BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS

Size: px
Start display at page:

Download "BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS"

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 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 information

Performance Evaluation of Queuing Systems

Performance 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 information

CPSC 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 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 information

QUEUING SYSTEM. Yetunde Folajimi, PhD

QUEUING 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 information

CDA5530: Performance Models of Computers and Networks. Chapter 4: Elementary Queuing Theory

CDA5530: 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 information

Lecture 20: Reversible Processes and Queues

Lecture 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 information

Introduction to queuing theory

Introduction 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 information

Data analysis and stochastic modeling

Data 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 information

Introduction to queuing theory

Introduction 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 information

Queueing systems. Renato Lo Cigno. Simulation and Performance Evaluation Queueing systems - Renato Lo Cigno 1

Queueing 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 information

PBW 654 Applied Statistics - I Urban Operations Research

PBW 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 information

57:022 Principles of Design II Final Exam Solutions - Spring 1997

57:022 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 information

λ λ λ In-class problems

λ 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 information

Queues and Queueing Networks

Queues 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 information

Analysis of A Single Queue

Analysis 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 information

Introduction to Queuing Networks Solutions to Problem Sheet 3

Introduction 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 information

6 Solving Queueing Models

6 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 information

Queuing Theory. Using the Math. Management Science

Queuing 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 information

4.7 Finite Population Source Model

4.7 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 information

M/G/1 and M/G/1/K systems

M/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 information

Chapter 3 Balance equations, birth-death processes, continuous Markov Chains

Chapter 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 information

Queueing Theory. VK Room: M Last updated: October 17, 2013.

Queueing 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 information

Computer Systems Modelling

Computer 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 information

Homework 1 - SOLUTION

Homework 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 information

Introduction to Queueing Theory

Introduction 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 information

Chapter 8 Queuing Theory Roanna Gee. W = average number of time a customer spends in the system.

Chapter 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

Queuing Theory. Richard Lockhart. Simon Fraser University. STAT 870 Summer 2011

Queuing 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 information

Chapter 1. Introduction. 1.1 Stochastic process

Chapter 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 information

The Behavior of a Multichannel Queueing System under Three Queue Disciplines

The 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 information

Systems Simulation Chapter 6: Queuing Models

Systems 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 information

Introduction to Markov Chains, Queuing Theory, and Network Performance

Introduction 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 information

Classification of Queuing Models

Classification 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 information

QUEUING MODELS AND MARKOV PROCESSES

QUEUING 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 information

GI/M/1 and GI/M/m queuing systems

GI/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 information

Kendall notation. PASTA theorem Basics of M/M/1 queue

Kendall 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 information

Review of Queuing Models

Review 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 information

Continuous Time Processes

Continuous 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 information

Readings: Finish Section 5.2

Readings: 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 information

RENEWAL PROCESSES AND POISSON PROCESSES

RENEWAL 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 information

Introduction to Queueing Theory

Introduction 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 information

Queueing Review. Christos Alexopoulos and Dave Goldsman 10/6/16. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA

Queueing 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 information

CS 798: Homework Assignment 3 (Queueing Theory)

CS 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 information

Queueing Systems: Lecture 3. Amedeo R. Odoni October 18, Announcements

Queueing 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 information

Networking = Plumbing. Queueing Analysis: I. Last Lecture. Lecture Outline. Jeremiah Deng. 29 July 2013

Networking = 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 information

Chapter 6 Queueing Models. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 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 information

Computer Networks More general queuing systems

Computer 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 information

Link Models for Packet Switching

Link 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 information

Elementary queueing system

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 information

Simple queueing models

Simple 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 information

Queueing Review. Christos Alexopoulos and Dave Goldsman 10/25/17. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA

Queueing 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 information

Queuing Analysis. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall

Queuing 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 information

Non Markovian Queues (contd.)

Non 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 information

Introduction to Queueing Theory

Introduction 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 information

Performance analysis of queueing systems with resequencing

Performance 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 information

Outline. 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. 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 information

Classical Queueing Models.

Classical 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 information

Performance Modelling of Computer Systems

Performance 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 information

IEOR 6711, HMWK 5, Professor Sigman

IEOR 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 information

5/15/18. Operations Research: An Introduction Hamdy A. Taha. Copyright 2011, 2007 by Pearson Education, Inc. All rights reserved.

5/15/18. 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 information

Part II: continuous time Markov chain (CTMC)

Part 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 information

The Transition Probability Function P ij (t)

The 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 information

Exercises Solutions. Automation IEA, LTH. Chapter 2 Manufacturing and process systems. Chapter 5 Discrete manufacturing problems

Exercises 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 information

Queueing. Chapter Continuous Time Markov Chains 2 CHAPTER 5. QUEUEING

Queueing. 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 information

Statistics 150: Spring 2007

Statistics 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 information

Queuing 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 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 information

P (L d k = n). P (L(t) = n),

P (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 information

Chapter 5: Special Types of Queuing Models

Chapter 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 information

Q = (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?

Q = (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 information

λ, µ, ρ, A n, W n, L(t), L, L Q, w, w Q etc. These

λ, 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 information

Buzen s algorithm. Cyclic network Extension of Jackson networks

Buzen 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 information

Continuous-Time Markov Chain

Continuous-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 information

YORK 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 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 information

Little s result. T = average sojourn time (time spent) in the system N = average number of customers in the system. Little s result says that

Little 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 information

Basic Queueing Theory

Basic 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 information

Exercises Stochastic Performance Modelling. Hamilton Institute, Summer 2010

Exercises 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 information

Figure 10.1: Recording when the event E occurs

Figure 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 information

Stochastic Models of Manufacturing Systems

Stochastic 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 information

Multiserver Queueing Model subject to Single Exponential Vacation

Multiserver 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 information

Computer Systems Modelling

Computer 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 information

2905 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES

2905 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 information

Queuing Theory. 3. Birth-Death Process. Law of Motion Flow balance equations Steady-state probabilities: , if

Queuing 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 information

Slides 9: Queuing Models

Slides 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 information

Photo: US National Archives

Photo: 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 information

Lecture 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 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 information

Analysis of Software Artifacts

Analysis 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

Since 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.

Since 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 information

λ 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 =

λ 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 information

Markov Processes and Queues

Markov 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 information

Queuing Theory. Queuing Theory. Fatih Cavdur April 27, 2015

Queuing 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 information

Retrial queue for cloud systems with separated processing and storage units

Retrial 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 information

All models are wrong / inaccurate, but some are useful. George Box (Wikipedia). wkc/course/part2.pdf

All 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 information

SOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012

SOLUTIONS 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 information

MAT SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions.

MAT 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 information

Page 0 of 5 Final Examination Name. Closed book. 120 minutes. Cover page plus five pages of exam.

Page 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 information

EE 368. Weeks 3 (Notes)

EE 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 information

Class 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis.

Class 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 information

Simulation of Discrete Event Systems

Simulation 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 information

1 IEOR 4701: Continuous-Time Markov Chains

1 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 information

Chapter 10. Queuing Systems. D (Queuing Theory) Queuing theory is the branch of operations research concerned with waiting lines.

Chapter 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