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:    • Poisson distribution (extra)  

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 https://www.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: https://en.wikipedia.org/wiki/M/M/1_q...
M/M/c queue: https://en.wikipedia.org/wiki/M/M/c_q...
M/M/∞ queue: https://en.wikipedia.org/wiki/M/M/%E2...
M/G/1 queue: https://en.wikipedia.org/wiki/M/G/1_q...
M/G/k queue: https://en.wikipedia.org/wiki/M/G/k_q...
G/M/1 queue: https://en.wikipedia.org/wiki/G/M/1_q...
G/G/1 queue: https://en.wikipedia.org/wiki/G/G/1_q...
Jackson Network: https://en.wikipedia.org/wiki/Jackson...
More general queues: https://en.wikipedia.org/wiki/Kendall...

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: https://www.statslab.cam.ac.uk/~james...
Birth-and-death chain: https://en.wikipedia.org/wiki/Birth%E...
Bessel functions (for the solution to the differential equations): https://en.wikipedia.org/wiki/Bessel_...

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