Základy teorie front II
|
|
- Phyllis Casey
- 5 years ago
- Views:
Transcription
1 Základy teorie front II Aplikace Poissonova procesu v teorii 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 v Praze Rudolf Blažek & Roman Kotecký, 2011 Statistika pro informatiku MI-SPI, ZS 2011/12, Přednáška 17 Evropský sociální fond Praha & EU: Investujeme do vaší budoucnos@
2 Introduction to Queueing Theory II The Poisson Process in Queueing Theory Mgr. Rudolf B. Blažek, Ph.D. prof. RNDr. Roman Kotecký, DrSc. Department of Computer Systems Department of Theoretical Informatics Faculty of Information Technologies Czech Technical University in Prague Rudolf Blažek & Roman Kotecký, 2011 Statistics for Informatics MI-SPI, ZS 2011/12, Lecture 17 The European Social Fund Prague & EU: We Invest in Your Future
3 Queueing Theory Review Queueing System Diagram Fronta Waiting Queue Obsluha Service Facility Customer Population Arriving Customers Vstupní tok požadavků Departing Customers Výstupní tok požadavků Prvky systému hromadné obsluhy 3
4 Queueing Theory Review Kendall Notation A / B / X / Y / Z A:( Customer arrival pattern ( ( (Interarrival time distribution) B:( Service pattern (Service time distribution) X:( Number of parallel servers Y: ( System capacity Z:( Queueing discipline Default values:( Y =, Z = FCFS Example: ( M / M / 3 = M / M / 3 / / FCFS ( ( ( ( (Poisson arrivals, Exp. service times, 3 servers) 4
5 Queueing Theory Review Characteristics of the Input Process Arriving patterns M: ( Markovian or Memoryless Poisson Process ( ( (I.e. exponential & independent interarrival times) D:( Ek:( G:( GI:( Deterministic, constant interarrival times Erlang distribution of order k of interarrival times General probability distribution of interarrival times General & Independent distribution of interarrival times Default Assumption: Poisson Process Charakteristiky vstupního toku požadavků 5
6 Queueing Theory Review Characteristics of the Output Process Service time distribution M: ( Markovian or Memoryless exponential service times D:( Deterministic, constant service times Ek:( Erlang distribution of order k of service times G:( General probability distribution of service times Default Assumption: Exponential service times Charakteristiky výstupního toku požadavků 6
7 Queueing Theory Review Queueing System M/M/m Infinite FCFS Queue m Parallel Servers w/ Exp(μ) service time Infinite Customer Population Arriving Customers Poisson Process w/ rate λ Departing Customers 7
8 Queueing Theory Review Web and Database Servers Example Pool of m application servers (e.g. Tomcat) submits a job to a central database server Application server 1 Application server 2 Central Database Server Application server m 8
9 Queueing Theory Review Web and Database Servers Example We assume Poisson arrival process (λ) for the requests. Scenario 1: Application servers can submit multiple requests We have m application servers Then we obtain a Poisson arrival process with the rate ( ( ( ( ( ( ( ( ( μ = m λ We will see later why... 9
10 Queueing Theory Review Web and Database Servers Example Case 1: Application servers can submit multiple requests Application server 1 Rate λ Application server 2 Rate λ Rate mλ Central Database Server Application server m Rate λ 10
11 Queueing Theory Review Web and Database Servers Example We assume Poisson arrival process (λ) for the requests. Scenario 2: Application servers must wait for their request to finish State k: If k servers are waiting for their requests to finish, then only (m-k) servers can submit requests Then we obtain a state-dependent Poisson arrival process with the rate (k) = (m k) k < m 0 k m 11
12 Queueing Theory Review Plan of Study We will focus on M/M/m systems We must therefore study The Exponential Distribution (interarrival & service times) The Poisson Process (interarrival times are Exponential) Birth & Death Markov chains with continuous time (the number of customers in the system) We will also look at a M/G/ system Poisson arrivals, General service time, many servers 12
13 The Poisson Process 13
14 Defining the Poisson Process by Number of arrivals N(t) = number of arrivals during (0,t) N(t)... Poisson Process with rate λ t3 ~ Exp(λ) t4 ~ Exp(λ) t1 ~ Exp(λ) t2 ~ Exp(λ) t5 ~ Exp(λ) t5 > t -T4 N(t) = 4 0 T1 T2 T3 T4 t Time t Tk(...( arrival time of the k th customer tk ~ Exp(λ)(...( independent exponential interarrival times 14
15 Exponential Distribution Definition!!!!!!!!!!!!!! T ~ Exp(λ) A random variable T has an exponential distribution with rate λ if its density is e t for t 0 f T (t) = 0 for t < 0. We can also write the definition in terms of the distribution function F T (t) =P(T apple t) =1 e t, 8t 0 or in terms of the survival function P(T > t) =e t, 8t 0. 15
16 Main Properties of the Exponential Distribution Properties of T ~ Exp(λ) If T ~ Exp(λ) then ET =1/ ET 2 =2/ 2 Var T = ET 2 (ET ) 2 =1/ 2 Example: Average arrival rate: λ = 10 arrivals per minute Average wait time between arrivals: 1/10 = 0.1 minutes 16
17 Exponential Distribution Lack of Memory Property Lemma Assume waiting time T ~ Exp(λ). Given that we already waited s units of time, the remaining waiting time T s has the same distribution as if we did not wait at all, i.e. Exp(λ). Abbreviated notation: T ~ Exp(λ) (T s T>s) ~ Exp(λ) We waited s, but no arrival Conditional remaining waiting time We say: Exponential distribution is Memoryless 17
18 Exponential Distribution Lack of Memory Property Lack of memory: Conditional remaining waiting time: (T-s T > s) ~ Exp(λ) Remaining waiting time: T-s Original waiting time: T ~ Exp(λ) 0 s Time t Fix time s and observe that there was no arrival... we know T > s 18
19 Lack of Memory & the Poisson Process Number of arrivals N(t) = number of arrivals during (0,t) N(t)... Poisson Process with rate λ t3 ~ Exp(λ) t4 ~ Exp(λ) t1 ~ Exp(λ) t2 ~ Exp(λ) t5 ~ Exp(λ) t5 > t -T4 N(t) = 4 0 T1 T2 T3 T4 Different notation: Tk( ( ( (...( arrival time of the k th customer t Time t tk ~ Exp(λ)(...( independent exponential interarrival times 19
20 Lack of Memory & the Poisson Process Number of arrivals Conditional remaining waiting time: (t5-s t5 > s)~ Exp(λ) Original waiting time: t5 ~ Exp(λ) Given: t5 > t -T4 Given: t5 > s s = t -T4 0 T4 T1 T2 T3 t Time t Tk( ( ( (...( arrival time of the k th customer tk ~ Exp(λ)(...( independent exponential interarrival times 20
21 Proof of the Lack of Memory Property We want to prove: T ~ Exp(λ) (T s T>s) ~ Exp(λ) It s easier to use the survival function: T Exp( ) if and only if P(T > t) =e t, 8t 0. P(T s > t T > s) 8s, t 0 = = P(T > t + s, T > s) P(T > s) P(T > t + s) = e (t+s) P(T > s) e s = e t... Exp( ). 21
22 Exponential Races Lemma Let S ~ Exp(λ) and T ~ Exp(μ) be independent. Then min(s, T ) Exp( + µ). Proof: P(min(S, T ) > t) =P(S > t, T > t) =P(S > t)p(t > t) = e t e µt = e ( +µ)t... Exp( + µ). 22
23 Exponential Races & Queueing Systems Line 1 Independent waiting times Waiting time for arrival S ~ Exp(λ) Waiting time for arrival V = min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 T ~ Exp(μ) Waiting time for arrival 23
24 Exponential Races & Queueing Systems E.g. average rate of arrivals λ = 3/min Line 1 Waiting S ~ Exp(λ) Independent waiting times Line 2 Waiting T ~ Exp(μ) E.g. average rate of arrivals μ = 5/min Waiting min(s, T ) ~ Exp(λ+μ) Then the average rate of arrivals is λ + μ = 3+5 = 8 / min 24
25 Exponential Races & Queueing Systems Line 1 Waiting S ~ Exp(λ) Independent waiting times Waiting min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 Waiting T ~ Exp(μ) OK, the first arrival has exponential waiting time But how about the next arrivals? Is it a Poisson Process? Are all interarrival times exponential? Are all interarrival times independent? 25
26 Exponential Races & Lack of Memory Waiting for arrival on line 1 or 2 Line 1 s1 ~ Exp(λ) s2 ~ Exp(λ)... independent Line 2 Merged Line t1 ~ Exp(μ) v1 =min(s1, t1) v1 ~ Exp(λ+ μ) 0 v t 1 = (t1-v t1 > v) ~ Exp(μ)... independent v2 = min(s2, t 1) ~ Exp(λ+ μ) Time t Independent interarrival times ~ Exp(λ+ μ) Poisson Process (λ+ μ) Arrival observed at time v. Assume it was from line 1. 26
27 Example Web and Database Servers Example Case 1: Application servers can submit multiple requests Application server 1 Rate λ Application server 2 Rate λ Rate mλ Central Database Server Application server m Rate λ If the Poisson Processes are independent then the we obtain a Poisson Process on the merged line 27
28 The Winner of an Exponential Race Lemma Let S ~ Exp(λ) and T ~ Exp(μ) be independent. Then the probability that S arrives first is P(S < T )= + µ P(T < S) = µ + µ 28
29 The Winner of an Exponential Race & Queueing Systems Line 1 Waiting S ~ Exp(λ) Independent waiting times Waiting min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 Waiting T ~ Exp(μ) An arrival is observed on the merged line. Where from? µ P(From Line 1) = + µ P(From Line 2) = + µ 29
30 The Winner of an Exponential Race & Queueing Systems Line 1 S ~ Exp(λ), λ = 3/min Independent waiting times λ + μ = 3+5 = 8 / min min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 T ~ Exp(μ), μ = 5/min An arrival is observed on the merged line. Where from? µ P(Line 1) = + µ = 3 8 P(Line 2) = + µ =
31 Exponential Races for n Variables Corollary Let Ti ~ Exp(λi) be independent, i = 1, 2,...,n. Then min(t 1,..., T n ) Exp( n ). Proof is very similar as for two variables 31
32 The Winner of an Exponential Race Corollary Let Ti ~ Exp(λi) be independent, i = 1, 2,...,n. Then the probability that Tk arrives first is P(T k = min(t 1,..., T n )) = k n 32
33 The Poisson Process Definition!!!!!! Poissonův Process s intenzitou λ Let ti ~ Exp(λ) be independent random variables, i = 1, 2,... Let Tn = t1 + t tn with T0 = 0, and define N(s) = max {n: Tn s} for all s 0. Then N(s) is called the Poisson Process with rate λ. ti ~ Exp(λ)(...( independent exponential interarrival times Tn( ( (...( arrival time of the n th customer N(s)( (...( number of arrivals during time interval (0,s) 33
34 Definition and Basic Properties Defining the Poisson Process by Number of arrivals N(s) = number of arrivals during (0,s) N(s)... Poisson Process with rate λ t1 ~ Exp(λ) t2 ~ Exp(λ) t3 ~ Exp(λ) t4 ~ Exp(λ) t5 ~ Exp(λ) t5 > s -T4 N(s) = max {n: Tn s} N(s) = max {1,2,3,4} N(s) = 4 0 T2 T4 T1 =t1 T3 =t1+t2+t3 (T2 =t1+t2) (T4 =t1+t2+t3+t4) s Time t ti ~ Exp(λ)(...( independent exponential interarrival times Tn( ( ( (...( arrival time of the n th customer 34
35 Poisson Processes Definition and Basic Properties The Poisson Distribution Lemma N(s) has a Poisson distribution with mean λs. Definition!!!!!!!!!!!! X ~ Poisson(μ) A random variable X has a Poisson distribution with mean μ if P(X = n) =e µ µn for n = 0, 1,... n! 35
Základy 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 informationMarkovské řetězce se spojitým parametrem
Markovské řetězce se spojitým parametrem Mgr. Rudolf B. Blažek, Ph.D. prof. RNDr. Roman Kotecký, DrSc. Katedra počítačových systémů Katedra teoretické informatiky Fakulta informačních technologií České
More informationBootstrap metody II Kernelové Odhady Hustot
Bootstrap metody II Kernelové Odhady Hustot Mgr. Rudolf B. Blažek, Ph.D. prof. RNDr. Roman Kotecký, DrSc. Katedra počítačových systémů Katedra teoretické informatiky Fakulta informačních technologií České
More informationStatistika pro informatiku
Statistika pro informatiku prof. RNDr. Roman Kotecký DrSc., Dr. Rudolf Blažek, PhD Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-SPI, ZS 2011/12, Přednáška 5 Evropský sociální
More informationStatistika pro informatiku
Statistika pro informatiku prof. RNDr. Roman Kotecký DrSc., Dr. Rudolf Blažek, PhD Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-SPI, ZS 2011/12, Přednáška 2 Evropský sociální
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 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 informationQuantum computing. Jan Černý, FIT, Czech Technical University in Prague. České vysoké učení technické v Praze. Fakulta informačních technologií
České vysoké učení technické v Praze Fakulta informačních technologií Katedra teoretické informatiky Evropský sociální fond Praha & EU: Investujeme do vaší budoucnosti MI-MVI Methods of Computational Intelligence(2010/2011)
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 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 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 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 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 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 informationStatistika pro informatiku
Statistika pro informatiku prof. RNDr. Roman Kotecký DrSc., Dr. Rudolf Blažek, PhD Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-SPI, ZS 2011/12, Přednáška 1 Evropský sociální
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 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 informationLECTURE #6 BIRTH-DEATH PROCESS
LECTURE #6 BIRTH-DEATH PROCESS 204528 Queueing Theory and Applications in Networks Assoc. Prof., Ph.D. (รศ.ดร. อน นต ผลเพ ม) Computer Engineering Department, Kasetsart University Outline 2 Birth-Death
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 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 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 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 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 informationQueuing Theory. Using the Math. Management Science
Queuing Theory Using the Math 1 Markov Processes (Chains) A process consisting of a countable sequence of stages, that can be judged at each stage to fall into future states independent of how the process
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 informationReadings: Finish Section 5.2
LECTURE 19 Readings: Finish Section 5.2 Lecture outline Markov Processes I Checkout counter example. Markov process: definition. -step transition probabilities. Classification of states. Example: Checkout
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 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 informationMarkov processes and queueing networks
Inria September 22, 2015 Outline Poisson processes Markov jump processes Some queueing networks The Poisson distribution (Siméon-Denis Poisson, 1781-1840) { } e λ λ n n! As prevalent as Gaussian distribution
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 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 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 informationIEOR 6711, HMWK 5, Professor Sigman
IEOR 6711, HMWK 5, Professor Sigman 1. Semi-Markov processes: Consider an irreducible positive recurrent discrete-time Markov chain {X n } with transition matrix P (P i,j ), i, j S, and finite state space.
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 informationPhoto: US National Archives
ESD.86. Markov Processes and their Application to Queueing II Richard C. Larson March 7, 2007 Photo: US National Archives Outline Little s Law, one more time PASTA treat Markov Birth and Death Queueing
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 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 informationCole s MergeSort. prof. Ing. Pavel Tvrdík CSc. Fakulta informačních technologií České vysoké učení technické v Praze c Pavel Tvrdík, 2010
Cole s MergeSort prof. Ing. Pavel Tvrdík CSc. Katedra počítačových systémů Fakulta informačních technologií České vysoké učení technické v Praze c Pavel Tvrdík, 2010 Pokročilé paralelní algoritmy (PI-PPA)
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 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 informationIntroduction to Queuing Networks Solutions to Problem Sheet 3
Introduction to Queuing Networks Solutions to Problem Sheet 3 1. (a) The state space is the whole numbers {, 1, 2,...}. The transition rates are q i,i+1 λ for all i and q i, for all i 1 since, when a bus
More informationThe Transition Probability Function P ij (t)
The Transition Probability Function P ij (t) Consider a continuous time Markov chain {X(t), t 0}. We are interested in the probability that in t time units the process will be in state j, given that it
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 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 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 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 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 informationChapter 3 Balance equations, birth-death processes, continuous Markov Chains
Chapter 3 Balance equations, birth-death processes, continuous Markov Chains Ioannis Glaropoulos November 4, 2012 1 Exercise 3.2 Consider a birth-death process with 3 states, where the transition rate
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 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 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 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 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 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 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 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 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 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 informationThe exponential distribution and the Poisson process
The exponential distribution and the Poisson process 1-1 Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T]
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 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 informationPoisson Processes. Stochastic Processes. Feb UC3M
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationComputational intelligence methods
Computational intelligence methods GA, schemas, diversity Pavel Kordík, Martin Šlapák Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-MVI, ZS 2011/12, Lect. 5 https://edux.fit.cvut.cz/courses/mi-mvi/
More informationContinuous-Time Markov Chain
Continuous-Time Markov Chain Consider the process {X(t),t 0} with state space {0, 1, 2,...}. The process {X(t),t 0} is a continuous-time Markov chain if for all s, t 0 and nonnegative integers i, j, x(u),
More informationKendall notation. PASTA theorem Basics of M/M/1 queue
Elementary queueing system Kendall notation Little s Law PASTA theorem Basics of M/M/1 queue 1 History of queueing theory An old research area Started in 1909, by Agner Erlang (to model the Copenhagen
More informationElementary 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 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 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 informationMI-RUB Testing Lecture 10
MI-RUB Testing Lecture 10 Pavel Strnad pavel.strnad@fel.cvut.cz Dept. of Computer Science, FEE CTU Prague, Karlovo nám. 13, 121 35 Praha, Czech Republic MI-RUB, WS 2011/12 Evropský sociální fond Praha
More informationContinuous-time Markov Chains
Continuous-time Markov Chains Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 23, 2017
More informationOther properties of M M 1
Other properties of M M 1 Přemysl Bejda premyslbejda@gmail.com 2012 Contents 1 Reflected Lévy Process 2 Time dependent properties of M M 1 3 Waiting times and queue disciplines in M M 1 Contents 1 Reflected
More informationThe story of the film so far... Mathematics for Informatics 4a. Continuous-time Markov processes. Counting processes
The story of the film so far... Mathematics for Informatics 4a José Figueroa-O Farrill Lecture 19 28 March 2012 We have been studying stochastic processes; i.e., systems whose time evolution has an element
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross
More informationAdvanced Queueing Theory
Advanced Queueing Theory 1 Networks of queues (reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's
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 informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More informationCS 798: Homework Assignment 3 (Queueing Theory)
1.0 Little s law Assigned: October 6, 009 Patients arriving to the emergency room at the Grand River Hospital have a mean waiting time of three hours. It has been found that, averaged over the period of
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 informationStatistics 253/317 Introduction to Probability Models. Winter Midterm Exam Monday, Feb 10, 2014
Statistics 253/317 Introduction to Probability Models Winter 2014 - Midterm Exam Monday, Feb 10, 2014 Student Name (print): (a) Do not sit directly next to another student. (b) This is a closed-book, closed-note
More informationNEW FRONTIERS IN APPLIED PROBABILITY
J. Appl. Prob. Spec. Vol. 48A, 209 213 (2011) Applied Probability Trust 2011 NEW FRONTIERS IN APPLIED PROBABILITY A Festschrift for SØREN ASMUSSEN Edited by P. GLYNN, T. MIKOSCH and T. ROLSKI Part 4. Simulation
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 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 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 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 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 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 informationMultiaccess Problem. How to let distributed users (efficiently) share a single broadcast channel? How to form a queue for distributed users?
Multiaccess Problem How to let distributed users (efficiently) share a single broadcast channel? How to form a queue for distributed users? z The protocols we used to solve this multiaccess problem are
More informationComputational Intelligence Methods
Computational Intelligence Methods Ant Colony Optimization, Partical Swarm Optimization Pavel Kordík, Martin Šlapák Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-MVI, ZS 2011/12,
More informationEstimation of arrival and service rates for M/M/c queue system
Estimation of arrival and service rates for M/M/c queue system Katarína Starinská starinskak@gmail.com Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics
More informationMI-RUB Testing II Lecture 11
MI-RUB Testing II Lecture 11 Pavel Strnad pavel.strnad@fel.cvut.cz Dept. of Computer Science, FEE CTU Prague, Karlovo nám. 13, 121 35 Praha, Czech Republic MI-RUB, WS 2011/12 Evropský sociální fond Praha
More informationChapter 5. Continuous-Time Markov Chains. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 5. Continuous-Time Markov Chains Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Continuous-Time Markov Chains Consider a continuous-time stochastic process
More informationλ λ λ In-class problems
In-class problems 1. Customers arrive at a single-service facility at a Poisson rate of 40 per hour. When two or fewer customers are present, a single attendant operates the facility, and the service time
More informationSolutions to Homework Discrete Stochastic Processes MIT, Spring 2011
Exercise 6.5: Solutions to Homework 0 6.262 Discrete Stochastic Processes MIT, Spring 20 Consider the Markov process illustrated below. The transitions are labelled by the rate q ij at which those transitions
More informationPage 0 of 5 Final Examination Name. Closed book. 120 minutes. Cover page plus five pages of exam.
Final Examination Closed book. 120 minutes. Cover page plus five pages of exam. To receive full credit, show enough work to indicate your logic. Do not spend time calculating. You will receive full credit
More informationChapter 8 Queuing Theory Roanna Gee. W = average number of time a customer spends in the system.
8. Preliminaries L, L Q, W, W Q L = average number of customers in the system. L Q = average number of customers waiting in queue. W = average number of time a customer spends in the system. W Q = average
More information(u v) = f (u,v) Equation 1
Problem Two-horse race.0j /.8J /.5J / 5.07J /.7J / ESD.J Solution Problem Set # (a). The conditional pdf of U given that V v is: The marginal pdf of V is given by: (u v) f (u,v) Equation f U V fv ( v )
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 information