The length of a queue at time
can be modelled by a simple birth and death process (people joining and leaving the queue respectively).
Let
then the general Kolmogorov equations are
where
For the simple queues
for all
and
for all
This gives
for
This is a difficult problem and since how a queue behaves in the long term is of more interest, we analyse the steady state solution
and all the parameters are constant. The Kolmogorov equations become
The last equation gives
which inductively says
for
hence
forwith
The sum of any probability distribution is 1 by definition hence
so that
Then
for