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CS433 Modeling and Simulation Lecture 12 Queueing Theory

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Al-Imam Mohammad Ibn Saud University

CS433Modeling and SimulationLecture 12Queueing Theory

Dr. Anis Koubâa

03 May 2008

Goals for Today

- Understand the Queuing Model and its applications
- Understand how to describe a Queue Model
- Lean the most important queuing models (Part 02)
- Single Queue
- Multiple Queues
- Multiple Servers

Course Outline

- The Queuing Model and Definitions
- Application of Queuing Theory
- Little’s Law
- Queuing System Notation
- Stationary Analysis of Elementary Queueing Systems
- M/M/1
- M/M/m
- M/M/1/K
- …

Queuing System

Queue

Server

The Queuing ModelClick for Queue Simulator

- Use Queuing models to
- Describe the behavior of queuing systems
- Evaluate system performance
- A Queue System is characterized by
- Queue (Buffer): with a finite or infinite size
- The state of the system is described by the Queue Size
- Server: with a given processing speed
- Events: Arrival (birth) or Departure (death) with given rates

Queuing theory definitions

- (Bose) “the basic phenomenon of queueing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.”
- (Wolff) “The primary tool for studying these problems [of congestions] is known as queueing theory.”
- (Kleinrock) “We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory." "Any system in which arrivals place demands upon a finite capacity resource may be termed a queueing system.”
- (Mathworld) “The study of the waiting times, lengths, and other properties of queues.”

http://www2.uwindsor.ca/~hlynka/queue.html

Applications of Queuing Theory

- Telecommunications
- Computer Networks
- Predicting computer performance
- Health services (eg. control of hospital bed assignments)
- Airport traffic, airline ticket sales
- Layout of manufacturing systems.

Example application of queuing theory

- In many stores and banks, we can find:
- multiple line/multiple checkout system →a queuing system where customers wait for the next available cashier
- We can prove using queuing theory that : throughput improves/increases when queues are used instead of separate lines

http://www.andrews.edu/~calkins/math/webtexts/prod10.htm#PD

Example application of queuing theory

http://www.bsbpa.umkc.edu/classes/ashley/Chaptr14/sld006.htm

Queuing theory for studying networks

- View network as collections of queues
- FIFO data-structures
- Queuing theory provides probabilistic analysis of these queues
- Examples:
- Average length
- Average waiting time
- Probability queue is at a certain length
- Probability a packet will be lost

The Little’s Law

The long-term average number of customers in a stable system N, is equal to the long-term average arrival rate, λ, multiplied by the long-term average time a customer spends in the system, T.

Little’s Law

Expected number of customers in the system

Expected time in the system

Arrival rate IN the system

λ

Aggregate Arrival rate

Generality of Little’s LawMean number tasks in system = mean arrival rate x mean response time

- Little’s Law is a pretty general result
- It does not depend on the arrival process distribution
- It does not depend on the service process distribution
- It does not depend on the number of servers and buffers in the system.
- Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks

Queueing

Network

Characteristics of queuing systems

- Arrival Process
- The distribution that determines how the tasks arrives in the system.
- Service Process
- The distribution that determines the task processing time
- Number of Servers
- Total number of servers available to process the tasks

Specification of Queueing Systems

- Arrival/Departure
- Customer arrivaland service stochastic models
- Structural Parameters
- Number of servers: What is the number of servers?
- Storage capacity: are buffer finite or infinite?
- Operating policies
- Customer class differentiation
- are all customers treated the same or do some have priority over others?
- Scheduling/Queueing policies
- which customer is served next
- Admission policies
- which/when customers are admitted

Kendall Notation A/B/m(/K/N/X)

- To specify a queue, we use the Kendall Notation.
- The First three parameters are typically used, unless specified
- A: Arrival Distribution
- B: Service Distribution
- m: Number of servers
- K: Storage Capacity (infinite if not specified)
- N: Population Size (infinite)
- X: Service Discipline (FCFS/FIFO)

http://en.wikipedia.org/wiki/Kendall's_notation

Kendall Notation of Queueing System

- Arrival Process
- M: Markovian
- D: Deterministic
- Er: Erlang
- G: General

- Service Process
- M: Markovian
- D: Deterministic
- Er: Erlang
- G: General

A/B/m/K/N/X

Number of servers m=1,2,…

Service Discipline

FIFO, LIFO, Round Robin, …

Storage Capacity

K= 1,2,…

(if ∞ then it is omitted)

Number of customers

N= 1,2,…

(for closed networks, otherwise it is omitted)

Distributions

- M: stands for "Markovian", implying exponential distribution for service times or inter-arrival times.
- D: Deterministic (e.g. fixed constant)
- Ek: Erlang with parameter k http://en.wikipedia.org/wiki/Erlang_distribution
- Hk: Hyper-exponential with parameter k
- G: General (anything)

Kendall Notation Examples

- M/M/1 Queue
- Poisson arrivals (exponential inter-arrival), and exponential service, 1 server, infinite capacity and population, FCFS (FIFO)
- the simplest ‘realistic’ queue
- M/M/m Queue
- Same, but m servers
- M/D/1 Queue
- Poisson arrivals and CONSTANT service times, 1 server, infinite capacity and population, FIFO.
- G/G/3/20/1500/SPF
- General arrival and service distributions, 3 servers, 17 queues (20-3), 1500 total jobs, Shortest Packet First

Performance Measures of Interest

- We are interested in steady state behavior
- Even though it is possible to pursue transient results, it is a significantly more difficult task.
- E[S]: average system (response) time (average time spent in the system)
- E[W]:average waiting time (average time spent waiting in queue(s))
- E[X]:average queue length
- E[U]: average utilization (fraction of time that the resources are being used)
- E[R]: average throughput (rate that customers leave the system)
- E[L]: average customer loss (rate that customers are lost or probability that a customer is lost)

λj-1

λ0

λ1

λj

λ2

λj-2

j-1

j

0

1

2

μj+1

μ3

μj-1

μj

μ1

μ2

Solution exists if

Recall the Birth-Death Chain Example- At steady state, we obtain

- In general

- Making the sum equal to 1

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