# Poisson Process - Courts

IITK sports facility has $4$ tennis courts. Players arrive at the courts at a Poisson rate of one pair per $10$ min and use a court for an exponentially distributed time with mean $40$ min. Suppose that a pair of players arrives and finds all courts busy and $k$ other pairs waiting in queue. How long will they have to wait to get a court on the average?

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They need to wait for $k+1$ pairs to finish before they can start.
Since the exponential distribution is memoryless, the expected time for a given pair to finish once started is $40$ minutes, and there are four courts, the expected time for any of the four courts to finish is $\frac{40}{4}=10$ minutes, and so their expected waiting time is $10(k+1)$ minutes.
Slightly more worryingly, since the service rate of the courts is equal to the arrival rate, the expected waiting time (in effect summing over $k$ weighted by the probability that there are $k$ queuing at any one time) is infinite.