Queuing Theory

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  • Topic: Poisson distribution, Queueing theory, Exponential distribution
  • Pages : 38 (3177 words )
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  • Published : December 2, 2012
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Waiting Line Models









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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