Consider a stationary M$/$M$/1$ queue with arrival rate $\backslash$lambda and service rate $\backslash$mu $.$ Let Yt denote the number of departures in time $[0,$ t$].$ Using a direct argument $($without invoking reversibility$),$ show that $\{$Yt$\}$ is a Poisson process of rate $\backslash$lambda $.$

$($Hint: Consider two cases $-$ when the queue is empty and nonempty$)$