A run that begins with an empty system spends its first slots somewhere the steady state is not, and every average taken from slot zero carries that as a bias — a wrongness in the expectation, which no number of replications removes. Carrying the whole distribution forward one slot at a time makes the bias exact rather than anecdotal, and a longer run only dilutes it while discarding a warm-up removes it.
Core ideas
Three ideas. The first is the distinction the whole page turns on, the second makes it computable, and the third is the fix that is usually reached for and is the wrong one.
Variance is how much the estimate moves when you reseed, and replications shrink it. Bias is where the estimate is aimed, and replications do nothing to it whatever: a thousand runs all started empty give a thousand estimates of the same wrong number, with a beautifully small standard error around it.
Starting from an empty system, the distribution of the number in system at slot t can be carried forward exactly, one slot at a time. For this queue the expected occupancy runs 0, 2/5, 3/5, 37/50, 17/20, … toward the stationary 12/5, and those are fractions rather than measurements. The bias is then a statement about numbers you can print.
The transient is a fixed quantity of wrongness at the front of the run, so its share of the average falls like 1/T. Quadrupling the run length from 100 slots to 400 divides the remaining bias by about 3.5; throwing away the first quarter of the four-hundred-slot run divides it by about 5.5. Deleting wins, and it wins on a quarter of the compute.
The material
Why the beginning of a run is not a sample of the thing being estimated, how much that is worth in numbers, and what actually removes it.
The initial transient is the part of a run during which the state distribution is still visibly moving away from the initial condition and toward the steady-state distribution. It is not a period of “noise”: the process is behaving correctly throughout, and it is simply not yet behaving like the system the long-run average describes.
An estimator T of a quantity θ has bias E[T] − θ. Bias is a property of the estimator, not of a run: it does not shrink with replications, it does not show up in a standard error, and an interval built on that standard error is centred in the wrong place.
That is the whole reason this page exists. Every technique after it attacks variance, and all of them are worthless on an estimator aimed at the wrong number. A narrow interval around a biased estimate is worse than a wide one, because it is more convincing.
The number in system at the end of a slot is a Markov chain: from state n a departure happens first with probability q, then an arrival with probability p, so the chain steps to n − 1, n or n + 1, and from 0 only to 0 or 1. Starting from the point mass at zero and pushing the whole distribution forward one slot at a time gives E[Nₜ] exactly, for every t.
Both of those averages are exact fractions shown to four places, not rounded numbers — they are averages of finitely many rationals. The bias of the naive estimator is 2.2443 − 2.4, about −0.1557; the bias after discarding a hundred slots is about −0.0283. The discard removes roughly four fifths of what was there, and it does so in theory, before any run has been made.
Let B be the total excess or shortfall contributed by the transient. Averaging over T slots contributes B/T to the estimate, which falls like 1/T. Averaging over slots w to T instead removes the part of B that lay before w outright. Lengthening the run is therefore a 1/T effect and discarding is not.
The numbers say the same thing. At 100 slots the expected average is 1.8624, a shortfall of 0.5376; at 400 slots it is 2.2443, a shortfall of 0.1557. Four times the run divided the remaining bias by about 3.5. Discarding the first hundred of those four hundred slots takes the shortfall to 0.0283 — a further factor of about 5.5, bought with a quarter of the data rather than with four times the compute.
The average over every slot is 169/80 = 2.1125; the average over the three hundred slots after a warm-up of one hundred is 361/150 ≈ 2.4067. The discard moved the estimate up by about 0.2942, which is the direction the theory says and rather more than the theory's 0.1274 — because one run is an observation of the expectation and not the expectation.
Reseed and the sign can change. The same settings at seed 1 give 1.3075 from slot zero and 1.2367 after the discard: the discard moved the estimate down, on a run that spent its whole length below the steady state. That is not a counterexample to the bias; it is what a single observation of a quantity with a standard error looks like, and it is exactly why the exact transient is printed beside the run rather than the run being left to make the argument.
Discard the warm-up, split what remains into b equal batches, and take the mean of each. A batch mean is far closer to independent of its neighbour than one slot is of the next, so the b batch means can be treated as replications: their average is the estimate and their sample variance is what an interval is built from.
On the run at seed 0 the four batches of seventy-five slots give means 34/25, 116/75, 19/25 and 149/25 — that is 1.36, 1.55, 0.76 and 5.96. Their average is 361/150, within 0.007 of the exact 12/5, and their sample variance is 10733/1875 ≈ 5.72. The point estimate could hardly be better and the four numbers behind it say the run has established almost nothing. Printing the second fact beside the first is the whole job.
That spread is also a warning about the batches themselves. A batch mean is only nearly independent of its neighbour if the batch is long compared with how fast the system forgets its state, and seventy-five slots of a queue running at 4/5 utilisation is not obviously long enough. The remedy is fewer, longer batches, and the diagnostic is to watch the batch variance as the batch count falls.
Interactive
Every average taken from an empty start is biased, and a longer run only dilutes it
Method
Five steps, and the fourth one is the check that most warm-up procedures skip.
An empty system is a choice, not a neutral default, and it is the choice that biases a queueing average downward. Starting in a typical state, if you have one, is a legitimate alternative and it moves the bias rather than removing it.
Not for the noise — for the drift. A trace that climbs for two hundred slots and then wanders is showing you a transient and then a steady state, and the elbow is roughly where the warm-up should end.
“The first hundred of four hundred slots” is a reportable procedure. A discard chosen after seeing which choice gave the nicest answer is a procedure that will not survive a second seed, and it is not honest to report only its output.
Increase the discard and watch the kept average. If it is still drifting, the warm-up is too short. If it has settled but the variance is climbing, you are throwing away data you needed, and the fix is a longer run rather than a longer discard.
Consecutive slots of a queue are strongly dependent, so the sample variance over slots understates the spread badly. Batch means are the standard repair and they come with their own condition: the batches must be long enough for their means to be nearly independent.
The panel's opening state: p = 2/5, q = 1/2, four hundred slots, a warm-up of one hundred, four batches, seed 0. The exact column is computed from the chain and does not depend on the seed at all.
The two exact averages are the spine of the page. 2.2443 and 2.3717 are what the estimator is aimed at with and without the discard, and both are exact fractions shown to four places. Against the stationary 2.4 they say the naive estimator is short by about 0.16 in expectation and the discarded one by about 0.03, before a single seed has been chosen.
The run then does its own thing, as runs do. It moved by 0.2942 where the theory says 0.1274, and at seed 1 it moves the other way entirely. Neither observation argues against the theory and neither confirms it; what confirms it is that the exact column does not move when the seed does.
For a faded rehearsal, drag the warm-up from 100 to 0 and then to 200. The supplied first move is that at a warm-up of zero the two averages are the same number by definition and the exact expected average reads 2.2443. Say what you expect the exact figure to be at a warm-up of 200, whether it can ever exceed 12/5, and what happens to the batch variance when only two hundred slots are left to split four ways.
You should be able to say what distinguishes bias from variance and which techniques touch which, read the exact expected occupancy of a chain started empty and turn it into the expected average over a window, compare a longer run against a discard on the numbers rather than by intuition, form batch means from one run and state the condition under which they may be treated as replications, and say why a single run moving the wrong way is not evidence against any of it.
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