Queueing theory (simple)

Summary

In 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

• 📚 Queuing theory is a mathematical approach used in industrial engineering and operations research to model and analyze the management of queues.
• 🔄 It involves the study of items, people, or parts that need to be processed by a server, which performs work that takes time.
• 📈 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.
• 📉 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).
• 📊 Queuing theory uses equations to describe system performance, including capacity utilization, number of items in the system, and time in the system.
• 🔢 The capacity utilization (ρ) is calculated as the arrival rate (λ) divided by the service rate (μ), indicating how busy the server is.
• 📏 The number of items in the system is determined by the formula (λ / μ) / (1 - λ / μ), which accounts for both waiting and processing items.
• ⏱ The 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.
• 🏭 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.
• 📐 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

00:00

📚 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.

05:01

🔍 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.

Keywords

💡Queuing Theory

Queuing Theory is a branch of operations research that deals with the analysis of waiting lines or queues. It is used to optimize service delivery by predicting the number of customers and their waiting times. In the video, Liz Thompson uses queuing theory to explain how items or people wait in line to be processed by a server, which is a fundamental concept in industrial engineering and operations management.

💡Server

In the context of queuing theory, a server refers to the entity that provides service to the items in the queue. It could be a person, a machine, or a system that processes the queue's elements. The video script describes a scenario where a milling machine acts as a server, processing parts that arrive at a certain rate.

💡Queue

A queue in queuing theory is the line or sequence of items or customers waiting to be served. The video script uses the term to describe the line that forms as items approach the server for processing. It's a key element in the analysis of service systems, as it helps determine the efficiency and waiting times.

💡First In First Out (FIFO)

FIFO is a scheduling discipline where the first item or customer in the queue is the first to be served. This principle is essential in maintaining order and fairness in service systems. The video mentions a one-server, one-line FIFO system as an example of a queuing configuration.

💡Arrival Rate (Lambda)

The arrival rate, denoted by lambda (λ), is the average number of items arriving per unit of time. It's a critical parameter in queuing theory as it influences the length of the queue and waiting times. In the video, the arrival rate of parts to the milling machine is given as 100 per hour, illustrating how this rate is calculated and used in analysis.

💡Service Rate (Mu)

The service rate, denoted by mu (μ), is the average number of items that can be served per unit of time. It's a measure of the server's capacity to process items. The video explains how to calculate the service rate based on the time it takes to process an item, which in the example is 30 seconds per part.

💡Exponential Distribution

The exponential distribution is a probability distribution that models the time between events in a Poisson process, such as arrivals in a queue. It's used in queuing theory to describe the variability in inter-arrival and service times. The video script mentions that both arrival and service times are exponentially distributed.

💡Capacity Utilization

Capacity utilization is the ratio of the arrival rate to the service rate, expressed as a percentage. It indicates how efficiently a server is being used. The video calculates capacity utilization for the milling machine example, showing that the machine is running at 83% of its capacity.

💡Waiting Time

Waiting time in queuing theory refers to the time an item spends in the queue before being served. It's a crucial measure of system performance and customer satisfaction. The video calculates the expected waiting time for parts in the system, which is approximately three minutes.

💡Poisson Process

A Poisson process is a statistical process that models the number of events occurring in a fixed interval of time or space. In queuing theory, it's often used to model the random arrivals of customers or items. The video mentions that the arrival of parts to the milling machine follows a Poisson process.

💡Facilities Design

Facilities design involves planning and designing physical spaces to optimize workflow and efficiency. The video uses queuing theory to inform facilities design, such as calculating the space needed for a waiting area based on the expected number of items in the queue.

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 (μ) 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 (ρ = λ/μ).

The number of items in the system is given by the formula (λ/μ) / (1 - λ/μ).

The waiting time in the system can be determined by the formula (λ/μ) / 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