Every queue on this course has assumed Poisson arrivals. Here is where they come from: cut one unit of time into n independent opportunities, each an arrival with probability λ/n. The count is binomial and that is exact rational arithmetic — at n = 40 the entries are fractions over sixty-five-digit denominators. Its limit as n grows is the Poisson distribution, and that is not exact and cannot be, because every entry carries a factor of e^(−λ), which is irrational for every rational λ other than 0. The exact column and the rounded column sit side by side and the difference between them is printed: on the worked numbers, 8.866e-3 at k = 3.
Core ideas
Three ideas. The first builds the arrival process, the second says exactly where the exactness stops, and the third gives the limit a name in terms of a quantity you can compute.
Split a unit of time into n slots and let each hold an arrival with probability λ/n, independently. The number of arrivals is Binomial(n, λ/n), whose mean is λ whatever n is. Letting n grow while holding the mean fixed is what the Poisson distribution is, and the whole content of the word “limit” is how far apart the two are at a given n.
e^(−λ) is irrational for every rational λ other than 0, so the Poisson column cannot be a fraction. It is computed by expNegApprox, shared with the System Design path, as the reciprocal of the positive-term series for e^λ — every term the same sign, so nothing cancels — and it is correct to the rounding of a double. The page names the function, names the method, and prints the difference from the exact column beside it.
The Poisson signature is that the variance equals the mean. The binomial's variance is λ(1 − λ/n), short of λ by exactly λ²/n. At λ = 3 and n = 40 that is 111/40 against 3, short by 9/40. So the distance to the limit is not a vague closeness; it is a fraction, it is printed, and it is what a window of finitely many opportunities costs.
The material
The Bernoulli construction, the exact binomial and the recurrence that makes it computable, the limit, the irrational factor, and the variance gap.
Divide one unit of time into n independent opportunities, each producing an arrival with probability p = λ/n. The number of arrivals is Binomial(n, p), with mean np = λ and variance np(1 − p) = λ(1 − λ/n).
As n → ∞ with λ fixed, P(k) → e^(−λ)λᵏ/k!, the Poisson distribution with mean λ. Its variance is λ, equal to its mean.
That construction is the reason Poisson arrivals are a sensible default and not a convenience. Many small independent chances of something happening, none of them individually likely, adding up to a fixed expected number: that is the shape of a great many arrival processes, and the limit says the count does not depend on how the time was sliced.
The binomial probabilities are computed as exact fractions, but not from C(n, k) pᵏ(1 − p)^(n−k) term by term: at n = 1000 the closed form's denominator per term runs to two thousand digits and the page never paints. The kit uses the same recurrence the System Design path uses for the same job, P(k+1) = P(k) · ((n − k)/(k + 1)) · p/(1 − p) starting from P(0) = (1 − p)ⁿ, and scripts/mathcheck.js holds the two implementations against each other on the same inputs so that the Subjects cannot drift apart about a binomial.
The left column is exact rational arithmetic and has no caveat attached to it. The right column is not exact and cannot be. Every Poisson entry carries a factor of e^(−λ), which is irrational whenever λ is a nonzero rational, so no amount of care makes that column a fraction. What the page can do, and does, is name the function that produced it, name the method, and print the difference rather than describing it.
expNegApprox is worth a sentence on its own. It is the reciprocal of a positive-term series for e^λ, which matters: the alternating series for e^(−λ) suffers catastrophic cancellation at moderate λ, where large terms of opposite sign subtract to leave a small answer, and the result can lose most of its digits. Summing positive terms and inverting at the end has no cancellation anywhere, so the answer is right to the rounding of a double — about sixteen significant figures, and at λ = 3 it differs from the platform's own exponential by 1.4e-17.
Two columns of that table say the same thing in different currencies. The largest gap falls roughly like 1/n, and the variance shortfall λ²/n falls exactly like 1/n; the second is an exact fraction and the first is a measurement in doubles. Either way “the binomial tends to the Poisson” has a size attached to it at every n, and at n = 8 that size is nearly six per cent.
The last line is the other half of the exactness story. Pushing n up makes the approximation better and the exact column more expensive: at four hundred opportunities P(0) is a fraction whose denominator has a thousand and forty-one digits, carried whole. That is the trade this library makes deliberately, and it is why the recurrence rather than the closed form is what ships.
Set λ = 8 with only twenty opportunities, so p = 2/5. The exact P(0) is 0.000036562 against the Poisson's 0.000335463, an order of magnitude out, and the largest difference across the table is 4.012e-2 at k = 8.
With p = 2/5 the individual chances are not small and the construction's premise has gone. The variance is 24/5 against a mean of 8, short by 16/5, so the binomial here is far tighter than any Poisson distribution and the shapes genuinely differ. The limit is a statement about large n and small p together.
At λ = 1/2 with twenty opportunities, P(0) = 0.602687680 and P(1) = 0.309070605, so more than ninety-one per cent of the mass is in the first two counts. The largest gap to the Poisson column is 5.805e-3 at k = 1.
This is the regime most arrival models live in per unit of time, and it is the one where the approximation is best and matters least: the shape is nearly all in the head of the distribution, and both columns agree about where the head is.
Every arrival process in the rest of this course is Poisson, which is why this is the page that says where the exactness stops. The cut equations, the balance systems, the steady states, the fundamental matrices and the policy values are all exact fractions; e^(−λ) is the one quantity on this half of the course that is not, and the footer of this Subject names it in public alongside the other three.
Interactive
a binomial in fractions, and a limit that carries an irrational number
Method
Four steps. The second and the fourth are about saying what kind of number you are holding.
The construction needs many independent chances, each unlikely, in the window you are counting over. If the chances are few or large — twenty slots at 2/5 each — the count is binomial and the Poisson shape is genuinely different, not merely approximated.
One recurrence from (1 − p)ⁿ gives every term as a fraction and costs one multiplication each. It is exact, it has no caveat, and it lets the approximation be measured instead of assumed.
The Poisson has them equal. The binomial's variance is short by λ²/n, which is an exact fraction and is the cheapest single number describing how far from Poisson your window is. At λ = 3 and n = 8 it is 9/8, more than a third of the mean.
“Correct to the rounding of a double, from a positive-term series” is a claim a reader can evaluate. “Approximately” is not. Everything else on this course is exact, and that distinction is only load-bearing if the word is reserved for the places it belongs.
One unit of time, n independent opportunities, each an arrival with probability 3/n. The mean is 3 in every row. Only the slicing changes.
The signs in the difference column are worth reading rather than skipping. The binomial is below the Poisson at k = 0 and k = 1, above it at k = 2 through k = 5, and below again from k = 6. That is the tighter variance showing: mass has been pulled out of both ends and into the middle, and the single number λ²/n = 9/40 is what the pulling amounts to.
The second block is the word “limit” turned into a table. Eight slices gives a largest error of nearly six per cent; four hundred gives eight parts in ten thousand. Nothing about the sequence is mysterious and nothing about it requires a limit theorem to describe at a fixed n — the shortfall in variance is 9/n, exactly, at every row.
And the price of the exact column is in the last entry. At four hundred opportunities the exact P(0) has a denominator of a thousand and forty-one digits. It is carried whole, the Poisson column beside it is a double, and the page is explicit about which is which. That is the arrangement this whole Subject is built on: exact everywhere it can be, labelled everywhere it cannot.
For a rehearsal, set λ = 1/2 and predict the largest gap at n = 20 before looking. The supplied first move is that the variance shortfall is λ²/n = 1/80, one fortieth of the mean, against 9/40 / 3 = 3/40 in the worked case. Say whether you expect the gap to be larger or smaller than 8.866e-3, and then check the 5.805e-3 the page reports at k = 1.
You should be able to build the Bernoulli construction and identify the count as binomial, compute its terms exactly from the recurrence, state the Poisson limit and name e^(−λ) as the irrational factor that makes it unrepresentable, quote the variance shortfall λ²/n as the distance to the limit, and describe how expNegApprox avoids cancellation.
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