Presented to :
Presented by :
Biswajit Bhattacharjee (19)
Bikash Choudhury (16)
Biswaraj Das Purkayastha(20)
Introduction to Queueing Theory
Agner Krarup Erlang
(1878-1929) the Danish
started applying principles of
queuing theory in the area of
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
Question: Should another MD position be instated?
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
– Ex. Inspection stations, conveyor belts, computer support …
• Social service systems
– Ex. Judicial process, hospital, waiting lists for organ transplants or student
dorm rooms …
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?
– 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
Problems of a customer
• Why does it always feel to us like all other queues move
• 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
– Average length (buffer)
– Average waiting time
– Probability queue is at a certain length
– Probability a packet will be lost
Model Queuing System
• Use Queuing models to
– Describe the behavior of queuing systems
– Evaluate system performance
• 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
Components of a Basic Queuing Process
The Queuing System
Principal Queue Parameters
Number of Servers
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
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
3. Service Process
• How long does it take to service a job or
– Distribution of arrival times?
– Rework or repair?
– Service center (machine) breakdown?
• Exponential service times often assumed
– Works well for maintenance or unscheduled
4. Number of Servers
• How many servers are available?
Single Server Queue
Multiple Server Queue
Example – Two Queue Configurations
Multiple vs Single Customer Queue
Multiple Line Advantages
1. The service provided can be
– Ex. Supermarket express lanes
2. Labor specialization possible
3. Customer has more flexibility
4. Balking behavior may be
– Several medium-length lines are
less intimidating than one very
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
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
• 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
• 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
“Last in First Out”
Elevators are a circumstance where this occurs.
“Service In Random Order”
• Like drawing tickets out of a pool of tickets for service.
Single-server Single-stage Queue
Multiple-server Single-stage Queue
Single-server Multiple-stage Queue
Pharmacy Conveyor System >>>>>
Multiple-server Multiple-Stage Queue
Types of Queues of Interest
• Analytical Models for Estimating Capacity and
– 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
• Single channel, constant service time (M/D/1)
• Multiple channel, exponential service time
• Multiple priority service, exponential service
• 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
• 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.
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
• 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
• 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.