Please solve manually, do not use excel. Thanks. 1 (10) Suppose jobs arrive at a

Question

Please solve manually, do not use excel. Thanks. 1 (10) Suppose jobs arrive at a single-machine workstation at a rate of 29 per hour and the average process time is 1.75 minutes What is the utilization of the machine? Suppose that inter-arrival and process times are exponential, a. b. i) What is the average time a job spends at the station (i.e. waiting plus process time)? ii) What is the average number ofjobs at the station? iii) What is the long run probability of finding more than three Jobs at the station? c. Process times are not exponential, but instead have a mean of 2 minutes and a standard deviation of five minutes, i) What is the average time a job spends at the station? i) What is the average number ofjobs at the station? ii) What is the average number of jobs in the queue?

Explanation / Answer

Arrival Rate = 29/hr

Process Time = 1.75 min = 1.75/60 hours.

Process Rate = 60/1.75 = 34.28/hr

A) Utilization = Arrival Rate/Service rate = 29/34.28 = 0.8459 =84.59%

B) Average Number of Jobs in queue = Utilization/1-Utilization = (0.8459)2/(1-0.8459) = 4.6433

Wait Time in Queue = Average Number of Jobs in queue/Arrival Rate = 4.6433/29 = 0.16 hour = 9.6 Minutes

1) Wait Time in System = Wait Time in Queue + Process Time = 9.6 + 1.75 = 11.35 minutes

2) Average Jobs at the station = Arrival Rate*Wait Time in System = 29*11.35/60 = 5.49

Probability of No jobs at the station = 1- Utilization = 0.1541 = 15.41%

Probability of 1 Job at the station = Utilization*Probability of No jobs at the station = 0.8459*0.1541 = 0.13 = 13%

Probability of 2 Job at the station = Utilization*Probability of 1 job at the station = 0.8459* 0.13 = 0.1099 = 10.99%

Probability of Having less than 3 jobs at the station = 15.41+13+10.99 = 39.4%

3) Probability of Having more than 3 jobs at the station = 1-Probability of Having less than 3 jobs at the station = 60.6%

C) Utilization = Arrival Rate/Mean Service Rate = 29/(60/2) = 0.9666 = 96.66%

3) Number of Jobs in Queue = (Arrival Rate2 * Service Time Std Dev?2 + Utilization2 )/(2*(1-Utilization)) = (29*29*0.0833*0.0833 + 0.9666*0.9666)/(2*(1-0.9666)) = 101.34

Wait Time in Queue = Number of Jobs in Queue/Arrival Rate = 3.494 hours = 209.68 minutes

1) Wait Time in System = Wait Time in Queue + Mean Process Time = 209.68 + 2 = 211.68 minutes

2) Number of Jobs in System = Wait Time in System*Arrival Rate = 102.31

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