The girl at the ticket booth serves a customer in 2 minutes, on average. Her servicing...

The girl at the ticket booth serves a customer in 2 minutes, on average. Her servicing time is distributed exponentially. Customers arrive at intervals distributed also exponentially. On average, next customer arrives in 2.5 minutes.
1. What is the arrival rate per hour?
2. What is the servicing rate per hour?
3. What is the traffic intensity?
4. What proportion of her time the girl is busy?
5. What is the probability that the waiting line is 3 people long?
6. What is the probability that the girl is idle?
7. What is the probability for the situation that the girl is busy but there is no waiting
  
  line?

Expert Solution

Arrival rate per hour, = 1 per 2.5 minutes = (1/2.5) per minutes = (60 / 2.5) per hour = 24 per hour

Servicing rate per hour , = 1 per 2 minutes = (1/2) per minutes = (60 / 2) per hour = 30 per hour

Traffic intensity = / = 24 / 30 = 0.8

Proportion of her time the girl is busy = = 0.8

Probability that the waiting line is 3 people long = Probability that there is only foure person in the system (3 in waiting and one in servicing) = (1 - ) = (1 - 0.8) * 0.8⁴ = 0.08192

Probability that the girl is idle = 1 - = 1 - 0.8 = 0.2

Probability for the situation that the girl is busy but there is no waiting line = Probability that there is only one person in the system = (1 - ) = (1 - 0.8) * 0.8 = 0.16

orchestra answered 2 years ago

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