# Probability and the Queing Theory

People arrive a a particular sales counter at the rate of 6 in any 15 minute period and are served on a "first come first served" basis. Arrivals occur randomly throughout the hour and the arrival rate is the same during the entire business day. It takes 5 minutes to service a single customer. How likely is it that exactly 1 customer will arrive in any 5 minutes?

Using the information from above: If there are no additional customers waiting (or walking up to the counter), when a particular customer begins her transaction, what is the probability that no one will be waiting in line when her transaction is finished?

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#### Solution Preview

Arrival Rate = 2 per 5 mins [ 6 per 15 mins converted into 2 per 5 mins]

P(n) customers arriving = e^-m* m^n/n!

P(1) ...

#### Solution Summary

This response determines the probability that there will be no line after a certain amount of time.