SlideShare a Scribd company logo
1 of 25
Download to read offline
“
“QUEUING THEORY”
QUEUING THEORY”
Presented By--
Presented By-- Anil Kumar
Anil Kumar
Avtar Singh
Avtar Singh
Queuing Theory
Queuing Theory

Queuing theory is the mathematics of waiting lines.
Queuing theory is the mathematics of waiting lines.
 It is extremely useful in predicting and evaluating
It is extremely useful in predicting and evaluating
system performance.
system performance.
 Queuing theory has been used for operations
Queuing theory has been used for operations
research, manufacturing and systems analysis.
research, manufacturing and systems analysis.
Traditional queuing theory problems refer to
Traditional queuing theory problems refer to
customers visiting a store, analogous to requests
customers visiting a store, analogous to requests
arriving at a device.
arriving at a device.
Applications of Queuing Theory
Applications of Queuing Theory
 Telecommunications
Telecommunications
 Traffic control
Traffic control
 Determining the sequence of computer
Determining the sequence of computer
operations
operations
 Predicting computer performance
Predicting computer performance
 Health services (e.g.. control of hospital bed
Health services (e.g.. control of hospital bed
assignments)
assignments)
 Airport traffic, airline ticket sales
Airport traffic, airline ticket sales
 Layout of manufacturing systems.
Layout of manufacturing systems.
Queuing System
Queuing System
 Model processes in which customers arrive.
Model processes in which customers arrive.
 Wait their turn for service.
Wait their turn for service.
 Are serviced and then leave.
Are serviced and then leave.
input
Server
Queue
output
Characteristics of Queuing
Systems
 Key elements
Key elements of queuing systems
of queuing systems
•
• Customer:--
Customer:-- refers to anything that arrives at a facility
refers to anything that arrives at a facility
and requires service, e.g., people, machines, trucks,
and requires service, e.g., people, machines, trucks,
emails.
emails.
•
• Server:--
Server:-- refers to any resource that provides the
refers to any resource that provides the
requested service, eg. repairpersons, retrieval machines,
requested service, eg. repairpersons, retrieval machines,
runways at airport.
runways at airport.
System Customers Server
Reception desk People Receptionist
Hospital Patients Nurses
Airport Airplanes Runway
Road network Cars Traffic light
Grocery Shoppers Checkout
station
Computer Jobs CPU, disk, CD
Queuing examples
Queuing examples
Components of a Queuing System
Components of a Queuing System
Arrival Process
Servers
Queue or
Waiting Line
Service Process
Exit
Parts of a Waiting Line
Parts of a Waiting Line
Dave’s
Dave’s
Car Wash
Car Wash
enter
enter exit
exit
Population of
Population of
dirty cars
dirty cars
Arrivals
Arrivals
from the
from the
general
general
population …
population …
Queue
Queue
(waiting line)
(waiting line)
Service
Service
facility
facility
Exit the system
Exit the system
Exit the system
Exit the system
Arrivals to the system
Arrivals to the system In the system
In the system
Arrival Characteristics
Arrival Characteristics
•Size of the population
Size of the population
•Behavior of arrivals
Behavior of arrivals
•Statistical distribution
Statistical distribution
of arrivals
of arrivals
Waiting Line
Waiting Line
Characteristics
Characteristics
•Limited vs. unlimited
Limited vs. unlimited
•Queue discipline
Queue discipline
Service
Service
Characteristics
Characteristics
•Service design
Service design
•Statistical
Statistical
distribution of
distribution of
service
service
1.
1. Arrival Process
Arrival Process
 According to source
According to source
 According to numbers
According to numbers
 According to time
According to time
2. Queue Structure
• First-come-first-served (FCFS)
• Last-come-first-serve (LCFS)
• Service-in-random-order (SIRO)
• Priority service
3.
3. Service system
Service system
1.
1. A single service system.
A single service system.
Queue
Arrivals
Arrivals
Service
facility
Departures
Departures
after service
after service
e.g- Your family dentist’s office, Library counter
e.g- Your family dentist’s office, Library counter
2. Multiple, parallel server, single
2. Multiple, parallel server, single
queue model
queue model
Queue
Service
facility
Channel 1
Service
facility
Channel 2
Service
facility
Channel 3
Arrivals
Arrivals
Departures
Departures
after service
after service
e.g- Booking at a service station
e.g- Booking at a service station
3. Multiple, parallel facilities with
3. Multiple, parallel facilities with
multiple queues Model
multiple queues Model
Service station Customers
leave
Queues
Arrivals
e.g.- Different cash counters in electricity office
4. Service facilities in a series
4. Service facilities in a series
Arrivals
Queues
Service station 1 Service station 2
Queues
Customers
leave
Phase 1 Phase 2
e.g.- Cutting, turning, knurling, drilling, grinding,
packaging operation of steel
Queuing Models
Queuing Models
1.
1. Deterministic queuing model
Deterministic queuing model
2.
2. Probabilistic queuing model
Probabilistic queuing model
• Deterministic queuing model
Deterministic queuing model :--
:--
λ
λ =
= Mean number of arrivals per time
Mean number of arrivals per time
period
period
µ
µ =
= Mean number of units served per
Mean number of units served per
time period
time period
Assumptions
Assumptions
• If
If λ
λ > µ, then waiting line shall be formed and
> µ, then waiting line shall be formed and
increased indefinitely and service system would fail
increased indefinitely and service system would fail
ultimately
ultimately
2. If
2. If λ
λ µ, there shall be no waiting line
µ, there shall be no waiting line
≤
≤
2.Probabilistic queuing model
2.Probabilistic queuing model
Probability that n customers will arrive in the
Probability that n customers will arrive in the
system in time interval T is
system in time interval T is
( ) ( )
!
n
e
t
n
P
t
n
t
λ
λ −
=
Single Channel Model
Single Channel Model
λ
λ =
= Mean number of arrivals per time
Mean number of arrivals per time
period
period
µ
µ =
= Mean number of units served per
Mean number of units served per
time period
time period
L
Ls
s =
= Average number of units
Average number of units
(customers) in the system (waiting and being
(customers) in the system (waiting and being
served)
served)
=
=
W
Ws
s =
= Average time a unit spends in the
Average time a unit spends in the
system (waiting time plus service time)
system (waiting time plus service time)
=
=
λ
λ
µ –
µ – λ
λ
1
1
µ –
µ – λ
λ
L
Lq
q =
= Average number of units waiting
Average number of units waiting
in the queue
in the queue
=
=
W
Wq
q =
= Average
Average time a unit spends
time a unit spends
waiting in the queue
waiting in the queue
=
=
p
p =
= Utilization factor for the system
Utilization factor for the system
=
=
λ
λ2
2
µ(µ –
µ(µ – λ
λ)
)
λ
λ
µ(µ –
µ(µ – λ
λ)
)
λ
λ
µ
µ
P
P0
0 =
= Probability of
Probability of 0
0 units in the
units in the
system (that is, the service unit is idle)
system (that is, the service unit is idle)
=
= 1 –
1 –
P
Pn > k
n > k =
= Probability of more than k units in the
Probability of more than k units in the
system, where n is the number of units in
system, where n is the number of units in
the system
the system
=
=
λ
λ
µ
µ
λ
λ
µ
µ
k
k + 1
+ 1
Single Channel Model Example
Single Channel Model Example
λ
λ =
= 2
2 cars arriving/hour
cars arriving/hour
µ
µ = 3
= 3 cars serviced/hour
cars serviced/hour
L
Ls
s = = = 2
= = = 2 cars
cars
in the system on average
in the system on average
W
Ws
s =
= = = 1
= = 1
hour average waiting time in
hour average waiting time in
the system
the system
L
Lq
q =
= = =
= =
1.33
1.33 cars waiting in line
cars waiting in line
λ
λ2
2
µ(µ –
µ(µ – λ
λ)
)
λ
λ
µ –
µ – λ
λ
1
1
µ –
µ – λ
λ
2
2
3 - 2
3 - 2
1
1
3 - 2
3 - 2
2
22
2
3(3 - 2)
3(3 - 2)
Cont…
Cont…
λ
λ =
= 2
2 cars arriving/hour,
cars arriving/hour, µ
µ = 3
= 3 cars
cars
serviced/hour
serviced/hour
W
Wq
q = =
= =
= 40
= 40 minute
minute
average waiting time
average waiting time
p
p =
= λ
λ/µ = 2/3 =
/µ = 2/3 =
66.6%
66.6% of time mechanic
of time mechanic
is busy
is busy
λ
λ
µ(µ –
µ(µ – λ
λ)
)
2
2
3(3 - 2)
3(3 - 2)
λ
λ
µ
µ
P
P0
0 = 1 - = .33
= 1 - = .33 probability
probability
there are
there are 0
0 cars in the system
cars in the system
Suggestions for Managing Queues
Suggestions for Managing Queues
1.
1. Determine an acceptable waiting time for
Determine an acceptable waiting time for
your customers
your customers
2.
2. Try to divert your customer’s attention when
Try to divert your customer’s attention when
waiting
waiting
3.
3. Inform your customers of what to expect
Inform your customers of what to expect
4.
4. Keep employees not serving the customers
Keep employees not serving the customers
out of sight
out of sight
5.
5. Segment customers
Segment customers
1.
1. Train your servers to be friendly
Train your servers to be friendly
2.
2. Encourage customers to come during the
Encourage customers to come during the
slack periods
slack periods
3.
3. Take a long-term perspective toward getting
Take a long-term perspective toward getting
rid of the queues
rid of the queues
Where the Time Goes
Where the Time Goes
In a life time, the average
person will spend :
SIX MONTHS Waiting at stoplights
EIGHT MONTHS Opening junk mail
ONE YEAR Looking for misplaced 0bjects
TWO YEARS Reading E-mail
FOUR YEARS Doing housework
FIVE YEARS Waiting in line
SIX YEARS Eating
ANY QUESTIONS
ANY QUESTIONS
PLEASE ??
PLEASE ??

More Related Content

Similar to queuingtheory-091005084417-phpapp01 (2).pdf

Similar to queuingtheory-091005084417-phpapp01 (2).pdf (20)

DES
DES DES
DES
 
Simulation chapter 4
Simulation chapter 4Simulation chapter 4
Simulation chapter 4
 
Unit 3 des
Unit 3 desUnit 3 des
Unit 3 des
 
Waiting line system
Waiting line systemWaiting line system
Waiting line system
 
Introduction to queueing theory
Introduction to queueing theoryIntroduction to queueing theory
Introduction to queueing theory
 
QUEUING THEORY
QUEUING THEORY QUEUING THEORY
QUEUING THEORY
 
Ramniwas final
Ramniwas finalRamniwas final
Ramniwas final
 
Queuing theory and its applications
Queuing theory and its applicationsQueuing theory and its applications
Queuing theory and its applications
 
Unit V - Queuing Theory
Unit V - Queuing TheoryUnit V - Queuing Theory
Unit V - Queuing Theory
 
Operations Research_18ME735_module 4 - queuing systems.pdf
Operations Research_18ME735_module 4 - queuing systems.pdfOperations Research_18ME735_module 4 - queuing systems.pdf
Operations Research_18ME735_module 4 - queuing systems.pdf
 
chapter11 liner model programing.pptx
chapter11 liner model programing.pptxchapter11 liner model programing.pptx
chapter11 liner model programing.pptx
 
Queuing theory
Queuing theoryQueuing theory
Queuing theory
 
Tn7 wtg.line+mgt
Tn7 wtg.line+mgtTn7 wtg.line+mgt
Tn7 wtg.line+mgt
 
Tn7 wtg.line mgt
Tn7 wtg.line mgtTn7 wtg.line mgt
Tn7 wtg.line mgt
 
Queuing Theory by Dr. B. J. Mohite
Queuing Theory by Dr. B. J. MohiteQueuing Theory by Dr. B. J. Mohite
Queuing Theory by Dr. B. J. Mohite
 
Georgia Tech: Performance Engineering - Queuing Theory and Predictive Modeling
Georgia Tech: Performance Engineering - Queuing Theory and Predictive ModelingGeorgia Tech: Performance Engineering - Queuing Theory and Predictive Modeling
Georgia Tech: Performance Engineering - Queuing Theory and Predictive Modeling
 
queueing problems in banking
queueing problems in bankingqueueing problems in banking
queueing problems in banking
 
Queuing_analysis
Queuing_analysisQueuing_analysis
Queuing_analysis
 
Chapter 02 simulation examples
Chapter 02   simulation examplesChapter 02   simulation examples
Chapter 02 simulation examples
 
Queing theory and delay analysis
Queing theory and delay analysisQueing theory and delay analysis
Queing theory and delay analysis
 

More from Aditya Mane

CONTRACT ADMINISTRATION Updated.pptx
CONTRACT ADMINISTRATION Updated.pptxCONTRACT ADMINISTRATION Updated.pptx
CONTRACT ADMINISTRATION Updated.pptxAditya Mane
 
Construction Disputes & Settlement.pptx
Construction Disputes & Settlement.pptxConstruction Disputes & Settlement.pptx
Construction Disputes & Settlement.pptxAditya Mane
 
CCCN GROUP ASSIGNMENT No-4pptx.pptx
CCCN GROUP ASSIGNMENT No-4pptx.pptxCCCN GROUP ASSIGNMENT No-4pptx.pptx
CCCN GROUP ASSIGNMENT No-4pptx.pptxAditya Mane
 
CCCN GROUP ASSIGNMENT No-4.pptx
CCCN GROUP ASSIGNMENT No-4.pptxCCCN GROUP ASSIGNMENT No-4.pptx
CCCN GROUP ASSIGNMENT No-4.pptxAditya Mane
 
Contracts PPT.pptx
Contracts PPT.pptxContracts PPT.pptx
Contracts PPT.pptxAditya Mane
 
Planning & Scheduling 222030008.pptx
Planning & Scheduling 222030008.pptxPlanning & Scheduling 222030008.pptx
Planning & Scheduling 222030008.pptxAditya Mane
 
WBS OF TUNNEL.pptx
WBS OF TUNNEL.pptxWBS OF TUNNEL.pptx
WBS OF TUNNEL.pptxAditya Mane
 
Module 3.1_Project Planning.pptx
Module 3.1_Project Planning.pptxModule 3.1_Project Planning.pptx
Module 3.1_Project Planning.pptxAditya Mane
 
Rehabilitation ppt.pptx
Rehabilitation ppt.pptxRehabilitation ppt.pptx
Rehabilitation ppt.pptxAditya Mane
 
ATAL TUNNEL.pptx
ATAL TUNNEL.pptxATAL TUNNEL.pptx
ATAL TUNNEL.pptxAditya Mane
 
ATAL TUNNEL.pptx
ATAL TUNNEL.pptxATAL TUNNEL.pptx
ATAL TUNNEL.pptxAditya Mane
 
Aptitude Gate.pdf
Aptitude Gate.pdfAptitude Gate.pdf
Aptitude Gate.pdfAditya Mane
 

More from Aditya Mane (12)

CONTRACT ADMINISTRATION Updated.pptx
CONTRACT ADMINISTRATION Updated.pptxCONTRACT ADMINISTRATION Updated.pptx
CONTRACT ADMINISTRATION Updated.pptx
 
Construction Disputes & Settlement.pptx
Construction Disputes & Settlement.pptxConstruction Disputes & Settlement.pptx
Construction Disputes & Settlement.pptx
 
CCCN GROUP ASSIGNMENT No-4pptx.pptx
CCCN GROUP ASSIGNMENT No-4pptx.pptxCCCN GROUP ASSIGNMENT No-4pptx.pptx
CCCN GROUP ASSIGNMENT No-4pptx.pptx
 
CCCN GROUP ASSIGNMENT No-4.pptx
CCCN GROUP ASSIGNMENT No-4.pptxCCCN GROUP ASSIGNMENT No-4.pptx
CCCN GROUP ASSIGNMENT No-4.pptx
 
Contracts PPT.pptx
Contracts PPT.pptxContracts PPT.pptx
Contracts PPT.pptx
 
Planning & Scheduling 222030008.pptx
Planning & Scheduling 222030008.pptxPlanning & Scheduling 222030008.pptx
Planning & Scheduling 222030008.pptx
 
WBS OF TUNNEL.pptx
WBS OF TUNNEL.pptxWBS OF TUNNEL.pptx
WBS OF TUNNEL.pptx
 
Module 3.1_Project Planning.pptx
Module 3.1_Project Planning.pptxModule 3.1_Project Planning.pptx
Module 3.1_Project Planning.pptx
 
Rehabilitation ppt.pptx
Rehabilitation ppt.pptxRehabilitation ppt.pptx
Rehabilitation ppt.pptx
 
ATAL TUNNEL.pptx
ATAL TUNNEL.pptxATAL TUNNEL.pptx
ATAL TUNNEL.pptx
 
ATAL TUNNEL.pptx
ATAL TUNNEL.pptxATAL TUNNEL.pptx
ATAL TUNNEL.pptx
 
Aptitude Gate.pdf
Aptitude Gate.pdfAptitude Gate.pdf
Aptitude Gate.pdf
 

Recently uploaded

Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...
Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...
Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...Sean Meyn
 
Renewable Energy & Entrepreneurship Workshop_21Feb2024.pdf
Renewable Energy & Entrepreneurship Workshop_21Feb2024.pdfRenewable Energy & Entrepreneurship Workshop_21Feb2024.pdf
Renewable Energy & Entrepreneurship Workshop_21Feb2024.pdfodunowoeminence2019
 
Vertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptx
Vertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptxVertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptx
Vertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptxLMW Machine Tool Division
 
Mohs Scale of Hardness, Hardness Scale.pptx
Mohs Scale of Hardness, Hardness Scale.pptxMohs Scale of Hardness, Hardness Scale.pptx
Mohs Scale of Hardness, Hardness Scale.pptxKISHAN KUMAR
 
cme397 surface engineering unit 5 part A questions and answers
cme397 surface engineering unit 5 part A questions and answerscme397 surface engineering unit 5 part A questions and answers
cme397 surface engineering unit 5 part A questions and answerskarthi keyan
 
Design Analysis of Alogorithm 1 ppt 2024.pptx
Design Analysis of Alogorithm 1 ppt 2024.pptxDesign Analysis of Alogorithm 1 ppt 2024.pptx
Design Analysis of Alogorithm 1 ppt 2024.pptxrajesshs31r
 
Carbohydrates principles of biochemistry
Carbohydrates principles of biochemistryCarbohydrates principles of biochemistry
Carbohydrates principles of biochemistryKomakeTature
 
Technical Management of cement industry.pdf
Technical Management of cement industry.pdfTechnical Management of cement industry.pdf
Technical Management of cement industry.pdfMadan Karki
 
cloud computing notes for anna university syllabus
cloud computing notes for anna university syllabuscloud computing notes for anna university syllabus
cloud computing notes for anna university syllabusViolet Violet
 
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS Bahzad5
 
Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...
Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...
Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...Amil baba
 
Introduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECHIntroduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECHC Sai Kiran
 
The relationship between iot and communication technology
The relationship between iot and communication technologyThe relationship between iot and communication technology
The relationship between iot and communication technologyabdulkadirmukarram03
 
Engineering Mechanics Chapter 5 Equilibrium of a Rigid Body
Engineering Mechanics  Chapter 5  Equilibrium of a Rigid BodyEngineering Mechanics  Chapter 5  Equilibrium of a Rigid Body
Engineering Mechanics Chapter 5 Equilibrium of a Rigid BodyAhmadHajasad2
 
UNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptxUNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptxrealme6igamerr
 
A Seminar on Electric Vehicle Software Simulation
A Seminar on Electric Vehicle Software SimulationA Seminar on Electric Vehicle Software Simulation
A Seminar on Electric Vehicle Software SimulationMohsinKhanA
 
Graphics Primitives and CG Display Devices
Graphics Primitives and CG Display DevicesGraphics Primitives and CG Display Devices
Graphics Primitives and CG Display DevicesDIPIKA83
 

Recently uploaded (20)

Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...
Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...
Quasi-Stochastic Approximation: Algorithm Design Principles with Applications...
 
Renewable Energy & Entrepreneurship Workshop_21Feb2024.pdf
Renewable Energy & Entrepreneurship Workshop_21Feb2024.pdfRenewable Energy & Entrepreneurship Workshop_21Feb2024.pdf
Renewable Energy & Entrepreneurship Workshop_21Feb2024.pdf
 
Vertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptx
Vertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptxVertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptx
Vertical- Machining - Center - VMC -LMW-Machine-Tool-Division.pptx
 
Mohs Scale of Hardness, Hardness Scale.pptx
Mohs Scale of Hardness, Hardness Scale.pptxMohs Scale of Hardness, Hardness Scale.pptx
Mohs Scale of Hardness, Hardness Scale.pptx
 
cme397 surface engineering unit 5 part A questions and answers
cme397 surface engineering unit 5 part A questions and answerscme397 surface engineering unit 5 part A questions and answers
cme397 surface engineering unit 5 part A questions and answers
 
Design Analysis of Alogorithm 1 ppt 2024.pptx
Design Analysis of Alogorithm 1 ppt 2024.pptxDesign Analysis of Alogorithm 1 ppt 2024.pptx
Design Analysis of Alogorithm 1 ppt 2024.pptx
 
Carbohydrates principles of biochemistry
Carbohydrates principles of biochemistryCarbohydrates principles of biochemistry
Carbohydrates principles of biochemistry
 
Technical Management of cement industry.pdf
Technical Management of cement industry.pdfTechnical Management of cement industry.pdf
Technical Management of cement industry.pdf
 
cloud computing notes for anna university syllabus
cloud computing notes for anna university syllabuscloud computing notes for anna university syllabus
cloud computing notes for anna university syllabus
 
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS GENERAL CONDITIONS  FOR  CONTRACTS OF CIVIL ENGINEERING WORKS
GENERAL CONDITIONS FOR CONTRACTS OF CIVIL ENGINEERING WORKS
 
計劃趕得上變化
計劃趕得上變化計劃趕得上變化
計劃趕得上變化
 
Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...
Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...
Popular-NO1 Kala Jadu Expert Specialist In Germany Kala Jadu Expert Specialis...
 
Introduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECHIntroduction to Machine Learning Unit-3 for II MECH
Introduction to Machine Learning Unit-3 for II MECH
 
Lecture 4 .pdf
Lecture 4                              .pdfLecture 4                              .pdf
Lecture 4 .pdf
 
The relationship between iot and communication technology
The relationship between iot and communication technologyThe relationship between iot and communication technology
The relationship between iot and communication technology
 
Engineering Mechanics Chapter 5 Equilibrium of a Rigid Body
Engineering Mechanics  Chapter 5  Equilibrium of a Rigid BodyEngineering Mechanics  Chapter 5  Equilibrium of a Rigid Body
Engineering Mechanics Chapter 5 Equilibrium of a Rigid Body
 
Présentation IIRB 2024 Chloe Dufrane.pdf
Présentation IIRB 2024 Chloe Dufrane.pdfPrésentation IIRB 2024 Chloe Dufrane.pdf
Présentation IIRB 2024 Chloe Dufrane.pdf
 
UNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptxUNIT4_ESD_wfffffggggggggggggith_ARM.pptx
UNIT4_ESD_wfffffggggggggggggith_ARM.pptx
 
A Seminar on Electric Vehicle Software Simulation
A Seminar on Electric Vehicle Software SimulationA Seminar on Electric Vehicle Software Simulation
A Seminar on Electric Vehicle Software Simulation
 
Graphics Primitives and CG Display Devices
Graphics Primitives and CG Display DevicesGraphics Primitives and CG Display Devices
Graphics Primitives and CG Display Devices
 

queuingtheory-091005084417-phpapp01 (2).pdf

  • 1. “ “QUEUING THEORY” QUEUING THEORY” Presented By-- Presented By-- Anil Kumar Anil Kumar Avtar Singh Avtar Singh
  • 2. Queuing Theory Queuing Theory  Queuing theory is the mathematics of waiting lines. Queuing theory is the mathematics of waiting lines.  It is extremely useful in predicting and evaluating It is extremely useful in predicting and evaluating system performance. system performance.  Queuing theory has been used for operations Queuing theory has been used for operations research, manufacturing and systems analysis. research, manufacturing and systems analysis. Traditional queuing theory problems refer to Traditional queuing theory problems refer to customers visiting a store, analogous to requests customers visiting a store, analogous to requests arriving at a device. arriving at a device.
  • 3. Applications of Queuing Theory Applications of Queuing Theory  Telecommunications Telecommunications  Traffic control Traffic control  Determining the sequence of computer Determining the sequence of computer operations operations  Predicting computer performance Predicting computer performance  Health services (e.g.. control of hospital bed Health services (e.g.. control of hospital bed assignments) assignments)  Airport traffic, airline ticket sales Airport traffic, airline ticket sales  Layout of manufacturing systems. Layout of manufacturing systems.
  • 4. Queuing System Queuing System  Model processes in which customers arrive. Model processes in which customers arrive.  Wait their turn for service. Wait their turn for service.  Are serviced and then leave. Are serviced and then leave. input Server Queue output
  • 5. Characteristics of Queuing Systems  Key elements Key elements of queuing systems of queuing systems • • Customer:-- Customer:-- refers to anything that arrives at a facility refers to anything that arrives at a facility and requires service, e.g., people, machines, trucks, and requires service, e.g., people, machines, trucks, emails. emails. • • Server:-- Server:-- refers to any resource that provides the refers to any resource that provides the requested service, eg. repairpersons, retrieval machines, requested service, eg. repairpersons, retrieval machines, runways at airport. runways at airport.
  • 6. System Customers Server Reception desk People Receptionist Hospital Patients Nurses Airport Airplanes Runway Road network Cars Traffic light Grocery Shoppers Checkout station Computer Jobs CPU, disk, CD Queuing examples Queuing examples
  • 7. Components of a Queuing System Components of a Queuing System Arrival Process Servers Queue or Waiting Line Service Process Exit
  • 8. Parts of a Waiting Line Parts of a Waiting Line Dave’s Dave’s Car Wash Car Wash enter enter exit exit Population of Population of dirty cars dirty cars Arrivals Arrivals from the from the general general population … population … Queue Queue (waiting line) (waiting line) Service Service facility facility Exit the system Exit the system Exit the system Exit the system Arrivals to the system Arrivals to the system In the system In the system Arrival Characteristics Arrival Characteristics •Size of the population Size of the population •Behavior of arrivals Behavior of arrivals •Statistical distribution Statistical distribution of arrivals of arrivals Waiting Line Waiting Line Characteristics Characteristics •Limited vs. unlimited Limited vs. unlimited •Queue discipline Queue discipline Service Service Characteristics Characteristics •Service design Service design •Statistical Statistical distribution of distribution of service service
  • 9. 1. 1. Arrival Process Arrival Process  According to source According to source  According to numbers According to numbers  According to time According to time 2. Queue Structure • First-come-first-served (FCFS) • Last-come-first-serve (LCFS) • Service-in-random-order (SIRO) • Priority service
  • 10. 3. 3. Service system Service system 1. 1. A single service system. A single service system. Queue Arrivals Arrivals Service facility Departures Departures after service after service e.g- Your family dentist’s office, Library counter e.g- Your family dentist’s office, Library counter
  • 11. 2. Multiple, parallel server, single 2. Multiple, parallel server, single queue model queue model Queue Service facility Channel 1 Service facility Channel 2 Service facility Channel 3 Arrivals Arrivals Departures Departures after service after service e.g- Booking at a service station e.g- Booking at a service station
  • 12. 3. Multiple, parallel facilities with 3. Multiple, parallel facilities with multiple queues Model multiple queues Model Service station Customers leave Queues Arrivals e.g.- Different cash counters in electricity office
  • 13. 4. Service facilities in a series 4. Service facilities in a series Arrivals Queues Service station 1 Service station 2 Queues Customers leave Phase 1 Phase 2 e.g.- Cutting, turning, knurling, drilling, grinding, packaging operation of steel
  • 14. Queuing Models Queuing Models 1. 1. Deterministic queuing model Deterministic queuing model 2. 2. Probabilistic queuing model Probabilistic queuing model • Deterministic queuing model Deterministic queuing model :-- :-- λ λ = = Mean number of arrivals per time Mean number of arrivals per time period period µ µ = = Mean number of units served per Mean number of units served per time period time period
  • 15. Assumptions Assumptions • If If λ λ > µ, then waiting line shall be formed and > µ, then waiting line shall be formed and increased indefinitely and service system would fail increased indefinitely and service system would fail ultimately ultimately 2. If 2. If λ λ µ, there shall be no waiting line µ, there shall be no waiting line ≤ ≤
  • 16. 2.Probabilistic queuing model 2.Probabilistic queuing model Probability that n customers will arrive in the Probability that n customers will arrive in the system in time interval T is system in time interval T is ( ) ( ) ! n e t n P t n t λ λ − =
  • 17. Single Channel Model Single Channel Model λ λ = = Mean number of arrivals per time Mean number of arrivals per time period period µ µ = = Mean number of units served per Mean number of units served per time period time period L Ls s = = Average number of units Average number of units (customers) in the system (waiting and being (customers) in the system (waiting and being served) served) = = W Ws s = = Average time a unit spends in the Average time a unit spends in the system (waiting time plus service time) system (waiting time plus service time) = = λ λ µ – µ – λ λ 1 1 µ – µ – λ λ
  • 18. L Lq q = = Average number of units waiting Average number of units waiting in the queue in the queue = = W Wq q = = Average Average time a unit spends time a unit spends waiting in the queue waiting in the queue = = p p = = Utilization factor for the system Utilization factor for the system = = λ λ2 2 µ(µ – µ(µ – λ λ) ) λ λ µ(µ – µ(µ – λ λ) ) λ λ µ µ
  • 19. P P0 0 = = Probability of Probability of 0 0 units in the units in the system (that is, the service unit is idle) system (that is, the service unit is idle) = = 1 – 1 – P Pn > k n > k = = Probability of more than k units in the Probability of more than k units in the system, where n is the number of units in system, where n is the number of units in the system the system = = λ λ µ µ λ λ µ µ k k + 1 + 1
  • 20. Single Channel Model Example Single Channel Model Example λ λ = = 2 2 cars arriving/hour cars arriving/hour µ µ = 3 = 3 cars serviced/hour cars serviced/hour L Ls s = = = 2 = = = 2 cars cars in the system on average in the system on average W Ws s = = = = 1 = = 1 hour average waiting time in hour average waiting time in the system the system L Lq q = = = = = = 1.33 1.33 cars waiting in line cars waiting in line λ λ2 2 µ(µ – µ(µ – λ λ) ) λ λ µ – µ – λ λ 1 1 µ – µ – λ λ 2 2 3 - 2 3 - 2 1 1 3 - 2 3 - 2 2 22 2 3(3 - 2) 3(3 - 2)
  • 21. Cont… Cont… λ λ = = 2 2 cars arriving/hour, cars arriving/hour, µ µ = 3 = 3 cars cars serviced/hour serviced/hour W Wq q = = = = = 40 = 40 minute minute average waiting time average waiting time p p = = λ λ/µ = 2/3 = /µ = 2/3 = 66.6% 66.6% of time mechanic of time mechanic is busy is busy λ λ µ(µ – µ(µ – λ λ) ) 2 2 3(3 - 2) 3(3 - 2) λ λ µ µ P P0 0 = 1 - = .33 = 1 - = .33 probability probability there are there are 0 0 cars in the system cars in the system
  • 22. Suggestions for Managing Queues Suggestions for Managing Queues 1. 1. Determine an acceptable waiting time for Determine an acceptable waiting time for your customers your customers 2. 2. Try to divert your customer’s attention when Try to divert your customer’s attention when waiting waiting 3. 3. Inform your customers of what to expect Inform your customers of what to expect 4. 4. Keep employees not serving the customers Keep employees not serving the customers out of sight out of sight 5. 5. Segment customers Segment customers
  • 23. 1. 1. Train your servers to be friendly Train your servers to be friendly 2. 2. Encourage customers to come during the Encourage customers to come during the slack periods slack periods 3. 3. Take a long-term perspective toward getting Take a long-term perspective toward getting rid of the queues rid of the queues
  • 24. Where the Time Goes Where the Time Goes In a life time, the average person will spend : SIX MONTHS Waiting at stoplights EIGHT MONTHS Opening junk mail ONE YEAR Looking for misplaced 0bjects TWO YEARS Reading E-mail FOUR YEARS Doing housework FIVE YEARS Waiting in line SIX YEARS Eating