The One Relationship Worth Memorising

Last timeWhen the Queue Fills Up

The number of things in a system equals the rate they arrive multiplied by the time each one stays. It assumes almost nothing, and it gives you whichever of the three you cannot measure.

The relationship

Everything in this course has been about one system or another. This is the one

statement that holds across all of them.

The number of items inside a system equals the rate at which items arrive

multiplied by the average time each one spends inside. Written the other way

round, the time something spends inside is the number inside divided by the

arrival rate.

FIG 1The relationship, arranged to give the time inside

: the average number of items inside the system
: the rate at which items arrive, and therefore also leave
: the average time an item spends inside

The relation being rearranged is that the number inside is the arrival rate multiplied by the time inside, which is the form it is usually quoted in. The reason to show this arrangement is that the time inside is the quantity that is hardest to measure and the one people most want. A queue observed to hold on average forty items, fed by twenty arrivals a second, is making each item wait two seconds, and that was obtained without timing a single item. The reverse is just as useful: four hundred arrivals a second each staying a quarter of a second means a hundred items inside at any moment, which is how many workers, connections or buffer slots the system needs.

The word system there is deliberately vague, and that is the strength of it. The

system can be one server, a queue, a queue and its servers together, a whole

service, a datacentre, or a company backlog of support tickets. Whatever

boundary is drawn, if things go in and come out and the amount inside is not

systematically growing, the relationship holds across that boundary.

Work in progress

The most common use is sizing. The amount of work inside a system is not

something anybody chooses, it is a consequence of the rate and the residence

time, and that turns several guesses into calculations.

The lesson stops here

4 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

The One Relationship Worth Memorising

The relationship

Work in progress

4 more paragraphs to go

This is the reading half

Read alongside