The school of business has 3 fax machines.The toner in each machine needs to be changed after about 5 hours of use.There is one unit secretary who is responsible for the fax machine maintenance.It takes him 15 minutes to replace the toner cartridge.What is the average waiting time in the queue?
A).0142 hour
B).1563 hour
C).0249 hour
D).2749 hour
E)None of the above

Question

Quizplus · Accepted Answer

We can use the M/M/1 queuing model to solve this problem, where arrivals are Poisson distributed and service times are exponentially distributed.

The arrival rate, λ, is the rate at which toner cartridges need to be changed, which is 1/5 cartridges per hour per fax machine since each machine needs a cartridge change after 5 hours of use. Therefore, the total arrival rate for all 3 fax machines is 3/5 cartridges per hour.

The service time, μ, is the time it takes to replace a toner cartridge, which is 15 minutes or 0.25 hours.

The utilization factor, ρ, is λ/μ = (3/5)/(0.25) = 2.4.

Using Little's Law, L = λW, where L is the average number of cartridges in the queue and W is the average waiting time in the queue, we can solve for W:

L = ρ2/(1-ρ) = (2.4)2/(1-2.4) = 14.4 cartridges
W = L/λ = 14.4/(3/5) = 8 hours

Therefore, the average waiting time in the queue is 8 hours, which is equivalent to 0.249 hours or approximately 0.0249 hour (choice C).

The School of Business Has 3 Fax Machines