Question

Q. The neighborhood kids set up an outdoor lemonade stand in Maryland in June. They find that their average customer arrival rate is 14 per hour even though they are capable of serving 1 customer every 2 minutes.

1. What is the average time a customer spends waiting in line to be served? Please provide your answer in seconds.

2. What is the probability that the kids will have MORE THAN 2 customers in the queueing system?

Answer 1

A) As arrival rate is 14 per hour, is 14

As service rate is 1 customer every 2 minutes or 30 customer per 60 minutes, = 60

Average time a customer spends waiting in line is given by below formula

So for the given data we get

In terms of second this is 0.0292*60*60=105.12 seconds

B) Probability that there are 0 customers in the system P₀ is given by below formula

In the given case this becomes

Now probability that there are n customers in the system P_n​​​​​ is given by

Thus we get P₁​​​​ = (14/30)*0.53=0.25

Thus we get P₂​​​​​​ = (14/30)²​​​​​​*0.53=0.12

Thus probability of having 2 or less customers in the system is 0.53+0.25+0.12 which is equal to 0.9. Probability of having more than 2 customers is 1-probability of having 2 or less customers

Thus probability of having 2 or more customers in this system is 1-0.9=0.1