Question

An insurance company has one adjuster in the branch office. People with claims against the company are found to arrive in a Poisson fashion during an 8 to 5 workday. Determine the hourly service and arrival rates using an 8-hour workday. The amount of time that the adjuster spends with a claimant is exponentially distributed. Claimants are processed in the order of their arrival.

You are the manager of this branch and you wanted to investigate service provided by your branch to claimants using waiting lines performance measures. You asked your adjuster to collect data on (1) time when clients arrived, and (2) how much time he (she) spent with each client for one typical 8-hour workday. Below are the data collected:

Customer Arrival Time	8:00	8:26	9:31	10:07	10:26	10:29	10:41	10:47	11:21	1:24	1:25	1:48	2:00	2:05	2:21	3:10	3:30	3:50	4:05	4:30
Length of Time with Claimant (Minutes)	13	13	12	10	12	13	18	24	20	12	14	21	3	14	15	10	24	13	14	15

Answer the following questions, assuming this observation day is very typical of the adjuster’s 8 to 5 workday. Assume also that the adjuster takes an hour-long lunch break when (s)he is idle so that there is no interruption on the service time of any customer arriving.

1. Show and explain how you obtained the arrival and service rates you will use for answering the following questions.

2. On a typical 8-hour workday, how many hours is the adjuster idle (no claimant)?

3. If a claimant arrives during a typical day for the adjuster, what is the probability that the claimant will have to wait for service?

4. On average, how many claimants will be waiting to be served by the adjuster?

5. If a claimant has to wait for service, how long (express your answer in minutes) will the wait be, on average?

6. What is the expected amount of time (in minutes) a claimant will be at the branch office?

7. The adjuster’s office has a waiting room that has only two chairs. What is the probability that one or more claimants will have to stand while awaiting service by the adjuster?

8. After looking at the performance measures (time of waiting, time of service, length of queue, etc.), you determined that the service needs to be improved. Suggest ways that you can implement to improve the line performance measures. Be specific.

9. Assume that you want the claimants to spend 15 minutes or less in the branch office. You decided that you will do this by hiring additional adjuster(s) who will have the same service speed as the current adjuster.
   1. What is the minimum number of adjusters that will meet this target?
   2. What queueing model do you recommend and why? (Show work, or show templates.)
   3. Describe the characteristics (i.e. measures of line effectiveness) of the line model you recommend in (b).

Answer 1

Dear student, according to our guidelines, we can answer only first four subparts at a time. Kindly, please post the remaining questions separately. Thank you for understanding.

1. Number of customers arriving in a 8 hours day = 20

average arrival rate lambda = 20/8 =2.5 per hour

Average service rate mu = AVERAGE of given length of time= 14.5 min that is 60/14.5  per hour =4.14 per hour

Utilisation rho = lambda /mu = 2.5 /4.14 =0.6039 = 60.39%

2. Idle time for adjuster in minutes = 480 x ( 1-2.5 /4.04) = 190.13 minutes

3. Probability of having no customer in system P(0) = 1-rho =1-0.6039 = 0.3961

4. Lq ( number of customers waiting = (lambda)² / mu ( mu-lambda)

= 2.5x2.5 / 4.14 x(4.14-2.5)

= 0.921

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