BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS
|
|
- Godwin Stafford
- 5 years ago
- Views:
Transcription
1 BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS Andrea Bobbio Anno Accademico
2 Queueing Systems 2 Notation for Queueing Systems /λ mean time between arrivals S = /µ ρ = λ/µ N mean service time traffic intensity Number of customers in the queue (including those in service) N Q Number of customers in the queue (excluding those in service) N S Number of customers in service R Response time (including the service time) W Waiting time ( = R S) U 0 Utilization factor T Throughput (Expected number of jobs completed in a time unit)
3 Queueing Systems 3 Birth-Death Processes Let us identify by state i the condition of the system in which there are i objects. Given the system is in state i, new elements arrive at rate λ i, and leave at rate µ i. The state space transition diagram is: )+*,-/.0 #$#$# %&('!" 2/34 Let N(t) be the number of elements in the system at time t, and E i (t) be the event N(t) = i. E i- birth E i no events E i E i+ death The figure shows the way in which the event E i (t + t) can be generated.
4 Queueing Systems 4 Birth-Death Processes By the theorem of the total probability, we can write for i > 0: P r{n(t + t) = i N(t) = i } = λ i t + O( t) P r{n(t + t) = i N(t) = i + } = µ i+ t + O( t) P r{n(t + t) = i N(t) = i} = λ i t µ i t + O( t) Where: lim t 0 O( t) t = 0 For i = 0, we can write: P r{n(t + t) = 0 N(t) = } = µ t + O( t) P r{n(t + t) = 0 N(t) = 0} = λ 0 t + O( t) Let us define: P i (t) = P r{n(t) = i}
5 Queueing Systems 5 Birth-Death Processes According to the above relations we can write: P 0 (t + t) = µ t P (t) + ( λ 0 t) P 0 (t) i = 0 P i (t + t) = λ i tp i (t) + µ i+ tp i+ (t) + ( λ i t µ i t) P i (t) i > 0 P 0 (t + t) P 0 (t) t P i (t + t) P i (t) t = λ 0 P 0 (t) + µ P (t) i = 0 = (λ i + µ i )P i (t) + λ i P i (t) + µ i+ P i+ (t) i > 0 Taking the limit t 0, the following set of linear differential equations is derived: d P 0 (t) d t d P i (t) d t = λ 0 P 0 (t) + µ P (t) i = 0 = (λ i + µ i ) P i (t) + λ i P i (t) + µ i+ P i+ (t) i > 0 () with initial conditions: P 0 (0) = i = 0 P i (0) = 0 i > 0
6 Queueing Systems 6 Transient Balance Equation Transient continuity (balance) equation in state i. The flow variation in state i equals the difference between the ingoing flow minus the outgoing flow. variation of flow = d P i(t) d t ingoing flow = λ i P i (t) + µ i+ P i+ (t) ; i 0 outgoing flow = (λ i + µ i ) P i (t)
7 Queueing Systems 7 Matrix representation of B/D processes Given the B/D process of the figure: )+*,-/.0 #$#$# %&('!" 2/34 we define a transition rate matrix Q and a state probability row vector p(t) at time t: Q = i i i + 0 λ 0 λ 0 µ (λ + µ ) λ µ 2. (λ 2 + µ 2 ) λ i i µ i (λ i + µ i ) λ i i +. p(t) = {p 0 p p 2... p i...} The solution equation of () can be written in matrix form: d p(t) d t = p Q
8 Queueing Systems 8 Steady-state of B/D processes For t, the B/D process may reach a steady-state (equilibrium) condition. Steady state means that the state probabilities do not depend on the time any more. If a steady-state solution exists, it is characterized by: lim t d P i (t) d t = 0 (i = 0,, 2,...) Let us denote: P i = lim P i (t). t become: The steady state equations 0 = λ 0 P 0 + µ P i = 0 0 = (λ i + µ i ) P i + λ i P i + µ i+ P i+ i > 0 that can be rewritten as balance equations (ingoing flow equals outgoing flow) as: λ 0 P 0 = µ P i = 0 (λ i + µ i ) P i = λ i P i + µ i+ P i+ i > 0
9 Queueing Systems 9 Steady-state of B/D processes The steady-state equation can be written as: λ 0 P 0 µ P = 0 λ P µ 2 P 2 = λ 0 P 0 µ P = λ i P i µ i+ P i+ = λ i P i µ i P i = From the above, the i-th term becomes: λ i P i = µ i P i = P i = λ i µ i P i (i ) P i = λ i µ i λ i 2 µ i P i 2 = λ 0 λ... λ i µ µ 2... µ i P 0 = P 0 i j=0 λ j µ j+ The following normalization condition must hold: Hence: P 0 = i 0 P i = + i i j=0 λ j µ j+ The steady state distribution exists, with P i > 0, if the series i λ j converges. µ j+ i j=0
10 Queueing Systems 0 Standard notation for queueing systems The standard notation to identify the main elements that define the structure of a queueing system is the following (due to Kendall): where: A/B/c/d/e A Is the distribution of the interarrival times; B Is the distribution of the service times; c Is the number of servers; d Is the storage capacity of the system (number of servers plus the storage capacity of the buffer); e Is the number of sources that provide clients. The usual assumption for the interarrival and service time distributions A and B is: M Markovian (or exponential); G General.
11 Queueing Systems M/M/ The M/M/ queueing system is a B/D process characterized by having the arrival rates λ and the service rates µ independent of the state. The usual picture for the M/M/ is: The state space of the M/M/ is: λ i = λ for i 0 ; µ i = µ for i By applying the general equilibrium results of a B/D process: P i = λ µ P i = P i = i λ µ P 0 By applying the normalization condition: P i = = P 0 = + λ i (2) j= µ Let us introduce a new parameter called the traffic intensity: ρ = λ µ
12 Queueing Systems 2 Steady state solution of a M/M/ The denominator of (2) is the geometric series: + ρ + ρ ρ i +... = If ρ <, the series (3) converges to the value ρ i = ρ ρ i (3) Hence, if ρ < a steady state solution exists, and the M/M/ is asymptotically stable. If ρ <, the state probabilities depend on λ and µ only through the traffic intensity ρ, and are given by: P 0 P = ρ = ( ρ) ρ = ( ρ) ρ i P i Since the state probabilities are known, the system is completely specified, and various measures can be computed.
13 Queueing Systems 3 M/M/: Probability vs ρ The state probability P i as a function of i and for various values of ρ is depicted in the figure: P i = ( ρ) ρ i. P i i
14 Queueing Systems 4 Expected number of customers in a M/M/ Server utilization factor (probability the server is busy): U 0 = i= P i = P 0 = ρ Expected number of customers E[N] Let N be the number of customers in the queue, including the one in service: the expected number of customers E[N] is given by: E[N] = i P i = 0 P 0 + ρ P ρ 2 P ρ 3 P = P 0 i ρ i = ( ρ) i ρ i = ρ ρ The above proof is based on the sum of the modified geometric series: i ρ i = ρ ρ = ρ ( ρ) 2 ρ i = ρ ρ ρ
15 Queueing Systems 5 M/M/: Little s formula The Little s formula states that the expected number of customers in the queue E[N] is equal to the arrival rate λ times the expected time spent in the system (the expected response time) E[R]. E[N] { E[R] E[N] = λ E[R] The expected response time for a M/M/ queue is obtained by applying Little s formula: E[R] = λ E[N] = λ ρ ρ = /µ ρ From the above formula, the expected response time E[R] can be interpreted as the ratio between the mean service time (/µ) and the probability of the sever to be idle ( ρ).
16 Queueing Systems 6 M/M/: Performance measures Expected waiting time Let us define the waiting time W = R S as the time a customer waits in the queue before service, where R is the response time and S the service time. The expected waiting time E[W ] is given by: E[W ] = E[R] E[S] = µ ( ρ) µ = ρ µ ( ρ) Expected number of customers in the line The expected number of customers in the line (awaiting for service) is obtained by applying Little s rule to the queue only: E[NQ] { E[W] E[N Q ] = λ E[W ] = Number of customers in service ρ2 ρ The expected number of customers in service is: E[N S ] = E[N] E[N Q ] = ρ From the Little s rule applied to the server, only: E[N S ] = λ E[S] = λ µ = ρ
17 Queueing Systems 7 Summary of results for the M/M/ λ arrival rate µ service rate ρ = λ/µ traffic intensity E[N] = ρ ρ Expected number of customers in the queue (including those in service) E[R] = /µ ρ E[S] = µ Expected response time Expected service time E[W ] = E[R] E[S] ρ = µ( ρ) Expected waiting time E[N Q ] = λ E[W ] = ρ2 ρ Expected number of waiting customers E[N S ] = E[N] E[N Q ] = λ E[S] = ρ Expected number of customers in service
18 Queueing Systems 8 M/M//K: finite storage K The storage capacity of the system is K (one customer in service and K customers in the waiting line) and the exceeding customers are refused. The general B/D process can be particularized as follows: λ i = λ i < K 0 i K The state probabilities satisfy i P i = P 0 j=0 ; µ i = µ λ µ = P 0 ρ i i K P i = 0 i > K From the normalization condition: P 0 = + K j= ρ j = + ρ( ρk ) ρ = ρ ρ K+
19 Queueing Systems 9 M/M//K: finite storage The M/M//K queue is stable for any positive value of the traffic intensity ρ. The state probabilities are: P i = ( ρ) ρi ρ K+ i K P i = 0 i > K For ρ the above formula is undefined. We find the limit resorting to De l Hospital rule: lim ρ P i = lim ρ ( ρ) ρ i ρ K+ = lim ρ ρ i + i ( ρ) ρ i (K + ) ρ K = K + Let us define the rejection probability as the probability of an arriving customer to find the queue full and to be rejected. Since the queue is full when in state K, the rejection probability is: P K = ( ρ) ρk ρ K+
20 Queueing Systems 20 M/M//K: finite storage Expected number of customers E[N] E[N] = K i P i = K = ρ ρ K+ = K i ( ρ) ρi ρ K+ i ρ i (4) ρ (K + ) ρ K + K ρ K+ ρ K+ ( ρ) The above formula (4) is based on the following finite series sum: K i ρ i = ρ ρ K i= ρ i = ρ ρ ρ ( ρ K ) ρ = ρ (K + ) ρk + K ρ K+ ( ρ) 2 From formula (4), it follows: lim ρ 0 E[N] = 0 ; lim ρ E[N] = K ; lim ρ E[N] = K 2 where the last limit (ρ ) is obtained by applying twice the De l Hospital rule.
21 Queueing Systems 2 M/M//: no waiting line The queue does not have a waiting line and the arriving customer enters service only if the server is idle. From the M/M//K case, we get: P 0 = P = + ρ = µ λ + µ ρ + ρ = λ λ + µ
22 % Queueing Systems 22 M/M/m - Queueing system with m servers The queue has one arrival line and m identical servers with service rate µ. The structure of the queue and its state space are represented in the figures: m!"$# &('. / ) *+-, 0( 2(3 The general B/D process can be particularized as follows: λ i = λ i 0 ; µ i = i µ m µ 0 < i < m i m The state probabilities satisfy: P i i = P 0 j=0 λ (j + ) µ = P 0 λ µ i i! i < m P i m = P 0 j=0 λ (j + ) µ i k=m λ m µ = P 0 i λ µ m! m i m i m
23 Queueing Systems 23 M/M/m - Queueing system with m servers Let us define the traffic intensity as ρ = The stability condition requires ρ <. λ m µ. By rewriting the state probabilities in terms of the traffic intensity, we obtain: P i = P 0 (m ρ) i i! P 0 ρ i m m m! i < m i m From the normalization condition, we obtain: P 0 = m (m ρ) i i! + i=m ρ i m m m! (5) The second sum in (5) can be written as: i=m ρ i m m m! = ρm m m m! k=0 ρ k = (m ρ)m m! ρ So that (5) becomes: P 0 = m (m ρ) i i! + (m ρ)m m! ρ
24 Queueing Systems 24 M/M/m - Queueing system with m servers Expected number of customers in the queue: E[N] = i P i = m ρ + ρ (m ρ)m m! P 0 ( ρ) 2 Expected number of busy servers: E[M] = m i P i + m i=m P i = m ρ = λ µ Probability that an arriving customer should join the queue (equal to the probability that an arriving customer finds all the servers busy): P [queue] = i=m P i = P m ρ = (m ρ)m m! P 0 ρ
25 % Queueing Systems 25 M/M/ : infinite number of servers The state space of the queue is represented in the figure:!"$# &('. / ) *+-, The general B/D process can be particularized as follows: The state probabilities become: λ i = λ i 0 µ i = i µ i 0 i P i = P 0 j= λ (j + ) µ = P 0 i! i λ µ The normalization condition provides: P 0 = + i= i! λ µ i = e λ/µ Hence, the state probabilities assume the following form and are Poisson distributed: λ/µ (λ/µ)i P i = e i! E[N] = λ/µ ; E[R] = E[N] λ = µ
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 "
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationComputer Systems Modelling
Computer Systems Modelling Computer Laboratory Computer Science Tripos, Part II Lent Term 2010/11 R. J. Gibbens Problem sheet William Gates Building 15 JJ Thomson Avenue Cambridge CB3 0FD http://www.cl.cam.ac.uk/
More informationHomework 1 - SOLUTION
Homework - SOLUTION Problem M/M/ Queue ) Use the fact above to express π k, k > 0, as a function of π 0. π k = ( ) k λ π 0 µ 2) Using λ < µ and the fact that all π k s sum to, compute π 0 (as a function
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 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 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 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 informationThe Behavior of a Multichannel Queueing System under Three Queue Disciplines
The Behavior of a Multichannel Queueing System under Three Queue Disciplines John K Karlof John Jenkins November 11, 2002 Abstract In this paper we investigate a multichannel channel queueing system where
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 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 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 informationQUEUING MODELS AND MARKOV PROCESSES
QUEUING MODELS AND MARKOV ROCESSES Queues form when customer demand for a service cannot be met immediately. They occur because of fluctuations in demand levels so that models of queuing are intrinsically
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 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 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 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 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 informationRENEWAL PROCESSES AND POISSON PROCESSES
1 RENEWAL PROCESSES AND POISSON PROCESSES Andrea Bobbio Anno Accademico 1997-1998 Renewal and Poisson Processes 2 Renewal Processes A renewal process is a point process characterized by the fact that the
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 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 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 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 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 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 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 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 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 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 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 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 informationQueuing Analysis. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Queuing Analysis Chapter 13 13-1 Chapter Topics Elements of Waiting Line Analysis The Single-Server Waiting Line System Undefined and Constant Service Times Finite Queue Length Finite Calling Problem The
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 informationPerformance analysis of queueing systems with resequencing
UNIVERSITÀ DEGLI STUDI DI SALERNO Dipartimento di Matematica Dottorato di Ricerca in Matematica XIV ciclo - Nuova serie Performance analysis of queueing systems with resequencing Candidato: Caraccio Ilaria
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 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 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 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 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 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 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 informationExercises Solutions. Automation IEA, LTH. Chapter 2 Manufacturing and process systems. Chapter 5 Discrete manufacturing problems
Exercises Solutions Note, that we have not formulated the answers for all the review questions. You will find the answers for many questions by reading and reflecting about the text in the book. Chapter
More informationQueueing. Chapter Continuous Time Markov Chains 2 CHAPTER 5. QUEUEING
2 CHAPTER 5. QUEUEING Chapter 5 Queueing Systems are often modeled by automata, and discrete events are transitions from one state to another. In this chapter we want to analyze such discrete events systems.
More informationStatistics 150: Spring 2007
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities p ij is irreducible and positive recurrent; then the limiting probabilities
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 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 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 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 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 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 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 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 informationLittle s result. T = average sojourn time (time spent) in the system N = average number of customers in the system. Little s result says that
J. Virtamo 38.143 Queueing Theory / Little s result 1 Little s result The result Little s result or Little s theorem is a very simple (but fundamental) relation between the arrival rate of customers, average
More informationBasic Queueing Theory
After The Race The Boston Marathon is a local institution with over a century of history and tradition. The race is run on Patriot s Day, starting on the Hopkinton green and ending at the Prudential Center
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 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 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 informationMultiserver Queueing Model subject to Single Exponential Vacation
Journal of Physics: Conference Series PAPER OPEN ACCESS Multiserver Queueing Model subject to Single Exponential Vacation To cite this article: K V Vijayashree B Janani 2018 J. Phys.: Conf. Ser. 1000 012129
More informationComputer Systems Modelling
Computer Systems Modelling Computer Laboratory Computer Science Tripos, Part II Michaelmas Term 2003 R. J. Gibbens Problem sheet William Gates Building JJ Thomson Avenue Cambridge CB3 0FD http://www.cl.cam.ac.uk/
More information2905 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES
295 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES 16 Queueing Systems with Two Types of Customers In this section, we discuss queueing systems with two types of customers.
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 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 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 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 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 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 informationλ 2 1 = 2 = 2 2 2P 1 =4P 2 P 0 + P 1 + P 2 =1 P 0 =, P 1 =, P 2 = ρ 1 = P 1 + P 2 =, ρ 2 = P 1 + P 2 =
Urban Operations Research Compiled by James S. Kang Fall 001 Quiz Solutions 1//001 Problem 1 (Larson, 001) 1 (a) One is tempted to say yes by setting ρ = Nµ = =.But = is not the rate at which customers
More informationMarkov Processes and Queues
MIT 2.853/2.854 Introduction to Manufacturing Systems Markov Processes and Queues Stanley B. Gershwin Laboratory for Manufacturing and Productivity Massachusetts Institute of Technology Markov Processes
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 informationRetrial queue for cloud systems with separated processing and storage units
Retrial queue for cloud systems with separated processing and storage units Tuan Phung-Duc Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo, Japan
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 informationSOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012
SOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012 This exam is closed book. YOU NEED TO SHOW YOUR WORK. Honor Code: Students are expected to behave honorably, following the accepted
More informationMAT SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions.
MAT 4371 - SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions. Question 1: Consider the following generator for a continuous time Markov chain. 4 1 3 Q = 2 5 3 5 2 7 (a) Give
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 informationEE 368. Weeks 3 (Notes)
EE 368 Weeks 3 (Notes) 1 State of a Queuing System State: Set of parameters that describe the condition of the system at a point in time. Why do we need it? Average size of Queue Average waiting time How
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 informationSimulation of Discrete Event Systems
Siulation of Discrete Event Systes Unit 9 Queueing Models Fall Winter 207/208 Prof. Dr.-Ing. Dipl.-Wirt.-Ing. Sven Tackenberg Benedikt Andrew Latos M.Sc.RWTH Chair and Institute of Industrial Engineering
More information1 IEOR 4701: Continuous-Time Markov Chains
Copyright c 2006 by Karl Sigman 1 IEOR 4701: Continuous-Time Markov Chains A Markov chain in discrete time, {X n : n 0}, remains in any state for exactly one unit of time before making a transition (change
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 information