Question

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.

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.

Answer 1

This is a M/C/1 (Constant service time) queue system

Current automated machine

Arrival rate, $\lambda$ = 7.8 per hour

Service rate, $\mu$ = 60/6 = 10 per hour

Average number of drivers in system, L = $\lambda$/$\mu$ + $\lambda$²/(2$\mu$*($\mu$-$\lambda$))

= 7.8/10 + 7.8²/(2*10*(10-7.8))

= 2.163

Total operating cost per hour = Cs + L*Cw

= 20+2.163*8

= $ 37.3

New efficient automated machine

Arrival rate, $\lambda$ = 7.8 per hour

Service rate, $\mu$ = 60/5 = 12 per hour

Average number of drivers in system, L = $\lambda$/$\mu$ + $\lambda$²/(2$\mu$*($\mu$-$\lambda$))

= 7.8/12 + 7.8²/(2*12*(12-7.8))

= 1.254

Total operating cost per hour = Cs + L*Cw

= 30+1.254*8

= $ 40.03

Total operating cost of the new efficient machine is higher.

Therefore, it is NOT worth replacing the current automated machine with the more efficient automated machine.