Queuing Theory Summary of results. 2 Notations Typical performance characteristics of queuing models are: L : Ave. number of customers in the system L.

Slides:

Advertisements

Similar presentations

Lecture 10 Queueing Theory. There are a few basic elements common to almost all queueing theory application. Customers arrive, they wait for service in.

Advertisements

Introduction to Queuing Theory

Chapter Queueing Notation

1 Chapter 8 Queueing models. 2 Delay and Queueing Main source of delay Transmission (e.g., n/R) Propagation (e.g., d/c) Retransmission (e.g., in ARQ)

Cheng-Fu Chou, CMLab, CSIE, NTU Basic Queueing Theory (I) Cheng-Fu Chou.

Queueing Theory: Recap

Queuing Analysis Based on noted from Appendix A of Stallings Operating System text 6/10/20151.

Previously Optimization Probability Review Inventory Models Markov Decision Processes.

Chap. 20, page 1051 Queuing Theory Arrival process Service process Queue Discipline Method to join queue IE 417, Chap 20, Jan 99.

HW # Due Day: Nov 23.

System Performance & Scalability i206 Fall 2010 John Chuang.

ECS 152A Acknowledgement: slides from S. Kalyanaraman & B.Sikdar

Multiple server queues In particular, we look at M/M/k Need to find steady state probabilities.

Queueing Theory: Part I

Waiting Line Management

Example 14.4 Queuing | 14.2 | 14.3 | 14.5 | 14.6 | 14.7 |14.8 | Background Information n Which system has the.

1 Queueing Theory H Plan: –Introduce basics of Queueing Theory –Define notation and terminology used –Discuss properties of queuing models –Show examples.

Queuing. Elements of Waiting Lines  Population –Source of customers Infinite or finite.

1 TCOM 501: Networking Theory & Fundamentals Lectures 9 & 10 M/G/1 Queue Prof. Yannis A. Korilis.

Rensselaer Polytechnic Institute © Shivkumar Kalvanaraman & © Biplab Sikdar1 ECSE-4730: Computer Communication Networks (CCN) Network Layer Performance.

HW # Due Day: Nov 23.

Introduction to Queuing Theory. 2 Queuing theory definitions  (Kleinrock) “We study the phenomena of standing, waiting, and serving, and we call this.

QUEUING MODELS Queuing theory is the analysis of waiting lines It can be used to: –Determine the # checkout stands to have open at a store –Determine the.

Flows and Networks Plan for today (lecture 5): Last time / Questions? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean.

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  …

On the variance curve of outputs for some queues and networks Yoni Nazarathy Gideon Weiss Yoav Kerner QPA Seminar, EURANDOM January 8, 2009.

Network Analysis A brief introduction on queues, delays, and tokens Lin Gu, Computer Networking: A Top Down Approach 6 th edition. Jim Kurose.

Introduction to Queuing Theory

CS433 Modeling and Simulation Lecture 13 Queueing Theory Dr. Anis Koubâa 03 May 2009 Al-Imam Mohammad Ibn Saud University.

Introduction to Queueing Theory

Queuing Theory Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues Chapter.

Queueing Theory What is a queue? Examples of queues: Grocery store checkout Fast food (McDonalds – vs- Wendy’s) Hospital Emergency rooms Machines waiting.

TexPoint fonts used in EMF.

1 Elements of Queuing Theory The queuing model –Core components; –Notation; –Parameters and performance measures –Characteristics; Markov Process –Discrete-time.

Lecture 3: 1 Introduction to Queuing Theory More interested in long term, steady state than in startup => Arrivals = Departures Little’s Law: Mean number.

M/M/1 queue λn = λ, (n >=0); μn = μ (n>=1) λ μ λ: arrival rate

CS433 Modeling and Simulation Lecture 12 Queueing Theory Dr. Anis Koubâa 03 May 2008 Al-Imam Mohammad Ibn Saud University.

1 Chapters 8 Overview of Queuing Analysis. Chapter 8 Overview of Queuing Analysis 2 Projected vs. Actual Response Time.

On the variance curve of outputs for some queues and networks Yoni Nazarathy Gideon Weiss Yoav Kerner CWI Amsterdam March 2009.

Modeling and Simulation Queuing theory

Queuing Theory and Traffic Analysis Based on Slides by Richard Martin.

M/M/1 Queues Customers arrive according to a Poisson process with rate. There is only one server. Service time is exponential with rate  j-1 jj+1...

The M/M/ N / N Queue etc COMP5416 Advanced Network Technologies.

CS352 - Introduction to Queuing Theory Rutgers University.

Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Model of the Nodes in the Packet Network Chapter 10.

Structure of a Waiting Line System Queuing theory is the study of waiting lines Four characteristics of a queuing system: –The manner in which customers.

Chapter 6 Queueing Models

Waiting Line Theory Akhid Yulianto, SE, MSc (log).

Introduction Definition M/M queues M/M/1 M/M/S M/M/infinity M/M/S/K.

Queuing Theory.  Queuing Theory deals with systems of the following type:  Typically we are interested in how much queuing occurs or in the delays at.

CS 4594 Broadband Intro to Queuing Theory. Kendall Notation Kendall notation: [Kendal 1951] A/B/c/k/m/Z A = arrival probability distribution (most often.

Queueing Theory. The study of queues – why they form, how they can be evaluated, and how they can be optimized. Building blocks – arrival process and.

Mohammad Khalily Islamic Azad University.  Usually buffer size is finite  Interarrival time and service times are independent  State of the system.

Simple Queueing Theory: Page 5.1 CPE Systems Modelling & Simulation Techniques Topic 5: Simple Queueing Theory  Queueing Models  Kendall notation.

Managerial Decision Making Chapter 13 Queuing Models.

WAITING LINES AND SIMULATION

Al-Imam Mohammad Ibn Saud University

CTMCs & N/M/* Queues.

Queueing Theory What is a queue? Examples of queues:

Queuing Theory Non-Markov Systems

Birth-Death Process Birth – arrival of a customer to the system

Introduction Notation Little’s Law aka Little’s Result

Chapter 20 Queuing Theory

TexPoint fonts used in EMF.

Queuing Theory By: Brian Murphy.

Queueing Theory 2008.

Carey Williamson Department of Computer Science University of Calgary

Waiting Line Models Waiting takes place in virtually every productive process or service. Since the time spent by people and things waiting in line is.

Kendall’s Notation ❚ Simple way of summarizing the characteristics of a queue. Arrival characteristics / Departure characteristics / Number of servers.

Presentation transcript:

Queuing Theory Summary of results

2 Notations Typical performance characteristics of queuing models are: L : Ave. number of customers in the system L Q : Ave. number of customers waiting in queue W : Ave. time customer spends in the system W Q : Ave. time customer spends waiting in the queue

3 Queue notation M/M/k/c Arrival process M = Markovian GI = General Departure process (Service time distribution) M = Markovian G = General Number of servers Capacity of the queue If nothing is specified, we assume infinite capacity

4 M/M/1 queue

5 M/M/1/N queue

6 Expected amount of time spent by the customer…. But, what is a customer? Do we include those who came but didn’t join the queue because it was full? Or, are we including only those who actually joined the system? Depending on our consideration, λ a = λ in the first case; whereas λ a =λ(1-P N ) in the second case. Then,

7 Tandem queues M/M/1

8 M/G/1 Derivation of these results is slightly more tedious because, unlike previous models where the Markov theory was extensively used, M/G/1 model requires Renewal Reward theory.

9 M/G/1: Priority queues Let there be two types of customers (Type 1 and 2). Type 1 being the priority class. Meaning, service can never begin for Type 2 customer, if Type 1 is waiting in queue. However, preemption is not allowed. Let arrival rate of Type 1 customers be λ 1 and that of Type 2 be λ 2 (both arrivals are Poisson processes). Respective service time distribution be G 1 and G 2. That is S 1 ~ G 1 and S 2 ~ G 2. Once again, we derive the performance characteristics based on the renewal reward theory.

10 M/G/1: Priority queues

11 M/M/k queues Two types of queues could be considered: 1.Loss function queues: Customer does not join the system if (s/)he sees all k servers busy. This queue is called Erlang’s Loss system. 2.Infinite capacity queues. These are exact extensions of M/M/1 queues.

12 M/M/k queues