Queuing theory and Poisson process

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.

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.

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:

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:

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[mathemaniac]

This video primarily serves as the prerequisite of the next video, which is going to be out soon. The prerequisite only includes Poisson distribution, but I thought it might be a little less textbook-like if I associate this with queuing theory.

[hamsterdam]

This video actually shows why, out of all things, Poisson distribution appears when studying radioactive decay

[johnchessant]

Mathemaniac and 3b1b both uploading probability videos on the same day? Awesome!

[ryancantpvp]

Normally I would watch the entire video before commenting, but I was so surprised by how the first few minutes of the video explained how probability works with time, and that’s something I’ve been wondering for a while. Great content as always!

[libberator]

I just have some video editing feedback. Some people (like myself) keep subtitles on. So don't use the bottom ~10% for important information, like the results of certain equations. It gets blocked, and having to toggle subtitles off and on just to see parts of the video can get annoying. Consider where the viewers' vision typically rest and put things you want to bring their attention to near the center of the screen. Having to look towards the peripherals can be distracting and just extra work, making information transfer harder. Anyways, keep up the good work!

[klikkolee]

The derivations at 9:40 neglect the possibility of a customer arriving and a customer leaving in the same time step. These events were assumed to be independent, so it cannot also be assumed that they cannot coincide. If the queue is in state 3, and a customer arrives and a customer leaves in the same time step, the queue is still in state 3 -- not state 2. The term should have a factor of 1-λ/n to stipulate that a customer did not arrive on that time step. Similarly, the term for the state 1 => 2 transition should have a factor of 1-μ/n to stipulate that a customer did not depart on that time step -- since otherwise the queue would still be in state 1.

And there should be another state 2 term for if a customer arrives and a customer departs in the same time step when already in state 2.

So the equation should be:

+ p_3(t)(μ/n)(1-λ/n) + p_1(t)(λ/n)(1-μ/n)

In state 0, the events cannot be independent -- a customer can only depart in that time step if they arrived in that time step (were served within a single time step). The only way to interpret the departure probability in this scenario is as the probability that a customer departs given that a customer arrives. This however results in the same term for the probability for their intersection as when they were independent.

The equation for state 0 becomes:

+ p_1(t)(μ/n)(1-λ/n)

[danielchin]

The trick of *invariant distribution* is so profoundly generalizable, once you get used to it, you'll start to use it every where.

[antonkot]

Probably the best math video I've seen in couple of years (with lambda over mu probability :) )

[ariebaudoin]

nice video, i have an exam on ODE's this friday, so i was immediately thinking about how to write the ODE in terms of a matrix, we didnt have infinitely big systems, so it was really fun to see one

[parsimoniousdialog]

Then, thinking about the nature of a battery, we might consider the common description: One side is plus (positive) +. The other side is minus (negative) -. Then the boundary condition along the number line is usually called zero (nothing) 0.

Then a formulation of the number line might follow: (-∞, -, 0, +, +∞).

This formulation resembles the object called a battery.

[dwivedys]

This is an unbelievably amazing video!!

[xX_swagger_Xx]

you explained this way better than my professor

[mohammedel-naggar]

great job in explaining it. I didn't understand it from MIT.

[ANunes]

All of this made perfect sense to me in class and I hated doing it with a passion. These feelings were unrelated. I just hate the procedure, despite how obvious it is.

[baonguyen-ctnj]

in the equation at 9:34, shouldn't the 2 departures from p2 to p1 and p3 at time = t be incorporated to the state at time = t+1? I don't understand why they are excluded.

[josephjepson]

I liked the piano music while doing math manipulation. It was relaxing.

[scalex]

At 03:35 you divide the numerator of the factorial fraction by n from (lambda/n)^k but it seems you forgot to apply the ^k to the denominator. Where is the (1/n^(k-1))??

[fresheFresse]

When evaluation state k and looking at probability contributions to pk(t + 1/n): Why is the contribution from state k+1 not pk+1(t)(u/n)(1-lambda/n)?
In other words when we are in state k+1 shouldn't there only be a contribution when a departure happens AND no new arrival comes in the interval?
Same for the contribution from k-1

[etienneparcollet]

The equations of transition are not valid at finite n, as multiple hops cases are not included. What Would have been needed is explicitly saying that the remainder is o(1/n).

[alejandrom.]

Wonderful explicative video!! I will do Probability and Statistics next semester so it will be handy in some months.