# Queuing Theory

**Topics:**Poisson distribution, Queueing theory, Exponential distribution

**Pages:**38 (3177 words)

**Published:**December 2, 2012

The Structure of a Waiting Line System

Queuing Systems

Queuing System Input Characteristics

Queuing System Operating Characteristics

Analytical Formulas

Single-Channel Waiting Line Model with Poisson

Arrivals and Exponential Service Times

Multiple-Channel Waiting Line Model with Poisson

Arrivals and Exponential Service Times

Economic Analysis of Waiting Lines

Slide 1

Structure of a Waiting Line System

Queuing theory is the study of waiting lines.

Four characteristics of a queuing system are:

•the manner in which customers arrive

•the time required for service

•the priority determining the order of service

•the number and configuration of servers in the

system.

Slide 2

Structure of a Waiting Line System

Distribution of Arrivals

•Generally, the arrival of customers into the system is

a random event.

•Frequently the arrival pattern is modeled as a

Poisson process.

Distribution of Service Times

•Service time is also usually a random variable.

•A distribution commonly used to describe service

time is the exponential distribution.

Slide 3

Structure of a Waiting Line System

Queue Discipline

•Most common queue discipline is first come, first

served (FCFS).

•An elevator is an example of last come, first served

(LCFS) queue discipline.

•Other disciplines assign priorities to the waiting

units and then serve the unit with the highest

priority first.

Slide 4

Structure of a Waiting Line System

Single Service Channel

Customer

arrives

Waiting line

Multiple Service Channels

System

S1

Customer

leaves

System

S1

Customer

arrives

Waiting line

S2

Customer

leaves

S3

Slide 5

Examples of Internal Service Systems

That Are Queueing Systems

Type of System

Customers

Server(s)

Secretarial services

Employees

Secretary

Copying services

Employees

Copy machine

Computer programming

services

Employees

Programmer

Mainframe computer

Employees

Computer

First-aid center

Employees

Nurse

Faxing services

Employees

Fax machine

Materials-handling

system

Loads

Materials-handling

unit

Maintenance system

Machines

Repair crew

Inspection station

Items

Inspector

Production system

Jobs

Machine

Semiautomatic machines

Machines

Operator

Tool crib

Machine

Clerk

Slide 6

Examples of Transportation Service Systems

That Are Queueing Systems

Type of System

Customers

Server(s)

Highway tollbooth

Cars

Cashier

Truck loading dock

Trucks

Loading crew

Port unloading area

Ships

Unloading crew

Airplanes waiting to take

off

Airplanes

Runway

Airplanes waiting to land

Airplanes

Runway

Airline service

People

Airplane

Taxicab service

People

Taxicab

Elevator service

People

Elevator

Fire department

Fires

Fire truck

Parking lot

Cars

Parking space

Ambulance service

People

Ambulance

Slide 7

Queuing Systems

A three part code of the form A/B/k is used to

describe various queuing systems.

A identifies the arrival distribution, B the service

(departure) distribution and k the number of

channels for the system.

Symbols used for the arrival and service processes

are: M - Markov distributions (Poisson/exponential),

D - Deterministic (constant) and G - General

distribution (with a known mean and variance).

For example, M/M/k refers to a system in which

arrivals occur according to a Poisson distribution,

service times follow an exponential distribution and

there are k servers working at identical service rates.

Slide 8

Queuing System Input Characteristics

=

1/ =

µ=

1/µ =

=

the average arrival rate

the average time between arrivals

the average service rate for each server

the average service time

the standard deviation of the service...

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