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Bruno dreams about opening a doughnuts franchise
and everyday, as he goes to his favorite doughnuts store
he keeps asking himself whether it's worth opening the business.
Bruno asks questions such as:
- "What is the arrival rate of clients to this store?"
- "How much do they buy per day?"
- "I have been coming to this store for many months
and I've noticed that, on average, I find 3 customers in line
and I take 3 minutes between entering and leaving the store."
Each person observes the number of customers in the store varying over time
and takes a certain amount of time to reach the head of the line to be serviced.
The quantities usually vary,
and for this reason, we call them random variables.
The quantities tracked by Bruno
are the averages of such random variables.
We indicate them by:
overline N: the mean number of customers inside the store,
and overline T: the mean residence time.
- "But I still want to know the arrival rate of clients!"
It is at this point that Little's result can help us.
It claims that the mean number of clients in the system (the store)
equals the product of the arrival rate of clients
times the mean residence time of clients in the system.
That is, clients arrive at rate 1 client per minute to the store.
To understand why this occurs
let us look to the plots that show
the transit of clients to the store
and reflect about the areas that they represent.
This plot shows, in the y axis,
the number of clients in the store over time,
which is represented in the x axis.
Notice that the client at the bottom of the plot
is being served at the moment.
This plot shows a picture of the store
in a time window (0,t).
During this time window
the number of clients varies over time,
as clients come in and out.
However, notice that if we rearrange the plot in a different way,
the area, which we denote by A, remains the same
and so does the time window.
The ratio between the total area and the window,
both constants,
is also a constant
which we refer to as overline N,
the mean number of customers in the store
during that time window.
Let us return to our original plot.
We can excahange the x and y axes
to represent the individual residence time of each client.
which includes the time during which the client is served,
and, possibly, the waiting time in line.
The red line shows
the mean time the users stay in the system.
One way to obtain such time
is through the following procedure:
Aligning in an horizontal line all these rectangles
we obtain the sum of the time spent by all clients in the store.
The area is still the same
and dividing it by n,
the total number of clients
that entered in the store during the time window,
we obtain a new constant, overline T,
which corresponds to the mean residence time of clients.
Comparing the two equations,
we obtain a third,
where n represents the number of clients that arrived to the system
in a time window (0,t).
Dividing this quantity by the window length
we obtain a value that expresses the quantity of clients
that arrives per time unit.
This does not mean
that clients arrived with this exact rate during the whole window.
In fact, this is the mean arrival rate.
And we refer to it as lambda.
All these equations
refer to what happens in the system during a time window (0,t).
As the window length t grows,
we obtain equations that describe
what happens in a week,
in a month,
in a year,
or during all eternity.
Little's result is very generic,
and can be applied with no restrictions
with respect to the distribution
of the time between client arrivals,
of the distribution of waiting time of clients,
or of the service times.
Hence its importance.
Little's result is widely applicable
and is one of pillars of queing theory.