Queueing theory (simple)
Summary
TLDRIn this video, Liz Thompson introduces queuing theory, a fundamental concept in industrial engineering and operations research. She explains the basics of queuing theory, focusing on a single server, single line system with a first-in, first-out approach. Key terms like arrival rate (lambda) and service rate (mu) are defined, along with their exponential distribution. Thompson demonstrates how to calculate capacity utilization, the number of items in the system, and waiting time using simple equations derived from complex mathematical models. She concludes with a practical example of applying queuing theory to a milling machine scenario, showing how to determine the machine's capacity utilization and the space needed for a waiting area.
Takeaways
- đ Queuing theory is a mathematical approach used in industrial engineering and operations research to model and analyze the management of queues.
- đ It involves the study of items, people, or parts that need to be processed by a server, which performs work that takes time.
- đ The theory includes concepts like arrival rates (lambda) and service rates (mu), which are used to describe the flow and processing of items in a queue.
- đ Queuing systems can be configured in various ways, such as single-server, multiple-server, first-in-first-out (FIFO), or last-in-first-out (LIFO).
- đ Queuing theory uses equations to describe system performance, including capacity utilization, number of items in the system, and time in the system.
- đą The capacity utilization (Ï) is calculated as the arrival rate (λ) divided by the service rate (ÎŒ), indicating how busy the server is.
- đ The number of items in the system is determined by the formula (λ / ÎŒ) / (1 - λ / ÎŒ), which accounts for both waiting and processing items.
- ⱠThe waiting time in the system is calculated as the number of items in the system divided by the arrival rate (λ), providing insight into the efficiency of the process.
- đ An example provided in the script involves parts arriving at a milling machine, with calculations demonstrating how queuing theory can be applied to real-world industrial scenarios.
- đ The script also discusses the practical application of queuing theory in facilities design, such as determining the necessary waiting area size based on the number of items and their footprint.
Q & A
What is queuing theory?
-Queuing theory is a branch of mathematics that deals with the study of waiting lines, or queues, and is used in industrial engineering and operations research to analyze and optimize service systems.
What is the basic concept of queuing theory?
-The basic concept of queuing theory involves items, people, or parts that need to be processed by a server, which performs work that takes time. These items form a line, or queue, and are served by the server one at a time.
What is the difference between arrival rate (lambda) and service rate (mu) in queuing theory?
-The arrival rate (lambda) is the expected number of arrivals per unit of time, while the service rate (mu) is the average number of items that can be served per unit of time. Lambda is associated with the input rate to the system, and mu with the output rate.
What does a 'first in, first out' (FIFO) queue mean?
-A 'first in, first out' (FIFO) queue means that the order in which items arrive is the same order in which they are served, ensuring that the first item to arrive is the first to be processed.
What is the significance of the capacity utilization (rho) in a queuing system?
-Capacity utilization (rho) is the ratio of the arrival rate (lambda) to the service rate (mu), indicating the proportion of time the server is busy. It helps to understand how efficiently the system is being used.
How is the number of items in a queuing system calculated?
-The number of items in a queuing system is calculated using the formula L = (lambda / (1 - rho)), where L is the average number of items in the system, lambda is the arrival rate, and rho is the capacity utilization.
What does the waiting time in a queuing system represent?
-The waiting time in a queuing system represents the average time an item spends in the queue before being served, which can be calculated as W = L / lambda, where W is the waiting time, L is the number of items in the system, and lambda is the arrival rate.
Why is it important to understand the distribution of arrival and service times in queuing theory?
-Understanding the distribution of arrival and service times is important because it allows for more accurate predictions of system performance, such as queue length and waiting times, and helps in making informed decisions about system design and resource allocation.
How can queuing theory be applied in an industrial setting?
-Queuing theory can be applied in industrial settings to optimize production lines, manage customer service queues, and improve the efficiency of resource allocation, leading to better utilization of machinery and personnel.
What is an example of how queuing theory can be used to calculate space requirements for a waiting area?
-In the provided example, queuing theory is used to calculate the space needed for a waiting area by determining the average number of items waiting and multiplying it by the space required per item, which in this case was 4.9 items times 2 square feet per item, resulting in a 10 square feet area.
Outlines
đ Introduction to Queuing Theory
The video introduces queuing theory, a fundamental concept in industrial engineering and operations research. Liz Thompson explains the basics of queuing theory, which involves items or people waiting in a line to be processed by a server. The video aims to simplify the concept and demonstrate its application through an example. Queuing theory is visualized as items forming a line (queue) and being processed by a server. The video emphasizes the importance of defining the type of queue, such as a single-server, single-line, first-in-first-out (FIFO) system. Key parameters like lambda (arrival rate) and mu (service rate) are introduced, with lambda representing the expected time between arrivals and mu indicating the service rate. These rates are assumed to follow an exponential distribution. The video promises to show how these parameters can be used to analyze the system's capacity, utilization, and other metrics using simple equations derived from complex mathematical models.
đ Queuing Theory Application Example
In the second paragraph, the video script delves into a practical example of applying queuing theory. The scenario involves parts arriving at a milling machine, with an arrival rate of 100 parts per hour, following a Poisson process. The service time, or the time taken to process each part, is exponentially distributed with an average of 30 seconds. The video calculates the capacity utilization of the machine, which is the proportion of time the machine is in use, using the formula lambda over mu. The result is 83%, indicating a high utilization rate. Further calculations are made to determine the number of items in the system, both waiting and being processed, which is approximately 4.9 items. The waiting time in the system for each part is also calculated to be about three minutes. Finally, the video addresses the space required for the waiting area, given the footprint of each part and the number of parts expected to be waiting. The example illustrates how queuing theory can be used to analyze and optimize processes in industrial settings.
Mindmap
Keywords
đĄQueuing Theory
đĄServer
đĄQueue
đĄFirst In First Out (FIFO)
đĄArrival Rate (Lambda)
đĄService Rate (Mu)
đĄExponential Distribution
đĄCapacity Utilization
đĄWaiting Time
đĄPoisson Process
đĄFacilities Design
Highlights
Queuing theory is a fundamental concept in industrial engineering and operations research.
It involves the study of items, people, or parts that need to be processed by a server.
The system is defined by a queue and a server, excluding items in transit to the system.
A one-server, one-line, first-in-first-out (FIFO) queue is a common type of queuing system.
Queuing theory uses mathematical models to analyze and predict system performance.
Lambda (λ) represents the arrival rate, and Mu (Ό) represents the service rate in the system.
Arrival and service rates are often modeled using exponential distributions.
Capacity utilization is calculated as the ratio of lambda to mu (Ï = λ/ÎŒ).
The number of items in the system is given by the formula (λ/Ό) / (1 - λ/Ό).
The waiting time in the system can be determined by the formula (λ/Ό) / λ.
Queuing theory can be applied to analyze real-world scenarios, such as parts arriving at a milling machine.
The arrival rate is determined by the number of parts per hour following a Poisson process.
Service rate is calculated by converting the service time per part into units per hour.
The example demonstrates calculating capacity utilization, number of items in the system, and waiting time.
Space allocation for the waiting area can be determined based on the number of items and their footprint.
Queuing theory provides insights into system efficiency and resource allocation.
The mathematical equations derived from queuing theory are complex but provide valuable system insights.
Transcripts
hi my name is liz thompson and this is a
quick video on an introduction to
queuing theory
um sort of the basics of using math in
industrial engineering
queuing theory is also a part of
operations research
so i'm going to explain what cueing
theory is in the simplest form and the
simplest
aspects of it i'm also going to then
work through an example
and show how it might be used in
analysis
so cuic theory is the concept of there's
items people or parts or
anything that needs to be processed by a
server
who does some work that takes some time
on the items or with the people
so the way we usually conceptualize this
is
these the three dots on the left is what
i'm saying are
the items that need to be served they
start to move towards the server and
form a line and then as that as the
server is available
um the process one person in the line or
one item in the line
is served by the server and then exits
the system
so we call this um line a cue and that's
what we call
um queuing theory and then the whole
system is we define as the line
and the server so we don't include the
items that are traveling to
the systems we draw the system this way
one of the important things about
queuing theory
is to really define the type of queue
you can imagine there's
all kinds of different types of queue
this particular type of queue is a one
server
one line first in first out so that
means
the first items to get in the line are
the next ones that are going to be
processed so we travel through the line
this way
now there's all kinds of other types of
configurations and
for queuing theory there could be
multiple server
one line first in first out there could
be
um multiple server one line last in
first
out so instead of going to the end of
the line the the next item goes to the
front of the line
now all of those indicate a different
kinds of calculation
but the calculations i'm going to talk
about right now are a one
one server one line last in first out
type
q and in this particular orientation
what we do is we define
a lamba and a mu which is the lambda
is the arrival rates and the arrival
rates what we
say is lambda is an expected time of a
arrival a arrival per hour
and that but these vary there's a
distribution associated with it and we
define the distribution as
exponential so that looks sort of like
this where we have a lambda that is
the the expected value but it's
distributed over time
mu is an indication of service rate and
that's
there's also an expected value and that
varies over time
so we can think about these things as
distributions like
sometimes um you there's two people that
arrive together
and then sometimes there's like 10
minutes until the next person arrives
and then in the service rate sometimes
the service takes
10 seconds and sometimes it takes three
minutes so there's a variety of
distribution of the service rate so what
we do is we have these lambda and mu
where the arrivals are expressed in
items per unit time
and the service is also expressed in
items per unit time
so the input in a queuing system is the
queuing type which
in this case we have a one line one
server
first and first out and then we also
have an arrival rate
and we have a service rate and that's
kind of all we need
and then through some um really kind of
neat math
that you will understand if you actually
study this in depth we can actually
describe things about the system in
pretty simple equations
and examples of the ways that we can
describe the system and there's actually
a lot of
ways to describe the system but examples
are we can talk about the capacity
utilization we can talk about the number
of people in the system
we can talk about the time that that
people or
items we can talk about the time in the
system so
simple equations like the capacity
utilization is lambda over mu pretty
simple equation
the number of people or number of items
in the system
is a row or the capacitor utilization
divided by one minus row
and then the um waiting time in the
system is
the the number of people in this system
divided by lambda
so we have these pretty simple equations
that are actually
complex to derive but they're based on
distribution of arrival rates
and um service time so it's really kind
of cool that you can
describe the system in this way so i'm
going to go through a really quick
example of this
well not maybe not maybe really quick
i'm going to go through an example
so in this example it's um parts are
being arriving to a mill a milling
machine
and they're going to be processed and
they arrive at a rate of a hundred per
hour
this arrival follows a poisson process
and the processing time for the items on
the mill have an exp
expected um time of processing of 30
seconds and the processing times are
exponentially distributed so the first
question is what's the capacity
utilization of the machine
that means how much of the time the
machine is being used
and then what is what are the number of
items in the system
both waiting and processing and what is
the time in the system for each part
like how long are the items going to be
waiting there
and if each part is large with a
footprint of two square feet how much
space should you assign to the waiting
area
so these are some of the questions we
can get from this simple
cueing problem so in this queuing
problem
we initially can just establish lamba
which is the arrival rate as a hundred
units per hour
the mu is a little bit harder because
what we see is it takes 30 seconds to
process the part that's the service time
and we have to make sure that the mu is
actually in
units per hour and the 30 seconds per
part is actually seconds
per part which is not the right units so
we have to take into account that
there's 60 seconds in a minute and 60
minutes in an hour
so matt that means it takes .0083
hours to process a unit but still that's
not quite the right units for mu
the right units per mu is one unit per
0.0083 hours
which means 100 uni 120 units per hour
so that's our lambda values that we got
so now we can do the calculation i mean
that's the mu
value so now we can do the calculation
of capacity utilization
by taking just lambda over mu for rho
and we get 100 over 120 which gives us
83
or 0.83 as the capacity utilization that
means the machine is running 83
of the time that's pretty good i mean
that's a pretty good utilization
um so the next thing we then can do
is we can calculate out the um the
length or the number of items in the
system so the system is both
weighting and processing and that's
going to be row
divided by 1 minus row and it and so
that calculation says there's going to
be 4.9
items in the system so about five units
are going to be waiting there'll be four
units waiting
and then there'll be um one being
processed
um and then the next thing we can
calculate is the
um the waiting and this is the time
waiting in the system
so that's l of s divided by lambda and
so that's 4.9 divided by 100
or 0.049 hours or about three minutes
in the system and that gives us an idea
of how quickly things are being
processed
the next thing we might want to do is
calculate what's the area
that we should be waiting and i assumed
there would be 4.9 units waiting
but actually one of those is probably on
the machine so it would
maybe be 3.9 but either way we make that
assumption that 4.9 times 2
feet 2 square feet means that we need
about 10
square feet available so we would if we
were doing a facilities design we would
allocate a space
that's about that that is about 10
square feet now i hope that gives you an
idea of
some of the ways that we can use
queuing theory to get some some good
analysis going
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