Question

Customers enter the camera department of a store at an average rate of five per hour. The department is staffed by one employee, who takes an average of 8.0 minutes to serve each arrival. Assume this is a simple Poisson arrival, exponentially distributed service time situation. (Use the Excel spreadsheet Queue Models.)

a-1. As a casual observer, how many people would you expect to see in the camera department (excluding the clerk)? (Round your answer to 2 decimal places.)

a-2. How long would a customer expect to spend in the camera department (total time)? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

b. What is the utilization of the clerk? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

c. What is the probability that there are more than two people in the camera department (excluding the clerk)?(Do not round intermediate calculations. Round your answer to 1 decimal place.)

d. Another clerk has been hired for the camera department who also takes an average of 8.0 minutes to serve each arrival. How long would a customer expect to spend in the department now? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

Answer 1

Let the arrival rate = a= 5 per hour

Let the service rate = s = 7.5 per hour

Total time a customer is expected to spend in camera department

= 5/7.5 x( 7.5 – 5)   + 1/ 7.5      hour

= 3/7.5 hour

= 24 minutes

CUSTOMER WOULD EXPECT 24 MINUTES IN THE CAMERA DEPARTMENT

Probability that there are 0 people in the camera department

= P0 = 1 – a/s = 1 – 5/7.5 = 1 – 0.666 = 0.334

Probability of 1 person in the camera department

= a/s x P0

= (5/7.5) x 0.334

= 0.666 x 0.334

= 0.222

Probability of 2 person in the camera department

= a/s x P0

= (5/7.5)^2 x 0.334

= 0.666x 0.666 x 0.334

= 0.148

Probability that there will be 2 or more customers

= 1 – Probability that there will be 0 customers – Probability that there will be 1 customer – Probability that there will be 2 persons in the camera department

= 1 – 0.334 – 0.222 – 0.148

= 0.296

Probability that there are more than 2 people in the camera department = 0.296

When there are 2 clerks , each of whom has a service rate of 8 minutes to serve a customer, effective service rate of the process becomes 4 minutes / customer on average.

Thus number of customers it can serve per hour = S1= 60/4 = 15

Revised time a customer is expected to spend in camera department now Total time a customer is expected to spend in camera department

= 5/15 x( 15 – 5)   + 1/ 15      hour

= 1.5/15 hour

= 1/10 hour

= 6 minutes

Now the service rate is S=10 per hour

The expected time to spend = 1/(S-A) = 1/(10-8) = 1/2 HOUR = 0.5 Minutes

CUSTOMER IS EXPECTED TO SPEND 0.5  MINUTES IN THE DEPARTMENT