Mathemaniac | Queuing theory and Poisson process @mathemaniac | Uploaded 1 year ago | Updated 2 hours ago
Queuing theory is indispensable, but here is an introduction to the simplest queuing model - an M/M/1 queue. Also included is the discussion on Poisson process, which is the underlying assumption for the M/M/1 queue.
Second channel video on variance of Poisson distribution: youtu.be/xiX1QoX8i6M
To me, this is mainly a "prequel" which serves as a prerequisite for the next video, even though the next video is not as long.
For the files created for this video, please visit mathemaniac.co.uk/download and enter the password:
queuingmodelM/M/1
and follow the instructions on the website. If you can't enter the website, watch the latest video! It always changes when a new video is up.
Sources:
Different queues:
M/M/1 queue: en.wikipedia.org/wiki/M/M/1_queue
M/M/c queue: en.wikipedia.org/wiki/M/M/c_queue
M/M/∞ queue: en.wikipedia.org/wiki/M/M/%E2%88%9E_queue
M/G/1 queue: en.wikipedia.org/wiki/M/G/1_queue
M/G/k queue: en.wikipedia.org/wiki/M/G/k_queue
G/M/1 queue: en.wikipedia.org/wiki/G/M/1_queue
G/G/1 queue: en.wikipedia.org/wiki/G/G/1_queue
Jackson Network: en.wikipedia.org/wiki/Jackson_network
More general queues: en.wikipedia.org/wiki/Kendall%27s_notation
The (transient) solution: Computer Networks and Systems. New York, NY: Springer New York. p. 72 (uses moment-generating function and Laplace transforms); for more details, see Gross, D. and Harris, C.M., Fundamentals of Queueing Theory, Wiley, New York, 1974, 1985. (Section 3.11.2)
Other related sources:
Markov Chains: statslab.cam.ac.uk/~james/Markov
Birth-and-death chain: en.wikipedia.org/wiki/Birth%E2%80%93death_process
Bessel functions (for the solution to the differential equations): en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I%CE%B1,_K%CE%B1
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
Queuing theory is indispensable, but here is an introduction to the simplest queuing model - an M/M/1 queue. Also included is the discussion on Poisson process, which is the underlying assumption for the M/M/1 queue.
Second channel video on variance of Poisson distribution: youtu.be/xiX1QoX8i6M
To me, this is mainly a "prequel" which serves as a prerequisite for the next video, even though the next video is not as long.
For the files created for this video, please visit mathemaniac.co.uk/download and enter the password:
queuingmodelM/M/1
and follow the instructions on the website. If you can't enter the website, watch the latest video! It always changes when a new video is up.
Sources:
Different queues:
M/M/1 queue: en.wikipedia.org/wiki/M/M/1_queue
M/M/c queue: en.wikipedia.org/wiki/M/M/c_queue
M/M/∞ queue: en.wikipedia.org/wiki/M/M/%E2%88%9E_queue
M/G/1 queue: en.wikipedia.org/wiki/M/G/1_queue
M/G/k queue: en.wikipedia.org/wiki/M/G/k_queue
G/M/1 queue: en.wikipedia.org/wiki/G/M/1_queue
G/G/1 queue: en.wikipedia.org/wiki/G/G/1_queue
Jackson Network: en.wikipedia.org/wiki/Jackson_network
More general queues: en.wikipedia.org/wiki/Kendall%27s_notation
The (transient) solution: Computer Networks and Systems. New York, NY: Springer New York. p. 72 (uses moment-generating function and Laplace transforms); for more details, see Gross, D. and Harris, C.M., Fundamentals of Queueing Theory, Wiley, New York, 1974, 1985. (Section 3.11.2)
Other related sources:
Markov Chains: statslab.cam.ac.uk/~james/Markov
Birth-and-death chain: en.wikipedia.org/wiki/Birth%E2%80%93death_process
Bessel functions (for the solution to the differential equations): en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I%CE%B1,_K%CE%B1
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
![Limitations of mathematical models; historical context of BGW process [PART III]](https://i.ytimg.com/vi/zjlpZ0xPnYI/hqdefault.jpg "Limitations of mathematical models; historical context of BGW process [PART III]
Part 3 of a series on a stochastic process approach to model the spread of coronavirus (COVID-19) as opposed to the compartmental deterministic SIR model. This model is generally known as branching process, but this video only focuses on the simplest type, called Bienaymé-Galton-Watson (BGW) process. This video will especially be on the inevitable limitations on the BGW process model, to illustrate the limitations of any mathematical model in general. There can be problems in applicability and difficulty in interpreting data from the predictions of the model, but we can always strive to make the model more realistic. But the cost would be more complicated mathematics, and higher computing power necessary.
The historical context of the Bienaymé-Galton-Watson (BGW) process will be also discussed in this video, and it is actually a bit more interesting than usual. It is not just that the three mathematicians got together and studied this process.
I currently dont have a solid plan for my future videos, but feel free to comment below about the topics that I can cover in a future video!
REFERENCES / SOURCES:
(1) Branching Processes: Their Role in Epidemiology [basis / limitations / improvement of the model]
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2872325/pdf/ijerph-07-01186.pdf
(2) Branching Processes Since 1873 [historical context of the Bienaymé-Galton-Watson process]
https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s1-41.1.385
(3) The Genealogy of Genealogy Branching Processes before (and after) 1873 [historical context of the Bienaymé-Galton-Watson process]
https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms/7.3.225
(4) The Educational Times (March 1873) [Galtons original famous Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF
(5) The Educational Times (August 1873) [Watsons (brief) answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF
(6) On the probability of extinction of families [Watsons paper / more detailed albeit partially incorrect answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://www.jstor.org/stable/pdf/2841222.pdf
(7) BBC news [breaking lockdown rules complaint]: https://www.bbc.com/news/uk-england-humber-53361096
(8) Guardian [KFC party breaking recent lockdown in Melbourne]:
https://www.theguardian.com/australia-news/2020/jul/10/kfc-birthday-party-in-melbourne-costs-26000-in-covid-19-fines-after-police-track-order
(9) Metro [18000 people breaking lockdown rules]: https://metro.co.uk/2020/06/28/18000-people-fined-breaking-lockdown-rules-since-march-12913927/
(10) Contact tracing map: https://ourworldindata.org/grapher/covid-contact-tracing?year=2020-07-10
(11) What is contact tracing (CDC): https://www.cdc.gov/coronavirus/2019-ncov/daily-life-coping/contact-tracing.html
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I dont use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #coronavirus #COVID_19 #SIRmodel #statistics #probability #epidemic #pandemic
Stay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
For my contact email, check my About page on a PC.
See you next time!")
![How likely is coronavirus (COVID-19) eradicated? [PART II]](https://i.ytimg.com/vi/zw8p9_NzSGU/hqdefault.jpg "How likely is coronavirus (COVID-19) eradicated? [PART II]
Part 2 of a series of videos on a stochastic process approach to model the spread of coronavirus (COVID-19) as opposed to the compartmental deterministic SIR model. This model is generally known as branching process, but this video only focuses on the simplest type, called Bienaymé-Galton-Watson (BGW) process. This video will explore how we can extract the extinction probability (probability that coronavirus will eventually get eradicated) from the BGW process using the cobwebbing technique. Although it is not a rigorous approach, it is a very nice visual way to see whats happening. We then, not too surprisingly, come up with the concept of basic reproduction number, a concept that is seen in both the BGW model and the SIR model.
This video will involve concepts like distribution, independence, expected value, generating function, and the cobwebbing techniques to visualise iteration processes and so on, but a basic understanding of the concept of probabilities will be good, and basic understanding on differentiation (definition as slope of a function, power rule and linearity of the differential operator) will be required.
The next video on the limitations / improvement and the historical context of the BGW process should be released in a couple of days, because they are already done.
REFERENCES / SOURCES (which I will explain in much more detail in later videos that I promised, but if you are impatient, you can read these):
(1) Branching Processes: Their Role in Epidemiology [basis / limitations / improvement of the model]
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2872325/pdf/ijerph-07-01186.pdf
(2) Branching Processes Since 1873 [historical context of the Bienaymé-Galton-Watson process]
https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s1-41.1.385
(3) The Genealogy of Genealogy Branching Processes before (and after) 1873 [historical context of the Bienaymé-Galton-Watson process]
https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms/7.3.225
(4) The Educational Times (March 1873) [Galtons original famous Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF
(5) The Educational Times (August 1873) [Watsons (brief) answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF
(6) On the probability of extinction of families [Watsons paper / more detailed albeit partially incorrect answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://www.jstor.org/stable/pdf/2841222.pdf
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I dont use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #coronavirus #COVID_19 #SIRmodel #statistics #probability #epidemic #pandemic
Stay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
For my contact email, check my About page on a PC.
See you next time!")
