Question

1. 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 6 customers per hour or 0.1 customers per minute. Also assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 54 customers per hour, or 0.9 customers per minute. Determine the probabilities of 0, 1, 2, and 3 customers in the system. Round your answers to 4 decimal places.

n	P(n)
0
1
2
3

What is the probability that more than three customers will be in the drive-up teller system at the same time? Round your answer to 4 decimal places.

2. The reference desk of a university library receives requests for assistance. Assume that a Poisson probability distribution with an arrival rate of 11 requests per hour can be used to describe the arrival pattern and that service times follow an exponential probability distribution with a service rate of 13 requests per hour.

1. What is the probability that no requests for assistance are in the system? If required, round your answer to four decimal places.
   
   P₀ =  
   
2. What is the average number of requests that will be waiting for service? If required, round your answer to four decimal places.
   
   L_q =  
   
3. What is the average waiting time in minutes before service begins? If required, round your answer to nearest whole number.
   
   W_q =  min
   
4. What is the average time at the reference desk in minutes (waiting time plus service time)? If required, round your answer to nearest whole number.
   
   W =  min
   
5. What is the probability that a new arrival has to wait for service? If required, round your answer to four decimal places.
   
   P_w =

3. Marty's Barber Shop has one barber. Customers have an arrival rate of 1.3 customers per hour, and haircuts are given with a service rate of 6 per hour. Use the Poisson arrivals and exponential service times model to answer the following questions:

1. What is the probability that no units are in the system? Round your answer to four decimal places.
   
   P₀ =  
   
2. What is the probability that one customer is receiving a haircut and no one is waiting? Round your answer to four decimal places.
   
   P₁ =  
   
3. What is the probability that one customer is receiving a haircut and one customer is waiting? Round your answer to four decimal places.
   
   P₂ =  
   
4. What is the probability that one customer is receiving a haircut and two customers are waiting? Round your answer to four decimal places.
   
   P₃ =  
   
5. What is the probability that more than two customers are waiting? Round your answer to four decimal places.
   
   P(More than 2 waiting) =  
   
6. What is the average time a customer waits for service? Round your answer to four decimal places.
   
   W_q =  minutes

4.

Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.6 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 12 minutes with each customer.

1. Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times. Round your answers to four decimal places. Do not round intermediate calculations.
   
   L_q =  
   
   L =  
   
   W_q =
   
   W =
   
   P_w =  
   
2. Service goals dictate that an arriving customer should not wait for service more than an average of 4 minutes. Is this goal being met? If not, what action do you recommend?
   
   No. Firm should increase the mean service rate u for the consultant or hire a second consultant.  
   
3. If the consultant can reduce the average time spent per customer to 8 minutes, what is the mean service rate? Round your answer to four decimal places. Do not round intermediate calculations.
   
   µ =  customers per hour
   
   W_q =  minutes
   
   Will the service goal be met?

4.

Pete's Market is a small local grocery store with only one checkout counter. Assume that shoppers arrive at the checkout lane according to a Poisson probability distribution, with an arrival rate of 9 customers per hour. The checkout service times follow an exponential probability distribution, with a service rate of 15 customers per hour. The manager’s service goal is to limit the waiting time prior to beginning the checkout process to no more than five minutes. Also the manager of Pete's Market wants to consider one of the following alternatives for improving service. Calculate the value of W_q for each alternative.

1. Hire a second person to bag the groceries while the cash register operator is entering the cost data and collecting money from the customer. With this improved single-server operation, the service rate could be increased to 19 customers per hour. Round your answer to three decimal places. Do not round intermediate calculations.
   
   W_q =  minutes
   
2. Hire a second person to operate a second checkout counter. The two-server operation would have a service rate of 15 customers per hour for each server. Note: Use P₀values from Table 11.4 to answer this question. Round your answer to three decimal places. Do not round intermediate calculations.
   
   W_q =

Answer 1

1.

Arrival rate, = 0.1 customers per minute

Service rate, = 0.9 customers per minute

Utilization = / = 0.1/0.9 = 0.1111

Probability that n customers are waiting is,

P(n) = (1 - )

For n = 0,   P(0) = (1 - ) = 1 - 0.1111 = 0.8889

For n = 1, P(1) = (1 - ) = (1 - 0.1111) * 0.1111 = 0.0988

For n = 2, P(1) = (1 - ) = (1 - 0.1111) * 0.1111² = 0.0110

For n = 3, P(1) = (1 - ) = (1 - 0.1111) * 0.1111³ = 0.0012

Probability that more than three customers will be in the drive-up teller system = P(N 4) = = 0.1111⁴ = 0.0002