The bumper cars amusement attraction has a problem with cars becoming disabled a

Question

The bumper cars amusement attraction has a problem with cars becoming disabled and in need of repair. Repair personel can be hired at a rate of $20 per hour. One repairer can fix cars in an average of 25 mins. While a car is disabled or being repaired, lost income is $40.00 per hour. Cars tend to break down at the rate of two per hour. Assume there is 1 per to repair, the arrival rate follows a Poisson Distribution and the service time follows an exponential distribution.

a - on average, how long is a disabled car waiting to be fixed?

b. on average how many disabled cars are out of service and not able to take riders on the car attraction

c. when a bumper car breaks down what is the probability that it will find that there are at least 3 cars already waiting to be fixed

d. the park decides to increase its repair capacity by hiring either one or two extra people. they will not work individually but as a team. this if two or three people are working they will work toegther on the same repair. one repair work can fix cars in an average of 25 mins. two repairers take 20 mins and three workers take 15 mins. What is the cost of the repairer operation for the two repair strategies (adding 1 or 2)? Consider the cost of the service with only other worke, would either of the options be prefereed to the one work option - explain.

Explanation / Answer

a) On average, how long is a disabled bumper car waiting to be serviced?
One repairer can fix cars in an average time of 25 minutes so the service time is (60/25) cars per hour = 2.4 cars per hour.

Cars tend to break down at the rate of two per hour so arrival rate is 2 cars per hour.

Average Waiting Time =

b) On average, how many disabled bumper cars are out of service waiting to be serviced or being serviced?

Average number of cars in the system =

c) When a bumper car becomes disabled, what is the probability that it will find that there are at least three cars already waiting to be repaired?
Here utilization rate = =5/6.

Now the probability that it will find that there are at least three cars already waiting to be repaired
= (utiliziation rate)3 = (5/6)3 = 0.5787
d) The amusement part has decided to increase its repair capacity by adding either one or two additional repair people. These will not work individually but they only work as one team. Thus if two or three people are working, they will work together on the same repair. One repair worker can fix cars in an average time of 25 minutes. Two repair workers working as a team take 20 minutes and three repair workers working as a team take 15 minutes. What is the cost of the repair operation for the two repair strategies (adding 1 or 2 repair workers) that it is considering?

For one more repair worker,

Cost for repair workers = 2*$20 = $40 per hour.

Average waiting time in the system = 1 hour thus the cost for that = $40

Hence total cost for the strategy of adding 1 more repair workers = $80 per hour.

For adding 2 more repair workers,

Cost of repair workers = $60 per hour.

Average waiting time in the system = 0.5.

Thus cost to wait = $40*0.5 = $20.

Hence total cost = $60+$20 = $80.

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