Cars arrive at Joe’s Service Station for an oil change every 15 minutes, and the

Question

Cars arrive at Joe’s Service Station for an oil change every 15 minutes, and the interarrival time has an exponential distribution. The service station is capable of serving up to 48 cars during an 8-hour period with no idle time. Assume that the service time is also a random variable with an exponential distribution.

Estimate or calculate

1.The value of .

2.The mean arrival rate.

3.The value of .

4.The mean service time.

5.The mean service rate.

6.The expected number of cars in the system.

7.The expected number of cars in the queue.

8.The expected waiting time.

9.The expected time in the queue.

10.The probability that the system is empty.

Explanation / Answer

This is a M/M/1 queue model

1.The value of = 60 minutes per hour / car arrival time of 15 minutes = 4 cars per hour

2.The mean arrival rate = 4 cars per hour

3.The value of = 48 cars / 8 hours period = 6 cars per hour

4.The mean service time = 60 minutes per hour / 6 cars per hour = 10 minutes per car

5.The mean service rate = 6 cars per hour

6.The expected number of cars in the system, L = /(-) = 4/(6-4) = 2 cars

7.The expected number of cars in the queue, Lq = 2/(*(-)) =  42/(6*(6-4)) = 1.33

8.The expected waiting time, Wq = Lq/ = 1.33/4 = 0.33 hour or 20 minutes

9.The expected time in the queue, Wq = 0.33 hour or 20 minutes

Expected time in system, W = L/ = 2/4 = 0.5 hour or 30 minutes  

10.The probability that the system is empty = 1-/ = 1 - 4/6 = 0.33

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