Queueing Theory

Presented to :Dr. Dibyojyoti Bhattacharjee

Presented by :Biswajit Bhattacharjee (19)

Bikash Choudhury (16)Biswaraj Das Purkayastha(20)

Kunal Sengupta(37)

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.

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.

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?

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• 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 …

Examples of Real World Queuing Systems?

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Problems of a store manger

• When and why do we get queues?

• What can we do about it?– More cash desks

• What if there are too many desks open?

– Let on more people in – First serve the people who need little service time– Limit the time during which someone is being served

– Too many people, too few desks

costs

e.g. limited treatment time per client at the doctor

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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?

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

• Use Queuing models to – Describe the behavior of queuing systems– Evaluate system performance

Model Queuing System

Queuing System

Queue Server

Customers

Time

Time

Arrival event

Delay

Begin service

Begin service

Arrival event

Delay

Activity

Activity

End service

End service

Customer n+1

Customer n

Interarrival

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

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

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Components of a Basic Queuing Process

Calling Population Queue

Service Mechanism

Input Source The Queuing System

Jobs

Arrival Process

Queue Configuration

Queue Discipline

Served Jobs

Service Process

leave the system

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Principal Queue Parameters

1. Calling Population2. Arrival Process3. Service Process4. Number of Servers5. Queue Discipline

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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

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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

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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

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4. Number of Servers

• How many servers are available?

Single Server Queue

Multiple Server Queue

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Example – Two Queue Configurations

Servers

Multiple Queues

Servers

Single Queue

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1. The service provided can be differentiated– Ex. Supermarket express lanes

2. Labor specialization possible3. Customer has more flexibility4. Balking behavior may be

deterred– Several medium-length lines are

less intimidating than one very long line

1. Guarantees fairness– FIFO applied to all arrivals

2. No customer anxiety regarding choice of queue

3. Avoids “cutting in” problems4. The most efficient set up for

minimizing time in the queue5. Jockeying (line switching) is

avoided

Multiple vs Single Customer Queue Configuration

Multiple Line Advantages Single Line Advantages

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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

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

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

Service Facility

CustomersIn queue

Arrival Stream

Multiple-server Single-stage Queue

Service Facilities

CustomersIn queue

Single-server Multiple-stage Queue

Service Facility

CustomersIn queue

Pharmacy Conveyor System >>>>>

Multiple-server Multiple-Stage Queue

Service Facilities

CustomersIn queue

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

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

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.

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• 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.

Limitations of Queuing theory

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-Probability-Statistics/dp/047179127X

• http://staff.um.edu.mt/jskl1/simweb/intro.htm