Willow Brook National Bank operates a drive-up teller window that allows custome

Question

Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 12 customers per hour or 0.2 customers per minute. In the same bank waiting line system, assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 18 customers per hour, or 0.3 customers per minute. Determine the following operating characteristics for the system:

The probability that no customers are in the system. If required, round your answer to four decimal places.

P0 =  

The average number of customers waiting. If required, round your answer to four decimal places.

Lq =  

The average number of customers in the system. If required, round your answer to four decimal places.

L =  

The average time a customer spends waiting. If required, round your answer to four decimal places.

Wq =  min

The average time a customer spends in the system. If required, round your answer to four decimal places.

W =  min

The probability that arriving customers will have to wait for service. If required, round your answer to four decimal places.

Pw =

Explanation / Answer

As per MM1 system, arrivals follow Poisson and service follows exponential = 12 per hour µ = 18 per hour 1. Probability that there are no customers in the system = P0 = (1 - p)p^0 p = / µ = 12/18 = 0.6667 P (0) = (1 - 12/18) * 0.6667^0 = 0.3333 2. Average number of customers waiting = Lq = pLs p = / µ = 12/18 = 0.6667 = 0.6667 * 2 = 1.3333 customers 3. Average number of customers in the system = Ls = /µ- = 12 / 18 - 12 = 2 customers 4. Average time a customer is in the queue = Wq = pWs p = / µ = 12/18 = 0.6667 = 0.6667 * 10 minutes = 6.6667 minutes 5. Average time a customer is in the system = Ws = 1 /µ- = 1 / 18-12 = 0.1667 hours or 10 minutes 6. Probability that arriving customers will have to wait for service = / µ = 12/18 = 0.6667

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