# Length of a queue

Let we have a queue of independent events. The events are processed in the order of arrival, processing one event takes $t$ time.

Events arrive in average $N$ events per identity time.

What it the average length of the queue?

I am waiting up to three days for a company to process my request and wonder why the queue is so long (three days), while obviously processing one item should not take much time (it can be done in a minute, I think). So I want to see a formula for this. I do not need a proof.

Events arrive at $N$ events per hour: so $\lambda =\frac{1}{N}$
Average processing time of $t$: so $\mu = \frac{1}{t}$
the probability, $p$, that the server will be busy is therefore: $\frac{t}{N}$
We can use the properties of the exponential distribution to make observations on the queue length $L$ which is
$$L = \frac{p^2}{1-p} = \frac{\left(\frac{t}{N}\right)^2}{1-\frac{t}{N}}.$$