Little's formula is one of the cornerstones of the queueing theory, because it applies to a wide class of systems. It states that the number of customers in the system is the product of the average arrival rate and the mean time a customer spends in the system. Like all fundamental things it allows different proofs and interprerations. Some of them are really convincing, involving Kolmogorov-Chapman equations and related apparatus.
I will present a rather "mechanical" approach which gives the result in just three lines, but requires some preparatory discussion. We shall consider a warehouse where unitized (boxes, pallets, containers) cargo is stored. Cargo unit is a "customer" in the queue,storage time is service time.
Warehouse operator charges for (amount of cargo)X(storage time). It is reasonable to call this magnitude "warehouse work": A, (box)X(day). In a similar way "transport work" is introduced as (cargo)X(distance). Consider a limit case of a warehouse X having 1 storage place, say for 1 box, and another warehouse Y capable of storing 2 boxes at the same time.Then amount of work of 2 box-days can be performed by X in 2 days, while Y can do the same in 1 day. It is therefore natural to develop analogues with mechanics and consider capacity E as "power" of the warehouse meaning the amount of work that can be performed in a unit of time.
Now we move on to the proof. Suppose Q units of cargo are brought to the warehouse every t days. Warehouse has capacity of E units and mean dwelling T time is known based on observations. What is the condition for the system to work in a stable way, i.e. to cope with the incoming flow?
Incoming Q units will "create" warehouse work of QT box-days. The warehouse must "process" them in t days (because then the next lot is coming) having at least capacity E, that is it needs to perform work of Et box-days. The balance equation (or "law of conservation", if you like it) then looks like this:
But is the average daily rate of the incoming flow. Therefore:
Which is the required result. In this interpretation it gives the minimum required capacity to put the cargo flow through the system.
Comments
mean time?
Why the mean time? If i have 100 customers and 99 take 10 seconds and one takes 10 minutes…?
This isn't something that I understand, and I'm basically poking you with a stick but…I'd be interested to know why the mean time mattered.
mean time?
Could be, however, if we are talking about a stochastically stable process, the impact of such annoying customers should be diminishing.
I have been making observations of container traffic through a certain port for more than a year now. Surely, there are units that stay on the terminal for months. Nevertheless, the mean time which is crucial in many formulas stabilized at 8-9 days after a couple of months of observations and is subject to only minor fluctuations.
I hope more posts devoted to mean time itself will follow ;)