Queuing Theory.

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

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

Continuous Time Markov Chains and Basic Queueing Theory

Nur Aini Masruroh Queuing Theory. Outlines IntroductionBirth-death processSingle server modelMulti server model.

Using one level of Cache:

Queuing Models Basic Concepts

Queuing Models Basic Concepts. QUEUING MODELS Queuing is the analysis of waiting lines It can be used to: –Determine the # checkout stands to have open.

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

Model Antrian By : Render, ect. Outline  Characteristics of a Waiting-Line System.  Arrival characteristics.  Waiting-Line characteristics.  Service.

System Performance & Scalability i206 Fall 2010 John Chuang.

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

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Simple queuing models (Sec )

RAIDs Performance Prediction based on Fuzzy Queue Theory Carlos Campos Bracho ECE 510 Project Prof. Dr. Duncan Elliot.

Consider the M/M/1 queue Arrival process: Poi( t) Service distribution: Poi(  t) One server, infinite queue length possible Prob(arrival in small interval.

Lec13, Ch.6, pp : Gap acceptance and Queuing Theory (Objectives)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ Chapter 14.

2003 Fall Queuing Theory Midterm Exam(Time limit：2 hours)

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

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.

Internet Queuing Delay Introduction How many packets in the queue? How long a packet takes to go through?

Queuing Networks. Input source Queue Service mechanism arriving customers exiting customers Structure of Single Queuing Systems Note: 1.Customers need.

Copyright ©: Nahrstedt, Angrave, Abdelzaher, Caccamo1 Queueing Systems.

Chapter 2 Machine Interference Model Long Run Analysis Deterministic Model Markov Model.

4/11: Queuing Models Collect homework, roll call Queuing Theory, Situations Single-Channel Waiting Line System –Distribution of arrivals –Distribution.

Lecture 10: Queueing Theory. Queueing Analysis Jobs serviced by the system resources Jobs wait in a queue to use a busy server queueserver.

Introduction to Queueing Theory

Network Design and Analysis-----Wang Wenjie Queueing System IV: 1 © Graduate University, Chinese academy of Sciences. Network Design and Analysis Wang.

IE 429, Parisay, January 2010 What you need to know from Probability and Statistics: Experiment outcome: constant, random variable Random variable: discrete,

TexPoint fonts used in EMF.

Queuing Queues are a part of life and waiting to be served is never really pleasant. The longer people wait the less likely they are to want to come back.

Patient Queuing Solution

Network Design and Analysis-----Wang Wenjie Queuing Theory III: 1 © Graduate University, Chinese academy of Sciences. Network Design and Performance Analysis.

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...

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.

Copyright ©: Nahrstedt, Angrave, Abdelzaher, Caccamo1 Queueing Systems.

Delays  Deterministic Assumes “error free” type case Delay only when demand (known) exceeds capacity (known)  Stochastic Delay may occur any time Random.

Random Variables r Random variables define a real valued function over a sample space. r The value of a random variable is determined by the outcome of.

QUEUING. CONTINUOUS TIME MARKOV CHAINS {X(t), t >= 0} is a continuous time process with > sojourn times S 0, S 1, S 2,... > embedded process X n = X(S.

The reading is 7.38 mm. The reading is 7.72 mm.

Queuing Theory. Model Customers arrive randomly in accordance with some arrival time distribution. One server services customers in order of arrival.

WAITING LINES AND SIMULATION

Chapter 1 Introduction.

MASS AND ENERGY ANALYSIS OF CONTROL VOLUMES

Al-Imam Mohammad Ibn Saud University

Queuing Theory Non-Markov Systems

Internet Queuing Delay Introduction

Demo on Queuing Concepts

Finite M/M/1 queue Consider an M/M/1 queue with finite waiting room.

Service Operations Management (SOM) Waiting Line Models

Internet Queuing Delay Introduction

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

Waiting Lines (Queuing Theory) Service Analysis Tutorial 1

Introduction Notation Little’s Law aka Little’s Result

Solutions Queueing Theory 1

IENG 362 Markov Chains.

TexPoint fonts used in EMF.

A Queue system based on the bank service

Queueing theory Birth-death Analysis

Native simulation of different scheduling policies

مهندسی مجدد فرآیندهای تجاری

Queuing Theory II.

Queuing Theory By: Brian Murphy.

Mitchell Jareo MAT4340 – Operations Research Dr. Bauldry

Queuing Theory III.

Model Antrian M/M/s.

Queuing Theory III.

Queuing Theory III.

Queuing Theory II.

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.

Chapter 2 Machine Interference Model

Presentation transcript:

Queuing Theory

M/M/1 Queue Rate Transitions 1 2 3 n-1 n E t times process enters state n L leaves ( ) # = - £ 1

M/M/1 Queue Rate Transitions 1 2 3 n-1 n lim ( ) t n E L ® ¥ - £ = 1 lim ( ) t n E L ® ¥ = rate in = rate out

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m In steady state, rate at which process enters a state = rate at which a process leaves a state state 0 l m p 1 =

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m State Balance Eq. l m p 1 =

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m State Balance Eq. 1 l m p 1 = l m p 2 1 + =

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m State Balance Eq. 1 n l m p 1 = l m p 2 1 + = l m p n 1 + = -

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m Balance Eq. p 1 = l m l m p 1 = 2 + n -

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m Balance Eq. p 1 = l m l m p 1 = 2 + n - l m p 2 + = ( )

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m l m p 2 + = - ( ) p 1 = l m

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m l m p 2 + = - ( ) / m p 1 = l m

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m l m p 2 + = - ( ) / m p 1 = l m l m p 2 = ( )

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m p n = ( ) l m In general,

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m p n = ( ) l m In general, We also know that in steady state, we must be somewhere. That is,

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m p n = ( ) l m In general, We also know that in steady state, we must be somewhere. That is, p n = ¥ å 1

M/M/1 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m m m m m m m ¥ p n = å 1 p n = ¥ å 1 ( ) l m p = - 1 ( ) l m

M/M/1 Queue p = - 1 ( ) l m p 1 = - ( ) l m 1 p n 1 = - ( )n ( ) l m

Performance Criteria m L average in system np n = - ¥ å # ( ) 1 l

Performance Criteria å å m L average in system np n = - # ( ) 1 l Re , ¥ å # ( ) 1 l Re , ( ) call nx x n = ¥ å - 2 1

Performance Criteria å L = - ( ) 1 l m Re , ( ) call nx x - 1 2 ¥ n 2 2 1

Performance Criteria l W average wait in system L = - m 1

Performance Criteria W avg wait time in system service E S = - [ ] ( ) q = - [ ] ( ) 1 m l

Performance Criteria L avg in queue w q = - # ( ) l m 2

Performance Criteria L W q = - l m 1 2 ( )