Question

PLEASE DO NOT USE HAND WRITING     PLEASE DO NOT USE HAND WRITING     PLEASE DO NOT USE HAND WRITING PLEASE DO NOT USE HAND WRITING

6. For the problem given in Question 5, suppose the current automated machine can be replaced by a more efficient automated machine which requires $30 per hour of leasing cost whereas the current automated machine is leased at $20 per hour. The more efficient automated machine can prepare a license in 5 minutes. If a driver’s time is considered to be worth $8 per hour, is it worth to replace the current automated machine by the more efficient automated machine? Calculate the total cost of the current automated machine and the drivers’ time and the total cost of the more efficient automated machine and the drivers’ time to answer this question.

QUESTION 5 WAS

5. Drivers who come to get their licenses at the Department of Motor Vehicles have their photograph taken by an automated machine that develops the photograph onto the license card. The machine requires 6 minutes to prepare a license. Drivers arrive at the machine at the mean rate of 7.8 per hour. Assume it is a single-server waiting line model.

(a) Determine the mean arrival rate and the mean service rate.

(b) Determine the probability that a driver will have an empty queue.

(c) Determine the probability that 3 drivers are in the queuing system.

(d) Determine the average number of drivers in the queue and the average number of drivers in the system.

(e) Determine the average waiting time in the queue and the average total time in the system for a driver.

(f) Find the utilization factor of the automated machine.

Answer 1

ANSWER:

Given that,

5)

(a) Mean Arrival rate, λ = 7.8 per hour

Mean service rate, μ = 1/6 minutes * 60 minutes per hour = 10 per hour

(b) Probability that a driver have an empty queue = 1 - λ/μ = 1 - 7.8/10 = 0.22

(c) Probability that 3  drivers are in the queuing system = (1-λ/μ)*(λ/μ)² = 0.22* = 0.1338

(d) Average number of drivers in the queue = λ²/μ/(μ-λ) =0.78² / (10-7.8) = 0.3101

Average number of drivers in the system = λ/(μ-λ) = 7.8/(10-7.8) = 3.54

(e) Average waiting time in the queue = λ/μ/(μ-λ) = 0.78/(10-7.8) = 0.3545 hour or 34 minutes

Average total time in system = 1/(μ-λ) = 1/(10-7.8) = 0.4545 hour or 40 minutes

(f) Utilization factor of the automated machine = 7.8/10 = 0.78

6)

Current automated machine

Total hourly cost of the current automated machine = Number of drivers in the system*cost of driver's time + leasing cost of current machine = 3.54*8 + 20 = $ 48.32

More efficient automated machine

Service rate, μ = 1/5*60 = 12 per hour

Number of drivers in the system = λ/(μ-λ) = 7.8/(12-7.8) = 1.85

Total hourly cost of the more efficient automated machine = Number of drivers in the system*cost of driver's time + leasing cost of current machine =1.85*12 + 30 = $ 52.2

We see that total hourly cost with new efficient automated machine is much less as compared to the current automated machine. Therefore it is worth to replace the current automated machine with the more efficient automated machine.