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Question: The Judy Gray Income Tax Service is analyzing its customer service operations during the month pr...
The Judy Gray Income Tax Service is analyzing its customer service operations during the month prior to the April filing deadline. On the basis of past data, it has been estimated that customers arrive according to a Poisson process with an average arrival rate of one every 12 minutes. The time to complete a return for a customer is exponentially distributed with a mean of 10 minutes. Assume that the customer has all of the information needed for Judy to complete the tax return at this one visit. When the person is being helped, she meets with Judy in her office. If the individual is waiting for help, she will be in the waiting area. Based on this information, answer the following questions. Do not round your results to integer numbers
a) On the average, how many customers should the waiting area be designed to hold?
b) What is the probability that an arriving customer would find at least four people in the waiting area waiting for help?
c) At this time, Judy is not going to add another person to help complete the tax returns. If the arrival rate remains unchanged but the average time in the system must be 40 minutes or less, what would need to be changed and what would the value of the ch ange need to be?
d) Judy is now considering adding a person to help her process the tax returns. This person will help Judy by checking the paper work for the customers but does not work directly with any customer. Judy believes that this will allow her (Judy) to complete the returns in six minutes 40 seconds on average. Judy would need to pay this person $50 per hour. Judy believes that she has to reduce her cost of the service by $15 per hour for every hour a customer is in her office either waiting for help or being helped. Should she add this person or continue working by herself? Base your analysis on the economics of the two options.
Expert Answer
Explanation / Answer
(a)
L = average arrival rate = 1 in 12 minutes = 5 per hour
M = average service rate = 1 in 10 minutes = 6 per hour
S = number of servers = 1
Lq = no. of customers waiting on an average = L2 / {M(M - L)} = 25 / (6*(6 - 5)) = 4.16 customers.
So, the waiting area should be designed for 5 persons waiting.
(b)
P(no. of customers waiting >= 4) = P(no. of customers in system >= 5) = (L / M)5 = (5/6)5 = 0.402
(c)
Ws = average time in the system = 1 / (M - L) = 40 minutes = 40/60 hours
L is same as before i.e. 5 per hour
So, 1 / (M - 5) = 40 / 60
or, 40M - 200 = 60
or, M = 260 / 40 = 6.5
So, the average service rate should be raised to 6.5 customers per hour. In other words, average service time for a customer should be reduced to 60 / 6.5 = 9.23 minutes.
(d)
Service time = 6 min 40 sec = 400 sec
So, M = 3600 / 400 = 9 per hour
Earlier Ws was = 1 / (M - L) = 1 / (6 - 5) hour = 1 hr.
Now, Ws will be = 1 / (9 - 5) hour = 0.25 hr.
Total cost earlier = Ws x L x $15 = 1 x 5 x $15 = $75 per hour
Total cost now = Ws x L x $15 + $50 = 0.25 x 5 x $15 + $50 = $68.75 per hour
So, it is worth considering the hiring of this additional helper.
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