CS433Modeling and Simulation

Lecture 12

Queueing Theory

Dr. Anis Koubâa03 May 2008

Al-Imam Mohammad Ibn Saud UniversityAl-Imam Mohammad Ibn Saud University

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 …

The Queuing Model

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

Queue Server

Queuing System

Click for Queue Simulator

Queuing theory definitions5

(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

6Applications of Queuing Theory

6

Applications of Queuing Theory

7

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

8

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

9

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

Queuing theory for studying networks

10

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

QNAP/ModlineExample of a Queue Simulator

11

12 The Little’s Law

12

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.

The Queuing Times

Queue Server

Queuing System

Queuing Time Service Time

Response Time (or Delay)

Little’s Law

E EN T Expected number of customers in the system

Expected time in the system

Arrival rate IN the system

Generality of Little’s Law

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

E EN T

Queueing Network

Queueing Network

λ

Aggregate Arrival rate

Mean number tasks in system = mean arrival rate x mean response time

16Specification of Queuing Systems

16

Characteristics of queuing systems

17

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 arrival and 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)

19

To specify a queue, we use the Kendall Notation.

The First three parameters are typically used, unless specified

1. A: Arrival Distribution2. B: Service Distribution3. m: Number of servers 4. K: Storage Capacity (infinite if not specified) 5. N: Population Size (infinite) 6. X: Service Discipline (FCFS/FIFO)

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

Kendall Notation of Queueing System

A/B/m/K/N/X

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

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

Number of servers m=1,2,…

Storage Capacity K= 1,2,… (if ∞ then it is omitted)

Number of customers N= 1,2,… (for closed networks, otherwise it is omitted)

Service DisciplineFIFO, LIFO, Round Robin, …

Distributions

CS352 Fall,2005

21

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 Examples22

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

23 Performance Measures

23

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)

Recall the Birth-Death Chain Example

At steady state, we obtain

λ0

0 1μ1

λ1

2μ2

λj-2

j-1μj-1

λj-1

jμjμ3

λ2λj

μj+

1

0 0 1 1 0 01 0

1

In general

1 1 1 1 0j jj j j j j 0

1 01 1

...

...j

jj

Making the sum equal to 1

0 1

01 1

...1 1

...j

j j

Solution exists if

0 1

1 1

...1

...j

j j

S

26 End of Part 01

26