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Journal ArticleDOI

An empirical study of queueing approximations based on phase-type distributions

01 Jan 1993-Stochastic Models (Marcel Dekker, Inc.)-Vol. 9, Iss: 4, pp 531-561
TL;DR: In this article, the GI/PH/1 model is used to explore the behavior of phase-type (PH) approximations of interarrival- and service-time distributions.
Abstract: The GI/PH/1 model (single-server queue with general independent interarrival times and phase-type service times) is used to explore the behavior of phase-type (PH) approximations of interarrival- and service-time distributions. Various PH approximating distributions are considered for five non-PH interarrival-time distributions and two PH service-time distributions. Simple two- and three-moment-matching approximations and, in some cases, approximations obtained by also considering distribution shape are compared. Each approximating distribution is evaluated on the basis of the quality of the corresponding approximations of steady-state measures of the congestion seen by an arriving customer. The traffic intensity and the variability of the distribution (interarrival-or service-time) that is not approximated are shown to have a substantial effect on the accuracy of the queueing approximations. Rules for selecting approximating distributions for queueing applications are suggested
Citations
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Book
13 Dec 1996
TL;DR: This chapter discusses the modeling of Manufacturing Systems with a focus on the assembly of single-Stage, single-product systems and the role of the Kanban Scheme in this system.
Abstract: 1. Introduction to the Modeling of Manufacturing Systems.- 1.1. Typical Decision Problems in Manufacturing Systems.- 1.2. Performance Evaluation.- 1.3. Models of Manufacturing Systems.- 1.4. Design of Manufacturing Systems.- References.- 2. Tools of Probability.- 2.1. Random Variables and Their Distributions.- 2.2. Joint Distributions.- 2.3. Conditional Distributions.- 2.4. Functions of Random Variables.- 2.5. Moments of Random Variables.- 2.6. Special Distributions.- 2.7. Stochastic Processes.- 2.8. Phase-Type Distributions.- Related Bibliography on Phase-Type Distributions.- References.- Problems.- 3. Analysis of Single Work Stations.- 3.1. Introduction.- 3.2. A Simple Work Station.- 3.3. Distribution of the Number in the System.- 3.4. Work Stations Subject to Failures.- 3.5. Distribution of the Number of Jobs in a Work Station Subject to Failures.- 3.6. The Process-Completion Time.- 3.7. Failure-Repair Policies.- 3.8. The Machine Interference Problem.- Related Literature.- References.- Problems.- 4. Analysis of Flow Lines.- 4.1. Introduction.- 4.2. Characteristics of Flow Lines.- 4.3. Analysis of Two-Station Flow Lines.- 4.4. Analysis of Flow Lines With More Than Two Stations.- 4.5. Deterministic Processing Times.- 4.6. Bounds on the Output Rate.- 4.7. Design Problems in Flow Lines.- Related Literature.- References.- Problems.- 5. Analysis of Transfer Lines.- 5.1. Introduction.- 5.2. A Two-Station Line with No Intermediate Buffer: A Regenerative Approach.- 5.3. Transfer Lines with Work-in-Process Buffers.- 5.4. Buffer Clearing Policies in Transfer Lines.- 5.5. Analysis of a Two-Machine Synchronous Line Under BCP1.- 5.6. Operation Under BCP2.- 5.7. Line Behavior and Design Issues.- 5.8. Analysis of Larger Systems.- 5.9. Transfer Lines with No Scrapping.- Related Bibliography.- References.- Problems.- 6. Assembly Systems.- 6.1. Introduction.- 6.2. An Asynchronous Assembly System.- 6.3. Approximate Analysis of the Two-Station Assembly System.- 6.4. Analysis of Larger Systems.- 6.5. Synchronous Assembly Systems.- 6.6. Relaxing the Assembly Assumption.- Related Bibliography.- References.- Problems.- 7. Pull-Type Manufacturing Systems.- 7.1. Introduction.- 7.2. Production Control Policies.- 7.3. Analysis of Single-Stage, Single-Product Systems.- 7.4. A Single-Stage System with Two Products.- 7.5. Multistage Pull-Type Systems.- 7.6. The Generalized Kanban Scheme.- 7.7. Manufacturing Systems with Inventory Procurement.- 7.8. A Two-Stage, MultiProduct System.- 7.9. The Look-Back Policy.- Related Bibliography.- References.- Problems.- Appendix: A Single-Stage P/I System with PH Delay, Set-up, and Processing Times.

305 citations

Journal ArticleDOI
TL;DR: The developed approximation turns out to be accurate, robust and computationally efficient in an iterative approximate manner.
Abstract: The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) k-limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multi-queue polling system is decomposed into single-queue vacation systems with k-limited service and state-dependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrix-analytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out to be accurate, robust and computationally efficient.

52 citations


Cites background from "An empirical study of queueing appr..."

  • ...Of course, also other phase-type distributions may be fitted on the mean and the coefficient of variation, but numerical experiments suggest that choosing other distributions only have a minor effect on the results, as shown in [19]....

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Book ChapterDOI
TL;DR: An efficient approximation method to determine performance characteristics such as the throughput and mean sojourn times is developed, based on decomposition into two-station subsystems, the parameters of which are determined by iteration.
Abstract: In this paper we study multi-server tandem queues with finite buffers and blocking after service. The service times are generally distributed. We develop an efficient approximation method to determine performance characteristics such as the throughput and mean sojourn times. The method is based on decomposition into two-station subsystems, the parameters of which are determined by iteration. For the analysis of the subsystems we developed a spectral expansion method. Comparison with simulation shows that the approximation method produces accurate results. So it is useful for the design and analysis of production lines.

47 citations


Cites result from "An empirical study of queueing appr..."

  • ...Of course, also other phase-type distributions may be fitted on the rate and coefficient of variation of Di, but numerical experiments suggest that other distributions only have a minor effect on the results, as shown in [ 10 ]....

    [...]

Journal ArticleDOI
TL;DR: Reading this paper provides readers the foundational knowledge needed to develop intuition and insights on the complexities of stochastic simple serial lines, and serves as a guide to better understand and manage the effects of variability and design factors related to improving serial production line performance.

32 citations

References
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Journal ArticleDOI
TL;DR: In this paper, a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond is presented, which is a book of applicable mathematics, which avoids the approach of listing only the techniques, followed by a few examples.
Abstract: This is a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond. It is a book of applicable mathematics. It avoids the approach of listing only the techniques, followed by a few examples, without explaining why the techniques work. Thus, it provides not only the know-how but also the know-why. Equally, the text has not been written as a book of pure mathematics with a list of theorems followed by their proofs. The authors' aim is to help students develop an understanding of mathematics and its applications. They have refrained from using clichés like “it is obvious” and “it can be shown”, which may be true only to a mature mathematician. On the whole, the authors have been generous in writing down all the steps in solving the example problems.The book comprises ten chapters. Each chapter contains several solved problems clarifying the introduced concepts. Some of the examples are taken from the recent literature and serve to illustrate the applications in various fields of engineering and science. At the end of each chapter, there are assignment problems with two levels of difficulty. A list of references is provided at the end of the book.This book is the product of a close collaboration between two mathematicians and an engineer. The engineer has been helpful in pinpointing the problems which engineering students encounter in books written by mathematicians.

2,846 citations

Journal ArticleDOI
David M. Lucantoni1
01 Jan 1991
TL;DR: This work generalizes results to the single server queue with the batch arrival process and emphasizes the resulting simplifications of the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process.
Abstract: The versatile Markovian point process was introduced by M. F. Neuts in 1979. This is a rich class of point processes whichcontains many familiar arrival process as very special cases. Recently, the Batch Markovian Arrival Process, a class of point processes which was subsequently shown to be equivalent to Neuts’ point process, has been studied using a more transparent notation. Recent results in the matrix-analytic approach to queueing theory have substantially reduced the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process. We generalize these results to the single server queue with the batch arrival process and emphasize the resulting simplifications. Algorithms for the special cases of the PH/G/l and MMPP/G/1 queues are highlighted as these models are receiving renewed attention in the literature and the new algorithms proposed here are simpler than existing ones. In particular, the PH/G/1 queue has additional structure which further enh...

1,038 citations

Journal ArticleDOI
TL;DR: This book uses realistic examples to explore a wide variety of applications, such as inventory and production control, reliability, maintenance, queueing computer and communication systems, and will be of considerable interest to practitioners and researchers in operations research, statistics, computer science and engineering.
Abstract: The main concern of this text is the application of stochastic models to practical situations involving uncertainty and dynamism. A unique feature is the integrated treatment of models and computational methods for stochastic design and stochastic optimization problems. The book uses realistic examples to explore a wide variety of applications, such as inventory and production control, reliability, maintenance, queueing computer and communication systems. Exercises and suggestions for further reading are provided at the end of each chapter. The book was written with advanced students in mind, however, as it contains a wealth of material not found in other texts, it will also be of considerable interest to practitioners and researchers in operations research, statistics, computer science and engineering.

822 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the first κ (finite) moments of any nondegenerate distribution with support on can be matched by a mixture of Erlang distributions of (sufficiently high) common order.
Abstract: One approach to the moment-matching problem for phase distributions to to restrict selection to an appropriate subset of phase distributions. We investigate the use of mixtures of Erlang distributions of common order to match moments feasible for distributions with support on . We show that, except for special cases, the first κ (finite) moments of any nondegenerate distribution with support on can be matched by a mixture of Erlang distributions of (sufficiently high) common order. Moreover, we show that any κ-tuple of first κ moments feasible for a mixture of n-stage Erlang distributions feasible for a mixture of or fewer . The three-moment-matching problem is considered in detail. The set of pairs of second and third standardized moments feasible for mixtures of is characterized. An analytic expression is derived for the minimum order, n, such that a given set of first three moments is feasible for a mixture of .Expressions are also given for the parameters of the unique mixture of two that matches a fe...

156 citations

Journal ArticleDOI
Ward Whitt1
TL;DR: This work calculates the set of possible values for the mean queue length in a GI/M/1 queue and shows how it depends on the traffic intensity and the second moment, and uses extremal distributions to compare alternative parameters for approximations.
Abstract: Many approximations for queueing characteristics such as the mean equilibrium queue length are based on two moments of the interarrivai and service times. To evaluate these approximations, we suggest looking at the set of all possible values of the queueing characteristics given the specified moment parameters. This set-valued function is useful for evaluating the accuracy of approximations. For several models, such as the GI/M/1 queue, the set of possible values for the mean queue length given limited-moment information can be conveniently described by simple extremal distributions. Here we calculate the set of possible values for the mean queue length in a GI/M/1 queue and show how it depends on the traffic intensity and the second moment. We also use extremal distributions to compare alternative parameters for approximations. The results provide useful insights about approximations for non-Markov networks of queues and other complex queueing systems. The general procedure is widely applicable to investigate the accuracy of approximations.

85 citations