During the campus Spring Fling, the bumper car amusement attraction has a proble

Question

During the campus Spring Fling, the bumper car amusement attraction has a problem with cars becoming disabled and in need of repair. Repair personnel can be hired at the rate of $20 per hour. One repairer can fix cars in an average time of 25 minutes. While a car is disabled or being repaired, lost income is $40 per hour. Cars tend to break down at the rate of two per hour. Assume that there is only one repair person, the arrival rate follows a Poisson distribution and the service time follows an exponential distribution.

a) On average, how long is a disabled bumper car waiting to be serviced?

b) On average, how many disabled bumper cars are out of service waiting and not able to take riders on the bumper car attraction?

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?  

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? Considering the cost of the service with only other worker, would either of the options be preferred to the one work operation? Explain.

Explanation / Answer

Cars tends to break down two per hours whereas repairer can fix it in 25 minutes. Therefore we can say that arrival rate, represented by m = 2/hour and service rate, represented by x = 2.4 (60/25 )/hour=12/5 per hour

a) On average, how long is a disabled bumper car waiting to be serviced?

It means average waiting time in the queue, which is = m/(x*(x-m)) = 2/(2.4*.4) =2.083 hours=125minutes

b) On average, how many disabled bumper cars are out of service waiting and not able to take riders on the bumper car attraction? Not able to take riders means average number of cars waiting and getting repaired after disabled, average number of cars in the queue system, which is = m/(x-m) = 2*(5/2) = 5

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? Probability of at least three cars already waiting means Prob. ( > 3+1) which is given as (m/x)3+2 =(10/12)5 =.40187, say 40% chances of having at least 3 cars already waiting for repairs

d) Under the option of two repair workers, service rate is 60/20 = 3cars per hour and it becomes 60/15 = 4cars per hour in case of three repair workers.

Two types of costs are given hired cost of repair worker as $20/hour and lost income due to disabled car as $40/hour

With one repair worker, lost income = average time spend in queue system* average arrival rate*40 =(5/2)*2*40=200

Total cost = repair worker cost + lost income = 200 +20= $220

With two repair workers, lost income = 1*2*40=80, therefore total cost = 80+2*20 = $120

With three repair workers, lost income = (1/2)*2*40=40, therefore total cost = 40 +3*20 = $100

Hence the option of three repar workers turns out to be the most preferred

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