Kendall notation. PASTA theorem Basics of M/M/1 queue
|
|
- Neal Charles
- 5 years ago
- Views:
Transcription
1 Elementary queueing system Kendall notation Little s Law PASTA theorem Basics of M/M/1 queue 1
2 History of queueing theory An old research area Started in 1909, by Agner Erlang (to model the Copenhagen telephone exchange in Denmark) Booming after 1950 s, to model computer/communication/ David G. Kendall introduced an A/B/C queueing notation in 1953 used in modern packet switching networks in the early 1960s by Leonard Kleinrock (UCLA, his book) A branch of operations research Agner Erlang , Denmark 2
3 Model and analysis tool Modeling and analyze many systems Telephony systems, exchanger of telephone lines Computer systems Communication systems Transportation systems Production systems Hospital/banks/ Limitation Too restrictive, some assumptions Use other alternative methods, simulation/tools, 3
4 A general queueing system A general queuing system Customer arrival Customer departure waiting room Service facility A generic model dlfor Machine, computer system, communication system, intersection of roads, etc. 4
5 The basic elements of queueing system Arrival process of customers Interarrival time i.i.d., e.g, Poisson arrival Arrival one by one, or in batches, etc. Bh Behavior of customers Patient or impatient; different type of customers Service time i.i.d., i e.g., exponential; load/state dependent; 5
6 The basic elements of queueing Service discipline system (cont.) FCFS: first come first serve, LCFS: last come first serve (stack), RS: randomly serve, priority, SRPT: shortest remaining processing time first, PS: processor sharing Service capacity (number of servers) Single server or a group of servers Waiting room (system capacity) Finite or infinite; buffer size design 6
7 Kendall notation A family of notation symbols for different categories of queues Proposed by David G. Kendall in 1953 Professor of Oxford Univ. ( ) and Cambridge Univ. ( ) David G. Kendall , England 7
8 Kendall notation (cont.) A family of notation symbols for different categories of queues A/B/C/K/N/D A: distribution of interarrival time, M or G or D B: distribution of service time, M or G or D C: number of servers K: system capacity, infinite by default N: number of total customers, infinite by default D: the service discipline For default, K =, N = and D = FCFS 8
9 Example of Kendall notation M/M/1; M/G/1; G/M/1; G/G/1; M/D/1; M/Er/1; PH/G/1; M/M/c; M/M/1/B; M/M/ //N M/M/c/K/N/ /K/N/ M/M/c/K//LCFS / / // M/M/1///PS 9
10 A small joke Different of type of queue. 10
11 Utilization factor For a queue with single server λ: the arrival rate of customers x Utilization factor: : the mean service time Physical meaning: ρ: = λ x Single server: time fraction that server is busy Multiple server: fraction of busy servers ρ = λx / m Traffic intensity is defined as r: = λx for single or multiple servers 11
12 Performance measures Distribution of performance measures Distribution of waiting time W and sojourn time T of customers; E{W}, Var{W}, Pr{W>t} Distribution of number of customers in the queue N or in the system N q ; E{N}, Var{N}, Pr{N>n} Distribution of busy period of the server BP; E{BP}, Var{BP}, Pr{BP>t} Mean of performance measures Mean waiting time, mean sojourn time/mean response time; Average customer number, average queue length Average length ofbusyperiod Throughput of the system Open system: equals the arrival rate; Closed system: needs calculation and analysis 12
13 The Little s Law Named after John Little, (1928 ), professor of MIT For a stable queueing system, we have L: average number of customers in the system λ: arrival rate, average number of arrivals per unit time T: mean response time/sojourn time of customers L = λt 13
14 The Little s Law (cont.) Apply it to the queue (excluding the server) L q: average number of customers in the queue λ: arrival rate, average number of arrivals per unit time W: mean waiting time of customers Lq = λw 14
15 The Little s Law (cont.) Apply it to the server only ρ: average number of customers in the server, λ: average number of arrivals to the server per unit time x : mean service time of customers ρ = λxλ For M/M/1, we have ρ = λ μ 15
16 The Little s Law (cont.) Applicability Very general, G/G/c / Applicable to any queue which is STABLE Applicable to any subsystem of the queue Limitation Inapplicable to unstable queue Only mean metrics, inapplicable to study the transient metrics 16
17 PASTA Theorem Poisson Arrivals See Time Averages For queues with Poisson arrivals, M/./. The arriving customers find the same mean measures as that observed by an outside observer at an arbitrary point of time Intuitively i explained li dby the fact that Poisson arrivals occurs completely random in time (purely random sampling) 17
18 PASTA Theorem (cont.) Applicable to any queues with Poisson arrival M/M/1, M/G/1, etc. Not valid for some queues, e.g., D/D/1 Empty pyat time 0, arrive at 1,3,5,, service time is 1. Arrivals see empty queue, while average number of customers is 1/2 Little s Law and PASTA theorem is very fundamental and important in queueing theory E.g., Mean Value Analysis (MVA) for queueing networks Calculate the mean performance metrics, L, W, T, etc. 18
19 M/M/1 queue System parameters Poisson arrival rate λ,, service rate μ Infinite capacity, FCFS Study the performance metrics ti Customer arrival Customer departure waiting room Service facility 19
20 Dynamics of M/M/1 / queue server queue C 1 C 2 C 3 Departure Start t C 1 C 2 C 3 C 4 service t t C 1 C 2 C 3 C 4 Arrival # of customer t 20
21 State transition rate diagram of M/M/1 State transition rate diagram state: the number of customers in the system λ λ λ λ λ λ k 1 k k+1 μ μ μ μ μ μ μ A simple birth death process 21
22 Equilibrium behavior of M/M/1 When the system reaches steady Global balance equation π λ = π μ 0 1 n + n n 1 n π ( λ+ μ ) = π μ+ π λ, = 1,2,... Local balance equation π nλ = πn+ 1 μ, n = 0,1,2,... Normalization equation n=00 π n = 1 22
23 Equilibrium behavior of M/M/1 Chalk writing (cont.) Derive the steady state distribution 1 n n π n = = (1 ) G ρ ρ ρ G is called the normalization constant, G=1/(1 ρ) 3 more ways to solve the local balance equation recursion, Z transform, direct approach of solving difference equation Very simple to solve the local balance equation 23
24 Key performance metrics of M/M/1 (time average metrics) Average number of customers in the system, L We have L nπn n(1 ) Variance is ρ/(1 ρ) 2 Pr{n k}: Pr{ n k} = k π = ρ ρ 1 n = = ρ ρ = n= 0 n= 0 1 ρ Use dev. to void integration by Pr{ } i i= k parts! Trick. Average queue length (excluding the customer being served) We have Chalk writing The curve of L w.r.t. ρ 2 ρ L = q ( n 1) π n = L ρ = n= 1 1 ρ 24
25 Key performance metrics of M/M/1 (customer average metrics) Mean response time of customers T By Little s law, we have T = L/ λ = 1 μ λ Mean waiting time of customers (excluding the service time) We have 1 λ ρ μ μ( μ λ) μ λ Chalk writing W = T = = = ρt The curve of T, W w.r.t. ρ Example, double λ and μ, how are L,T,W changing? 25
26 Another way to calculate mean performance metrics (MVA) Mean Value Analysis (MVA) Combine the Little s law and PASTA theorem No need to know the distribution of steady state Analysis process Seen by an arriving customer, mean response time is (should be equivalent to T by PASTA) T=L/μ+1/μ Little s law: L=λT Called arrival relation Combine to obtain: T= 1/(μ λ), L= ρ/(1 ρ), etc. 26
27 Distribution of sojourn time We focus on an arriving customer S: the sojourn time of the arriving customer L q : # of customers in the system seen by the arrival B k1 k : service time of the kth customer, k=1,, L q We have a L + 1 k = 1 S = B k Since B k and L q are independent, we further have a L + 1 n+ 1 k k k= 1 n= 0 k= 1 a P( S > t) = P B > t = P B > t P( L = n) 27
28 Distribution of sojourn time (cont.) By PASTA theorem, a n P( L = n) = π = (1 ρρ ) n+ 1 P B k > t is a n+1 stage Erlang distribution So, k = 1 n The sojourn time is exponentially distributed with parameter μ(1 ρ). Interesting! 28
29 Distribution of sojourn time (cont.) Another easy way is using Laplace transform Since we have The above is an exponential distribution with parameter μ(1 ρ). ) (1 ) t PS ( > t) = e μ ρ 29
30 Distribution of waiting time, W Since S=W+B, so We have W=0 with probability 1 ρ; W is exponentially distributed with parameter μ(1 ρ) with probability ρ. 30
31 Busy period # of customer t Busy Idle period period IP(idle period) is exponentially distributed with parameter λ Mean busy period: 1/ μ EBP ( ) = 1 ρ So, it equals T. EBP ( ) EBP ( ) = = ρ EBP ( ) + EIP ( ) EBP ( ) + 1/ λ 31
32 Distribution of busy period It is complicated use Laplace transform and recursion analysis Omitted for simplicity pdf of BP: Where I 1 (.) is the modified Bessel function of the first kind of order 1, 32
Elementary queueing system
Elementary queueing system Kendall notation Little s Law PASTA theorem Basics of M/M/1 queue M/M/1 with preemptive-resume priority M/M/1 with non-preemptive priority 1 History of queueing theory An old
More informationPart II: continuous time Markov chain (CTMC)
Part II: continuous time Markov chain (CTMC) Continuous time discrete state Markov process Definition (Markovian property) X(t) is a CTMC, if for any n and any sequence t 1
More informationPerformance Evaluation of Queuing Systems
Performance Evaluation of Queuing Systems Introduction to Queuing Systems System Performance Measures & Little s Law Equilibrium Solution of Birth-Death Processes Analysis of Single-Station Queuing Systems
More informationQueueing Theory I Summary! Little s Law! Queueing System Notation! Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K "
Queueing Theory I Summary Little s Law Queueing System Notation Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K " Little s Law a(t): the process that counts the number of arrivals
More informationIntroduction to Queueing Theory
Introduction to Queueing Theory Raj Jain Washington University in Saint Louis Jain@eecs.berkeley.edu or Jain@wustl.edu A Mini-Course offered at UC Berkeley, Sept-Oct 2012 These slides and audio/video recordings
More informationCPSC 531: System Modeling and Simulation. Carey Williamson Department of Computer Science University of Calgary Fall 2017
CPSC 531: System Modeling and Simulation Carey Williamson Department of Computer Science University of Calgary Fall 2017 Motivating Quote for Queueing Models Good things come to those who wait - poet/writer
More informationOutline. Finite source queue M/M/c//K Queues with impatience (balking, reneging, jockeying, retrial) Transient behavior Advanced Queue.
Outline Finite source queue M/M/c//K Queues with impatience (balking, reneging, jockeying, retrial) Transient behavior Advanced Queue Batch queue Bulk input queue M [X] /M/1 Bulk service queue M/M [Y]
More informationCDA5530: Performance Models of Computers and Networks. Chapter 4: Elementary Queuing Theory
CDA5530: Performance Models of Computers and Networks Chapter 4: Elementary Queuing Theory Definition Queuing system: a buffer (waiting room), service facility (one or more servers) a scheduling policy
More informationNon Markovian Queues (contd.)
MODULE 7: RENEWAL PROCESSES 29 Lecture 5 Non Markovian Queues (contd) For the case where the service time is constant, V ar(b) = 0, then the P-K formula for M/D/ queue reduces to L s = ρ + ρ 2 2( ρ) where
More informationIntroduction to Queueing Theory
Introduction to Queueing Theory Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: 30-1 Overview Queueing Notation
More informationQueueing systems. Renato Lo Cigno. Simulation and Performance Evaluation Queueing systems - Renato Lo Cigno 1
Queueing systems Renato Lo Cigno Simulation and Performance Evaluation 2014-15 Queueing systems - Renato Lo Cigno 1 Queues A Birth-Death process is well modeled by a queue Indeed queues can be used to
More informationFigure 10.1: Recording when the event E occurs
10 Poisson Processes Let T R be an interval. A family of random variables {X(t) ; t T} is called a continuous time stochastic process. We often consider T = [0, 1] and T = [0, ). As X(t) is a random variable
More informationAnalysis of A Single Queue
Analysis of A Single Queue Raj Jain Washington University in Saint Louis Jain@eecs.berkeley.edu or Jain@wustl.edu A Mini-Course offered at UC Berkeley, Sept-Oct 2012 These slides and audio/video recordings
More informationClassical Queueing Models.
Sergey Zeltyn January 2005 STAT 99. Service Engineering. The Wharton School. University of Pennsylvania. Based on: Classical Queueing Models. Mandelbaum A. Service Engineering course, Technion. http://iew3.technion.ac.il/serveng2005w
More informationQueueing Systems: Lecture 3. Amedeo R. Odoni October 18, Announcements
Queueing Systems: Lecture 3 Amedeo R. Odoni October 18, 006 Announcements PS #3 due tomorrow by 3 PM Office hours Odoni: Wed, 10/18, :30-4:30; next week: Tue, 10/4 Quiz #1: October 5, open book, in class;
More informationQueues and Queueing Networks
Queues and Queueing Networks Sanjay K. Bose Dept. of EEE, IITG Copyright 2015, Sanjay K. Bose 1 Introduction to Queueing Models and Queueing Analysis Copyright 2015, Sanjay K. Bose 2 Model of a Queue Arrivals
More informationQUEUING SYSTEM. Yetunde Folajimi, PhD
QUEUING SYSTEM Yetunde Folajimi, PhD Part 2 Queuing Models Queueing models are constructed so that queue lengths and waiting times can be predicted They help us to understand and quantify the effect of
More informationGI/M/1 and GI/M/m queuing systems
GI/M/1 and GI/M/m queuing systems Dmitri A. Moltchanov moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/tlt-2716/ OUTLINE: GI/M/1 queuing system; Methods of analysis; Imbedded Markov chain approach; Waiting
More informationM/G/1 and M/G/1/K systems
M/G/1 and M/G/1/K systems Dmitri A. Moltchanov dmitri.moltchanov@tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Description of M/G/1 system; Methods of analysis; Residual life approach; Imbedded
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/6/16. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 24 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/6/16 2 / 24 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationIntroduction to Queueing Theory
Introduction to Queueing Theory Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse567-11/
More informationChapter 10. Queuing Systems. D (Queuing Theory) Queuing theory is the branch of operations research concerned with waiting lines.
Chapter 10 Queuing Systems D. 10. 1. (Queuing Theory) Queuing theory is the branch of operations research concerned with waiting lines. D. 10.. (Queuing System) A ueuing system consists of 1. a user source.
More informationBIRTH DEATH PROCESSES AND QUEUEING SYSTEMS
BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS Andrea Bobbio Anno Accademico 999-2000 Queueing Systems 2 Notation for Queueing Systems /λ mean time between arrivals S = /µ ρ = λ/µ N mean service time traffic
More informationQueueing Theory II. Summary. ! M/M/1 Output process. ! Networks of Queue! Method of Stages. ! General Distributions
Queueing Theory II Summary! M/M/1 Output process! Networks of Queue! Method of Stages " Erlang Distribution " Hyperexponential Distribution! General Distributions " Embedded Markov Chains M/M/1 Output
More informationLink Models for Circuit Switching
Link Models for Circuit Switching The basis of traffic engineering for telecommunication networks is the Erlang loss function. It basically allows us to determine the amount of telephone traffic that can
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/25/17. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 26 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/25/17 2 / 26 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationComputer Networks More general queuing systems
Computer Networks More general queuing systems Saad Mneimneh Computer Science Hunter College of CUNY New York M/G/ Introduction We now consider a queuing system where the customer service times have a
More informationData analysis and stochastic modeling
Data analysis and stochastic modeling Lecture 7 An introduction to queueing theory Guillaume Gravier guillaume.gravier@irisa.fr with a lot of help from Paul Jensen s course http://www.me.utexas.edu/ jensen/ormm/instruction/powerpoint/or_models_09/14_queuing.ppt
More informationBuzen s algorithm. Cyclic network Extension of Jackson networks
Outline Buzen s algorithm Mean value analysis for Jackson networks Cyclic network Extension of Jackson networks BCMP network 1 Marginal Distributions based on Buzen s algorithm With Buzen s algorithm,
More informationSlides 9: Queuing Models
Slides 9: Queuing Models Purpose Simulation is often used in the analysis of queuing models. A simple but typical queuing model is: Queuing models provide the analyst with a powerful tool for designing
More informationNetworking = Plumbing. Queueing Analysis: I. Last Lecture. Lecture Outline. Jeremiah Deng. 29 July 2013
Networking = Plumbing TELE302 Lecture 7 Queueing Analysis: I Jeremiah Deng University of Otago 29 July 2013 Jeremiah Deng (University of Otago) TELE302 Lecture 7 29 July 2013 1 / 33 Lecture Outline Jeremiah
More informationIntroduction to queuing theory
Introduction to queuing theory Queu(e)ing theory Queu(e)ing theory is the branch of mathematics devoted to how objects (packets in a network, people in a bank, processes in a CPU etc etc) join and leave
More informationChapter 6 Queueing Models. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter 6 Queueing Models Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Purpose Simulation is often used in the analysis of queueing models. A simple but typical queueing model: Queueing
More informationPBW 654 Applied Statistics - I Urban Operations Research
PBW 654 Applied Statistics - I Urban Operations Research Lecture 2.I Queuing Systems An Introduction Operations Research Models Deterministic Models Linear Programming Integer Programming Network Optimization
More informationDerivation of Formulas by Queueing Theory
Appendices Spectrum Requirement Planning in Wireless Communications: Model and Methodology for IMT-Advanced E dite d by H. Takagi and B. H. Walke 2008 J ohn Wiley & Sons, L td. ISBN: 978-0-470-98647-9
More informationClassification of Queuing Models
Classification of Queuing Models Generally Queuing models may be completely specified in the following symbol form:(a/b/c):(d/e)where a = Probability law for the arrival(or inter arrival)time, b = Probability
More informationContents Preface The Exponential Distribution and the Poisson Process Introduction to Renewal Theory
Contents Preface... v 1 The Exponential Distribution and the Poisson Process... 1 1.1 Introduction... 1 1.2 The Density, the Distribution, the Tail, and the Hazard Functions... 2 1.2.1 The Hazard Function
More informationSystems Simulation Chapter 6: Queuing Models
Systems Simulation Chapter 6: Queuing Models Fatih Cavdur fatihcavdur@uludag.edu.tr April 2, 2014 Introduction Introduction Simulation is often used in the analysis of queuing models. A simple but typical
More informationCS418 Operating Systems
CS418 Operating Systems Lecture 14 Queuing Analysis Textbook: Operating Systems by William Stallings 1 1. Why Queuing Analysis? If the system environment changes (like the number of users is doubled),
More informationIntroduction to Queuing Theory. Mathematical Modelling
Queuing Theory, COMPSCI 742 S2C, 2014 p. 1/23 Introduction to Queuing Theory and Mathematical Modelling Computer Science 742 S2C, 2014 Nevil Brownlee, with acknowledgements to Peter Fenwick, Ulrich Speidel
More informationIntroduction to Markov Chains, Queuing Theory, and Network Performance
Introduction to Markov Chains, Queuing Theory, and Network Performance Marceau Coupechoux Telecom ParisTech, departement Informatique et Réseaux marceau.coupechoux@telecom-paristech.fr IT.2403 Modélisation
More informationLecture 7: Simulation of Markov Processes. Pasi Lassila Department of Communications and Networking
Lecture 7: Simulation of Markov Processes Pasi Lassila Department of Communications and Networking Contents Markov processes theory recap Elementary queuing models for data networks Simulation of Markov
More informationQueueing Theory. VK Room: M Last updated: October 17, 2013.
Queueing Theory VK Room: M1.30 knightva@cf.ac.uk www.vincent-knight.com Last updated: October 17, 2013. 1 / 63 Overview Description of Queueing Processes The Single Server Markovian Queue Multi Server
More informationChapter 5: Special Types of Queuing Models
Chapter 5: Special Types of Queuing Models Some General Queueing Models Discouraged Arrivals Impatient Arrivals Bulk Service and Bulk Arrivals OR37-Dr.Khalid Al-Nowibet 1 5.1 General Queueing Models 1.
More informationINDEX. production, see Applications, manufacturing
INDEX Absorbing barriers, 103 Ample service, see Service, ample Analyticity, of generating functions, 100, 127 Anderson Darling (AD) test, 411 Aperiodic state, 37 Applications, 2, 3 aircraft, 3 airline
More informationPerformance Modelling of Computer Systems
Performance Modelling of Computer Systems Mirco Tribastone Institut für Informatik Ludwig-Maximilians-Universität München Fundamentals of Queueing Theory Tribastone (IFI LMU) Performance Modelling of Computer
More informationIntroduction to queuing theory
Introduction to queuing theory Claude Rigault ENST claude.rigault@enst.fr Introduction to Queuing theory 1 Outline The problem The number of clients in a system The client process Delay processes Loss
More informationM/G/1 and Priority Queueing
M/G/1 and Priority Queueing Richard T. B. Ma School of Computing National University of Singapore CS 5229: Advanced Compute Networks Outline PASTA M/G/1 Workload and FIFO Delay Pollaczek Khinchine Formula
More informationContinuous Time Processes
page 102 Chapter 7 Continuous Time Processes 7.1 Introduction In a continuous time stochastic process (with discrete state space), a change of state can occur at any time instant. The associated point
More informationJ. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY
J. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY SECOND EDITION ACADEMIC PRESS An imprint of Elsevier Science Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Contents
More information5/15/18. Operations Research: An Introduction Hamdy A. Taha. Copyright 2011, 2007 by Pearson Education, Inc. All rights reserved.
The objective of queuing analysis is to offer a reasonably satisfactory service to waiting customers. Unlike the other tools of OR, queuing theory is not an optimization technique. Rather, it determines
More informationChapter 2 Queueing Theory and Simulation
Chapter 2 Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University,
More informationWaiting Line Models: Queuing Theory Basics. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1
Waiting Line Models: Queuing Theory Basics Cuantitativos M. En C. Eduardo Bustos Farias 1 Agenda Queuing system structure Performance measures Components of queuing systems Arrival process Service process
More informationClass 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis.
Service Engineering Class 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis. G/G/1 Queue: Virtual Waiting Time (Unfinished Work). GI/GI/1: Lindley s Equations
More informationλ, µ, ρ, A n, W n, L(t), L, L Q, w, w Q etc. These
Queuing theory models systems with servers and clients (presumably waiting in queues). Notation: there are many standard symbols like λ, µ, ρ, A n, W n, L(t), L, L Q, w, w Q etc. These represent the actual
More informationThe effect of probabilities of departure with time in a bank
International Journal of Scientific & Engineering Research, Volume 3, Issue 7, July-2012 The effect of probabilities of departure with time in a bank Kasturi Nirmala, Shahnaz Bathul Abstract This paper
More informationP (L d k = n). P (L(t) = n),
4 M/G/1 queue In the M/G/1 queue customers arrive according to a Poisson process with rate λ and they are treated in order of arrival The service times are independent and identically distributed with
More informationAn M/M/1/N Queuing system with Encouraged Arrivals
Global Journal of Pure and Applied Mathematics. ISS 0973-1768 Volume 13, umber 7 (2017), pp. 3443-3453 Research India Publications http://www.ripublication.com An M/M/1/ Queuing system with Encouraged
More informationSQF: A slowdown queueing fairness measure
Performance Evaluation 64 (27) 1121 1136 www.elsevier.com/locate/peva SQF: A slowdown queueing fairness measure Benjamin Avi-Itzhak a, Eli Brosh b,, Hanoch Levy c a RUTCOR, Rutgers University, New Brunswick,
More information6 Solving Queueing Models
6 Solving Queueing Models 6.1 Introduction In this note we look at the solution of systems of queues, starting with simple isolated queues. The benefits of using predefined, easily classified queues will
More informationIntroduction to Queueing Theory with Applications to Air Transportation Systems
Introduction to Queueing Theory with Applications to Air Transportation Systems John Shortle George Mason University February 28, 2018 Outline Why stochastic models matter M/M/1 queue Little s law Priority
More informationAnalysis of Software Artifacts
Analysis of Software Artifacts System Performance I Shu-Ngai Yeung (with edits by Jeannette Wing) Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 2001 by Carnegie Mellon University
More informationSimple queueing models
Simple queueing models c University of Bristol, 2012 1 M/M/1 queue This model describes a queue with a single server which serves customers in the order in which they arrive. Customer arrivals constitute
More informationPart I Stochastic variables and Markov chains
Part I Stochastic variables and Markov chains Random variables describe the behaviour of a phenomenon independent of any specific sample space Distribution function (cdf, cumulative distribution function)
More information4.7 Finite Population Source Model
Characteristics 1. Arrival Process R independent Source All sources are identical Interarrival time is exponential with rate for each source No arrivals if all sources are in the system. OR372-Dr.Khalid
More informationIOE 202: lectures 11 and 12 outline
IOE 202: lectures 11 and 12 outline Announcements Last time... Queueing models intro Performance characteristics of a queueing system Steady state analysis of an M/M/1 queueing system Other queueing systems,
More informationStochastic process. X, a series of random variables indexed by t
Stochastic process X, a series of random variables indexed by t X={X(t), t 0} is a continuous time stochastic process X={X(t), t=0,1, } is a discrete time stochastic process X(t) is the state at time t,
More informationQueuing Theory and Stochas St t ochas ic Service Syste y ms Li Xia
Queuing Theory and Stochastic Service Systems Li Xia Syllabus Instructor Li Xia 夏俐, FIT 3 618, 62793029, xial@tsinghua.edu.cn Text book D. Gross, J.F. Shortle, J.M. Thompson, and C.M. Harris, Fundamentals
More informationAll models are wrong / inaccurate, but some are useful. George Box (Wikipedia). wkc/course/part2.pdf
PART II (3) Continuous Time Markov Chains : Theory and Examples -Pure Birth Process with Constant Rates -Pure Death Process -More on Birth-and-Death Process -Statistical Equilibrium (4) Introduction to
More informationExercises Stochastic Performance Modelling. Hamilton Institute, Summer 2010
Exercises Stochastic Performance Modelling Hamilton Institute, Summer Instruction Exercise Let X be a non-negative random variable with E[X ]
More informationName of the Student:
SUBJECT NAME : Probability & Queueing Theory SUBJECT CODE : MA 6453 MATERIAL NAME : Part A questions REGULATION : R2013 UPDATED ON : November 2017 (Upto N/D 2017 QP) (Scan the above QR code for the direct
More informationQueueing Theory and Simulation. Introduction
Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University, Japan
More informationZáklady teorie front
Základy teorie front Mgr. Rudolf B. Blažek, Ph.D. prof. RNDr. Roman Kotecký, DrSc. Katedra počítačových systémů Katedra teoretické informatiky Fakulta informačních technologií České vysoké učení technické
More informationLink Models for Packet Switching
Link Models for Packet Switching To begin our study of the performance of communications networks, we will study a model of a single link in a message switched network. The important feature of this model
More informationAdvanced Computer Networks Lecture 3. Models of Queuing
Advanced Computer Networks Lecture 3. Models of Queuing Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/13 Terminology of
More informationQueuing Theory. Queuing Theory. Fatih Cavdur April 27, 2015
Queuing Theory Fatih Cavdur fatihcavdur@uludag.edu.tr April 27, 2015 Introduction Introduction Simulation is often used in the analysis of queuing models. A simple but typical model is the single-server
More informationQueuing Theory. 3. Birth-Death Process. Law of Motion Flow balance equations Steady-state probabilities: , if
1 Queuing Theory 3. Birth-Death Process Law of Motion Flow balance equations Steady-state probabilities: c j = λ 0λ 1...λ j 1 µ 1 µ 2...µ j π 0 = 1 1+ j=1 c j, if j=1 c j is finite. π j = c j π 0 Example
More informationReview of Queuing Models
Review of Queuing Models Recitation, Apr. 1st Guillaume Roels 15.763J Manufacturing System and Supply Chain Design http://michael.toren.net/slides/ipqueue/slide001.html 2005 Guillaume Roels Outline Overview,
More informationStochastic Models of Manufacturing Systems
Stochastic Models of Manufacturing Systems Ivo Adan Systems 2/49 Continuous systems State changes continuously in time (e.g., in chemical applications) Discrete systems State is observed at fixed regular
More information11 The M/G/1 system with priorities
11 The M/G/1 system with priorities In this chapter we analyse queueing models with different types of customers, where one or more types of customers have priority over other types. More precisely we
More informationGlossary availability cellular manufacturing closed queueing network coefficient of variation (CV) conditional probability CONWIP
Glossary availability The long-run average fraction of time that the processor is available for processing jobs, denoted by a (p. 113). cellular manufacturing The concept of organizing the factory into
More informationYORK UNIVERSITY FACULTY OF ARTS DEPARTMENT OF MATHEMATICS AND STATISTICS MATH , YEAR APPLIED OPTIMIZATION (TEST #4 ) (SOLUTIONS)
YORK UNIVERSITY FACULTY OF ARTS DEPARTMENT OF MATHEMATICS AND STATISTICS Instructor : Dr. Igor Poliakov MATH 4570 6.0, YEAR 2006-07 APPLIED OPTIMIZATION (TEST #4 ) (SOLUTIONS) March 29, 2007 Name (print)
More informationChapter 1. Introduction. 1.1 Stochastic process
Chapter 1 Introduction Process is a phenomenon that takes place in time. In many practical situations, the result of a process at any time may not be certain. Such a process is called a stochastic process.
More informationNICTA Short Course. Network Analysis. Vijay Sivaraman. Day 1 Queueing Systems and Markov Chains. Network Analysis, 2008s2 1-1
NICTA Short Course Network Analysis Vijay Sivaraman Day 1 Queueing Systems and Markov Chains Network Analysis, 2008s2 1-1 Outline Why a short course on mathematical analysis? Limited current course offering
More informationLecture 20: Reversible Processes and Queues
Lecture 20: Reversible Processes and Queues 1 Examples of reversible processes 11 Birth-death processes We define two non-negative sequences birth and death rates denoted by {λ n : n N 0 } and {µ n : n
More informationQueuing Theory. Richard Lockhart. Simon Fraser University. STAT 870 Summer 2011
Queuing Theory Richard Lockhart Simon Fraser University STAT 870 Summer 2011 Richard Lockhart (Simon Fraser University) Queuing Theory STAT 870 Summer 2011 1 / 15 Purposes of Today s Lecture Describe general
More informationQueuing Theory. The present section focuses on the standard vocabulary of Waiting Line Models.
Queuing Theory Introduction Waiting lines are the most frequently encountered problems in everyday life. For example, queue at a cafeteria, library, bank, etc. Common to all of these cases are the arrivals
More informationIntro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks. Queuing Networks. Florence Perronnin
Queuing Networks Florence Perronnin Polytech Grenoble - UGA March 23, 27 F. Perronnin (UGA) Queuing Networks March 23, 27 / 46 Outline Introduction to Queuing Networks 2 Refresher: M/M/ queue 3 Reversibility
More information10.2 For the system in 10.1, find the following statistics for population 1 and 2. For populations 2, find: Lq, Ls, L, Wq, Ws, W, Wq 0 and SL.
Bibliography Asmussen, S. (2003). Applied probability and queues (2nd ed). New York: Springer. Baccelli, F., & Bremaud, P. (2003). Elements of queueing theory: Palm martingale calculus and stochastic recurrences
More informationQueuing Networks: Burke s Theorem, Kleinrock s Approximation, and Jackson s Theorem. Wade Trappe
Queuing Networks: Burke s Theorem, Kleinrock s Approximation, and Jackson s Theorem Wade Trappe Lecture Overview Network of Queues Introduction Queues in Tandem roduct Form Solutions Burke s Theorem What
More informationSince D has an exponential distribution, E[D] = 0.09 years. Since {A(t) : t 0} is a Poisson process with rate λ = 10, 000, A(0.
IEOR 46: Introduction to Operations Research: Stochastic Models Chapters 5-6 in Ross, Thursday, April, 4:5-5:35pm SOLUTIONS to Second Midterm Exam, Spring 9, Open Book: but only the Ross textbook, the
More informationWaiting Time Analysis of A Single Server Queue in an Out- Patient Clinic
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 3 Ver. V (May - Jun. 2015), PP 54-58 www.iosrjournals.org Waiting Time Analysis of A Single Server Queue in
More informationIntro to Queueing Theory
1 Intro to Queueing Theory Little s Law M/G/1 queue Conservation Law 1/31/017 M/G/1 queue (Simon S. Lam) 1 Little s Law No assumptions applicable to any system whose arrivals and departures are observable
More informationON THE LAW OF THE i TH WAITING TIME INABUSYPERIODOFG/M/c QUEUES
Probability in the Engineering and Informational Sciences, 22, 2008, 75 80. Printed in the U.S.A. DOI: 10.1017/S0269964808000053 ON THE LAW OF THE i TH WAITING TIME INABUSYPERIODOFG/M/c QUEUES OPHER BARON
More informationNATCOR: Stochastic Modelling
NATCOR: Stochastic Modelling Queueing Theory II Chris Kirkbride Management Science 2017 Overview of Today s Sessions I Introduction to Queueing Modelling II Multiclass Queueing Models III Queueing Control
More information57:022 Principles of Design II Final Exam Solutions - Spring 1997
57:022 Principles of Design II Final Exam Solutions - Spring 1997 Part: I II III IV V VI Total Possible Pts: 52 10 12 16 13 12 115 PART ONE Indicate "+" if True and "o" if False: + a. If a component's
More information7 Variance Reduction Techniques
7 Variance Reduction Techniques In a simulation study, we are interested in one or more performance measures for some stochastic model. For example, we want to determine the long-run average waiting time,
More informationBulk input queue M [X] /M/1 Bulk service queue M/M [Y] /1 Erlangian queue M/E k /1
Advanced Markovian queues Bulk input queue M [X] /M/ Bulk service queue M/M [Y] / Erlangian queue M/E k / Bulk input queue M [X] /M/ Batch arrival, Poisson process, arrival rate λ number of customers in
More informationChapter 2. Poisson Processes. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 2. Poisson Processes Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Outline Introduction to Poisson Processes Definition of arrival process Definition
More informationA Study on Performance Analysis of Queuing System with Multiple Heterogeneous Servers
UNIVERSITY OF OKLAHOMA GENERAL EXAM REPORT A Study on Performance Analysis of Queuing System with Multiple Heterogeneous Servers Prepared by HUSNU SANER NARMAN husnu@ou.edu based on the papers 1) F. S.
More information