An investigation of phase-distribution moment-matching algorithms for use in queueing models

Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

On Markovian Multi-Class,Multi-Server Queueing

Multi-class multi-server queueing problems are a generalisation of the well-known M/M/k queue to arrival processes with clients of N types that require exponentially distributed service with different average service times. In this paper, we give a procedure to construct exact solutions of the stationary state equations using the special structure of these equations. Essential in this procedure is the reduction of a part of the problem to a backward second order difference equation with constant coefficients. It follows that the exact solution can be found by eigenmode decomposition. In general eigenmodes do not have a simple product structure as one might expect intuitively. Further, using the exact solution, all kinds of interesting performance measures can be computed and compared with heuristic approximations (insofar available in the literature). We provide some new approximations based on special multiplicative eigenmodes, including the dominant mode in the heavy traffic limit. We illustrate our methods with numerical results. It turns out that our approximation method is better for higher moments than some other approximations known in the literature. Moreover, we demonstrate that our theory is useful to applications where correlation between items plays a role, such as spare parts management.     

On the inapproximability of <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">K</Emphasis>: why two moments of job size distribution are not enough

The M/G/K queueing system is one of the oldest models for multiserver systems and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (nonnumerical) approximations in the literature are based on (at most) the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions, and we provide a lower bound on the inapproximability ratio, which we refer to as “the gap.” This is the first such result in the literature to address “the gap.” The proof technique behind this result is novel as well and combines mean value analysis, sample path techniques, scheduling, regenerative arguments, and asymptotic estimates. Finally, our work provides insight into the effect of higher moments of the job size distribution on the mean waiting time.     

Higher order approximations for tandem queueing networks

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.     

Behaviour of queueing approximations based on sample moments

We investigate moment–based queueing approximations in the presence of sampling error. Let L be the steady–state mean number in the system for a GI/M/1 queue. We focus on the estimation of L under the assumption that only sample moments of the interarrival–time distribution are known. A simulation experiment is carried out for several interarrival–time distributions. For each case, sample moments from the interarrival–time distribution are matched to an approximating phase–type distribution and the corresponding estimate L is obtained. We show that the sampling error in the moments induces bias as well as variability in L. Based on our simulation experiment, we suggest matching only two moments when the sample coefficient of variation is low or when sample size is low; otherwise, matching three moments is preferable.     

Nonstationary queues with Interrrupted Poisson arrivals and unreliable/repairable servers

A queueing model having a nonstationary Interrupted Poisson arrival process (IPP(t)),s time-dependent exponential unreliable/repairable servers and finite capacityc is introduced, and an approximation method for analysis of it is developed and tested. Approximations are developed for the time-dependent queue length moments and the system viewpoint waiting time distributions and moments. The approximation involves state-space partitioning and numerically integrating partial-moment differential equations (PMDEs). Surrogate distribution approximations (SDA's) are used to close the system of PMDEs. The approximations allow for analysis using only (s + 1)(s + 6) differential equations for the queue length moments rather than the 2(c + 1)(s +1) equations required by the classic method of numerically integrating the full set of Kolmogorov-forward equations. Effectively hours of cpu time are reduced to minutes for even modest capacity systems. Approximations for waiting time distributions and moments are developed.This research was partially funded by National Science Foundation grant ECS-8404409.     

Queueing systems with inventory management with random lead times and with backordering

We investigate M/M/1/∞-systems with inventory management, continuous review, exponentially distributed lead times and backordering. We compute performance measures and derive optimality conditions under different order policies. For performance measures, which are not explicitly at hand, we present an approximation scheme for all possible parameter combinations. Although we cannot completely determine analytically the steady state probabilities for the system we are able to derive functional relations between interesting probabilities and show surprising insensitivity properties of several performance measures. For the approximations we develop an algorithm adapted to the system structure which suggests easy adaption to other systems.Work supported by Deutscher Akademischer Austauschdienst and KBN, Poland, Project D/02/32206.     

Numerical approximations for the steady‐state waiting times in a GI/G/1 queue

This paper focuses on easily computable numerical approximations for the distribution and moments of the steadystate waiting times in a stable GI/G/1 queue. The approximation methodology is based on the theory of Fredholm integral equations and involves solving a linear system of equations. Numerical experimentation for various M/G/1 and GI/M/1 queues reveals that the methodology results in estimates for the mean and variance of waiting times within ±1% of the corresponding exact values. Comparisons with competing approaches establish that our methodology is not only more accurate, but also more amenable to obtaining waiting time approximations from the operational data. Approximations are also obtained for the distributions of steadystate idle times and interdeparture times. The approximations presented in this paper are intended to be useful in roughcut analysis and design of manufacturing, telecommunications, and computer systems as well as in the verification of the accuracies of inequalities, bounds, and approximations.     

An interpolation approximation for the GI/G/1 queue based on multipoint Padé approximation

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy traffic conditions. In this paper, we propose a new approximation technique, which combines the light and heavy traffic characteristics. This interpolation approximation is based on the theory of multipoint Padé approximation which is applied at two points: light and heavy traffic. We show how this can be applied for estimating the waiting time moments of the ^GI/G/1 queue. The light traffic derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy traffic limits of the ^GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of traffic intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. This revised version was published online in June 2006 with corrections to the Cover Date.     

Average Approximations and Moments of Measures

We investigate average approximations of infinite dimensional mappings and related problems connected with moments of measures on linear spaces. A conjecture stated by J. F. Traub and A. G. Werschulz (1994, Math. Intelligencer16, 42–48) is settled. Several positive results concerning average approximations of Banach space valued mappings are obtained. Some related open problems are discussed.     

Approximating the Mean Waiting Time in the <Emphasis Type="Italic">GI/G/s</Emphasis> Queue

We provide two distribution-dependent approximations for the mean waiting time in a GI/G/s queue. Both approximations are weighted combinations of the exact mean waiting times for the GI/M/s and M/D/s queues each of which has the same mean service time and traffic intensity as in the approximating GI/G/s queue. The weights in the approximations are expressed by the service-time c.d.f. and the first two moments of interarrival and service times. To examine the performance of our approximations, they are numerically compared with exact solutions and previous two-moment approximations for various cases. Extensive numerical comparisons indicate that the relative percentage errors of the approximations are of the order of 5% in moderate traffic and 1% in heavy traffic, except for extreme cases.     

Waiting and sojourn times in a multi-server queue with mixed priorities

We consider a multi-server queue with K priority classes. In this system, customers of the P highest priorities (P<K) can preempt customers with lower priorities, ejecting them from service and sending them back into the queue. Service times are assumed exponential with the same mean for all classes. The Laplace–Stieltjes transforms of waiting times are calculated explicitly and the Laplace–Stieltjes transforms of sojourn times are provided in an implicit form via a system of functional equations. In both cases, moments of any order can be easily calculated. Specifically, we provide formulae for the steady state means and the second moments of waiting times for all priority classes. We also study some approximations of sojourn-time distributions via their moments. In a practical part of our paper, we discuss the use of mixed priorities for different types of Service Level Agreements, including an example based on a real scheduling problem of IT support teams.      

Approximations for multi-server queues: System interpolations

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.     

Discrete approximations to spherically symmetric distributions

Summary We consider the approximation of spherically symmetric distributions in ^d by linear combinations of Heaviside step functions or Dirac delta functions. The approximations are required to faithfully reproduce as many moments as possible. We discuss stable methods of computing such approximations, taking advantage of the close connection with Gauss-Christoffel quadrature. Numerical results are presented for the distributions of Maxwell, Bose-Einstein, and Fermi-Dirac.Dedicated to Fritz Bauer on the occasion of his 60th birthdayWork supported in part by the National Science Foundation under Grant MCS-7927158A1     

Moment Solutions for the State Exiting Counting Processes of a Markov Renewal Process

Important performance measures for many Markov renewal processes are the counts of the exits from each state. We present solutions for the conditional first, second, and covariance moments of the state exiting counting processes for a Markov renewal process, and solutions for the unconditional equilibrium versions of the moments. We demonstrate the relationship between the conditional first moments for the state exiting and the state entering counting processes. For analytical and illustrative purposes, we concentrate on the two state case. Two asymptotic expansions for the moment functions are proposed and evaluated both analytically and empirically. The two approximations are shown to be competitive in terms of absolute relative error, but the second approximation has a simpler analytical form which is useful in analyzing more complex stochastic processes having an underlying MRP structure.     

On an Approximation to the Mean Response Times of Priority Classes in a Stable G/G/c/PR Queue

Analytic approximations are proposed for the mean response-times of R(≥ 2) priority classes in a stable G/G/c/PR queue with general class interarrival and service time distributions and c(≥ 2) parallel servers under pre-emptive resume (PR) scheduling. The generalized exponential (GE) distributional model is used to represent general distributions with known first two moments per class. The analysis is based on the extension of known heuristic arguments and earlier results regarding the study of the stable GE/GE/c/FCFS (c ≥ 1, single class) and GE/G/1/PR queues. Numerical examples illustrate the accuracy of the proposed approximations in relation to simulations involving different interarrival and service time distributions per class. Moreover, GE-type performance bounds on the system response time per class are defined. Comments on the role of the new mean response time expressions towards the approximation of the joint and marginal queue length distributions of a stable G/G/c/PR queue are included.     

A new point estimation method for statistical moments based on dimension-reduction method and direct numerical integration

Estimation of statistical moments of structural response is one of the main topics for analysis of random systems. The balance between accuracy and efficiency remains a challenge. After investigating of the existing point estimation method (PEM), a new point estimate method based on the dimension-reduction method (DRM) is presented. By introducing transformations, a system with general variables is transformed into the one with independent variables. Then, the existing PEMs based on the DRMs are investigated. Based on the qualitative analysis of difference in the approximations for response function and moment function, a new PEM is proposed, in which the response function is decomposed directly and the moments are calculated by high dimensional integral directly. Compared with the existing PEM based on univariate DRM, the proposed method is more friendly and easier to implement without loss of accuracy and efficiency; as compared with the PEM based on the generalized DRM, the proposed method is of better precision at the cost of nearly the same efficiency and computational complexity, further, it does hold that the even-order moments are nonnegative. Finally, several examples are investigated to verify the performance of the new method.     

The m/g/1 retrial queue with nonpersistent customers

We consider anM/G/1 retrial queue in which blocked customers may leave the system forever without service. Basic equations concerning the system in steady state are established in terms of generating functions. An indirect method (the method of moments) is applied to solve the basic equations and expressions for related factorial moments, steady-state probabilities and other system performance measures are derived in terms of server utilization. A numerical algorithm is then developed for the calculation of the server utilization and some numerical results are presented.     

Simple Approximations for the GI/G/c Queue-II: The Moments,the Inverse Distribution Function and the Loss Function of the Number in the System and of the Queue Delay

Based on results obtained in part I of this paper, approximations for the first four moments of the number in the system are developed and thence used to approximate the inverse distribution function (IDF) and the loss functions (LF), employing Shore's general approximations. Existing approximations for the first two moments of queueing time in a GI/G/l queue serve to approximate the IDF and the LF of queueing time in the corresponding GI/G/c queue. The accuracy attained is generally satisfactory, while a remarkable algebraic simplicity is preserved. A numerical example demonstrates the applicability of some of the new approximations to solve optimization problems.