Consider a single server queue with a Poisson arrival process at rate $,$

and exponentially distributed service times with rate $.$ All interarrival

times and service times are independent of each other. This is similar

to the standard $M|M|1$ queue, but in this queue, as the queue size

increases, arrivals are more and more likely to decide not to join it$.$ If

an arrival finds $n$ people already in the queue ahead of them $($including

anyone being served$),$ then they join with probability $\frac{1}{n+1}.$ Let

$N(t)$ be the number in the queue at time $t.$

$($a$)$ Draw the transition diagram for this queue, including the transi$-$

tion rates.

$($b$)$ Write down equations that the equilibrium distribution satisfies.

$($c$)$ Find the equilibrium distribution for this queue, when it exists.

$($d$)$ What are conditions on $$ and $$ under which the equilibrium

distribution exists?