A Better Numerical Approach for Finding the Steady- State Waiting Time and the Average Queue Length of a System for the Arithmetic Gi/G/1 Queue
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Queueing-Theoretic Solution Methods for Models of Parallel and Distributed Systems·
1 Queueing-Theoretic Solution Methods for Models of Parallel and Distributed Systems· Onno Boxmat, Ger Koolet & Zhen Liui t CW/, Amsterdam, The Netherlands t INRIA-Sophia Antipolis, France This paper aims to give an overview of solution methods for the performance analysis of parallel and distributed systems. After a brief review of some important general solution methods, we discuss key models of parallel and distributed systems, and optimization issues, from the viewpoint of solution methodology. 1 INTRODUCTION The purpose of this paper is to present a survey of queueing theoretic methods for the quantitative modeling and analysis of parallel and distributed systems. We discuss a number of queueing models that can be viewed as key models for the performance analysis and optimization of parallel and distributed systems. Most of these models are very simple, but display an essential feature of dis tributed processing. In their simplest form they allow an exact analysis. We explore the possibilities and limitations of existing solution methods for these key models, with the purpose of obtaining insight into the potential of these solution methods for more realistic complex quantitative models. As far as references is concerned, we have restricted ourselves in the text mainly to key references that make a methodological contribution, and to sur veys that give the reader further access to the literature; we apologize for any inadvertent omissions. The reader is referred to Gelenbe's book [65] for a general introduction to the area of multiprocessor performance modeling and analysis. Stochastic Petri nets provide another formalism for modeling and perfor mance analysis of discrete event systems. -
Italic Entries Indicatefigures. 319-320,320
Index A Analytic hierarchy process, 16-24, role of ORIMS, 29-30 17-18 Availability,31 A * algorithm, 1 absolute, relative measurement Averch-lohnson hypothesis, 31 Acceptance sampling, 1 structural information, 17 Accounting prices. 1 absolute measurement, 23t, 23-24, Accreditation, 1 24 B Active constraint, 1 applications in industry, Active set methods, 1 government, 24 Backward chaining, 33 Activity, 1 comments on costlbenefit analysis, Backward Kolmogorov equations, 33 Activity-analysis problem, 1 17-19 Backward-recurrence time, 33 Activity level, 1 decomposition of problem into Balance equations, 33 Acyclic network, 1 hierarchy, 21 Balking, 33 Adjacency requirements formulation, eigenvector solution for weights, Banking, 33-36 facilities layout, 210 consistency, 19-20, 20t banking, 33-36 Adjacent, 1 employee evaluation hierarchy, 24 portfolio diversification, 35-36 Adjacent extreme points, 1 examples, 21-24 portfolio immunization, 34-35 Advertising, 1-2, 1-3 fundamental scale, 17, 18, 18t pricing contingent cash flows, competition, 3 hierarchic synthesis, rank, 20-21 33-34 optimal advertising policy, 2-3 pairwise comparison matrix for BarChart, 37 sales-advertising relationship, 2 levell, 21t, 22 Barrier, distance functions, 37-39 Affiliated values bidding model, 4 random consistency index, 20t modified barrier functions, 37-38 Affine-scaling algorithm, 4 ranking alternatives, 23t modified interior distance Affine transformation, 4 ranking intensities, 23t functions, 38-39 Agency theory, 4 relative measurement, 21, 21-22t, Basic feasible solution, 40 Agriculture 21-24 Basic solution, 40 crop production problems at farm structural difference between Basic vector, 40, 41 level,429 linear, nonlinear network, 18 Basis,40 food industry and, 4-6 structuring hierarchy, 20 Basis inverse, 40 natural resources, 428-429 synthesis, 23t Batch shops, 41 regional planning problems, 429 three level hierarchy, 17 Battle modeling, 41-44 AHP, 7 Animation, 24 attrition laws, 42-43 AI, 7. -
Markovian Bulk-Arrival and Bulk-Service Queues with General State-Dependent Control
Queueing Systems (2020) 95:331–378 https://doi.org/10.1007/s11134-020-09660-0 Markovian bulk-arrival and bulk-service queues with general state-dependent control Anyue Chen1,2 · Xiaohan Wu1 · Jing Zhang3 Received: 17 March 2019 / Revised: 25 May 2020 / Published online: 5 July 2020 © The Author(s) 2020 Abstract We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined, and the Laplace transform of the queue length distribution is presented. The important ques- tions regarding hitting times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regard- ing the busy period distributions including expectation and conditional expectation of busy periods are also explored. Keywords Bulk-arrival and bulk-service queues · State-dependent control · Equilibrium distribution · Queue length distributions · Busy period distributions Mathematics Subject Classification Primary 60J27; Secondary 60J80 B Anyue Chen [email protected]; [email protected] Xiaohan Wu [email protected] Jing Zhang [email protected] 1 Department of Mathematics, The Southern University of Science and Technology, Shenzhen 518055, Guangdong, China 2 Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, UK 3 Institute for Data and Decision Analytics, The Chinese University of Hong Kong, Shenzhen 518172, Guangdong, China 123 332 Queueing Systems (2020) 95:331–378 1 Introduction Markovian queues occupy a significant niche in applied probability. -
Approximating Heavy Traffic with Brownian Motion
APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION HARVEY BARNHARD Abstract. This expository paper shows how waiting times of certain queuing systems can be approximated by Brownian motion. In particular, when cus- tomers exit a queue at a slightly faster rate than they enter, the waiting time of the nth customer can be approximated by the supremum of reflected Brow- nian motion with negative drift. Along the way, we introduce fundamental concepts of queuing theory and Brownian motion. This paper assumes some familiarity of stochastic processes. Contents 1. Introduction 2 2. Queuing Theory 2 2.1. Continuous-Time Markov Chains 3 2.2. Birth-Death Processes 5 2.3. Queuing Models 6 2.4. Properties of Queuing Models 7 3. Brownian Motion 8 3.1. Random Walks 9 3.2. Brownian Motion, Definition 10 3.3. Brownian Motion, Construction 11 3.3.1. Scaled Random Walks 11 3.3.2. Skorokhod Embedding 12 3.3.3. Strong Approximation 14 3.4. Brownian Motion, Variants 15 4. Heavy Traffic Approximation 17 Acknowledgements 19 References 20 Date: November 21, 2018. 1 2 HARVEY BARNHARD 1. Introduction This paper discusses a Brownian Motion approximation of waiting times in sto- chastic systems known as queues. The paper begins by covering the fundamental concepts of queuing theory, the mathematical study of waiting lines. After waiting times and heavy traffic are introduced, we define and construct Brownian motion. The construction of Brownian motion in this paper will be less thorough than other texts on the subject. We instead emphasize the components of the construction most relevant to the final result—the waiting time of customers in a simple queue can be approximated with Brownian motion. -
Queueing Systems
Queueing Systems Ivo Adan and Jacques Resing Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands March 26, 2015 Contents 1 Introduction7 1.1 Examples....................................7 2 Basic concepts from probability theory 11 2.1 Random variable................................ 11 2.2 Generating function............................... 11 2.3 Laplace-Stieltjes transform........................... 12 2.4 Useful probability distributions........................ 12 2.4.1 Geometric distribution......................... 12 2.4.2 Poisson distribution........................... 13 2.4.3 Exponential distribution........................ 13 2.4.4 Erlang distribution........................... 14 2.4.5 Hyperexponential distribution..................... 15 2.4.6 Phase-type distribution......................... 16 2.5 Fitting distributions.............................. 17 2.6 Poisson process................................. 18 2.7 Exercises..................................... 20 3 Queueing models and some fundamental relations 23 3.1 Queueing models and Kendall's notation................... 23 3.2 Occupation rate................................. 25 3.3 Performance measures............................. 25 3.4 Little's law................................... 26 3.5 PASTA property................................ 27 3.6 Exercises..................................... 28 4 M=M=1 queue 29 4.1 Time-dependent behaviour........................... 29 4.2 Limiting behavior............................... -
A Survey on Performance Analysis of Warehouse Carousel Systems
A Survey on Performance Analysis of Warehouse Carousel Systems N. Litvak∗, M. Vlasiou∗∗ August 6, 2018 ∗ Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands. ∗∗ Eurandom and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected], [email protected] Abstract This paper gives an overview of recent research on the performance evaluation and design of carousel systems. We discuss picking strategies for problems involving one carousel, consider the throughput of the system for problems involving two carousels, give an overview of related problems in this area, and present an extensive literature review. Emphasis has been given on future research directions in this area. Keywords: order picking, carousels systems, travel time, throughput AMS Subject Classification: 90B05, 90B15 1 Introduction A carousel is an automated storage and retrieval system, widely used in modern warehouses. It consists of a number of shelves or drawers, which are linked together and are rotating in a closed loop. It is operated by a picker (human or robotic) that has a fixed position in front of the carousel. A typical vertical carousel is given in Figure1. arXiv:1404.5591v1 [math.PR] 22 Apr 2014 Carousels are widely used for storage and retrieval of small and medium-sized items, such as health and beauty products, repair parts of boilers for space heating, parts of vacuum cleaners and sewing machines, books, shoes and many other goods. In e-commerce companies use carousel to store small items and manage small individual orders. -
Queueing-Theoretic End-To-End Latency Modeling of Future Wireless Networks Schulz
Queueing-Theoretic End-to-End Latency Modeling of Future Wireless Networks Schulz Technische Universität Dresden Queueing-Theoretic End-to-End Latency Modeling of Future Wireless Networks Philipp Schulz der Fakultät Elektrotechnik und Informationstechnik der Technischen Universität Dresden zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) genehmigte Dissertation Vorsitzender: Prof. Dr.-Ing. habil. Leon Urbas Gutachter: Prof. Dr.-Ing. Dr. h.c. Gerhard Fettweis Univ. Prof. Dr. techn. Markus Rupp Tag der Einreichung: 01. Oktober 2019 Tag der Verteidigung: 13. Januar 2020 Philipp Schulz Queueing-Theoretic End-to-End Latency Modeling of Future Wireless Networks Dissertation, 13. Januar 2020 Vodafone Chair Mobile Communications Systems Institut für Nachrichtentechnik Fakultät Elektrotechnik und Informationstechnik Technische Universität Dresden 01062 Dresden Abstract The fifth generation (5G) of mobile communication networks is envisioned to enable a variety of novel applications. These applications demand requirements from the network, which are diverse and challenging. Consequently, the mobile network has to be not only capable to meet the demands of one of these appli- cations, but also be flexible enough that it can be tailored to different needs of various services. Among these new applications, there are use cases that require low latency as well as an ultra-high reliability, e. g., to ensure unobstructed pro- duction in factory automation or road safety for (autonomous) transportation. In these domains, the requirements are crucial, since violating them may lead to financial or even human damage. Hence, an ultra-low probability of failure is necessary. Based on this, two major questions arise that are the motivation for this thesis. First, how can ultra-low failure probabilities be evaluated, since experiments or simulations would require a tremendous number of runs and, thus, turn out to be infeasible. -
A Bibliography on Semi-Markov Processes BIBLIOGRAPHY 004
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector A bibliography on semi-Markov processes BIBLIOGRAPHY 004 Jozef L. Teugels (*) INTRODUCTION At the appearance of this bibliography we want to thank warmly all those who have helped compiling this edition by sending us titles of publications, preprints, reprints and other re- levant information. Especially Cheong C. Kong and Jos H.A. de Smit have contributed at an early stage. All publications by the same author have been arranged by the year of publication. This will be convenient if in the future we should decide to publish a supplement to this bibliography. The present version contains about 600 papers written by some 300 authors. We have indicated to the best of our kwowledge where a publication has been reviewed or summarized. We write ST for Statistical Theory and Method Abstracts, MR for Mathematical Reviews and Zbl for Zentralblatt fiir Mathematik und ihre Grenzgebiete. The first number refers to the volume and the second to the number of the page (for Zbl) or review/abstract (for ST and MR). Thus MR 27-1999 refers to review number" 1999 of volume 27 of Mathe- matical Reviews.. We follow MR in our abbreviations and transliterations for the names of journals. The reader's attention is drawn on translations made of journals published in Russian. The index volumes of MR should be consulted. New information can always be forwarded to the author. BIBLIOGRAPHY Akimov A.P. and Pogosjan I.A. Andronov A.M. and Gertsbakh I.G. On structural reservation systems with repair and cold reser- On properties of multidimensional functionals on semi-Mar- vation 1974 kov processes with finite state space. -
Modeling an M/M/1 Queue with Unreliable Service and a Working Vacation
Modeling an M/M/1 Queue with unreliable service and a working vacation by Joshua K. Patterson Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The University of Texas at Arlington May, 2019 i Modeling an M/M/1 Queue with unreliable service and a working vacation by Joshua K. Patterson Department of Mathematics The University of Texas at Arlington Abstract We define the new term ’unreliable service’ where the service itself is unreliable (i.e. may fail). We discuss how this differs from the current literature, and give examples showing just how common this phenomena is in many real-world scenarios. We first consider the classic M/M/1 queue with unreliable service and find some striking similarities with the well studied M/M/1 derivation. Next, we consider the M/M/1 queue with unreliable service and a working vacation. In each of these cases, surprising explicit results are found including positive recurrence conditions, the stationary queue length distribution, and a decomposition of both the queue length and waiting time. We also purpose a number of ideas for future research based on this newly defined phenomenon. ii Acknowledgements I would like to thanks my advisor Professor Andrzej Korzeniowski for his expertise and vast knowledge of queue theory and stochastic processes, the history of these topics, and major contributors therein. In addition, I would like to extend my gratitude towards my committee members, Drs. Suvra Pal, Shan Sun-Mitchell, and Jonghyun Yun. I would also like to acknowledge my mother, Brenda; father, Ken; grandmother, Brenda; step-grandfather, James; aunt, Amy; uncle, Brad; brothers Michael, Aaron; and step-brother David for their ongoing love and support from the very beginning–and I mean the very beginning. -
A Complete Bibliography of Publications in Statistics and Probability Letters: 1990–1999
A Complete Bibliography of Publications in Statistics and Probability Letters: 1990{1999 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 22 March 2018 Version 1.06 Title word cross-reference (N;d) [WC97]. (n; f; k) [CCH99]. 0 <r<1=2 [San91]. 1 [MH96]. 2 [EK96b, Ete99, SN96]. 2β [PP92b]. 2k [Kou96c]. 2m [Arc98a]. 3k [Kou96c]. 4 [Kab99]. 5 [Shi99]. a [PP92a]. A [Brz96, MH90]. α [AM94a, Alb97, CR92, Lie99, Mas96, Sam94, Sap99, Sha93, ZW97a, ZW97b]. L¯ [BS96a]. X¯ [KT93]. β [Hil99]. modn [DH97]. c [R¨us96]. C(α) [Wei92]. χ2 [Cha90d]. D [BH90, Brz96, Cha99c, CJ98, Kou96b, Kou96f, MZ97, Sch97c, Won95, MV95b]. D[0; 1] [BP93]. D+ [Fie90]. δ [Fal94, Kau95]. E [Brz95, Brz96, MZ97]. E(XjX + W )[Beh99].ER [Koz96]. [Bor96, Wu92]. F [Ext91, Run93, Sil96a, Zog94]. F α [Bal94]. G [Imh97, Won95, Kra98]. I0 [Fie90, RT93]. k [BP99b, BCQY97, BGP94, CSC93, Cha90b, CAGM98, HMG98, HT91, NNR97, PA90, PAT90, RC94, SB95a, SS95, Tso96, Vas90]. k(≥ 2) [MvdM97]. K × K × K [EZ99]. L [Abr93, BSW97, Brz96, Don98, Ger93, GR93a, PJ94, RW94, Sha91, Xia95]. 1 2 P p L [Lin95, Web92]. L (0; 1) [OS98]. L1 [AM94b, BW97, Dav93, EH99b, Hub97, KP94b, LM92, ST96]. L2 [LH98a]. L1 [CAGM98]. Lp [Ahm97, Che96, GKL96, Leb96, Wu90, Hei93]. Lq [Joh91]. Ls [FPMP97]. λ3 [HK91]. M [Arc94b, Arc98b, BlZ93, BKR92, Cro94, Fer98, Gen98, Jon94, LZ91, RZ92a, RC94, WW90b, Zha97a, Bo98, Che97b, GB96, Sco93, Wan99d, Wie96]. -
Continuous-Service M/M/1 Queuing Systems
Article Continuous-Service M/M/1 Queuing Systems Song Chew Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026, USA; [email protected]; Tel.: +1-618-650-3016 Received: 29 March 2019; Accepted: 24 April 2019; Published: 28 April 2019 Abstract: In this paper, we look into a novel notion of the standard M/M/1 queueing system. In our study, we assume that there is a single server and that there are two types of customers: real and imaginary customers. Real customers are regular customers arriving into our queueing system in accordance with a Poisson process. There exist infinitely many imaginary customers residing in the system. Real customers have service priority over imaginary customers. Thus, the server always serves real (regular) customers one by one if there are real customers present in the system. After serving all real customers, the server immediately serves, one at a time, imaginary customers residing in the system. A newly arriving real customer presumably does not preempt the service of an imaginary customer and hence must wait in the queue for their service. The server immediately serves a waiting real customer upon service completion of the imaginary customer currently under service. All service times are identically, independently, and exponentially distributed. Since our systems are characterized by continuous service by the server, we dub our systems continuous-service M/M/1 queueing systems. We conduct the steady-state analysis and determine common performance measures of our systems. In addition, we carry out simulation experiments to verify our results. We compare our results to that of the standard M/M/1 queueing system, and draw interesting conclusions. -
June 1985 LIDS-P-1481 the BULK SERVICE QUEUE with a GENERAL CONTROL STRATEGY
June 1985 LIDS-P-1481 THE BULK SERVICE QUEUE WITH A GENERAL CONTROL STRATEGY: THEORETICAL ANALYSIS AND A NEW COMPUTATIONAL PROCEDURE Warren B. Powell* Pierre Humblet** EES-83-5 Revised May, 1985 *Department of Civil Engineering, Princeton University. Research supported by NSF under grant No. CEE-8203476. **Research was conducted at the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technology with partial support provided by NSF under grant No. NSF-ECS-8310698. This paper has been accepted by Operations Research. Abstract A general framework is developed for analyzing a wide class of vehicle dispatching strategies for bulk arrival, bulk service queues. A simple derivation of the queue length transform for the imbedded Markov chain is provided and a new computational procedure is developed for finding the moments of the queue length distribution. Extensive computational tests are reported which demonstrate that the new procedure is significantly faster and more stable than the standard method referred to in the literature, which requires solving a set of simultaneous linear equations. Formulas for the mean and variance of the length of the queue are provided for the general case of compound Poisson arrivals, random batch capacities, general service times and a general control strategy. We consider the problem of analyzing bulk arrival, bulk service queues in steady state where customers arrive in groups at a point and wait for the next available vehicle. Under the simplest policy, termed here the bulk queue with no control, the vehicle, on arriving, will accept customers up to the capacity of the vehicle and then leave, independent of the number of customers waiting.