Traffic measurements from communication networks have shown that many quantities characterizing network performance have long-tail 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 long-tail distributions, being well described by distributions such as the Pareto and Weibull. It is known that long-tail distributions can have a dramatic effect upon performance, e.g., long-tail service-time distributions cause long-tail waiting-time distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component long-tail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a long-tail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...

Approximating general distributions by phase-type (PH) distributions is a popular technique in queueing analysis, since the Markovian property of PH distributions often allows analytical tractability.

informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queue-length 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 steady-state distribution of that diffusion process to obtain approximations for steady-state performance measures of the queueing model, focusing especially upon the steady-state delay probability. The approximations are based on heavy-traffic 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 queue-length 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 service-time 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

versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, Perron-Frobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steady-state 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 heavy-traffic limit. The coefficients of these heavy-traffic expansions depend on the moments of the service-time distribution and the derivatives of the Perron-Frobenius 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 time-average 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 multi-service 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 problem of matching moments to phase-type (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 third-moment bounds, which allow to go beyond existing low-order moment-fitting techniques - with respect to either the range of applicability or the precision or the order of the resulting PH distributions.

Approximating general distributions by phase-type (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 closed-form 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

This note describes a simulation experiment involving nine exponential queues in series with a non-Poisson arrival process, which demonstrates that the heavy-traffic bottleneck phenomenon can occur in practice (at reasonable traffic intensities) as well as in theory (in the limit). The results reveal limitations in customary two-moment approximations for open queueing networks.

Abstract: 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.

A common analytical technique involves using a Coxian distribution to model a general distribution G, where the Coxian distribution agrees with G on the rst three moments. This technique is motivated by the analytical tractability of the Coxian distribution. Algorithms for mapping an input distribution G to a Coxian distribution largely hinge on knowing a priori the necessary and sucient number of phases in the representative Coxian distribution. In this paper, we formally characterize the set of distributions G which are well-represented by an n-phase Coxian distribution, in the sense that the Coxian distribution matches the rst three moments of G. We also discuss a few common, practical examples.

We propose an enhancement to the parametric-decomposition method for calculating approximate steady-state performance measures of open queueing networks with non-Poisson arrival processes and non-exponential service-time distributions. Instead of using a variability parameter c a 2 for each arrival process, we suggest using a variability function c a 2 (ρ) , 0 < ρ < 1, for each arrival process; i.e., the variability parameter should be regarded as a function of the traffic intensity ρ of a queue to which the arrival process might go. Variability functions provide a convenient representation of different levels of variability in different time scales for arrival processes that are not nearly renewal processes. Variability functions enable the approximations to account for long-range effects in queueing networks that cannot be addressed by variability parameters. For example, the variability functions provide a way to address the heavy-traffic bottleneck phenomenon, in which exceptional variability (either high or low) in the input has little impact in a series of queues with low-to-moderate traffic intensities, and then has a big impact when it reaches a later queue with a relatively high traffic intensity. The variability functions also enable the approximations to characterize irregular periodic deterministic external arrival processes in a reasonable way; i.e., if there are no batches, then c a 2 (ρ) should be 0 for ρ near 0 or 1, but c a 2 (ρ) can assume arbitrarily large values for appropriate intermediate ρ. We present a full network algorithm with variability functions, showing that the idea is implementable. We also show how simulations of single queues can be effectively exploited to determine variability functions for difficult external arrival processes. Key words: queueing networks, tandem queues, approximations, parametric-decomposition approximations, two-moment approximations, heavy traffic, squared coefficient of variation.