The Westland Title Insurance Company has a copying machine that is used by the 8 agents...

The Westland Title Insurance Company has a copying machine that is used by the 8 agents at their office. Each agent needs to use the machine 2 times every hour on average and uses it for an average of 4 minutes. Assume the interarrival times and copying times are exponentially distributed.

What is the probability that a person arriving to use the machine will find it idle?
What is the utilization of this machine?
On average, how long will a person have to wait before getting to use the machine?
What is the probability at least 2 agents will be in line waiting for their turn to use the machine?
On average how many agents will be waiting to use the machine?
Suppose that the people who use the copy machine are paid an average of $30 per hour. The company is worried about the loss of productivity. It will cost $15 per hour to rent another copy machine. Should it rent another machine?

Solutions

Expert Solution

Arrival rate: λ = 2/60 = 0.033 minutes

Service rate: µ = 1/4 = 0.25 minutes

No. of agents = 8

a) What is the probability that a person arriving to use the machine will find it idle?

P0 = 1 − ρ = 1-(λ/µ) = 1-(0.033/0.25) = 0.868

b) What is the utilization of this machine?

ρ = λ/µ = 0.033/0.25 = 0.132

c) On average, how long will a person have to wait before getting to use the machine?

Waiting time to use the machine = Wq = Lq/λ

Lq = ρ2/(1 − ρ) = 0.8682/(1-0.868) = 5.71

Wq = 5.71/0.033 = 172.96 minutes

d) What is the probability at least 2 agents will be in line waiting for their turn to use the machine?

n = 2

Pn: Probability of n customers in waiting = (1 − ρ)/ρn

P2 = (1-0.868)/0.8682 = 0.175

e) On average how many agents will be waiting to use the machine?

Lq = ρ2/(1 − ρ) = 0.8682/(1-0.868) = 5.71 ~ 6

f) Suppose that the people who use the copy machine are paid an average of $30 per hour. The company is worried about the loss of productivity. It will cost $15 per hour to rent another copy machine. Should it rent another machine?

Average time spent by agent = 1/(µ-λ) = 1/(0.25-.033) = 4.61 minutes

Curent cost = 4.61 * (30/60) * 8 = 18.44 (8 agents)

Cost of new copy machine = 8*4.61*(15/60) = 9.22

Yes, it should rent another machine

orchestra answered 2 years ago

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