The Winterpeak ski resort operates a popular ski lift that transports skiers from the base to the top of the mountain. On a busy winter day, the resort observes that, on average, there are $90$ skiers either waiting in line for the ski lift or riding the lift at any given time. The ski lift operates continuously, and it has been noted that a skier arrives at the lift queue every $30$ seconds on average.

What is the average time a skier spends in the system, from joining the queue to getting off the lift at the top.

$40$ minutes

$48$ minutes

$50$ minutes

$45$ minutes