A drive-in market has one checkout counter where one employee operates the cash

Question

A drive-in market has one checkout counter where one employee operates the cash register. The combination of the cash register and the operator is the server in this queuing system; the customers who line up at the counter to pay for their selections form the waiting line. Customers arrive at a rate of 24 per hour according to a Poisson distribution, and service times are exponentially distributed with a mean rate of 30 customers per hour. What are the operating characteristics for this waiting line system?

Explanation / Answer

Probability of zero customers = 1-24/30 = 0.20

20% of the time no customer would be coming

Average number of customer in the queue = (24/30)-24 = 4

Average number of customers in the waiting line = (24*24)/30*(30-24) = 3.2

Average time spend by a customer = 1/30 - 24 = 0.167 hr ~ 10 minutes

Average waiting time by a customer = 24/30(30-24) = 0.133 ~ 8 minutes

Utilization = 24/30= 0.80 = 80%

Idle time = 1-.8 = .20

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