bitran and D. Tirupati (1988). For example, the effect of classdependent service times is considered here. With all procedures proposed here, the approximate variability parameter of the departure process of each class is a linear function of the variability parameters of the arrival processes of all the classes served at that queue, thus ensuring that the final arrival variability parameters in a general open network can be calculated by solving a system of linear equations.
Heavytraffic asymptotic expansions for the asymptotic decay rates
 in the BMAP/G/1 queue. Stochastic Models
, 1994
"... versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, PerronFrobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steadystate distributions in the BMAP / G /1 queue have asymptotically exponential tai ..."
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Cited by 18 (11 self)
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versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, PerronFrobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steadystate distributions in the BMAP / G /1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavytraffic limit. The coefficients of these heavytraffic expansions depend on the moments of the servicetime distribution and the derivatives of the PerronFrobenius eigenvalue δ(z) of the BMAP matrix generating function D(z) at z = 1. We give recursive formulas for the derivatives δ (k) ( 1). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting timeaverage of the factorial cumulant generating function (the logarithm of the generating function) of the arrival counting process, and the derivatives δ (k) ( 1) coincide with the asymptotic factorial cumulants of the arrival counting process. This insight is important for admission control schemes in multiservice networks based in part on asymptotic decay rates. The interpretation helps identify appropriate statistics to compute from network traffic data in order to predict performance. 1.
The Impact of Dependent Service Times on LargeScale Service Systems
"... This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic ..."
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Cited by 16 (11 self)
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This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic limit, allowing dependence among the service times. That limit shows that the number of customers in the system at any time is approximately Gaussian, where the timevarying mean is unaffected by the dependence, but the timevarying variance is affected by the dependence. As a consequence, required staffing to meet customary qualityofservice targets in a largescale service system with finitely many servers based on a normal approximation is primarily affected by dependence among the service times through this timevarying variance. This paper develops formulas and algorithms to quantify the impact of the dependence among the service times upon that variance. The approximation applies directly to infiniteserver models, but also indirectly to associated finiteserver models, exploiting approximations based on the peakedness (the ratio of the variance to the mean in the infiniteserver model). Comparisons with simulations confirm that the approximations can be useful to assess the impact of the dependence.