In queue theory, a theorem goes
$a_{n} = \pi\left ( n \right )$
Here, $a_{n}$ denotes the limiting fraction of arriving customers that sees a queue of size n.
$\pi\left ( n \right )$ is the limiting fraction of time that there are n individuals in the queue.
I'd like to obtain the explicit form for each of the term on both side of the equivalence.
Since $\pi\left ( n \right )$ is the limiting fraction of time that there are n individuals in the queue, we seek the limit of n individuals in the queue as time t tends to infinity.
For any agent $i$ that enters the system, they are first queued for a time interval before being served with a service time.
Now, if $\lambda_{a}$ is the rate of agent arrivals into the system, then, for an expected waiting time $W_{Q}$ in the queue for any agent, $\lambda_{a} W_{Q}$ gives the expected total number of agent in the queue. In particular, this is the greatest number of agent that can be in the queue before they are served by server(s). For n number of agents in the queue, there must exists some time $t$ such that $\lambda_{a} t=n$
There limiting fraction of time that there are n individuals in the queue is, I think,
$\lim_{t\rightarrow \infty}\frac{\lambda_{a}t}{\lambda_{a}W_{Q} }$
How should I think about $a_{n}$?
Any help is appreciated. Thanks in advance.