1. Variability 8/24/04 Paul A. Jensen
Operations Research Models and Methods
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2. Example A single machine performs an operation for a unit of product.
The mean operation time is 30 seconds.
Units arrive at the station with an average time between arrivals of 40 seconds.
There is room for three waiting units.

3. Arrival and service processes are constant
Arrival rate: 1.5/minute
Service rate: 2/minute
Percent utilization
Average delay
Average WIP

4. Arrival process is random Service process is constant
Arrival rate: 1.5/minute
Average Service rate: 2/minute
Percent utilization
Average delay
Average WIP

5. Arrival and service processes are random
Average Arrival rate: 1.5/minute
Average Service rate: 2/minute
Percent utilization
Average delay
Average WIP

6. Conclusion:

7. Analytical Determination of System Characteristicsu
Unit flow t
operation time U = ut
Unit time V
Production volume s
Number of machines

8. For analytical purposes
Assume service and interarrival times have exponential distributions

9. We want to compute: State Probabilities
Average Number and Time in the Queue
Average Number and Time in the System
Percent Utilization

10. The State Probabilities
The probability that the system is empty
The probability of n in the system for n ≤ s
The probability of n in the system for n > s

11. Average Number and Time
In the queue
In the system
Utilization

12. Queuing Networks

13. Equivalence Property
Assume: All stations have exponential service times and unlimited queues and all inputs to the system are Poisson processes.
Then: Each station can be analyzed independently with queuing analysis.
Then: System characteristics can be determined by summing station characteristics.

14. Consider the Job Shop

15. Queuing Analysis

16. System Characteristics

17. Stations Performing More than One Operation

18. Items Processed in Lots

19. The effects of producing in lots

20. Individual vs. lot production (neglecting setup time)
The minimum number of stations is the same
The traffic intensity is the same
The average number in the queue is the same
But: For individual production Lq is in units
For lot production Lq is in lots
For lot production, WIP is Q times greater
For lot production, W and Wq is Q times greater

21. Effects of setup time
Setup time increases the minimum number of machines
Setup time increases the traffic intensity
The effects are reduced by increasing the lot size
But, increasing the lot size increases WIP and throughput time by a factor of Q

22. Example
Production of A is per minute, and production of B is per minute.
Oper. 1 2 3 4 5 6 7 8 9 10
Time A
min. 18
Time B 12 15
Unit
Flow
1.272
1.111
WIP A
2.65
2.12
8.332
2.315
WIP B
10.6
12.72
9.258
13.89

23. Analyze Station 3 for flows in individual units

24. Determining the number of machines

25. Analyze Station 3 for flows in lots of 10