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Queueing Theory

Presented to :
Dr. Dibyojyoti
Bhattacharjee
Presented by :
Biswajit Bhattacharjee (19)
Bikash Choudhury (...
Introduction to Queueing Theory
A pioneer:
Agner Krarup Erlang
(1878-1929) the Danish
telecommunication engineer
started a...
What is queueing theory?
• Queueing theory is the mathematical study of waiting lines, or
queues. In queueing theory a mod...
Why is queueing theory important?
• Capacity problems are very common in industry and one
of the main drivers of process r...
Examples of Real World Queuing Systems?
• Commercial Queuing Systems
– Commercial organizations serving external customers...
Problems of a store manger
• When and why do we get queues?
– Too many people, too few desks

• What can we do about it?
–...
Problems of a customer
• Why does it always feel to us like all other queues move
faster?
• Which queue should I take? Whe...
Queuing theory for studying networks
• View network as collections of queues
– FIFO data-structures
• Queuing theory provi...
Model Queuing System

Customers
Queue

Server

Queuing System

• Use Queuing models to
– Describe the behavior of queuing ...
Customer n
Arrival
event

Delay

Begin
service

Activity

End
service
Time

Interarrival

Arrival
event

Begin
service
Del...
Assumptions
• Independent arrivals
• Exponential distributions
• Customers do not leave or change queues.
• Large queues d...
Measuring the Queue Performance
There are a number of measure that can help a
manager to balance the capacity and waiting ...
Components of a Basic Queuing Process
Input Source

Calling
Population

The Queuing System

Jobs

Service
Mechanism

Queue...
Principal Queue Parameters
1.
2.
3.
4.
5.

Calling Population
Arrival Process
Service Process
Number of Servers
Queue Disc...
1. The Calling Population
• Population of customers or jobs
• The size can be finite or infinite
– The latter is most comm...
2. Arrival Process
• In what pattern do jobs / customers arrive to
the queueing system?
– Distribution of arrival times?
–...
3. Service Process
• How long does it take to service a job or
customer?
– Distribution of arrival times?
– Rework or repa...
4. Number of Servers
• How many servers are available?
Single Server Queue

Multiple Server Queue

18
Example – Two Queue Configurations
Multiple Queues
Servers

Single Queue
Servers

19
Multiple vs Single Customer Queue
Configuration
Multiple Line Advantages
1. The service provided can be
differentiated
– E...
5. Queue Discipline
• How are jobs / customers selected from the
queue for service?
– First Come First Served (FCFS)
– Sho...
Three queuing disciplines used in
Telephone Networks
• First In First Out – This principle states that
customers are serve...
FIFO
“First In First Out”
LIFO
“Last in First Out”
Elevators are a circumstance where this occurs.
SIRO
“Service In Random Order”
• Like drawing tickets out of a pool of tickets for service.
Single-server Single-stage Queue

Arrival Stream

Customers
In queue

Service
Facility
Multiple-server Single-stage Queue

Customers
In queue
Service
Facilities
Single-server Multiple-stage Queue

Customers
In queue

Service
Facility

Pharmacy Conveyor System >>>>>
Multiple-server Multiple-Stage Queue

Customers
In queue

Service
Facilities
Types of Queues of Interest
• Analytical Models for Estimating Capacity and
Related Metrics
– Single Server
• M/M/1, M/G/1...
Infinite-Source Queuing Models
• Single channel, exponential service time
(M/M/1)
• Single channel, constant service time ...
BUSINESS APPLICATIONS
• It is a practical operations management technique that is commonly used to
determine staffing, sch...
Limitations of Queuing theory
• The assumptions of classical queuing theory may be too restrictive to be able to

model re...
Primary References
•

http://en.wikipedia.org/wiki/Queueing_theory

•

http://www.eventhelix.com/realtimemantra/congestion...
Queueing Theory and its BusinessS Applications
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Queueing Theory and its BusinessS Applications

Queueing Theory and its BusinessS Applications

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Queueing Theory and its BusinessS Applications

  1. 1. Queueing Theory Presented to : Dr. Dibyojyoti Bhattacharjee Presented by : Biswajit Bhattacharjee (19) Bikash Choudhury (16) Biswaraj Das Purkayastha(20) Kunal Sengupta(37)
  2. 2. Introduction to Queueing Theory A pioneer: Agner Krarup Erlang (1878-1929) the Danish telecommunication engineer started applying principles of queuing theory in the area of telecommunications.
  3. 3. What is queueing theory? • Queueing theory is the mathematical study of waiting lines, or queues. In queueing theory a model is constructed so that queue lengths and waiting times can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. • Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange. The ideas have since seen applications including telecommunications, traffic engineering, computing and the design of factories, shops, offices and hospitals.
  4. 4. Why is queueing theory important? • Capacity problems are very common in industry and one of the main drivers of process redesign – Need to balance the cost of increased capacity against the gains of increased productivity and service • Queuing and waiting time analysis is particularly important in service systems – Large costs of waiting and of lost sales due to waiting Example – Hospital • Patients arrive by ambulance or by their own accord • One doctor is always on duty • More and more patients seeks help longer waiting times  Question: Should another MD position be instated?
  5. 5. Examples of Real World Queuing Systems? • Commercial Queuing Systems – Commercial organizations serving external customers – Ex. Dentist, bank, ATM, gas stations, plumber, garage … • Transportation service systems – Vehicles are customers or servers – Ex. Vehicles waiting at toll stations and traffic lights, trucks or ships waiting to be loaded, taxi cabs, fire engines, buses … • Business-internal service systems – Customers receiving service are internal to the organization providing the service – Ex. Inspection stations, conveyor belts, computer support … • Social service systems – Ex. Judicial process, hospital, waiting lists for organ transplants or student dorm rooms … 5
  6. 6. Problems of a store manger • When and why do we get queues? – Too many people, too few desks • What can we do about it? – More cash desks • What if there are too many desks open?  costs – Let on more people in  – First serve the people who need little service time – Limit the time during which someone is being served  e.g. limited treatment time per client at the doctor 6
  7. 7. Problems of a customer • Why does it always feel to us like all other queues move faster? • Which queue should I take? Where shall I append? How long do I have to wait? – – – – – Where there are the fewest people in queue? Where the people have the least products to dispatch? Where the fastest cashier is? Where one can only pay cash? Where someone helps me bagging? 7
  8. 8. 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 (buffer) – Average waiting time – Probability queue is at a certain length – Probability a packet will be lost
  9. 9. Model Queuing System Customers Queue Server Queuing System • Use Queuing models to – Describe the behavior of queuing systems – Evaluate system performance
  10. 10. Customer n Arrival event Delay Begin service Activity End service Time Interarrival Arrival event Begin service Delay End service Activity Time Customer n+1
  11. 11. Assumptions • Independent arrivals • Exponential distributions • Customers do not leave or change queues. • Large queues do not discourage customers. Many assumptions are not always true, but queuing theory gives good results anyway
  12. 12. Measuring the Queue Performance There are a number of measure that can help a manager to balance the capacity and waiting costs: – – – – – – Average time in a queue Average length of a queue Average customer time in the system Number of customers in a queue Probability of numbers in a queue Probability of system being unused The last two above, looking at probability is where most of the work on queue theory goes on You will need to understand more about statistics, particularly Poisson distribution) to delve deeper into this
  13. 13. Components of a Basic Queuing Process Input Source Calling Population The Queuing System Jobs Service Mechanism Queue Served Jobs leave the system Queue Discipline Arrival Process Queue Configuration Service Process 13
  14. 14. Principal Queue Parameters 1. 2. 3. 4. 5. Calling Population Arrival Process Service Process Number of Servers Queue Discipline 14
  15. 15. 1. The Calling Population • Population of customers or jobs • The size can be finite or infinite – The latter is most common • Can be homogeneous – Only one type of customers/ jobs • Or heterogeneous – Several different kinds of customers/jobs 15
  16. 16. 2. Arrival Process • In what pattern do jobs / customers arrive to the queueing system? – Distribution of arrival times? – Batch arrivals? – Finite population? – Finite queue length? • Poisson arrival process often assumed – Many real-world arrival processes can be modeled using a Poisson process 16
  17. 17. 3. Service Process • How long does it take to service a job or customer? – Distribution of arrival times? – Rework or repair? – Service center (machine) breakdown? • Exponential service times often assumed – Works well for maintenance or unscheduled service situations 17
  18. 18. 4. Number of Servers • How many servers are available? Single Server Queue Multiple Server Queue 18
  19. 19. Example – Two Queue Configurations Multiple Queues Servers Single Queue Servers 19
  20. 20. Multiple vs Single Customer Queue Configuration Multiple Line Advantages 1. The service provided can be differentiated – Ex. Supermarket express lanes 2. Labor specialization possible 3. Customer has more flexibility 4. Balking behavior may be deterred – Several medium-length lines are less intimidating than one very long line Single Line Advantages 1. Guarantees fairness – FIFO applied to all arrivals 2. No customer anxiety regarding choice of queue 3. Avoids “cutting in” problems 4. The most efficient set up for minimizing time in the queue 5. Jockeying (line switching) is avoided 20
  21. 21. 5. Queue Discipline • How are jobs / customers selected from the queue for service? – First Come First Served (FCFS) – Shortest Processing Time (SPT) – Earliest Due Date (EDD) – Priority (jobs are in different priority classes) • FCFS default assumption for most models 21
  22. 22. Three queuing disciplines used in Telephone Networks • First In First Out – This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first. • Last In First Out – This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. • Processor Sharing – Customers are served equally. Network capacity is shared between customers and they all effectively experience the same delay
  23. 23. FIFO “First In First Out”
  24. 24. LIFO “Last in First Out” Elevators are a circumstance where this occurs.
  25. 25. SIRO “Service In Random Order” • Like drawing tickets out of a pool of tickets for service.
  26. 26. Single-server Single-stage Queue Arrival Stream Customers In queue Service Facility
  27. 27. Multiple-server Single-stage Queue Customers In queue Service Facilities
  28. 28. Single-server Multiple-stage Queue Customers In queue Service Facility Pharmacy Conveyor System >>>>>
  29. 29. Multiple-server Multiple-Stage Queue Customers In queue Service Facilities
  30. 30. Types of Queues of Interest • Analytical Models for Estimating Capacity and Related Metrics – Single Server • M/M/1, M/G/1, M/D/1, G/G/1 – Multiple Server • M/M/c, M/G/∞ etc. – Multiple Stage • Markov Chain models
  31. 31. Infinite-Source Queuing Models • Single channel, exponential service time (M/M/1) • Single channel, constant service time (M/D/1) • Multiple channel, exponential service time (M/M/S) • Multiple priority service, exponential service time
  32. 32. BUSINESS APPLICATIONS • It is a practical operations management technique that is commonly used to determine staffing, scheduling and calculating inventory levels. • To improve customer satisfaction. • Six Sigma professionals – through their knowledge of probability distributions, process mapping and basic process improvement techniques – can help organizations design and implement robust queuing models to create this competitive advantage. • Timeliness: Businesses conduct studies using mathematical models and formulas to determine the best way of serving the greatest number of customers, given their staffing resources. In retail businesses, the volume of transactions is extremely important in maximizing revenues and profitability • Remove Inefficiencies: for eg., bank, needs to stick to its model once it's been determined that maximum efficiency can be achieved, both in labor costs and customers served, by using a centralized queue based on staffing at least three tellers during peak hours.
  33. 33. Limitations of Queuing theory • The assumptions of classical queuing theory may be too restrictive to be able to model real-world situations exactly. • The complexity of production lines with product-specific characteristics cannot be handled with those models. Often, although the bounds do exist, they can be safely ignored. • Because the differences between the real-world and theory is not statistically significant, as the probability that such boundary situations might occur is remote compared to the expected normal situation. • Furthermore, several studies show the robustness of queuing models outside their assumptions. • In other cases the theoretical solution may either prove intractable or insufficiently informative to be useful. •Alternative means of analysis have thus been devised in order to provide some insight into problems that do not fall under the scope of queuing theory, •Although they are often scenario-specific because they generally consist of computer analysis of experimental data. 33
  34. 34. Primary References • http://en.wikipedia.org/wiki/Queueing_theory • http://www.eventhelix.com/realtimemantra/congestioncontrol/queueing_theory.htm • http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html • http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html • http://www.merriam-webster.com/dictionary/queuing%20theory • http://www.amazon.com/Fundamentals-Queueing-Theory-ProbabilityStatistics/dp/047179127X • http://staff.um.edu.mt/jskl1/simweb/intro.htm

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