$\textbf{Problem}$: An athletic facility has $5$ tennis courts. Pairs of players arrive at the courts and use a court for an exponentially distributed time with mean $40$ minutes. Suppose a pair of players arrives and finds all courts busy and k other pairs waiting in queue. What is the expected waiting time to get a court?
$\textbf{Answer}$: As long as the pair of players are waiting, all five courts are occupied by other players. When all five courts are occupied, the time until a court is freed up is exponentially distributed with mean $40/5=8 $ minutes. For our pair of players to get a court, a court must be freed up $k+1$ times. Thus, the expected waiting time is $8(k + 1)$.
The issue that I'm having in understanding is why does "The time until a court is freed up is exponentially distributed with mean $40/5=8$ minutes" rather than simply $40$ minutes. Since every court is independent and using the memoryless property.
On the contrary, if I look at $$Z=min(X_{1}, X_{2}, X_{3}, X_{4}, X_{5})$$. where $X_{i}$ is time taken by a pair to play in court $i$ and is exponentially distributed time with mean $40$ minutes.
Then $Z$ itself becomes and exponentially distributed with mean $40*5 = 200$ minutes.
I am unable to understand it intuitively. Any help is appreciated.