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

Ans:

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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Waiting Lines M/M/1 (Single Server Model) Data Results Arrival rate (l) 2.00 Average server utilization(r) 0.833333 Service rate (m) 2.40 Average number of customers in the queue(Lq) 4.166667 Number of servers 1 Average number of customers in the system(L) 5 Server cost $/time) 20.00 Average waiting time in the queue(Wq) 2.083333 Waiting cost ($/time) 40.00 Average time in the system(W) 2.5 Probability (% of time) system is empty (P0) 0.166667 Cost - based on waiting 186.6667 Cost - based on system 220 Probabilities Number Probability Cumulative Probability 0 0.166667 0.166667 1 0.138889 0.305556 2 0.115741 0.421296 3 0.096451 0.517747 4 0.080376 0.598122 5 0.066980 0.665102 6 0.055816 0.720918 7 0.046514 0.767432 8 0.038761 0.806193 9 0.032301 0.838494 10 0.026918 0.865412 11 0.022431 0.887843 12 0.018693 0.906536 13 0.015577 0.922113 14 0.012981 0.935095 15 0.010818 0.945912 16 0.009015 0.954927 17 0.007512 0.962439 18 0.006260 0.968699 19 0.005217 0.973916 20 0.004347 0.978263

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