Previously Optimization Probability Review Inventory Models Markov Decision Processes.

Slides:

Advertisements

Similar presentations

OPSM 301: Operations Management Session 12: Service processes and flow variability Koç University Graduate School of Business MBA Program Zeynep Aksin.

Advertisements

S. D. Deshmukh OM V. Capacity Planning in Services u Matching Supply and Demand u The Service Process u Performance Measures u Causes of Waiting u Economics.

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)

Queueing Theory: Recap

Queueing Model 박희경.

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

Queuing Models Basic Concepts

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.

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

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.

1 Analysis Of Queues For this session, the learning objectives are:  Learn the fundamental structure of a queueing system.  Learn what needs to be specified.

1 Performance Evaluation of Computer Networks Objectives  Introduction to Queuing Theory  Little’s Theorem  Standard Notation of Queuing Systems  Poisson.

Previously Optimization Probability Review Inventory Models Markov Decision Processes Queues.

Previously Optimization Probability Review Inventory Models Markov Decision Processes.

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

Modeling and Analysis of Manufacturing Systems Session 2 QUEUEING MODELS January 2001.

Previously Optimization Probability Review Inventory Models Markov Decision Processes Queues.

ELEN 602 Lecture 9 Review of last lecture Little’s Formula M/M/1 Queue

Queuing Analysis Based on noted from Appendix A of Stallings Operating System text 6/28/20151.

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

7/3/2015© 2007 Raymond P. Jefferis III1 Queuing Systems.

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

Model Antrian By : Render, ect. M/M/1 Example 2 Five copy machines break down at UM St. Louis per eight hour day on average. The average service time.

1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Terminology: The characteristics of a queuing system is captured by five parameters:

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.

Previously Optimization Probability Review Inventory Models Markov Decision Processes.

Chapter 9: Queuing Models

Lecture 14 – Queuing Systems

Queuing Models and Capacity Planning

Introduction to Queuing Theory

Queueing Theory Models Training Presentation By: Seth Randall.

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

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

Probability Review Thinh Nguyen. Probability Theory Review Sample space Bayes’ Rule Independence Expectation Distributions.

Queueing Analysis of Production Systems (Factory Physics)

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

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.

Chapter 16 Capacity Planning and Queuing Models

1 Queuing Systems (2). Queueing Models (Henry C. Co)2 Queuing Analysis Cost of service capacity Cost of customers waiting Cost Service capacity Total.

The production inventory problem What is the expected inventory level? What is expected backorder level? What is the expected total cost? What is the optimal.

OPSM 301: Operations Management Session 19: Flow variability Koç University Zeynep Aksin

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

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

Computer Networking Queueing (A Summary from Appendix A) Dr Sandra I. Woolley.

Chapter 1 Introduction. “Wait-in-line” is a common phenomenon in everywhere. Reason: Demand is more than service. “How long must a customer wait?” or.

Chapter 6 Queueing Models

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

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

Chapter 5 Elementary Stochastic Analysis Prof. Ali Movaghar.

Queuing Models.

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.

QUEUING THEORY 1.  - means the number of arrivals per second   - service rate of a device  T - mean service time for each arrival   = ( ) Utilization,

Chapter 1 Introduction.

Queueing Theory What is a queue? Examples of queues:

Chapter 9: Queuing Models

Demo on Queuing Concepts

Our favorite simple stochastic process.

Queueing Theory Carey Williamson Department of Computer Science

Queuing Theory By: Brian Murphy.

Mitchell Jareo MAT4340 – Operations Research Dr. Bauldry

24 February 2019 Model Antrian M/D/1.

Carey Williamson Department of Computer Science University of Calgary

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

Presentation transcript:

Previously Optimization Probability Review Inventory Models Markov Decision Processes

Agenda Queues

Performance Measures T time in system T q waiting time (time in queue) N #customers in system N q #customers in queue system arrivals departures queue servers W = E[T] W q = E[T q ] L = E[N] L q = E[N q ]  fraction of time servers are busy (utilization)

Plain-Vanilla Queue 1 queue, 1 class of customers identical servers time between customer arrivals is independent mean rate of arrivals, rate of service constant

What Are We Ignoring? Rush-hour effects Priority classes Balking, Reneging, Jockeying Batching Multi-step processes Queue capacity

Parameters mean arrival rate of customers (per unit time) c servers µ mean service rate (per unit time)

Some Relations  = /( cµ) utilization c  = / µ average # of busy servers W = W q + 1/ µ L = L q + c  Little’s Law : L q = W q andL = W

So What is L q ? Depends on details.  LqLq M/M/1 queue ( exponential arrival times, exponential processing times, 1 server)

Qualitatively Dependence on   1 means L q  Increased variability (arrival / service times) –L q increases Pooling queues –L q decreases

Queue Notation M / M / 1 M = ‘Markov’ exponential distribution D = ‘Deterministic’constant G = ‘General’ other distribution of the time between arrivals distribution of the processing time number of servers: 1, 2, … M/M/1 D/M/1 M/G/3 …

Back to L q … L q =E[N q ] M/G/1 –  2 = variance of the service time –M/D/1 –M/M/1

M/M/1 N+1 has distribution Geometric(1-  ) –Var[N q ] = (1+  -  2 )  2 /(1-  ) 2 –StDev[N q ] > E[N q ] /  Pooling queues decreases L q Ex. (p501) 3 clinics with 1 nurse each (M/M/1) =4/hr, µ =1/13min  =87%, L q =5.6, W q =1.4hr Consolidated (M/M/3) =12/hr, µ =60/13hr,  =87% L q =4.9, W q =0.4